64c3814
==== TABLE OF CONTENTS ====
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//bibliography.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//conclusion.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//derived.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//fixed.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//introduction.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//nonrank.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//paper.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//proof-flow.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//rankone.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//reductions.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//relation.tex
Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//unicode-map.tex
===========================
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//bibliography.tex ====
\begin{thebibliography}{AJLOP21}
\bibitem[AJLOP21]{AJLOP}
N.~Arbesfeld, D.~Johnson, W.~Lim, D.~Oprea, and R.~Pandharipande,
\emph{The virtual $K$-theory of Quot schemes of surfaces},
J. Geom. Phys. \textbf{164} (2021), article 104154, 36 pp.
\href{https://doi.org/10.1016/j.geomphys.2021.104154}{doi:10.1016/j.geomphys.2021.104154};
\href{https://arxiv.org/abs/2008.10661v2}{arXiv:2008.10661v2}.
\bibitem[Ban26]{BanerjeeRankOne}
K.~Banerjee,
\emph{Bloch's conjecture on surfaces with $p_g=q=0,\ K^2=9$},
preprint, version of 19 September 2026,
\href{https://arxiv.org/abs/2508.13594v2}{arXiv:2508.13594v2}.
\bibitem[Bar85]{Barlow}
R.~Barlow,
\emph{Rational equivalence of zero cycles for some more surfaces with $p_g=0$},
Invent. Math. \textbf{79} (1985), 303--308.
\href{https://doi.org/10.1007/BF01388975}{doi:10.1007/BF01388975}.
\bibitem[Bau14]{BauerInoue}
I.~Bauer,
\emph{Bloch's conjecture for Inoue surfaces with $p_g=0$, $K^2=7$},
Proc. Amer. Math. Soc. \textbf{142} (2014), no.~10, 3335--3342.
\href{https://doi.org/10.1090/S0002-9939-2014-12246-5}{doi:10.1090/S0002-9939-2014-12246-5};
\href{https://arxiv.org/abs/1210.4287v2}{arXiv:1210.4287v2}.
Numbered results are cited according to the linked preprint.
\bibitem[BF97]{BF}
K.~Behrend and B.~Fantechi,
\emph{The intrinsic normal cone},
Invent. Math. \textbf{128} (1997), no.~1, 45--88.
\href{https://doi.org/10.1007/s002220050136}{doi:10.1007/s002220050136}.
\bibitem[Blo79]{BlochRoitman}
S.~Bloch, \emph{Torsion algebraic cycles and a theorem of Roitman},
Compositio Math. \textbf{39} (1979), no.~1, 107--127.
\href{https://www.numdam.org/item/CM_1979__39_1_107_0.pdf}{Primary full text}.
\bibitem[Blo10]{BlochLectures}
S.~Bloch, \emph{Lectures on Algebraic Cycles}, second edition,
New Mathematical Monographs, vol.~16, Cambridge University Press,
Cambridge, 2010.
\href{https://doi.org/10.1017/CBO9780511760693}{doi:10.1017/CBO9780511760693}.
First edition: Duke University Mathematics Series, vol.~IV,
Duke University, Mathematics Department, Durham, NC, 1980.
\bibitem[BKL76]{BKL}
S.~Bloch, A.~Kas, and D.~Lieberman,
\emph{Zero cycles on surfaces with $p_g=0$},
Compositio Math. \textbf{33} (1976), no.~2, 135--145.
\href{https://www.numdam.org/item/CM_1976__33_2_135_0.pdf}{Primary full text}.
\bibitem[BS83]{BlochSrinivas}
S.~Bloch and V.~Srinivas,
\emph{Remarks on correspondences and algebraic cycles},
Amer. J. Math. \textbf{105} (1983), no.~5, 1235--1253.
\href{https://doi.org/10.2307/2374341}{doi:10.2307/2374341}.
\bibitem[CO12]{CarlssonOkounkov}
E.~Carlsson and A.~Okounkov,
\emph{Exts and Vertex Operators},
Duke Math. J. \textbf{161} (2012), no.~9, 1797--1815.
\href{https://arxiv.org/abs/0801.2565v3}{arXiv:0801.2565v3}.
The construction locator refers to this version.
\bibitem[CKY17]{CKY}
D.~I. Cartwright, V.~Koziarz, and S.-K. Yeung,
\emph{On the Cartwright--Steger surface},
J. Algebraic Geom. \textbf{26} (2017), no.~4, 655--689.
\href{https://doi.org/10.1090/jag/696}{doi:10.1090/jag/696};
\href{https://www.maths.usyd.edu.au/u/donaldc/cs-surface/cs-surface.pdf}{author manuscript}.
\bibitem[CK13]{ChangKiem}
H.-L. Chang and Y.-H. Kiem,
\emph{Poincar\'e invariants are Seiberg--Witten invariants},
Geom. Topol. \textbf{17} (2013), no.~2, 1149--1163.
\href{https://doi.org/10.2140/gt.2013.17.1149}{doi:10.2140/gt.2013.17.1149}.
\bibitem[CFK01]{CFK}
I.~Ciocan-Fontanine and M.~Kapranov,
\emph{Derived Quot schemes},
Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{34} (2001), no.~3, 403--440.
\href{https://doi.org/10.1016/S0012-9593(01)01064-3}{doi:10.1016/S0012-9593(01)01064-3};
\href{https://www.numdam.org/item/ASENS_2001_4_34_3_403_0.pdf}{Primary full text}.
\bibitem[dCM02]{deCataldoMigliorini}
M.~A.~A. de Cataldo and L.~Migliorini,
\emph{The Chow groups and the motive of the Hilbert scheme of points on a surface},
J. Algebra \textbf{251} (2002), no.~2, 824--848.
\href{https://doi.org/10.1006/jabr.2001.9105}{doi:10.1006/jabr.2001.9105};
\href{https://arxiv.org/abs/math/0005249v1}{arXiv:math/0005249v1}.
Numbered results are cited according to the linked preprint.
\bibitem[EGL01]{EGL}
G.~Ellingsrud, L.~G\"ottsche, and M.~Lehn,
\emph{On the cobordism class of the Hilbert scheme of a surface},
J. Algebraic Geom. \textbf{10} (2001), no.~1, 81--100.
\href{https://arxiv.org/abs/math/9904095v1}{arXiv:math/9904095v1}.
\bibitem[Fog68]{Fogarty}
J.~Fogarty, \emph{Algebraic families on an algebraic surface},
Amer. J. Math. \textbf{90} (1968), no.~2, 511--521.
\href{https://doi.org/10.2307/2373541}{doi:10.2307/2373541}.
\bibitem[Ful98]{Fulton}
W.~Fulton, \emph{Intersection Theory}, second edition,
Ergebnisse der Mathematik und ihrer Grenzgebiete, third series, vol.~2,
Springer, 1998.
\href{https://doi.org/10.1007/978-1-4612-1700-8}{doi:10.1007/978-1-4612-1700-8}.
\bibitem[GP99]{GP}
T.~Graber and R.~Pandharipande, \emph{Localization of virtual classes},
Invent. Math. \textbf{135} (1999), no.~2, 487--518.
\href{https://doi.org/10.1007/s002220050293}{doi:10.1007/s002220050293};
\href{https://arxiv.org/abs/alg-geom/9708001v2}{arXiv:alg-geom/9708001v2}.
Numbered results are cited according to the linked preprint.
\bibitem[Gul25]{Guletskii}
V.~Guletskii, \emph{The Bloch conjecture}, preprint, 13 December 2025,
\href{https://arxiv.org/abs/2512.12451v1}{arXiv:2512.12451v1}.
\href{https://doi.org/10.48550/arXiv.2512.12451}{doi:10.48550/arXiv.2512.12451}.
\bibitem[GuP03]{GuletskiiPedrini}
V.~Guletski\u\i{} and C.~Pedrini,
\emph{Finite-dimensional motives and the conjectures of Beilinson and Murre},
K-Theory \textbf{30} (2003), no.~3, 243--263.
\href{https://doi.org/10.1023/B:KTHE.0000019787.69435.89}{doi:10.1023/\allowbreak B:KTHE.\allowbreak 0000019787.\allowbreak 69435.89};
\href{https://arxiv.org/abs/math/0303170v1}{arXiv:math/0303170v1}.
Numbered results are cited according to the linked preprint.
\bibitem[IM79]{InoseMizukami}
H.~Inose and M.~Mizukami,
\emph{Rational equivalence of $0$-cycles on some surfaces of general type with $p_g=0$},
Math. Ann. \textbf{244} (1979), 205--217.
\href{https://doi.org/10.1007/BF01420343}{doi:10.1007/BF01420343}.
\bibitem[KMP07]{KMP}
B.~Kahn, J.~P. Murre, and C.~Pedrini,
\emph{On the transcendental part of the motive of a surface},
in \emph{Algebraic Cycles and Motives}, vol.~2,
J.~Nagel and C.~Peters (eds.), Cambridge University Press, 2007,
143--202.
\href{https://doi.org/10.1017/CBO9781107325968.009}{doi:10.1017/CBO9781107325968.009};
\href{https://webusers.imj-prg.fr/~bruno.kahn/preprints/surfaces4.pdf}{author manuscript}.
Numbered results are cited according to the linked author manuscript.
\bibitem[KL13]{KiemLi}
Y.-H. Kiem and J.~Li, \emph{Localizing virtual cycles by cosections},
J. Amer. Math. Soc. \textbf{26} (2013), no.~4, 1025--1050.
\href{https://doi.org/10.1090/S0894-0347-2013-00768-7}{doi:10.1090/S0894-0347-2013-00768-7};
\href{https://arxiv.org/abs/1007.3085v2}{arXiv:1007.3085v2}.
Numbered results are cited according to the linked preprint.
\bibitem[Kim05]{Kimura}
S.-I.~Kimura,
\emph{Chow groups are finite dimensional, in some sense},
Math. Ann. \textbf{331} (2005), no.~1, 173--201.
\href{https://doi.org/10.1007/s00208-004-0577-3}{doi:10.1007/s00208-004-0577-3}.
\bibitem[Man12]{Manolache}
C.~Manolache, \emph{Virtual pull-backs},
J. Algebraic Geom. \textbf{21} (2012), no.~2, 201--245.
\href{https://doi.org/10.1090/S1056-3911-2011-00606-1}{doi:10.1090/S1056-3911-2011-00606-1};
\href{https://arxiv.org/abs/0805.2065v2}{arXiv:0805.2065v2}.
Numbered results are cited according to the linked preprint.
\bibitem[MOP17]{MOP}
A.~Marian, D.~Oprea, and R.~Pandharipande,
\emph{Segre classes and Hilbert schemes of points},
Ann. Sci. \'Ec. Norm. Sup\'er. (4) \textbf{50} (2017), no.~1, 239--267.
\href{https://doi.org/10.24033/asens.2320}{doi:10.24033/asens.2320};
\href{https://arxiv.org/abs/1507.00688v2}{arXiv:1507.00688v2}.
\bibitem[Mum69]{Mumford}
D.~Mumford, \emph{Rational equivalence of $0$-cycles on surfaces},
J. Math. Kyoto Univ. \textbf{9} (1969), no.~2, 195--204.
\href{https://doi.org/10.1215/kjm/1250523940}{doi:10.1215/kjm/1250523940}.
\bibitem[OP21]{OP}
D.~Oprea and R.~Pandharipande,
\emph{Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics},
Geom. Topol. \textbf{25} (2021), no.~7, 3425--3505.
\href{https://doi.org/10.2140/gt.2021.25.3425}{doi:10.2140/gt.2021.25.3425};
\href{https://arxiv.org/abs/1903.08787}{arXiv:1903.08787}.
\bibitem[PW16]{PedriniWeibel}
C.~Pedrini and C.~Weibel,
\emph{Some surfaces of general type for which Bloch's conjecture holds},
in \emph{Recent Advances in Hodge Theory: Period Domains, Algebraic Cycles,
and Arithmetic}, M.~Kerr and G.~Pearlstein (eds.),
London Mathematical Society Lecture Note Series, vol.~427,
Cambridge University Press, 2016, 308--330.
\href{https://doi.org/10.1017/CBO9781316387887.014}{doi:10.1017/CBO9781316387887.014};
\href{https://arxiv.org/abs/1304.7523v1}{arXiv:1304.7523v1}.
\bibitem[Roi80]{Roitman}
A.~A. Roitman,
\emph{The torsion of the group of $0$-cycles modulo rational equivalence},
Ann. of Math. (2) \textbf{111} (1980), no.~3, 553--569.
\href{https://doi.org/10.2307/1971109}{doi:10.2307/1971109}.
\bibitem[STV15]{STV}
T.~Sch\"urg, B.~To\"en, and G.~Vezzosi,
\emph{Derived algebraic geometry, determinants of perfect complexes,
and applications to obstruction theories for maps and complexes},
J. Reine Angew. Math. \textbf{702} (2015), 1--40.
\href{https://doi.org/10.1515/crelle-2013-0037}{doi:10.1515/crelle-2013-0037};
\href{https://flore.unifi.it/bitstream/2158/800683/1/crelle.pdf}{prepublication version}.
Numbered results are cited according to the linked prepublication version.
\bibitem[Ser73]{SerreArithmetic}
J.-P. Serre, \emph{A Course in Arithmetic},
Graduate Texts in Mathematics, vol.~7, Springer, New York, 1973.
\href{https://doi.org/10.1007/978-1-4684-9884-4}{doi:10.1007/978-1-4684-9884-4}.
\bibitem[Stacks]{Stacks}
The Stacks Project Authors, \emph{The Stacks Project},
\href{https://stacks.math.columbia.edu/tag/02JO}{Tag 02JO},
\href{https://stacks.math.columbia.edu/tag/0DHV}{Tag 0DHV}, and
\href{https://stacks.math.columbia.edu/tag/068X}{Tag 068X},
accessed 9 September 2026.
\bibitem[TV07]{TV}
B.~To\"en and M.~Vaqui\'e, \emph{Moduli of objects in dg-categories},
Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{40} (2007), no.~3, 387--444.
\href{https://doi.org/10.1016/j.ansens.2007.05.001}{doi:10.1016/j.ansens.2007.05.001}.
\bibitem[Voi14]{Voisin2014}
C.~Voisin,
\emph{Bloch's conjecture for Catanese and Barlow surfaces},
J. Differential Geom. \textbf{97} (2014), 149--175.
\href{https://arxiv.org/abs/1210.3935v2}{arXiv:1210.3935v2}.
\end{thebibliography}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//bibliography.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//conclusion.tex ====
\section{The integral conclusion and motivic consequences}\label{sec:conclusion}
\subsection{The integral zero-cycle theorem}\label{subsec:integral-proof}
Both numerical branches now supply the nonzero coefficient required
by Corollary~\ref{rel:correspondence}. We assemble the rational
vanishing and then use Roitman's theorem to remove torsion.
\begin{proof}[Proof of Theorem~\ref{thm:bloch}]
For a surface of Kodaira dimension below two, the assertion follows
from Bloch--Kas--Lieberman as in
Proposition~\ref{prop:reduction}. For a surface of general type,
pass to a smooth minimal model \(X_0\). If \(K_{X_0}^2\le8\),
Proposition~\ref{prop:nonrank} gives a nonzero diagonal coefficient
in Theorem~\ref{thm:chow-relation} on a point blowup of \(X_0\).
If \(K_{X_0}^2=9\), Proposition~\ref{prop:rankone} gives such a
coefficient on \(X_0\) itself. In either case
Corollary~\ref{rel:correspondence} and Lemma~\ref{lem:birational} yield
\[
\CH_0(S)^0\otimes_{\mathbb Z}\mathbb Q=0.
\]
Let \(z\in\CH_0(S)^0\). Vanishing after tensoring with
\(\mathbb Q\) means that some nonzero integer annihilates \(z\).
Roitman's torsion Theorem makes the Albanese map injective on torsion
zero-cycles over \(\mathbb C\)
\cite[Theorem~4.2]{BlochRoitman}; see also \cite{Roitman}.
But \(\Alb(S)=0\), since \(q(S)=0\). Therefore \(z=0\).
Thus the integral degree map is injective. Any complex point of
\(S\) has degree one, so it is also surjective.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:bloch-pg-zero}]
If \(S\) is not of general type, Bloch, Kas, and Lieberman prove that
the kernel of \(\operatorname{alb}_S\) vanishes with integral
coefficients \cite[pp.~135--136]{BKL}. Surjectivity gives the claim.
Suppose \(S\) is of general type, and let \(S_0\) be a smooth minimal
model. The inequalities \(K_{S_0}^2>0\) and \(\deg c_2(S_0)>0\),
together with Noether's formula, give
\(1-q(S_0)+p_g(S_0)>0\) \cite[Remark~1.3.1]{PedriniWeibel}.
By Lemma~\ref{lem:birational}, \(p_g(S_0)=p_g(S)=0\) and
\(q(S_0)=q(S)\), so the inequality forces \(q(S)=0\).
Theorem~\ref{thm:bloch} now gives \(\CH_0(S)^0=0\), while
\(\Alb(S)(\mathbb C)=0\). Thus \(\operatorname{alb}_S\) is an
isomorphism.
\end{proof}
\subsection{Rational Chow motives}\label{subsec:motivic-consequences}
We now pass from integral zero-cycles to rational Chow motives.
Corollary~\ref{cor:bloch-pg-zero} supplies the Albanese-kernel
vanishing for every surface with \(p_g=0\); the additional hypothesis
\(q=0\) enters only in the pure-Tate conclusions below.
\begin{proof}[Proof of Corollary~\ref{cor:tate-motives}]
Let \(T(S)\) be the kernel of the rational Albanese map
\[
\CH_0(S)^0\otimes_{\mathbb Z}\mathbb Q
\longrightarrow \Alb(S)(\mathbb C)\otimes_{\mathbb Z}\mathbb Q.
\]
Corollary~\ref{cor:bloch-pg-zero} gives \(T(S)=0\).
Kahn--Murre--Pedrini's equivalence
\cite[Corollaries~7.4.9 and~7.6.11]{KMP} then gives \(t_2(S)=0\)
and Kimura finite-dimensionality of \(h(S)_{\mathbb Q}\). Their
hypotheses hold over \(\mathbb C\), an algebraically closed
characteristic-zero field of infinite transcendence degree over
\(\mathbb Q\). This step does not require \(q(S)=0\).
Now assume \(q(S)=0\). Kahn--Murre--Pedrini's surface
Chow--K\"unneth decomposition
\cite[Propositions~7.2.1 and~7.2.3]{KMP} is
\[
h(S)_{\mathbb Q}\simeq
\mathbf 1\oplus h_1(S)\oplus\mathbb L^{\oplus\rho(S)}
\oplus t_2(S)\oplus h_3(S)\oplus\mathbb L^2,
\]
where \(\rho(S)\) is the Picard number. The endomorphism
identifications for \(h_1(S)\) and \(h_3(S)\) with those of the
Albanese and Picard varieties show that these two motives vanish when
\(q(S)=0\). The Betti realization of the middle decomposition, with
\(t_2(S)=0\), gives \(\rho(S)=b_2(S)\). This gives the asserted
decomposition. Tensoring it proves the assertion for all powers of
\(S\).
For a partition \(\nu=1^{a_1}\cdots n^{a_n}\) of \(n\ge1\), put
\(\ell(\nu)=\sum_j a_j\), \(\Sigma_\nu=\prod_j\mathfrak S_{a_j}\),
and \(S^{(\nu)}=S^{\ell(\nu)}/\Sigma_\nu\).
The formula of de Cataldo--Migliorini
\cite[Theorem~6.2.1]{deCataldoMigliorini} gives
\[
h(S^{[n]})_{\mathbb Q}\simeq
\bigoplus_{\nu\vdash n}h(S^{(\nu)})_{\mathbb Q}
\otimes\mathbb L^{\,n-\ell(\nu)}.
\]
Here \(h(S^{(\nu)})_{\mathbb Q}\) is the rational quotient motive of
the possibly singular symmetric-product quotient. In the quotient
formalism of \cite[Section~4.3, Lemma~4.3.1, and Remark~5.4.5]{deCataldoMigliorini},
it is the image on \(h(S^{\ell(\nu)})_{\mathbb Q}\) of the averaging
projector
\[
e_\nu=\frac{1}{|\Sigma_\nu|}
\sum_{\sigma\in\Sigma_\nu}\Gamma_\sigma.
\]
Since \(\Hom(\mathbb L^a,\mathbb L^b)\) is zero for \(a\ne b\) and
is \(\mathbb Q\) for \(a=b\), every idempotent summand of a finite
Tate sum is again a finite Tate sum. Each term in the formula is
therefore pure Tate, as is \(h(S^{[n]})_{\mathbb Q}\). The case
\(n=0\) is immediate.
Finally, on a summand \(\mathbb L^a\), the rational Chow group and
even Betti cohomology each have one generator in codimension \(a\)
and vanish in other codimensions; cycle class identifies these
generators, and odd Betti cohomology vanishes. Compatibility of cycle
class with correspondences transfers these assertions to the stated
pure Tate motives. Thus the displayed cycle-class maps are
isomorphisms of rational vector spaces. This uses the decomposition
in rational Chow motives, not finite-dimensionality alone.
\end{proof}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//conclusion.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//derived.tex ====
\section{The determinant-fixed Quot obstruction theory}\label{sec:derived}
We make the virtual construction in two settings:
\begin{enumerate}[label=(\roman*)]
\item $X$ is a smooth connected projective complex surface with $p_g=q=0$;
\item $X$ is a smooth connected projective complex surface with $p_g=q=1$,
$K=K_X$ ample, and $D=3K$.
\end{enumerate}
Write $K=K_X$. In both settings $\chi(\mathcal O_X)=1$.
The second setting is used only to determine universal coefficients in
Section~\ref{sec:rankone}. No positivity assumption on $K$
is imposed in setting~(i). All virtual classes and equivariant Chow groups
in this Section have rational coefficients.
Our goal is an equivariant perfect obstruction theory on the
determinant-fixed Quot scheme, together with its actual restriction to
the fixed locus. Surface Quot obstruction theories and localization
relations for a trivial target bundle were developed by
Marian, Oprea, and Pandharipande \cite[Section~1]{MOP}; subsequent
intersection-theoretic and $K$-theoretic developments appear in
\cite{OP,AJLOP}. Here the based determinant fiber must also work on
the irregular auxiliary surface. We construct that fiber as a derived
moduli problem so that its obstruction morphism, and not only its
virtual tangent class, is specified.
Fix line bundles $L_1,L_2,D$, and put $V=L_1\oplus L_2$ and $A=L_1-L_2$.
For an integer $n$, let $B$ be the scheme of quotients
\begin{equation}\label{der:quot-sequence}
\begin{gathered}
0\longrightarrow S'\xrightarrow{i}V\longrightarrow Q\longrightarrow0,
\\
\rk Q=0,\quad \det Q=D,\quad
\int_X\ch_2(Q)=n+L_1D-\frac{D^2}{2}.
\end{gathered}
\end{equation}
Here the determinant condition is equality in the Picard scheme.
Choose a polarization on $X$. The stated Chern data determine a Hilbert
polynomial, and $B$ is the determinant fiber in the corresponding projective
Quot scheme. In particular, $B$ is projective. The group
$G=\mathbb G_m$ acts on $V$ with characters $+1,-1$ on its two summands,
and hence acts on $B$. We write $t=c_1^G(\mathbb C_1)$ for the equivariant
parameter. Later we specialize $t$ to $1/2$, so the evaluated weights of
the two summands are $w_1=1/2$ and $w_2=-1/2$.
Our complexes use cohomological grading: a sheaf $F[-1]$ lies in degree
one. Pullbacks and tensor products of complexes are derived unless
specified otherwise. If $Y$ is a parameter scheme and
$\pi:X\times Y\to Y$ is projection,
we abbreviate
\[
\RHom(F,G)=R\pi_*R\mathcal{H}om(F,G).
\]
Thus Hom complexes in families are complexes on the parameter scheme.
The notation $\RGamma(F)$ for an object on $X$ denotes its derived sections,
pulled back as a constant complex when parameters are present.
\subsection{Derived arrows and classical families}
Ciocan-Fontanine and Kapranov constructed a derived Quot scheme
whose truncation is the classical Quot scheme and whose tangent
cohomology is given by Ext groups from the kernel to the quotient
\cite[Theorem~4.3.2]{CFK}. Here we use a perfect-arrow moduli
description to retain the maps needed for the based determinant fiber.
We use the derived moduli stack of perfect complexes and its morphism
stack as constructed in \cite[Theorem~3.6, Definition~3.18, and Corollaries~3.17,
3.19, and~3.29]{TV}. For smooth proper $X$ these stacks are locally
geometric and locally of finite presentation. The morphism stack with
target fixed to $V$ also has a direct description: over a perfect source
$F$ its fiber is the derived linear mapping stack associated to
$\RHom(F,V)$. This complex is perfect, and its formation commutes with
derived base change, because $X$ is smooth and proper.
\begin{lemma}[The open Quot enhancement]\label{der:arrow}
The fixed-target derived arrow stack has an open substack $\mathfrak U$
whose truncation $U$ is the ordinary Quot scheme with the chosen rank and
Hilbert polynomial, before fixing determinant. On $U$, its tangent
complex is naturally
\begin{equation}\label{der:unrestricted-tangent}
\begin{aligned}
T_{\mathfrak U}|_U
&\simeq\Cone\bigl(\RHom(S',S')\xrightarrow{i\circ-}\RHom(S',V)\bigr)\\
&\simeq\RHom(S',Q).
\end{aligned}
\end{equation}
It is perfect of Tor-amplitude $[0,1]$. These identifications are
compatible with forgetting the source or target, with direct sums of
arrows, and with base change. The same assertions hold for a fixed
vector-bundle target of any rank and torsion quotients.
\end{lemma}
\begin{proof}
First take the locus where both the source and its cone are sheaves in
degree zero on classical fibers, with the prescribed Hilbert polynomials.
We explain the family conditions in this description. For a perfect
complex on $X\times Y$, the locus where it has relative Tor-amplitude zero
over $Y$ is open. Locally one represents the complex by vector bundles.
At a fiber on which it is exact away from degree zero, the positive end
can be split off. At the negative end, fiberwise injectivity between
modules flat over the base implies injectivity with base-flat cokernel,
by the local criterion of flatness. Successively replacing the negative
end by these cokernels gives a sheaf flat over the base near that fiber.
Openness of flatness gives an open neighborhood in $X\times Y$; properness
of $X\times Y\to Y$ then gives the required open neighborhood in $Y$.
Applying this to the source and cone proves the asserted openness.
Conversely, let $V_Y\twoheadrightarrow Q$ be a flat quotient family over
a noetherian base $Y$. Its kernel $S'$ is flat over $Y$. Both $Q$ and
$S'$ are perfect on $X\times Y$, even if $Y$ is singular. Indeed, resolve
either sheaf locally by finite-rank free modules. Every successive
syzygy remains flat over $Y$. On a surface fiber the local rings are
regular of dimension at most two, so after two steps the syzygy is locally
free on that fiber. A finitely presented, base-flat module with locally
free fibers is locally free near the point, again by the local flatness
criterion. This truncates the resolution. This argument suffices over
noetherian bases, and finite presentation extends the moduli
identification to arbitrary classical test schemes.
Flat approximation and the perfection criteria in
\cite[Tags 02JO, 0DHV, and 068X]{Stacks} also give this last passage:
a finitely presented base-flat module is pseudo-coherent, and the
regular surface fibers give absolute Tor-amplitude contained in $[-2,0]$.
On our open locus the cone sequence is consequently an exact sequence
of sheaves with flat quotient. An automorphism $a$ of its source
satisfying $i\circ a=i$ is the identity, since $i$ is injective.
Negative Ext groups between these sheaves vanish as well, so no higher
automorphisms remain in the classical moduli problem. The truncation
is therefore the Quot scheme, including its functor on nonreduced bases.
Here is a tangent calculation that also records the relevant maps.
Allow both vertices of an arrow $F\to G$ to vary. Its derived
endomorphism complex is
\[
\fib\left\{
\RHom(F,F)\oplus\RHom(G,G)
\xrightarrow{(a,b)\mapsto f a-b f}\RHom(F,G)
\right\}.
\]
One can compute this complex with the two-term resolution of a one-arrow
diagram by diagrams induced from its vertices. More explicitly, if
$P_1=(\mathcal O_X\xrightarrow{\id}\mathcal O_X)$ and
$P_2=(0\to\mathcal O_X)$, the resolution has terms
$P_2\otimes F$ and $(P_1\otimes F)\oplus(P_2\otimes G)$.
Its map at the target vertex sends $s$ to $(s,-f(s))$.
Thus perfect vertices give a perfect diagram. The one-arrow path algebra
is finite dimensional and has this finite projective resolution; the
usual perfect-object tangent formula applies to its diagrams on $X$.
The tangent of their moduli is the displayed endomorphism complex
shifted by $[1]$.
Fixing the target takes the tangent fiber over
$\RHom(V,V)[1]$. The result is the first cone in
\eqref{der:unrestricted-tangent}; applying $\RHom(S',-)$ to
\eqref{der:quot-sequence} gives the second identification.
In particular, forgetting the arrow and remembering its source has
differential the boundary
\begin{equation}\label{der:source-boundary}
\RHom(S',Q)\longrightarrow\RHom(S',S')[1].
\end{equation}
These are identifications of perfect complexes in families. For
completeness, perfection and functoriality can also be checked before
fixing a vertex by taking pointed loops. The relative loop stack at a
perfect family is its self-equivalence stack, open in the linear mapping
stack of its perfect endomorphism complex. Its cotangent at the identity
is the dual endomorphism complex. The cotangent triangle for the loop
fiber product identifies this with the pulled-back moduli cotangent
shifted by $[1]$. Forgetting vertices and taking direct sums act on these
loop tangents by projection and block-diagonal sum. Taking the fixed-target
fiber preserves these identifications. This proves the claimed
functoriality, including the specified boundary.
Finally, at a geometric point $S'$ is torsion-free and $Q$ is torsion.
Surface Serre duality gives
\[
\Ext^2(S',Q)^\vee=\Hom(Q,S'(K))=0.
\]
Negative Ext groups vanish for sheaves; groups above degree two vanish
on a smooth surface. Consequently the derived tangent has geometric
fibers in degrees $[0,1]$. For a perfect complex the geometric-fiber
criterion is precisely the criterion for Tor-amplitude in this range.
\end{proof}
We will use the following comparison in both the absolute and relative
constructions. We include the infinitesimal argument to specify which
cotangent morphism supplies the obstruction theory.
\begin{lemma}[Truncation comparison]\label{der:truncation}
Let $\mathfrak Z$ be a locally geometric derived stack, locally of finite
presentation, whose classical truncation $j:Z\to\mathfrak Z$ is a scheme.
Suppose $E_Z=j^*L_{\mathfrak Z}$ is perfect of Tor-amplitude $[-1,0]$.
The canonical map
\[
\phi_Z:E_Z\longrightarrow L_Z
\]
has cone in $D^{\leq-2}(Z)$, and is a perfect obstruction theory.
For a map of such enhancements, the induced cotangent map commutes with
the corresponding maps $\phi_Z$.
\end{lemma}
\begin{proof}
The cotangent map is induced by $j$, so its functoriality is that of
cotangent complexes. The comparison statement is
\cite[Proposition~1.2 and Corollary~1.4]{STV}; the following argument gives
it directly in the present scheme-truncation situation.
Work on an affine open $x:\operatorname{Spec}R\to Z$, and write
$E=x^*E_Z$. For every ordinary $R$-module $N$, split square-zero
extensions and agreement of $Z$ and $\mathfrak Z$ on classical rings give
an isomorphism
\[
\Hom(L_R,N)\xrightarrow{\sim}\Hom(E,N).
\]
There is also an injection
\[
\Hom(L_R,N[1])\longrightarrow\Hom(E,N[1]).
\]
Indeed, let a derivation class $\delta:R\to R\oplus N[1]$ map to zero
on the right. The two points of $\mathfrak Z(R\oplus N[1])$ obtained
from $x$ by $\delta$ and by the zero derivation are then homotopic, over
$x$. Infinitesimal cartesianness of a locally geometric derived stack
produces a lift of $x$ over the homotopy pullback
\[
R'=R\mathbin{\times^h_{R\oplus N[1]}}R,
\]
whose two maps are $\delta$ and zero. This is the classical square-zero
extension of $R$ by $N$ classified by $\delta$. The lift is a classical
point of $Z$ and factors through the same open $\operatorname{Spec}R$;
it therefore gives a splitting of $R'\to R$. The extension class
$\delta$ is zero, proving injectivity.
Put $C=\Cone(E\to L_R)$. The two tests, together with connectivity,
give $\Hom(C,N)=\Hom(C,N[1])=0$ for every $N$. Since $C\in D^{\leq0}$,
the first equality implies $h^0(C)=0$, and the second then implies
$h^{-1}(C)=0$. Thus $C\in D^{\leq-2}$. Together with the
given perfect amplitude, this is the definition of a perfect obstruction
theory in \cite[Definitions~4.4 and~5.1]{BF}.
\end{proof}
\subsection{The based determinant and its amplitude}
The unrestricted Quot enhancement is now available. To impose the
determinant condition without introducing scalar automorphisms, we
first normalize line bundles at one point and then take a derived
fiber. The remaining amplitude check concerns its obstruction map.
Choose $o\in X$. Let $\mathfrak P^o$ be the derived Picard stack of line
bundles on $X$ trivialized at $o$: it is the fiber, over the trivial line,
of evaluation at $o$ from the derived line-bundle stack to
$B\mathbb G_m$. Its truncation is the Picard scheme $\Pic(X)$.
Indeed, every line $M$ on $X\times Y$ has the normalized form
\[
\nu(M)=M\otimes\pi^*(M|_{\{o\}\times Y})^{-1},
\]
with its canonical trivialization at $o$. Two lines with the same
Picard class differ by a line from the base, which disappears on
normalization. Moreover $\pi_*\mathcal O_{X\times Y}=\mathcal O_Y$,
so every line automorphism is a unit from the base; one preserving the
trivialization is the identity. These observations identify the
truncation as a functor, not only on complex points.
The evaluation tangent is $\RGamma(\mathcal O_X)[1]\to\mathbb C[1]$.
Constant functions split it, and hence
\begin{equation}\label{der:based-picard}
T_{\mathfrak P^o}=\RGamma(\mathcal I_o)[1],\qquad
\RGamma(\mathcal O_X)=\mathbb C\oplus\RGamma(\mathcal I_o).
\end{equation}
In particular its cohomology lies in degrees zero and one, with groups
$H^1(\mathcal O_X)$ and $H^2(\mathcal O_X)$ respectively.
The derived determinant morphism of \cite[Section~3.1,
Proposition~3.2]{STV}, followed by $\nu$, gives a morphism
$\mathfrak U\to\mathfrak P^o$. Set
\begin{equation}\label{der:derived-fiber}
\mathfrak B=
\mathfrak U\mathbin{\times^h_{\mathfrak P^o}}
\{\nu(L_1+L_2-D)\}.
\end{equation}
Its truncation is $B$: the fiber fixes the Picard class of $\det S'$
and hence of $\det Q$. It makes no choice of trivialization of the
unnormalized determinant of the family at $o$.
The determinant differential is trace at geometric points
\cite[Proposition~3.2]{STV}. We need this identification on the
universal complexes over classical parameter schemes, including
nonreduced ones, and justify that family statement here.
Let $Y$ be such a quotient parameter scheme. The perfect source on
$X\times Y$ classifies a map to the derived stack of perfect complexes.
Pull back the differential of its determinant morphism to the
line-bundle stack $B\mathbb G_m$. The perfect-object tangent formula
gives an internal map
\[
R\mathcal{H}om(S',S')[1]\longrightarrow\mathcal O_{X\times Y}[1].
\]
The determinant on complexes on $X$ is obtained by applying this
morphism to families on $X$; its differential is the derived pushforward
of this internal map.
Perfect trace duality identifies the internal map with an endomorphism
$a$ of the sheaf $S'$. On the locus where $S'$ is locally free, the
ordinary determinant differential gives $a=\id_{S'}$.
To extend this equality over $Y$, let $Z$ be the closed non-locally-free
locus of $S'$. Base flatness and the fiber criterion for local freeness
show that $Z$ has finite fibers over $Y$. Near a point of $Z$, choose
$f$ in its ideal whose image in the regular local surface fiber is a
nonzero element. That image is a nonzerodivisor; the local flatness
criterion makes $f$ a nonzerodivisor on the total space as well.
The image of $a-\id_{S'}$, viewed in $V_Y$, is coherent and supported
on $Z$, so locally a power of $f$ annihilates it. Since $V_Y$ is locally
free, this image is zero. Thus $a=\id_{S'}$ everywhere, proving the
trace identity as a map of complexes over $Y$. This argument applies
to the universal noetherian parameter schemes used below; naturality
and derived pullback preserve the identity on their base changes.
The tangent map defining \eqref{der:derived-fiber} is consequently the
composition of \eqref{der:source-boundary}, trace, and the projection
in \eqref{der:based-picard}. We denote this composition by $d\det^o$.
The two complexes in this tangent fiber have amplitude $[0,1]$.
Their long exact sequence shows that its only possible extra group is
\[
H^2(T_{\mathfrak B}|_b)=
\coker\!\left\{\Ext^1(S',Q)\longrightarrow H^2(\mathcal O_X)\right\}
\]
at a geometric point $b$. The following rank bound makes this group
vanish.
\begin{lemma}[Surjectivity of the determinant obstruction map]
\label{der:trace-surjectivity}
Let $0\to F\to V_0\to Q_0\to0$ be an exact sequence on $X$, where
$V_0$ is a vector bundle of rank $r$ and $Q_0$ is torsion. The map
\[
\Ext^1(F,Q_0)\longrightarrow\Ext^2(F,F)
\xrightarrow{\operatorname{tr}}H^2(\mathcal O_X)
\]
is surjective if $p_g=0$. It is also surjective if $K$ is ample and
$Kc_1(Q_0)>rK^2$.
\end{lemma}
\begin{proof}
There is nothing to prove when $H^2(\mathcal O_X)=0$. Otherwise Serre
duality and the exact sequence show that a section
$\omega\in H^0(K)$ annihilates the image precisely when
$\omega\id_F\in\Hom(F,F(K))$ lifts to a map
\[
a:V_0\longrightarrow F(K),\qquad a|_F=\omega\id_F.
\]
Writing $i:F\to V_0$, the maps $i(K)a$ and $\omega\id_{V_0}$ agree
where $F=V_0$, a dense open because $Q_0$ has rank zero. They are maps
between vector bundles, so they agree everywhere. Composing with the
quotient to $Q_0(K)$ gives $\omega Q_0=0$.
Suppose now that $\omega\ne0$, and write its effective divisor as
$C=(\omega)$. The quotient $Q_0$ factors through $V_0|_C$. At the
generic point of any integral curve $C_j\subset X$ the local ring is a
DVR. If $\omega$ has order $a_j$ there, a quotient of
$(\mathcal O_{X,C_j}/(\omega))^r$ has length at most $ra_j$.
The divisorial cycle of a torsion sheaf is the sum of these generic
lengths, and represents its first Chern class: the elementary-divisor
decomposition over a DVR identifies this length with the order of the
determinant. Therefore
\[
[Q_0]_{\mathrm{div}}\leq rC
\quad\hbox{componentwise},\qquad
Kc_1(Q_0)\leq rK^2.
\]
The latter inequality uses ampleness of $K$. It contradicts the
hypothesis. The argument allows embedded points and nonreduced curve
support, since it uses only generic lengths. Thus no nonzero
$\omega$ annihilates the image, proving surjectivity.
\end{proof}
\begin{proposition}[The determinant-fixed Quot theory]\label{prop:quot-pot}
In either setting~\textup{(i)} or~\textup{(ii)}, the enhancement
$\mathfrak B$ induces a perfect obstruction theory
$E_B\to L_B$, with virtual tangent
\begin{equation}\label{eq:quot-tangent}
E_B^\vee\simeq
\fib\left\{
\RHom(S',Q)\xrightarrow{d\det^o}
\RGamma(\mathcal I_o)[1]\otimes\mathcal O_B
\right\}.
\end{equation}
It is $G$-equivariant. Its virtual dimension is
\begin{equation}\label{eq:vdim}
v=2n+(A-K)D.
\end{equation}
\end{proposition}
\begin{proof}
The cotangent triangle of the derived fiber gives
\eqref{eq:quot-tangent} as an identity of perfect complexes.
Both its input and target have Tor-amplitude $[0,1]$; the input has
this amplitude by Lemma~\ref{der:arrow}. Their fiber has no negative
cohomology and has no cohomology above degree two. Its degree-two
cohomology vanishes exactly when
\[
\Ext^1(S',Q)\longrightarrow H^2(\mathcal O_X)
\]
is surjective at every geometric point. In setting~(i) its target is
zero. In setting~(ii), Lemma~\ref{der:trace-surjectivity} applies with
$r=2$, since $KD=3K^2>2K^2$. The geometric-fiber criterion for perfect
complexes now gives amplitude $[0,1]$. Lemma~\ref{der:truncation}
therefore supplies the specified obstruction morphism.
Postcomposition with the given $G$-action on $V$ defines an action on
the arrow stack, leaving the source determinant map invariant. It
thus acts on the fiber \eqref{der:derived-fiber}, and naturality of the
cotangent map makes its obstruction theory equivariant. Equivalently,
one can construct the truncation map on the quotient stacks relative
to $BG$, obtaining exactly this equivariant complex and morphism.
The Euler characteristic of the determinant target is $q-p_g=0$.
If $q_2=\int_X\ch_2(Q)$, Riemann--Roch gives
\[
\chi(V,Q)=2q_2-(K+L_1+L_2)D,\qquad
\chi(Q,Q)=-D^2.
\]
For the second identity, the only degree-four term in
$\ch(Q^\vee)\ch(Q)\td(X)$ is $-D^2$, since $Q$ has rank zero;
here $Q^\vee$ is the derived dual.
Consequently
\begin{align*}
v&=\chi(S',Q)-(q-p_g)
=\chi(V,Q)-\chi(Q,Q)\\
&=2\left(n+L_1D-\frac{D^2}{2}\right)
-(K+L_1+L_2)D+D^2
=2n+(A-K)D.
\end{align*}
\end{proof}
\subsection{Equivariant resolutions and the fixed theory}
The determinant-fixed obstruction theory is constructed. To apply
virtual localization, we still need global equivariant presentations
and a comparison identifying the fixed obstruction morphism itself.
\begin{lemma}[Equivariant global presentations]\label{der:global-resolution}
The scheme $B$ has a $G$-equivariant closed embedding in a smooth
projective scheme. Its obstruction theory has a global presentation
by two $G$-equivariant vector bundles. The morphism to the truncated
cotangent complex can be represented by a morphism of two-term
complexes for this embedding.
\end{lemma}
\begin{proof}
The Grassmannian construction of the projective Quot scheme is
equivariant for the action on $V$. The determinant fiber is closed and
invariant, so restriction and a Pl\"ucker embedding give an equivariant
embedding $B\hookrightarrow M$ with $M$ smooth projective. In particular
$B$ has a linearized ample bundle $\mathcal O_B(1)$.
Here is the global resolution argument for $E=E_B$. For $m$ sufficiently
large, $h^0(E)(m)$ is generated by sections and
$H^a(h^{-1}(E)(m))=0$ for every $a>0$. The hypercohomology sequence then lifts its
sections to $\mathbb H^0(E(m))$. These are finite-dimensional torus
representations; linear reductivity allows an equivariant choice of
lifts. Evaluation gives an equivariant bundle
$V_0=\mathcal O_B(-m)\otimes W_0$ and a map $V_0\to E$ surjective on
$h^0$. Its derived fiber $V_1$ is perfect. At every geometric point
$b$, the same map is surjective on $h^0$, since
$h^0(E\otimes^L k(b))=h^0(E)\otimes k(b)$. Its fiber has cohomology
only in degree zero. Thus $V_1$ is a vector bundle, and
$E\simeq[V_1\to V_0]$ in degrees $[-1,0]$.
Let $J$ be the conormal sheaf of $B\hookrightarrow M$, and set
$\Omega=\Omega_M|_B$. The truncated cotangent complex
$\tau_{\geq-1}L_B$ is represented by $C=[J\to\Omega]$ in degrees
$[-1,0]$. Increase $m$ above so that
$\Ext^1(V_0,J)=0$ as well; this is Serre vanishing for $J(m)$.
The triangle $J\to\Omega\to C\to J[1]$ then lifts
$V_0\to E\to C$ to $V_0\to\Omega$, equivariantly. The map
$E\to C$ includes a nullhomotopy on its restriction from $V_1$.
Together with the chosen lift, that nullhomotopy gives a map
$V_1\to J=\fib(\Omega\to C)$. These are ordinary sheaf maps and give
a commuting square
\[
\begin{array}{ccc}
V_1&\longrightarrow&V_0\\
\downarrow&&\downarrow\\
J&\longrightarrow&\Omega
\end{array}
\]
representing the original map to $C$. All choices can be made in the
equivariant category, since taking torus invariants is exact.
\end{proof}
On the classical fixed locus, an invariant subsheaf of
$L_1\oplus L_2$ decomposes as the direct sum of its two character
subsheaves. This holds over every classical base with trivial
$G$-action: the coaction decomposes a module into its weight modules,
and the inclusion in $V$ admits only the two indicated weights.
The kernels and quotients in these summands are flat because they are
direct summands of flat families. Thus a fixed arrow is precisely a pair
\[
S'_1\longrightarrow L_1,\qquad S'_2\longrightarrow L_2,
\]
of rank-one kernel arrows with torsion quotients $Q_1,Q_2$, subject to
the total data and determinant constraint. There are finitely many
discrete summand Hilbert polynomials on the fixed scheme: they are
locally constant, and the fixed scheme is of finite type.
Write $\mathfrak U_i$ for the unrestricted derived arrow Quot of
Lemma~\ref{der:arrow} with target $L_i$. For each choice of the discrete
data, let
\[
\mathfrak W=(\mathfrak U_1\times\mathfrak U_2)
\mathbin{\times^h_{\mathfrak P^o}}
\{\nu(L_1+L_2-D)\},
\]
where the map to $\mathfrak P^o$ takes the normalized determinant of
the direct sum of the sources. Its truncation $W$ is the corresponding
part of $B^G$, by the family decomposition just proved.
\begin{proposition}[The actual fixed obstruction theory]\label{prop:fixed-pot}
The enhancement $\mathfrak W$ induces a perfect obstruction theory
$E_W\to L_W$. For $k:W\hookrightarrow B$, there is a natural
isomorphism of obstruction theories
\begin{equation}\label{der:fixed-comparison}
(Lk^*E_B)^{\mathrm{fix}}\xrightarrow{\sim}E_W,
\end{equation}
compatible with their maps to $L_W$. Its virtual tangent is
\begin{equation}\label{der:fixed-tangent}
E_W^\vee\simeq
\fib\left\{
\bigoplus_{i=1}^2\RHom(S'_i,Q_i)
\longrightarrow\RGamma(\mathcal I_o)[1]\otimes\mathcal O_W
\right\}.
\end{equation}
The moving virtual normal complex is
\begin{equation}\label{der:moving-normal}
N_W^{\vir}=\bigoplus_{i\ne j}\RHom(S'_i,Q_j),
\end{equation}
where the $(i,j)$ summand has the difference between the characters on
$L_j$ and $L_i$, or evaluated weight $w_j-w_i$.
\end{proposition}
\begin{proof}
Direct sum is an actual morphism of perfect-arrow moduli problems.
The determinant of a direct sum is the tensor product of determinants,
compatibly with normalization. Consequently direct sum induces a
derived map $\mathfrak k:\mathfrak W\to\mathfrak B$ lifting $k$.
This map is coherently equivariant. Over the classifying stack $BG$,
tensor the source and target of each summand arrow with the same
character line of weight $+1$ or $-1$. Normalizing the determinant
cancels each base character line against its restriction at $o$.
These canonical cancellations respect tensor products and the chosen
homotopy in the determinant fiber. Thus direct sum defines
$BG\times\mathfrak W\to[\mathfrak B/G]$ over $BG$, with trivial
$G$-action on $\mathfrak W$. Relative cotangents over $BG$ give its
equivariant cotangent morphism.
Its cotangent morphism, restricted to truncations, gives
\[
Lk^*E_B\longrightarrow E_W\longrightarrow L_W.
\]
By functoriality of the truncation comparison, this composite equals
$Lk^*E_B\to Lk^*L_B\to L_W$.
To identify this map, use the tangent formula for arrows. On $W$,
\[
\RHom(S',Q)=\bigoplus_{i,j}\RHom(S'_i,Q_j).
\]
Direct sum maps the two input tangents to the diagonal blocks. Trace
vanishes on off-diagonal blocks, and the based Picard tangent has weight
zero. To display the comparison, abbreviate these complexes by
$\mathcal A_0=\bigoplus_i\RHom(S'_i,Q_i)$,
$\mathcal A=\bigoplus_{i,j}\RHom(S'_i,Q_j)$, and
$\mathcal P=\RGamma(\mathcal I_o)[1]\otimes\mathcal O_W$.
The derivative gives the morphism of fiber triangles
\[
\begin{array}{ccccc}
E_W^\vee&\longrightarrow&\mathcal A_0&\longrightarrow&\mathcal P\\
\downarrow&&\downarrow&&\Vert\\
Lk^*E_B^\vee&\longrightarrow&\mathcal A&\longrightarrow&\mathcal P,
\end{array}
\]
whose middle map is diagonal inclusion. After taking weight zero, the
middle and right maps are identities. The left map is therefore an
isomorphism of perfect complexes in
families. The remaining summands are exactly
\eqref{der:moving-normal}.
In particular \eqref{der:fixed-tangent} has Tor-amplitude $[0,1]$,
as a direct summand of $Lk^*E_B^\vee$. Lemma~\ref{der:truncation}
makes $E_W\to L_W$ a perfect obstruction theory. Dualizing the
fiber-triangle comparison gives \eqref{der:fixed-comparison}. The
compatibility already established shows that this is the fixed
obstruction morphism used in \cite[Proposition~1]{GP}, and not just an
identity of virtual tangent classes.
\end{proof}
The hypotheses of virtual localization are now in place: $B$ is
projective, Lemma~\ref{der:global-resolution} supplies an equivariant
embedding and global bundle presentations, and
Proposition~\ref{prop:fixed-pot} identifies the fixed obstruction
morphism. Hence \cite[Section~3]{GP} gives
\begin{equation}\label{der:localization}
[B]^{\vir}=\sum_W k_*\frac{[W]^{\vir}}{\eu_G(N_W^{\vir})}
\quad\hbox{in }\CH_*^G(B)\otimes_{\mathbb Q[t]}\mathbb Q(t).
\end{equation}
Here $[W]^{\vir}$ is the ordinary virtual class just constructed, given
its trivial equivariant lift. To see this explicitly, $G$ acts trivially
on $W$, and every equivariant coherent sheaf there decomposes into its
weights. A weight-zero perfect theory is therefore the ordinary theory
with trivial weights. On the mixed quotients defining equivariant Chow
groups, its intrinsic cone and zero-section Gysin construction are the
ordinary ones pulled back from $W$. This identifies its equivariant
virtual class with that lift. Every weight in the normal complex is
nonzero, so the Euler denominator is invertible after localization.
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//derived.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//fixed.tex ====
\section{Fixed virtual classes and the divisor cosection}\label{sec:fixed}
We compute the ordinary fixed virtual classes furnished by
Proposition~\ref{prop:fixed-pot}. Throughout this section, a symbol
\(\Div^{P}(X)\) denotes the scheme of effective divisors whose line bundle
lies in the specified component of \(\Pic(X)\); when that component is a
point, this is the complete linear system \(|P|\).
The universal divisor is denoted by \(\mathcal C\), and \(\pi\) always
denotes projection along the surface \(X\).
All derived Hom complexes below include this derived projection.
We retain the two settings of Section~\ref{sec:derived}; in particular,
\(\chi(\mathcal O_X)=1\) in both, and we put
\[
m_P=\frac{P(P-K)}2=\chi(P)-1.
\]
The calculation has two different forms. When \(p_g=q=0\), each
Picard component is a point and the based Picard tangent vanishes.
The divisor factors are then complete linear systems, and the fixed
loci are smooth; we compute their virtual classes from their
obstruction bundles. On the auxiliary surface the divisor lines can
vary in a Picard curve. We retain this variation in the obstruction
theory, localize one divisor class to the empty or canonical divisor,
and only then restrict the other factor and the insertions to the
resulting fiber. The rank-one family decomposition and point-ideal
comparison below provide the common input to both calculations.
\subsection{Rank-one quotients in families}
The divisor-and-point description and its tautological obstruction
factors appear for a trivial rank-one target on a simply connected
surface in \cite[Section~5.2]{OP}. We record the factorization in
arbitrary families and the obstruction-theory comparisons needed here.
We write \(X^{[r]}\) for the Hilbert scheme of length-\(r\)
zero-dimensional subschemes of \(X\).
\begin{lemma}[Divisor and point factorization]\label{lem:rank-one-family}
Let \(R\) be a noetherian parameter scheme and let
\(L_R\twoheadrightarrow Q\) be a flat family of rank-zero quotients of a
line bundle on \(X\times R\). Its kernel has a unique factorization
\[
S'=L_R(-\mathcal C)\mathcal I_Z,
\]
where \(\mathcal C\) is a relative effective Cartier divisor and
\(Z\subset X\times R\) is finite and flat over \(R\).
Both factors commute with arbitrary base change. Consequently, the
rank-one Quot functor with specified divisor component and point length
\(r\) is represented by
\begin{equation}\label{fix:rank-one-product}
U=\Div^P(X)\times X^{[r]},
\qquad S'=L(-\mathcal C)\mathcal I_Z.
\end{equation}
The assertion includes nonreduced parameter schemes.
\end{lemma}
\begin{proof}
Untwist by \(L_R\), and write \(I\subset\mathcal O_{X\times R}\) for the
kernel. It is flat over \(R\), because the source and quotient are flat.
Every geometric fiber \(I_u\) is torsion-free of rank one on a smooth
surface and has projective dimension at most one. Locally choose a
surjection from a vector bundle \(G\) onto \(I\). Its kernel \(E\) is
flat over \(R\) and has locally free geometric fibers, so is locally
free by the local criterion of flatness. Thus
\[
0\longrightarrow E\longrightarrow G\longrightarrow I\longrightarrow0
\]
is a resolution by vector bundles, and remains a resolution after any
base change.
Set \(M=\det G\otimes(\det E)^{-1}=\det I\). If \(E\) has rank \(e\),
wedging a section of \(G\) with the image of \(\bigwedge^eE\) gives the
maximal-minor map \(G\to M\); it kills \(E\) and induces \(I\to M\).
On the locus where \(I\) is invertible it is the canonical determinant
identification. These local maps glue. To justify this last assertion
over a nonreduced base, the fiber criterion for local freeness shows
that the closed noninvertible locus has finite fibers over \(R\);
properness makes this locus finite over \(R\).
Any closed support with finite fibers in the smooth relative
surface has grade at least two locally: choose two elements of its
ideal which form a regular sequence on the surface fiber and lift
them; successive applications of the local flatness criterion show
that they remain a regular sequence in the total space and that the
successive quotients are base-flat. Powers of these elements
annihilate any coherent sheaf with that support; the regular-sequence
calculation therefore makes its sheaf \(\mathcal{H}om\) and
\(\mathcal Ext^1\) into a line bundle vanish. In particular
a section of a line bundle supported there is zero. Hence two local
minor maps which agree on the invertible locus agree everywhere.
On each geometric fiber the resulting map \(I_u\to M_u\) is the usual
inclusion of a rank-one torsion-free sheaf into its determinant line:
it is injective, and its cokernel has finite support. Fiberwise
injectivity between base-flat modules now implies that \(I\to M\) is
injective with base-flat cokernel. After twisting by \(M^{-1}\) it
is an ideal \(J\subset\mathcal O_{X\times R}\), and
\(\mathcal O/J\) is base-flat with finite fibers. Properness of \(X\)
makes its support finite over \(R\). It therefore defines a finite
flat subscheme \(Z\), with \(I=M\mathcal I_Z\).
The original inclusion \(I\hookrightarrow\mathcal O_{X\times R}\)
extends uniquely to a map \(M\to\mathcal O_{X\times R}\).
Indeed the grade-two observation, applied to the support of
\(M/I=M\mathcal O_Z\), gives
\[
\mathcal{H}om(M\mathcal O_Z,\mathcal O_{X\times R})=
\mathcal Ext^1(M\mathcal O_Z,\mathcal O_{X\times R})=0.
\]
Applying \(\mathcal{H}om(-,\mathcal O_{X\times R})\) to
\(0\to I\to M\to M\mathcal O_Z\to0\) proves the extension and uniqueness.
Its restriction to every surface fiber is nonzero, hence injective.
The same flatness criterion makes it an injection with base-flat
cokernel. It is therefore a relative effective Cartier divisor
inclusion, \(M=\mathcal O(-\mathcal C)\).
The determinant resolutions, minor maps, and short exact sequences
all survive base change, the last because their quotients are
base-flat. The regular sequences used for uniqueness also survive
base change, because their successive quotients are base-flat.
Thus uniqueness identifies the resulting factorization after every
base change, including a nonnoetherian one.
Conversely, multiplying a relative Cartier ideal by a finite flat
point ideal gives a flat quotient: its quotient is an extension of
\(\mathcal O_{\mathcal C}\) by \(\mathcal O_Z(-\mathcal C)\), both
base-flat. This construction inverts the preceding one. More
explicitly, any other line extending \(I\) with relatively finite
cokernel is uniquely identified with \(M\): the same grade-two
extension argument extends the identity on \(I\) in both directions,
and uniqueness makes the composites identities. This proves the
functorial factorization, and hence the asserted isomorphism of
parameter schemes.
\end{proof}
Apply the Lemma to the two weight summands of a fixed quotient. Write
\(D_1=T\), \(D_2=H\), and \(T+H=D\) in the Picard scheme. If their
point lengths are \(n_1,n_2\), comparison of the second Chern characters
gives
\[
n_1+n_2+L_1T+L_2H-\frac{T^2+H^2}{2}
=n+L_1D-\frac{D^2}{2}.
\]
Since \(A=L_1-L_2\), this is
\begin{equation}\label{eq:length}
n_1+n_2=n-(D-A)H+H^2.
\end{equation}
These labels form a finite set on the fixed locus under consideration:
the Picard components, Hilbert polynomials, and point lengths are
locally constant there, and the fixed locus is of finite type.
\subsection{The point-ideal complexes}
The fixed locus has been expressed in divisor and point factors.
We next identify the point contribution to its obstruction theory;
this will separate the Hilbert-scheme tangent from the excess bundle.
For a line bundle \(P\), write
\(P^{[r]}=\pi_*\mathcal O_Z(P)\) for its tautological bundle
of rank \(r\) on \(X^{[r]}\).
For the universal ideal \(\mathcal I\), the trace-free endomorphism
complex means the fiber of
\(\operatorname{tr}:\RHom(\mathcal I,\mathcal I)
\to\RGamma(\mathcal O_X)\).
\begin{lemma}[Trace splitting and the point obstruction]\label{fix:trace}
On \(X^{[r]}\), with universal ideal \(\mathcal I\), there is a
quasi-isomorphism
\begin{equation}\label{fix:hilbert-splitting}
\RHom(\mathcal I,\mathcal I)
\simeq \RGamma(\mathcal O_X)\oplus T_{X^{[r]}}[-1].
\end{equation}
The two maps to the right-hand side are the trace and the inverse of
the trace-free Kodaira--Spencer isomorphism.
On \(\Div^P(X)\times X^{[r]}\), define
\begin{equation}\label{fix:point-bundle}
F=\bigl(\pi_*\mathcal O_Z(K-\mathcal C)\bigr)^\vee.
\end{equation}
It is a vector bundle of rank \(r\), and relative Serre duality gives
\begin{equation}\label{fix:point-duality}
\RHom(\mathcal O_Z,\mathcal O(\mathcal C))=F[-2].
\end{equation}
\end{lemma}
\begin{proof}
The scalar map \(\RGamma(\mathcal O_X)\to
\RHom(\mathcal I,\mathcal I)\) is split by trace, since \(\mathcal I\)
has rank one. At a geometric point \(Z\), extending across its
codimension-two support gives
\[
\Hom(I_Z,I_Z)=\mathbb C,\qquad
\Hom(I_Z,I_Z(K))=H^0(K).
\]
For example, an endomorphism extends to the determinant line and is
there multiplication by a global section; conversely multiplication
preserves the ideal. By Serre duality the trace-free
\(\Ext^2(I_Z,I_Z)\) vanishes, as does the trace-free \(\Ext^0\).
Riemann--Roch gives
\(\chi(I_Z,I_Z)=\chi(\mathcal O_X)-2r\); hence the trace-free
\(\Ext^1\) has dimension \(2r\).
The boundary map of the point-ideal sequence embeds
\[
T_ZX^{[r]}=\Hom(I_Z,\mathcal O_Z)
\longrightarrow \Ext^1(I_Z,I_Z).
\]
Indeed every map \(I_Z\to\mathcal O_X\) is a scalar multiple of
inclusion, so its composite with \(\mathcal O_X\to\mathcal O_Z\)
is zero. This boundary is the Kodaira--Spencer map of the universal
ideal. Its trace vanishes, because the universal point ideal has
constant determinant. Fogarty's smoothness Theorem
\cite[Theorem~2.4]{Fogarty} gives \(\dim X^{[r]}=2r\), so this is an isomorphism
onto the trace-free part. The family complexes are perfect and
commute with derived base change. A map between them is a
quasi-isomorphism if it is so on geometric fibers, proving
\eqref{fix:hilbert-splitting}.
For the other assertion, \(Z\) is finite flat of length \(r\), so
\(\pi_*\mathcal O_Z(K-\mathcal C)\) is locally free of that rank,
with arbitrary base change and no higher direct images. Relative
Serre duality for the smooth proper surface projection identifies
the dual of \(\RHom(\mathcal O_Z,\mathcal O(\mathcal C))\) with
\(\pi_*\mathcal O_Z(K-\mathcal C)[2]\).
Dualizing gives precisely \eqref{fix:point-duality}, including its
shift and the indicated dual bundle.
\end{proof}
For later calculations it follows that the virtual tangent class of
a rank-one arrow is
\begin{equation}\label{fix:arrow-class}
[T_U^{\vir}]
=[R\pi_*\mathcal O(\mathcal C)]-[F]
-[\RGamma(\mathcal O_X)]+[T_{X^{[r]}}].
\end{equation}
In fact it is the cone of
\(\RHom(\mathcal I,\mathcal I)\to
\RHom(\mathcal I,\mathcal O(\mathcal C))\), and the ideal sequence
and \eqref{fix:point-duality} give the formula.
\begin{lemma}[Euler class for a smooth morphism]\label{fix:smooth-euler}
Suppose \(g:Y\to Z\) is smooth and carries a relative perfect
obstruction theory \(E_g\to L_{Y/Z}\) of amplitude \([-1,0]\).
Then \(\Ob_g=h^1(E_g^\vee)\) is a vector bundle,
\[
[\Ob_g]=[T_{Y/Z}]-[E_g^\vee]\quad\hbox{in }K^0(Y),
\]
and its virtual pullback is
\begin{equation}\label{fix:smooth-operation}
g^!\gamma=\eu(\Ob_g)\cap g^*\gamma.
\end{equation}
The formula holds over arbitrary base cycles. For \(Z\) a point,
it says \([Y]^\vir=\eu(\Ob_Y)\cap[Y]\).
\end{lemma}
\begin{proof}
Here \(L_{Y/Z}=\Omega_{Y/Z}\) is a vector bundle in degree zero.
Locally split the surjection from a two-term presentation of \(E_g\)
onto this bundle. The remaining acyclic degree-zero terms can be
cancelled; one obtains
\[
E_g\simeq \Omega_{Y/Z}\oplus\Ob_g^\vee[1],
\]
with comparison map the projection. This shows local freeness
and the stated \(K\)-class, without requiring a global splitting.
The relative intrinsic normal cone of a smooth map is
\([0/T_{Y/Z}]\). The dual comparison morphism gives its global
inclusion into \(h^1/h^0(E_g^\vee)\), with quotient bundle
\(\Ob_g\). Locally this is the inclusion into
\([\Ob_g/T_{Y/Z}]\) supplied by the splitting above; globally the
subbundle excess formula gives its zero-section Gysin image as
the Euler class of \(\Ob_g\). Applying this excess formula over
each base cycle proves \eqref{fix:smooth-operation}.
\end{proof}
\subsection{The fixed class when \texorpdfstring{\(p_g=q=0\)}{p\_g=q=0}}
\begin{proposition}\label{prop:fixed-pg0}
Assume \(p_g=q=0\). A nonempty fixed part with lines \(T,H\)
and lengths satisfying \eqref{eq:length} is the smooth product
\[
W=|T|\times|H|\times X^{[n_1]}\times X^{[n_2]}.
\]
Let \(x_i=c_1(\mathcal O_{|D_i|}(1))\), with \(D_1=T,D_2=H\).
The fixed virtual class is
\begin{equation}\label{fix:pg0-class}
[W]^\vir=
\prod_{i=1}^2\bigl(x_i^{h^1(D_i)}\eu(F_i)\bigr)\cap[W],
\qquad
F_i=\bigl((K-D_i)^{[n_i]}\bigr)^\vee\otimes\mathcal O(1_i).
\end{equation}
A part with either \(m_T<0\) or \(m_H<0\) contributes zero.
For the other parts, after including \(\eu(F_1)\eu(F_2)\),
pushforward over the linear systems is coefficient extraction at
\([x_1^{m_T}x_2^{m_H}]\).
\end{proposition}
\begin{proof}
The Picard scheme is discrete and reduced, and
\(\RGamma(\mathcal I_o)=0\). Thus the determinant condition
leaves just the indicated products, with the sum of their two
rank-one obstruction theories. For an effective line \(P\),
multiplication by any nonzero section of \(P\) injects
\(H^0(K-P)\) into \(H^0(K)=0\), so \(H^2(P)=0\).
On its linear system,
\(\mathcal O(\mathcal C)=P\boxtimes\mathcal O(1)\).
The Euler sequence and \eqref{fix:arrow-class} therefore give
\[
[T_{|P|\times X^{[r]}}]-[T_U^\vir]
= [H^1(P)\otimes\mathcal O(1)]+[F].
\]
Apply Lemma~\ref{fix:smooth-euler}. Equality of these vector
bundle classes gives equality of their Chern classes, and hence
\eqref{fix:pg0-class}; it does not assert a global splitting of
the obstruction bundle.
Finally,
\[
\dim|P|-h^1(P)=h^0(P)-1-h^1(P)=m_P.
\]
If \(m_P<0\), the factor \(x^{h^1(P)}\) is already zero in
\(\CH^*(|P|)\). Otherwise integration against that factor
replaces extraction at degree \(\dim|P|\) by extraction at degree
\(m_P\). The localized moving factors are expanded with nonzero
equivariant constant terms, so introduce no negative hyperplane
powers. Thus this argument applies also to the localized
insertions used below.
\end{proof}
\subsection{A compatible relative theory in the auxiliary setting}
We now assume \(p_g=q=1\), that \(K\) is ample, and that \(D=3K\).
We impose the determinant constraint in one rank-one factor and apply
the divisor cosection to the other.
On each fixed part choose an index \(a\) such that
\(KD_a>K^2\), which is possible since
\(KD_1+KD_2=3K^2\). Call it the high index, and call the other
index \(b\) the low index. If both satisfy the inequality,
choose either one. The ensuing vanishing argument applies to
the resulting low index as well.
\begin{lemma}[Relative determinant pullback]\label{fix:relative-pot}
For the projection \(f:W\to U_b\) to the unrestricted low
rank-one Quot, there is a relative perfect obstruction theory
whose virtual tangent is
\begin{equation}\label{fix:relative-tangent}
T_f^\vir=
\fib\!\left\{\RHom(S'_a,Q_a)
\longrightarrow\RGamma(\mathcal I_o)[1]\right\}.
\end{equation}
It is compatible with the absolute theories of \(W\) and \(U_b\),
and
\begin{equation}\label{fix:relative-class}
[W]^\vir=f^![U_b]^\vir.
\end{equation}
\end{lemma}
\begin{proof}
Use the actual projection of enhancements
\(\mathfrak W\to\mathfrak U_b\) from
Proposition~\ref{prop:fixed-pot}. The tangent of its determinant
fiber in the high direction is \eqref{fix:relative-tangent}.
The map there is the high arrow's boundary followed by trace
and based normalization. Lemma~\ref{der:trace-surjectivity},
with rank one and \(KD_a>K^2\), makes its degree-one map onto
\(H^2(\mathcal O_X)\) surjective, so the fiber has amplitude
\([0,1]\). Explicitly, an annihilating canonical form would
annihilate \(Q_a\); at each generic curve point a quotient of
one generator annihilated by that form has length at most its
vanishing order. Thus \(KD_a\le K^2\), a contradiction.
Restrict the derived cotangent transitivity triangle to \(W\).
It has a natural morphism to the classical transitivity
triangle, giving
\[
\begin{array}{ccccc}
Lf^*E_{U_b}&\longrightarrow&E_W&\longrightarrow&E_f\\
\downarrow&&\downarrow&&\downarrow\\
Lf^*L_{U_b}&\longrightarrow&L_W&\longrightarrow&L_{W/U_b}.
\end{array}
\]
Each row continues by a shift of its first term.
Here the absolute comparisons have cones in degrees at most
\(-2\), by Lemma~\ref{der:truncation}. If these cones are
\(C_{U_b}\) and \(C_W\), and the relative comparison cone is
\(C_f\), the diagram gives a triangle
\[
Lf^*C_{U_b}\longrightarrow C_W\longrightarrow C_f
\longrightarrow Lf^*C_{U_b}[1].
\]
Derived pullback preserves the upper cohomological bound, and
the shift \([1]\) lowers that bound. Hence \(C_f\) is also
concentrated in degrees at most \(-2\). Combined with the
amplitude calculation, this proves that \(E_f\) is a relative
perfect obstruction theory.
We have thus produced the compatible triple, including its
morphisms to the cotangent triangle, required in
\cite[Definition~4.5]{Manolache}. The morphisms are between
quasiprojective schemes, so are of Deligne--Mumford type and
have the required stratifications by global quotients.
Virtual-pullback functoriality
\cite[Theorem~4.8 and Corollary~4.9]{Manolache} gives
\eqref{fix:relative-class}.
\end{proof}
\subsection{The point obstruction as an excess bundle}
Let \(\mathfrak D^P\) denote the divisor enhancement: the open
rank-one arrow stack with invertible source and divisor
quotient. Its classical truncation is \(\Div^P(X)\).
For a divisor \(C\), with source \(M=L(-C)\), its tangent is
\[
\RHom(M,L/M)=R\pi_*\mathcal O_C(C).
\]
It has amplitude \([0,1]\), and its Euler characteristic is
\(\chi(P)-\chi(\mathcal O_X)=m_P\). Thus it induces a divisor
virtual class of dimension \(m_P\).
\begin{lemma}[Compatible point excess]\label{fix:excess}
On the product \(U_b=\Div^{D_b}(X)\times X^{[n_b]}\), the
rank-one Quot virtual class is
\begin{equation}\label{fix:low-product}
[U_b]^\vir=
\eu(F_b)\cap
\bigl([\Div^{D_b}(X)]^\vir\times[X^{[n_b]}]\bigr).
\end{equation}
\end{lemma}
\begin{proof}
There is a morphism of derived moduli problems
\begin{equation}\label{fix:product-enhancement-map}
g:\mathfrak D^{D_b}\times X^{[n_b]}\longrightarrow\mathfrak U_b,
\qquad
(M_b\to L_b,\mathcal I_b)\longmapsto
(M_b\mathcal I_b\to M_b\to L_b).
\end{equation}
The ordinary smooth Hilbert scheme here carries its smooth
enhancement, and its universal ideal is perfect.
Tensoring and composing perfect families define the displayed
map on derived bases as well. Lemma~\ref{lem:rank-one-family}
identifies its truncation with the identity of \(U_b\).
To compute its differential, first forget the arrows and
compare their source moduli. On the common truncation the map is
\begin{equation}\label{fix:source-tangent-map}
\RGamma(\mathcal O_X)[1]\oplus T_{X^{[n_b]}}
\longrightarrow
\RHom(\mathcal I_b,\mathcal I_b)[1].
\end{equation}
The first summand acts by scalars and the second is the
Kodaira--Spencer map of the ideal, so
Lemma~\ref{fix:trace} makes this a quasi-isomorphism.
Indeed, on each geometric fiber the scalar map identifies the trace
summands, and the Hilbert map induces an isomorphism onto the
trace-free \(\Ext^1\). The source derivative is therefore triangular
with isomorphic diagonal blocks. Its perfect cone has zero geometric
fibers, so vanishes; no choice of a global splitting is needed here.
The scalar assertion follows directly by differentiating
tensoring with a line. Before taking sections it is
\(\mathcal O_X[1]\to
R\mathcal{H}om(\mathcal I_b,\mathcal I_b)[1]\),
and corresponds to the identity endomorphism.
The relative arrow tangent map over these source moduli is
\[
\RHom(M_b,L_b)\longrightarrow
\RHom(M_b\mathcal I_b,L_b).
\]
The point-ideal sequence and
\eqref{fix:point-duality} identify its cone with
\[
\RHom(\mathcal O_{Z_b},\mathcal O(\mathcal C_b))[1]
=F_b[-1].
\]
The comparison of the two source tangent triangles, using
\eqref{fix:source-tangent-map}, therefore proves
\begin{equation}\label{fix:excess-triangle}
T_{\mathrm{prod}}^\vir\longrightarrow T_{U_b}^\vir
\longrightarrow F_b[-1]
\longrightarrow T_{\mathrm{prod}}^\vir[1].
\end{equation}
Crucially, dualizing this is a morphism of perfect obstruction
theories \(E_{U_b}\to E_{\mathrm{prod}}\), compatible with
their maps to the same \(L_{U_b}\), because it came from
\eqref{fix:product-enhancement-map}. Thus the induced map
of obstruction bundle stacks is an inclusion
\[
h^1/h^0(T_{\mathrm{prod}}^\vir)
\longhookrightarrowactual
h^1/h^0(T_{U_b}^\vir)
\]
with quotient vector bundle \(F_b\). For precision, on an
affine open the extension in \eqref{fix:excess-triangle}
splits: its class lies in
\(\Hom(F_b,T_{\mathrm{prod}}^\vir[2])=0\), since \(F_b\)
is locally free and the tangent has amplitude \([0,1]\).
This describes the inclusion locally as a vector bundle
substack; the quotient is globally the displayed \(F_b\).
Compatibility of the obstruction morphisms implies that the
intrinsic normal cone embedded using the Quot theory is the
image of its embedding using the product theory.
The subbundle excess formula for zero-section Gysin maps
therefore multiplies the latter virtual class by \(\eu(F_b)\).
One can see the formula on any base cycle by pulling to the
larger bundle stack: the smaller stack is the zero locus of
the quotient-bundle coordinate, so its inclusion multiplies
the pullback class by the top Chern class of that quotient.
This is the zero-section excess formula of
\cite[\S6.3]{Fulton}; its vector-bundle form is also
\cite[Lemma~2.5]{KiemLi}. We apply it to the bundle-stack presentations
of the intrinsic cone in \cite{BF}.
The product theory has class
\([\Div^{D_b}(X)]^\vir\times[X^{[n_b]}]\).
This proves \eqref{fix:low-product} on the possibly singular
divisor scheme.
\end{proof}
\subsection{Localization on the divisor scheme}
The excess comparison separates the low point factor from its divisor
virtual class. We now determine the support of that divisor class
and the sign of its canonical contribution, using the divisor cosection
studied by Chang--Kiem \cite[Section~3.2]{ChangKiem}.
\begin{lemma}[Support of the divisor class]\label{fix:cosection-support}
In the auxiliary setting a divisor component with \(m_P<0\)
has zero virtual class. If \(m_P\ge0\), its virtual class
is supported, by cosection localization, on the empty
divisor or on the unique canonical divisor \(z\).
The empty-divisor class is the reduced point with coefficient
one. Only the Picard component containing \(K\) can give
the canonical contribution.
\end{lemma}
\begin{proof}
The first assertion is the vanishing of a Chow group in
negative dimension. For the second, the universal divisor
sequence yields a cosection
\begin{equation}\label{fix:cosection}
\sigma:\Ob_{\Div^P}
=R^1\pi_*\mathcal O_{\mathcal C}(\mathcal C)
\longrightarrow H^2(\mathcal O_X)\otimes\mathcal O
\simeq\mathcal O.
\end{equation}
At a divisor \(C\) of line class \(P\) the long exact
cohomology sequence ends in
\[
H^1(\mathcal O_C(C))\xrightarrow{\sigma_C}H^2(\mathcal O_X)
\longrightarrow H^2(P)\longrightarrow0.
\]
The obstruction fibers agree with these groups because the
perfect tangent has amplitude \([0,1]\). Thus
\(\coker(\sigma_C)=H^2(P)=H^0(K-P)^\vee\).
The cosection is not surjective exactly where \(K-P\) is
effective. Kiem--Li localization
\cite[Theorem~1.1]{KiemLi} expresses the ordinary virtual
class as the pushforward of a class on this degeneracy locus.
Suppose a degeneracy point exists and put \(r=KP\).
Both \(P\) and \(K-P\) are effective. Since \(K\) is ample
and \(m_P\ge0\), Hodge index gives
\[
0\le r\le K^2,\qquad
r\le P^2\le\frac{r^2}{K^2}.
\]
For \(0<r<K^2\) these inequalities contradict one another.
If \(r=0\), ampleness forces the effective divisor \(C\)
to be empty. If \(r=K^2\), any effective divisor in
\(K-P\) has zero intersection with \(K\), so is empty.
In the latter case \(P\cong K\) as line bundles, and
\(h^0(K)=1\) gives exactly one divisor \(z\).
For the empty divisor the tangent complex
\(R\pi_*\mathcal O_\varnothing\) is zero.
Its parameter scheme is therefore a reduced point with
the ordinary point class. A nontrivial numerically
trivial line has no effective divisor, again by ampleness.
This proves all the support assertions.
\end{proof}
\begin{lemma}[Canonical multiplicity and sign]\label{fix:canonical-sign}
For the divisor component whose Picard component contains
\(K\), there is an integer \(s>0\) such that
\begin{equation}\label{eq:canonical-cycle}
[\Div^{K}(X)]^\vir=-s[z].
\end{equation}
Here \(\Div^{K}(X)\) allows variation of the line in its
Picard component; \(z\) is the unique effective canonical
divisor.
\end{lemma}
\begin{proof}
Let \(C_{\Pic}\) be the Picard component containing \(K\);
it is a smooth projective curve because \(q=1\).
Choose an open neighborhood \(V\) of the point \(K\).
Upper semicontinuity, followed by shrinking, gives
\(h^0(P)\le1\) for \(P\in V\).
For \(P\ne K\) in this component, \(K-P\) is a nontrivial
numerically trivial line, so \(h^2(P)=0\).
Riemann--Roch gives \(\chi(P)=1\), and consequently
\(h^0(P)=1+h^1(P)\ge1\).
At \(K\), \(h^0(K)=1\) by assumption.
Thus \(h^0(P)\) is constantly one on \(V\).
We verify the resulting scheme structure, including
infinitesimal families. Represent the derived direct image
of the based universal line over \(V\), locally, by vector
bundles
\[
E^0\xrightarrow{d^0}E^1\xrightarrow{d^1}E^2.
\]
There are no negative terms because this is the direct image
of a line bundle, and relative dimension two bounds the
positive terms. The kernel of the fiber of \(d^0\) has
constant dimension one. Since \(V\) is reduced, all minors
above the constant rank vanish scheme-theoretically.
After inverting a maximal nonzero minor, elementary row
and column operations turn \(d^0\) into an identity block
and a zero block. Its kernel is therefore a line
subbundle and commutes with arbitrary, including
nonreduced, base change. It is the bundle of sections of
the universal line. The divisor functor over \(V\) is
its projective bundle of nonzero sections up to scalar,
and hence is \(V\) itself. Thus the divisor scheme is
a smooth curve in a neighborhood of \(z\).
The divisor theory has virtual dimension \(m_K=0\).
Lemma~\ref{fix:smooth-euler} makes its obstruction sheaf
an obstruction line \(\mathcal E\) on this neighborhood.
At \(z\) the map \(H^2(\mathcal O_X)\to H^2(K)\) in
the divisor sequence is an isomorphism: its dual maps
\(1\) to the nonzero canonical section.
Hence the cosection \(\mathcal E\to\mathcal O_V\)
vanishes at \(z\). By Lemma~\ref{fix:cosection-support}
it is nonzero at every other point nearby. On the smooth
curve its germ is multiplication by a unit times a
uniformizer to a finite positive power \(s\).
It follows that the line with its cosection is isomorphic to
\[
\bigl(\mathcal O_V(-sz),\
\mathcal O_V(-sz)\hookrightarrow\mathcal O_V\bigr).
\]
Shrink to an affine neighborhood of \(z\).
As in Lemma~\ref{fix:smooth-euler}, the perfect
obstruction theory there splits, with its comparison map, as
\(\Omega_V\oplus\mathcal E^\vee[1]\to\Omega_V\).
The localized intrinsic-cone calculation is thus the
localized Euler class of the zero section in
\(\mathcal E=\mathcal O_V(-sz)\).
For clarity, in the regularizing formula
\cite[Definition~2.3 and equation~(2.1)]{KiemLi},
take the identity modification of \(V\), the quotient line
\(\mathcal O_V(D)\) with \(D=-sz\), and its zero kernel.
The lifted zero cone is \(V\), so the formula gives
\[
[D]\cap[V]=-s[z].
\]
This determines the coefficient of the global localized class:
localized Gysin maps commute with restriction to an open
neighborhood of the degeneracy locus, by restricting the
regularizing modification, its cone cycles, and the ordinary
Gysin maps. This excision is also stated in
\cite[Proposition~2.3]{ChangKiem}.
The support lemma shows that \(z\) is the only degeneracy
point on the whole divisor component. Pushing its localized
class forward, by \cite[Theorem~1.1]{KiemLi}, proves
\eqref{eq:canonical-cycle}.
The argument used smoothness of the Picard curve; it imposed no
smoothness or reducedness condition on the canonical divisor itself.
\end{proof}
Chang and Kiem determine the canonical-divisor virtual coefficient more
precisely as $(-1)^{\chi(\mathcal O_X)}$ for minimal surfaces of general
type with $p_g>0$ \cite[Theorem~1.3 and Corollary~3.4]{ChangKiem}.
Their divisor theory has obstruction sheaf
\(R^1\pi_*\mathcal O_{\mathcal C}(\mathcal C)\)
\cite[Theorem~3.1]{ChangKiem}, and its cosection is
\eqref{fix:cosection} after identifying \(H^2(\mathcal O_X)\) with
\(\mathbb C\) using a nonzero canonical form
\cite[Section~3.2]{ChangKiem}.
On the smooth neighborhood \(V\) above, both theories therefore give
the localized Euler class of the same obstruction line and cosection.
Excision identifies their coefficients, so \(s=1\) here.
The local argument above retains only the sign and nonvanishing needed
here. This sign comes from the divisor cosection; the moving Euler
classes have their own weight signs, computed separately in
Section~\ref{sec:relation}.
\subsection{The high factor and restriction to its fibers}
\begin{proposition}\label{prop:fixed-auxiliary}
Assume \(p_g=q=1\), \(K\) ample, and \(D=3K\).
Every fixed contribution vanishes except those obtained
from a low divisor equal to the empty divisor or to the
canonical divisor. In either surviving case the
contribution is computed on the actual fiber
\[
|D_a|\times X^{[n_a]}\times X^{[n_b]},
\]
with the factors \(\eu(F_a)\eu(F_b)\).
The low hyperplane is \(x_b=0\) and \(m_{D_b}=0\);
pushforward over \(|D_a|\) is extraction at
\(x_a^{m_{D_a}}\).
The overall multiplier is \(1\) for the empty divisor and
\(-s\), with the same \(s>0\) as in
\eqref{eq:canonical-cycle}, for the canonical divisor.
The statement applies to operational insertions in
universal perfect complexes, also after adjoining marked
surface factors.
\end{proposition}
\begin{proof}
By Lemmas~\ref{fix:relative-pot} and \ref{fix:excess},
the fixed class is obtained by applying \(f^!\) to
\[
\eu(F_b)\cap
\bigl([\Div^{D_b}(X)]^\vir\times[X^{[n_b]}]\bigr).
\]
The support and canonical-sign Lemmas make this input
zero unless its divisor class is the point class of the
empty divisor or \(-s[z]\).
Over each of the corresponding low Picard components,
the high line determined by the determinant has
\(D_a-K\) numerically \(2K\) or \(K\), respectively.
It is therefore ample. Kodaira vanishing gives
\(H^j(D_a)=0\) for \(j>0\), uniformly in that component.
Use the based universal low line to obtain the based high
line as \(D-D_b\). Its pushforward is a vector bundle
of rank \(\chi(D_a)=1+m_{D_a}\), with arbitrary base
change. Lemma~\ref{lem:rank-one-family} consequently
expresses \(f\) here as its relative projective bundle
of high divisors, followed by the product with the smooth
Hilbert scheme \(X^{[n_a]}\). In particular \(f\) is smooth,
even if the low divisor scheme away from the degeneracy
point is singular.
Formula \eqref{fix:relative-tangent}, together with
\eqref{fix:arrow-class}, gives
\[
\begin{aligned}
[T_f^\vir]
={}&[R\pi_*\mathcal O(\mathcal C_a)]-[F_a]
+[T_{X^{[n_a]}}]\\
&-[\RGamma(\mathcal O_X)]
-[\RGamma(\mathcal I_o)[1]].
\end{aligned}
\]
The evaluation sequence at \(o\) implies
\[
[\RGamma(\mathcal O_X)]
+[\RGamma(\mathcal I_o)[1]]=1.
\]
Thus the two determinant and scalar terms together
subtract exactly one scalar. Higher cohomology vanishes
for the high line, and the relative projective Euler
sequence now identifies
\[
[T_f^\vir]=[T_f]-[F_a].
\]
Lemma~\ref{fix:smooth-euler} proves, on all base cycles,
\begin{equation}\label{fix:high-operation}
f^!\gamma=\eu(F_a)\cap f^*\gamma.
\end{equation}
Here the use of the \(K\)-class is legitimate: the
compatible relative perfect obstruction theory has
already been constructed, and smoothness identifies
its virtual operation with the Euler class of a vector
bundle. Its Chern classes are determined by the
displayed \(K\)-class.
Apply \eqref{fix:high-operation} to the point-supported
low input class. Compatibility of smooth pullback with
closed pushforward restricts it to the fiber over that
low divisor point, while retaining \(X^{[n_b]}\).
The projection formula similarly restricts every
operational insertion to the fiber. The universal
divisors there are \(D_i\boxtimes\mathcal O(1_i)\),
with the low factor a point; in particular \(x_b=0\).
The high projective space has dimension
\(h^0(D_a)-1=m_{D_a}\), proving the extraction rule.
The universal point ideals, quotient complexes, and
their derived Hom complexes restrict by derived base
change for the smooth proper surface projection.
All of these operations commute with adjoining the
marked surface factors. This proves the asserted
fiber formula with the multipliers already calculated.
\end{proof}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//fixed.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//introduction.tex ====
\section{Introduction}\label{sec:introduction}
For a smooth connected projective complex surface \(S\), write
\[
p_g(S)=h^0(S,\omega_S),\qquad q(S)=h^1(S,\mathcal O_S).
\]
The Chow group $\CH_0(S)$ is the group of integral zero-cycles modulo
rational equivalence, where the relations are generated by divisors
of rational functions on curves in $S$. The degree of a complex point is one, and
$\CH_0(S)^0$ denotes the kernel of the degree map.
The Albanese homomorphism
\(\operatorname{alb}_S:\CH_0(S)^0\to\Alb(S)(\mathbb C)\) is
surjective. Bloch's conjecture predicts that it is an isomorphism when
\(p_g(S)=0\). We prove the conjecture. The main theorem treats the
case \(q=0\), when the Albanese variety is a point and the prediction
is that any two complex points are rationally equivalent.
\begin{theorem}\label{thm:bloch}
Let \(S\) be a smooth connected projective surface over \(\mathbb C\)
with \(p_g(S)=q(S)=0\). Then
\[
\deg:\CH_0(S)\longrightarrow\mathbb Z
\]
is an isomorphism, where \(\CH_0(S)\) has integral coefficients.
\end{theorem}
Theorem~\ref{thm:bloch} imposes no minimality, fundamental-group, or
surface-construction hypothesis. Combining it with the classical
non-general-type case gives the full statement.
\begin{corollary}[Bloch's conjecture for surfaces with \(p_g=0\)]\label{cor:bloch-pg-zero}
Let \(S\) be a smooth connected projective surface over \(\mathbb C\)
with \(p_g(S)=0\). Then the Albanese homomorphism
\[
\operatorname{alb}_S:\CH_0(S)^0\longrightarrow\Alb(S)(\mathbb C)
\]
is an isomorphism of abelian groups, where \(\CH_0(S)\) has integral
coefficients.
\end{corollary}
For a surface of general type, $p_g=0$ forces $q=0$, so
Theorem~\ref{thm:bloch} applies. The other cases are the theorem of
Bloch--Kas--Lieberman \cite{BKL}. We give the precise reduction in
Section~\ref{subsec:integral-proof}.
These zero-cycle statements also have consequences for rational Chow
motives. For $p_g=q=0$, every rational Chow class on a power of $S$
or a Hilbert scheme of points is determined by its cohomology class.
We state the precise consequences and their motivic antecedents after
the proof overview.
\subsection*{History and antecedents}
Mumford showed that a nonzero holomorphic two-form prevents the
zero-cycles on a surface from admitting a finite-dimensional
parametrization modulo rational equivalence \cite{Mumford}.
Bloch's conjecture predicts the converse at the level of the Albanese
kernel: if $p_g=0$, degree and the Albanese image should account for
all rational-equivalence classes of zero-cycles \cite{BlochLectures}.
Bloch, Kas, and Lieberman proved this integral assertion for every
surface of Kodaira dimension less than two \cite[pp.~135--136]{BKL}.
Thus the remaining general-type problem concerns $p_g=q=0$, where
the Albanese variety carries no information and the degree map is
predicted to determine a zero-cycle completely.
Several successful approaches use additional geometry of the surface.
The group-action method of Inose--Mizukami \cite{InoseMizukami} and
its extension by Barlow \cite{Barlow} compare zero-cycles with their
images on suitable quotient surfaces. Bauer's treatment of Inoue
surfaces with $K^2=7$ gives a later application, expressed through
relations among the quotient correspondences
\cite[Theorem~0.2 and Section~1]{BauerInoue}.
Pedrini--Weibel recast involution arguments using the transcendental
motive: for a bidouble cover, vanishing of the transcendental motives
of the three intermediate quotients forces its own vanishing
\cite[Theorem~2.5]{PedriniWeibel}. Their applications include
numerical Godeaux and classical Campedelli examples, Keum--Naie,
Burniat, and Inoue surfaces, giving examples for each $1\le K^2\le8$.
A different route uses families of surfaces and the actions of
correspondences on zero-cycles. For a family of regular surfaces
whose fibered self-product has a rationally connected smooth projective
compactification, Voisin shows that a relative correspondence acting
trivially on holomorphic two-forms acts nilpotently on degree-zero
zero-cycles. She applies this theorem to Catanese surfaces
\cite[Theorems~0.3 and~0.6]{Voisin2014}. Passing to suitable quotients
then gives Bloch's conjecture for Barlow surfaces. This extends the
earlier result for Barlow surfaces carrying extra symmetries to the
general members of that family. The present argument requires no
prescribed family, covering construction, or automorphism of the
surface.
Recent preprints propose proofs with overlapping scope.
Guletski\u\i's Theorem~A assumes algebraicity of total cohomology
and claims vanishing of the Albanese kernel over algebraically closed
fields of characteristic zero \cite[Theorem~A and Section~2.1]{Guletskii}.
Over $\mathbb C$, this cohomological assumption is equivalent to
$p_g=q=0$, by Hodge theory and the Lefschetz $(1,1)$ theorem, so it
covers the full scope of Theorem~\ref{thm:bloch}. His proposed argument studies a hypersurface section through the
diagonal in the square of the surface, together with a nilpotence
argument for correspondences.
Banerjee proposes a separate proof for minimal general-type surfaces
with $p_g=q=0$ and $K^2=9$, using pluricanonical curves and their
Jacobians \cite[Theorems~1.1 and~2.1]{BanerjeeRankOne}.
We cite these as proposed proofs; the argument developed here obtains
its Chow relation from virtual localization on a Quot scheme.
The use of the diagonal to control zero-cycles has its antecedents in
Bloch--Srinivas \cite{BlochSrinivas}. Surface Quot obstruction theories
and localization relations were developed by Marian--Oprea--Pandharipande
\cite{MOP}, with subsequent virtual intersection theory in \cite{OP}
and its $K$-theoretic counterpart in \cite{AJLOP}.
Ciocan-Fontanine--Kapranov constructed derived Quot schemes \cite{CFK}.
We use the derived moduli and determinant constructions of
To\"en--Vaqui\'e and Sch\"urg--To\"en--Vezzosi \cite{TV,STV}
to impose the determinant constraint. The Chow calculation adapts the
nested-Hilbert-scheme recursion of Ellingsrud--G\"ottsche--Lehn
\cite{EGL} by leaving two surface factors unintegrated throughout.
This is the step that produces a correspondence identity rather than
only a numerical intersection formula.
\subsection*{The mechanism of the proof}
The rational cohomology of a surface with $p_g=q=0$ is generated by algebraic
classes, but cohomological identities alone do not settle rational
equivalence. After the classical and birational reductions, we work on
a minimal surface of general type or a point blowup $X$ of that surface.
The objective is a relation
\begin{equation}\label{eq:intro-diagonal}
0=c[\Delta_X]+\Gamma\quad\text{in }\CH^2(X\times X)_{\mathbb Q},
\end{equation}
where $c\ne0$ and $\Gamma$ is a sum of external products of cycles.
The diagonal acts as the identity on zero-cycles, whereas every external
product of total codimension two kills degree-zero zero-cycles.
Thus the relation gives $\CH_0(X)^0\otimes\mathbb Q=0$.
Constructing this Chow relation and detecting its coefficient are the
two main stages.
For line bundles $L_1,L_2,D$, consider rank-zero quotients of
$L_1\oplus L_2$ whose determinant has Picard class $D$.
Opposite torus weights on the summands give a localization formula.
We construct its obstruction theory and the compatible fixed theory
using a derived determinant fiber normalized at a point of $X$
(Sections~\ref{sec:derived} and~\ref{sec:fixed}). A fixed quotient
splits into a quotient of each line bundle, each described by an effective
divisor and a zero-dimensional subscheme. The divisor line bundles
$T,H$ satisfy $T+H=D$; on an irregular surface this determinant
condition also relates their Picard parameters.
Two second-Chern-character insertions of the universal quotient leave
a class on $X\times X$. We choose the virtual dimension so that
equivariant degree forces its codimension-two part to vanish. To calculate it in Chow, we retain both output copies
of $X$ throughout the Hilbert-scheme recursion of
Ellingsrud--G\"ottsche--Lehn~\cite{EGL}. The recursion replaces the
point incidence classes by diagonals identifying surface coordinates.
Terms joining both outputs contribute a multiple of $[\Delta_X]$;
terms with separated outputs contribute external products. For each
fixed fiber expression, this procedure chooses its diagonal coefficient
from the intersection numbers of its line bundles and the Chern
numbers of the surface. The total $c$ is the weighted sum over the
fixed contributions that occur (Section~\ref{sec:relation}). The rule
does not require a diagonal decomposition on an individual surface
to be unique.
For a minimal general-type surface with $p_g=q=0$ and canonical
class $K$, one has $b_2=10-K^2$. When $K^2\le8$, its lattice has a direction orthogonal
to the canonical class. A lattice construction and at most three
point blowups provide a class $\alpha$ with $\alpha^2<0$ and Quot
data with the following property. For each contributing fixed pair
with $\alpha H=0$, pairing with $\alpha\boxtimes\alpha$ kills
its external terms. Every remaining contributing pair has zero
point lengths, so its chosen
diagonal coefficient is zero, while its pairing is $(\alpha H)^2$.
Consequently the full relation gives
\[
0=c\alpha^2+\sum_{\alpha H\ne0}(\alpha H)^2.
\]
The sum runs over a finite set of effective fixed pairs, and a separate
effectivity argument shows that at least one occurs. Since
$\alpha^2<0$, this proves $c>0$ (Section~\ref{sec:nonrank}).
When $K^2=9$, the minimal surface has no such orthogonal direction.
We choose $D=3K$ and arrange that only the numerical divisor pairs
$(3K,0)$ and $(2K,K)$ can contribute to the diagonal coefficient.
Write $c_{\mathrm{out}}$ and $c_{\mathrm{mid}}$ for their respective
contributions, including both orders and all admissible point lengths.
They can be compared on the Cartwright--Steger surface, which has the same
Chern numbers but $p_g=q=1$. The real part of its holomorphic two-form
has nonzero square and is orthogonal to all divisor classes, so pairing
detects the diagonal coefficient. The divisor cosection of
Chang--Kiem~\cite{ChangKiem}, with Kiem--Li
localization~\cite{KiemLi}, reduces its fixed contributions to fibers
over the empty and canonical divisors. These have virtual
multiplicities $1$ and $-s$ with $s>0$. After restricting to those
fibers, the universal rule compares the same two coefficients on both
surfaces and gives $c_{\mathrm{out}}=s c_{\mathrm{mid}}$.
A length-two Hilbert-scheme calculation gives
$c_{\mathrm{mid}}=12$. On the original surface the two coefficients
occur with nonnegative multiplicities, and the outer type occurs once,
so its total coefficient is positive (Section~\ref{sec:rankone}).
Finally, rational vanishing makes each integral degree-zero class
torsion. Roitman's torsion theorem, together with $q=0$, makes it
zero~\cite{BlochRoitman,Roitman}.
The technical constructions used to obtain
\eqref{eq:intro-diagonal} are recorded separately. They include
compatible obstruction theories for the based determinant constraint
(Propositions~\ref{prop:quot-pot} and~\ref{prop:fixed-pot}),
an excess comparison for rank-one Quot schemes
(Lemma~\ref{fix:excess}), and a Chow-valued
universality statement with two marked surface factors. The latter
applies to the tautological pushforwards specified in
Proposition~\ref{prop:universal}, independently of the numerical choices
made for the zero-cycle application.
Figure~\ref{fig:proof-flow} records how the numerical branches feed
into the same diagonal criterion.
\input{figures/proof-flow}
\subsection*{Motivic consequences}
The full zero-cycle statement also has consequences for rational Chow
motives. Write \(h(Y)_{\mathbb Q}\) for the rational Chow motive of a
smooth projective complex variety \(Y\), and \(t_2(S)\) for the
transcendental degree-two summand defined by Kahn--Murre--Pedrini.
Let \(\mathbb L\) denote the effective Lefschetz motive, whose Betti
realization is \(\mathbb Q(-1)\) in degree two. A rational Chow motive
is \emph{pure Tate} if it is a finite direct sum of powers of
\(\mathbb L\).
\begin{corollary}[Motivic consequences]\label{cor:tate-motives}
Let \(S\) be a smooth connected projective surface over \(\mathbb C\)
with \(p_g(S)=0\). Then \(h(S)_{\mathbb Q}\) is finite-dimensional in
the sense of Kimura and \(t_2(S)=0\).
If, in addition, \(q(S)=0\), then
\[
h(S)_{\mathbb Q}\simeq
\mathbf 1\oplus\mathbb L^{\oplus b_2(S)}\oplus\mathbb L^2.
\]
In this case, for every \(m,n\ge0\), the rational Chow motives of the
power \(S^m\) and the Hilbert scheme \(S^{[n]}\) of length-\(n\)
zero-dimensional subschemes are pure Tate, with
\(S^0=S^{[0]}=\operatorname{Spec}\mathbb C\).
For \(Y=S^m\) or \(Y=S^{[n]}\), the rational cycle-class maps satisfy
\[
\operatorname{cl}_Y^i:\CH^i(Y)_{\mathbb Q}
\xrightarrow{\;\sim\;}H^{2i}(Y,\mathbb Q),
\qquad H^{2i+1}(Y,\mathbb Q)=0
\quad\text{for every }i\ge0.
\]
\end{corollary}
These are consequences of established motivic criteria. In the framework
of Kimura's finite-dimensional motives \cite{Kimura},
Guletski\u\i--Pedrini proved the equivalence with Bloch's conjecture
for surfaces with $p_g=0$ and recorded the resulting Tate decomposition when
$q=0$ \cite[Theorem~27 and Remark~28]{GuletskiiPedrini}.
The surface decomposition of Kahn--Murre--Pedrini identifies vanishing
of the Albanese kernel with $t_2(S)=0$ \cite{KMP}; the Hilbert-scheme
assertion then uses
the motive formula of de Cataldo--Migliorini \cite{deCataldoMigliorini}.
In particular, every rational Chow class on the stated powers and Hilbert
schemes is determined by its cohomology class. This conclusion uses
the Tate decomposition, not finite-dimensionality alone.
Section~\ref{subsec:motivic-consequences} gives the proof.
Section~\ref{sec:reductions}
first passes to minimal surfaces of general type.
Section~\ref{sec:derived} constructs the determinant-fixed theory
and its fixed part; Section~\ref{sec:fixed} computes the resulting
virtual classes and the divisor cosection. Section~\ref{sec:relation}
proves the marked Chow relation and its action on zero-cycles.
Sections~\ref{sec:nonrank} and~\ref{sec:rankone} prove the two
nonvanishing statements. Section~\ref{subsec:integral-proof} gives the
integral conclusion, and Section~\ref{subsec:motivic-consequences}
derives the motivic consequences.
\subsection*{Conventions}
All schemes and cohomology groups are over \(\mathbb C\).
The definitions of \(\CH_0(S)\) and \(\operatorname{alb}_S\) above,
Theorem~\ref{thm:bloch}, Corollary~\ref{cor:bloch-pg-zero} and its proof,
Lemma~\ref{lem:birational}, and Section~\ref{subsec:integral-proof} use
integral Chow groups. Elsewhere Chow groups have rational coefficients;
in particular, all Chow motives in Section~\ref{subsec:motivic-consequences}
have rational coefficients.
We use cohomological indexing for complexes. Line bundles are written
additively; in intersections the same symbol denotes their first
Chern class. For a perfect complex, its Euler class with a nonzero
torus weight is interpreted in localized equivariant Chow theory.
Statements about virtual classes always use the specified perfect
obstruction theories. A comparison of their classes in \(K\)-theory
is used to compute Chern classes only after the relevant obstruction
bundle or compatible map of theories has been identified.
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//introduction.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//nonrank.tex ====
\section{Nonvanishing when the minimal intersection lattice has rank at least two}
\label{sec:nonrank}
Let $X_0$ be a minimal surface of general type with $p_g=q=0$, and write
$K_0=K_{X_0}$ and $k_0=K_0^2$. By Proposition~\ref{prop:reduction},
$K_0$ is nef and big and $1\leq k_0\leq9$. This Section treats
$k_0\leq8$.
\begin{proposition}\label{prop:nonrank}
Suppose $1\leq k_0\leq8$. Blow up
\[
e=\max\{k_0-5,0\}
\]
distinct points of $X_0$, and call the resulting surface $X$.
There are choices of $L_1,L_2,D,n,p$ on $X$ for which
Theorem~\ref{thm:chow-relation} gives a relation with $c>0$.
Consequently the degree map on $\CH_0(X_0)_{\mathbb Q}$ is an isomorphism.
\end{proposition}
The proof pairs the Chow relation with $\alpha\boxtimes\alpha$, for a
class $\alpha\in H^2(X,\mathbb R)$ with $\alpha K_X=0$ and
$\alpha^2<0$. We will take $L_1,L_2,D$ to be integral multiples of
$K_X$. A fixed contribution indexed by $T+H=D$ with $\alpha H=0$
then pairs to $c_*\alpha^2$, since its external products pair to zero.
We arrange that every remaining contribution has zero point lengths,
hence $c_*=0$, and has positive pairing. At least one such contribution
will occur, so the vanishing of the full pairing will force $c>0$.
The lattice calculation supplies $\alpha$; the subsequent numerical
bounds isolate the contributions with zero point lengths.
\subsection{A characteristic-vector calculation}
The exponential sequence and $H^1(\mathcal O_{X_0})=
H^2(\mathcal O_{X_0})=0$ identify $\Pic(X_0)$ with
$H^2(X_0,\mathbb Z)$. In particular every integral cohomology class is
the first Chern class of a line bundle. The free quotient
\[
\Lambda=H^2(X_0,\mathbb Z)/\mathrm{torsion}
\]
is unimodular by Poincar\'e duality. Its signature is $(1,b)$ with
$b=9-k_0$, by Hodge index and $b_2(X_0)=10-k_0$.
Riemann--Roch gives
\[
x^2\equiv K_0x\pmod2\qquad(x\in\Lambda),
\]
so $K_0$ is a characteristic vector.
We use the classification of indefinite unimodular integral forms
\cite[Chapter~V, \S2.1, Corollary~1 to Theorem~2, and \S2.2,
Theorems~4--6]{SerreArithmetic}:
an odd form of signature $(1,b)$ is
$I_{1,b}=\langle1\rangle\oplus\langle-1\rangle^{\oplus b}$;
the signature difference of an even unimodular form is divisible by
eight. Since $1\leq b\leq8$, the even case has $b=1$ and is the
hyperbolic plane.
\begin{lemma}\label{nr:lattice-class}
If $\Lambda$ is odd, there is a line bundle $h_0$ on $X_0$ with
\[
h_0^2=-1,\qquad K_0h_0=1.
\]
If $\Lambda$ is even, then $k_0=8$ and there is a line bundle $h_0$ with
\[
h_0^2=0,\qquad K_0h_0=2.
\]
\end{lemma}
\begin{proof}
In the odd case choose diagonal coordinates and write
$K_0=(u;v_1,\ldots,v_b)$. The characteristic property says that all
these integers are odd. Coordinate sign changes and permutations allow
us to assume
\[
u>0,\qquad v_1\geq\cdots\geq v_b>0,
\qquad u^2=9-b+\sum_{i=1}^b v_i^2.
\]
If $v_i=1$, the class $-e_i$, for the corresponding negative basis
vector $e_i$, has square $-1$ and pairing $1$ with $K_0$.
We show by descent on $u$ that this
case can always be reached by integral lattice isometries.
For $b=1$, the equation
$(u-v_1)(u+v_1)=8$ forces $(u,v_1)=(3,1)$: both factors are positive
even integers and the second is strictly larger than the first.
Suppose next that $b\geq3$ and that every $v_i\geq3$. Put
$\ell=v_1+v_2+v_3$. The ordering of the $v_i$ gives
\begin{align*}
\ell^2-u^2
&=2(v_1v_2+v_1v_3+v_2v_3)
-\sum_{i=4}^b v_i^2-(9-b)\\
&\geq (9-b)(v_3^2-1)>0.
\end{align*}
On the other hand,
$\ell^2\leq3(v_1^2+v_2^2+v_3^2)<3u^2<4u^2$.
The vector $z=(1;1,1,1,0,\ldots,0)$ has square $-2$.
Its integral reflection $x\mapsto x+(xz)z$ changes the positive
coordinate of $K_0$ to
\[
u'=2u-\ell,\qquad 0<u'<u.
\]
For $b=2$ and $v_1,v_2\geq3$, put $\ell=v_1+v_2$. Here
\[
\ell^2-u^2=2v_1v_2-7>0,
\qquad \ell^2\leq2(u^2-7)<2u^2<\frac94u^2.
\]
The vector $z=(1;1,1)$ has square $-1$, so its reflection is the
integral isometry $x\mapsto x+2(xz)z$. It changes the positive
coordinate to
\[
u'=3u-2\ell,\qquad 0<u'<u.
\]
In either case the reflected vector remains characteristic and of square
$9-b$. Rechoose the signs and order of its negative coordinates and
repeat. The positive integer $u$ strictly decreases, so the process
must reach a coordinate $v_i=1$. Pulling the corresponding vector back
through the isometries gives the asserted class, which lifts to a line
bundle.
In the even case choose an isotropic basis $f,g$ with $fg=1$.
Writing $K_0=af+bg$, the characteristic property makes $a,b$ even,
and $2ab=K_0^2=8$. After changing both basis signs if necessary,
$a=b=2$. Either $f$ or $g$ gives $h_0$.
\end{proof}
Let $\pi:X\to X_0$ be the blowup in Proposition~\ref{prop:nonrank},
with exceptional curves $E_1,\ldots,E_e$. We usually suppress $\pi^*$
on line and cohomology classes. Thus
\[
K=K_X=\pi^*K_0+\sum_{j=1}^e E_j,
\qquad k=K^2=k_0-e=\min\{k_0,5\}>0.
\]
In the odd case put $h'=\pi^*h_0$. In the even case $k_0=8$ and
$e=3$, so we may put $h'=\pi^*h_0+E_1$. Both choices satisfy
\begin{equation}\label{nr:h-prime}
(h')^2=-1,\qquad Kh'=1.
\end{equation}
The symbol $h'$ denotes a line bundle and is distinct from the insertion
class $h$ on the Quot scheme.
Set $a=K_0h_0$, so $a=1$ in the odd case and $a=2$ in the even case,
and define
\begin{equation}\label{nr:alpha}
\alpha=\pi^*\!\left(h_0-\frac{a}{k_0}K_0\right)
\in H^2(X,\mathbb R).
\end{equation}
Pullback classes are orthogonal to the exceptional curves, whence
\begin{equation}\label{nr:alpha-identities}
\alpha K=0,\qquad
\alpha h'=\alpha^2=h_0^2-\frac{a^2}{k_0}<0.
\end{equation}
\subsection{Positive canonical pairings on the blowup}
The following bound will apply to each of the two effective divisor
classes in a fixed component.
\begin{lemma}\label{nr:positive-pairing}
Let $P$ be an effective line bundle on $X$ such that $m_P\geq0$ and
$\alpha P\ne0$. Then $KP>k$.
\end{lemma}
\begin{proof}
Write
\[
P=\pi^*P_0+\sum_{j=1}^e a_jE_j,
\qquad r_0=K_0P_0,
\]
where the $a_j$ are integers. Pushing forward an effective divisor
representing $P$ shows that $P_0$ is effective; consequently
$r_0\geq0$ by nefness of $K_0$.
The inequality $2m_P=P^2-KP\geq0$ gives
\begin{equation}\label{nr:exceptional-bound}
0\leq\sum_{j=1}^e a_j(a_j-1)
\leq P_0^2-r_0
<\frac{r_0^2}{k_0}-r_0.
\end{equation}
The first inequality holds for all integral $a_j$, including negative
ones. The last is strict Hodge index: $\alpha P\ne0$ implies that
$P_0$ is not numerically proportional to $K_0$. Together with
$r_0\geq0$, this forces $r_0>k_0$.
We next prove $KP>0$. If $e=0$, this is immediate. Otherwise put
$A_0=\sum_j a_j$ and suppose $KP=r_0-A_0\leq0$. By Cauchy--Schwarz,
\[
\sum_j a_j(a_j-1)
\geq\frac{A_0^2}{e}-A_0
\geq\frac{r_0^2}{e}-r_0
>\frac{r_0^2}{k_0}-r_0.
\]
Indeed $A_0\geq r_0>k_0>e$, and $u\mapsto u^2/e-u$ is increasing
on $[r_0,\infty)$. This contradicts
\eqref{nr:exceptional-bound}. Thus $KP>0$.
Since $K^2=k>0$, the intersection form is negative definite on
$K^\perp$. Also $\alpha P\ne0$ and $\alpha K=0$ show that $P$
is not numerically proportional to $K$. Therefore
\[
0<KP\leq P^2<\frac{(KP)^2}{k},
\]
where the middle inequality is $m_P\geq0$. Dividing by $KP>0$
proves $KP>k$. This application of Hodge index requires the positive
square of $K$; all effectivity bounds above used the nef class $\pi^*K_0$.
\end{proof}
\subsection{All contributions with nonzero divisor pairing}
We choose the Quot data so that every fixed contribution detected by
the divisor pairing with \(\alpha\) has zero point lengths. This will
make those contributions explicit, while the orthogonal terms account
for the diagonal coefficient. Set
\begin{equation}\label{nr:data}
\begin{array}{c|cc}
&d&p\\\hline
k=1&4&0\\
2\leq k\leq5&3&k-2
\end{array}
\qquad
D=dK,\quad L_1=0,\quad L_2=K,\quad n=d(k+1).
\end{equation}
Here $A=-K$, so \eqref{eq:vdim} gives $v=2d$.
For $k=1$ we have $v=8>2=p+2$; for $2\leq k\leq5$ we have
$v=6>k=p+2$. Thus Theorem~\ref{thm:chow-relation} applies.
This degree inequality is the reason for using blowups to reduce $k$
to at most five.
By Proposition~\ref{prop:fixed-pg0}, only effective line decompositions
$T+H=D$ with $m_T,m_H\geq0$ need be retained. Write
\[
r=KH,\qquad s=H^2,\qquad N=n_1+n_2.
\]
For these data the length constraint \eqref{eq:length} is
\begin{equation}\label{nr:length-specialized}
N=d(k+1)-(d+1)r+s\geq0.
\end{equation}
\begin{lemma}\label{nr:surviving-numerics}
Every such fixed contribution with $\alpha H\ne0$ satisfies
\begin{equation}\label{nr:survivors}
KH=H^2=k+1,\qquad
n_1=n_2=0,\qquad m_H=0,\qquad m_T=p.
\end{equation}
\end{lemma}
\begin{proof}
Since $\alpha T=-\alpha H\ne0$, Lemma~\ref{nr:positive-pairing}
applies to both $H$ and $T$. It gives
\[
r>k,\qquad dk-r>k,\qquad r\leq s<\frac{r^2}{k}.
\]
Furthermore $s-r$ is even by Riemann--Roch.
If $k=1$, then $d=4$ and the first two bounds force $r=2$.
The inequalities $2\leq s<4$ and $s\equiv2\pmod2$ force $s=2$.
Equation~\eqref{nr:length-specialized} gives $N=0$.
For $2\leq k\leq5$, write $r=k+j$. Since $d=3$, the two strict
canonical bounds give $1\leq j\leq k-1$. Use the parity condition
by writing $s=r+2m$, where $m=m_H$ is a nonnegative integer.
The strict Hodge bound gives
\begin{equation}\label{nr:length-bound}
2m<\frac{r^2}{k}-r=j+\frac{j^2}{k}<2j.
\end{equation}
Consequently $m\leq j-1$. Substituting into
\eqref{nr:length-specialized} yields
\begin{equation}\label{nr:length-parity}
0\leq N=3(1-j)+2m\leq1-j.
\end{equation}
It follows that $j=1$, then $m=0$ and $N=0$, so $r=s=k+1$.
This integral bound excludes every $j\geq2$, including the boundary
values for $k=3$ and $k=4$, without a separate numerical enumeration.
In every case $r=s=k+1$ and $N=0$. Since $n_1,n_2\geq0$, they
both vanish. Finally
\[
2m_T=(dK-H)^2-K(dK-H)
=d(d-1)k-2(d-1)(k+1),
\]
which equals $0$ for $(k,d)=(1,4)$ and $2(k-2)$ for $d=3$.
This proves all assertions.
\end{proof}
For reference, the possibilities proved above are
\begin{equation}\label{nr:numerical-table}
\begin{array}{c|cc|cc|c|cc}
k&d&p&KH&H^2&n_1+n_2&m_T&m_H\\\hline
1&4&0&2&2&0&0&0\\
2&3&0&3&3&0&0&0\\
3&3&1&4&4&0&1&0\\
4&3&2&5&5&0&2&0\\
5&3&3&6&6&0&3&0
\end{array}.
\end{equation}
\subsection{An effective pair and the sign of its contribution}
\begin{lemma}\label{nr:effective-pair}
The actual line bundles
\[
H=K+h',\qquad T=(d-1)K-h'
\]
are effective and satisfy \eqref{nr:survivors}, with $\alpha H\ne0$.
\end{lemma}
\begin{proof}
Equation~\eqref{nr:h-prime} gives
$KH=H^2=k+1$. The calculations in the preceding proof then give
$m_H=0$, $m_T=p$, and $N=0$. Consequently
\[
\chi(H)=1,\qquad \chi(T)=p+1>0.
\]
Let $B_0=\pi^*K_0$, a nef divisor on $X$. Its pairings with the
Serre-dual line bundles are
\[
B_0(K-H)=-a<0,\qquad
B_0(K-T)=(2-d)k_0+a<0.
\]
For the second inequality, in the odd case its value is $-1$ when
$k_0=1$ and $1-k_0<0$ when $k_0\geq2$; in the even case it is
$-8+2=-6$. Thus neither $K-H$ nor $K-T$ has a section, since an
effective divisor has nonnegative intersection with a nef divisor.
Serre duality gives $H^2(H)=H^2(T)=0$, and Riemann--Roch now yields
\[
h^0(H)=\chi(H)+h^1(H)>0,\qquad
h^0(T)=\chi(T)+h^1(T)>0.
\]
Both line bundles are therefore effective. Finally
$\alpha H=\alpha h'=\alpha^2<0$ by
\eqref{nr:alpha-identities}.
Thus this pair occurs in the fixed locus with both point subschemes
empty, by Lemma~\ref{lem:rank-one-family}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:nonrank}]
Use the data \eqref{nr:data} and pair the Chow relation of
Theorem~\ref{thm:chow-relation} with $\alpha\boxtimes\alpha$ in
cohomology. We compute the contributions using
\eqref{rel:fiber-contribution} and \eqref{rel:pairing}.
For any contribution with $\alpha H=0$, the class $\alpha$ is
orthogonal to $K,L_1,L_2,T,H$. Thus its external-product part has
zero pairing, and it contributes $c_*\alpha^2$ by
Proposition~\ref{prop:universal}.
For every remaining contribution, Lemma~\ref{nr:surviving-numerics}
gives $n_1=n_2=0$. Its chosen diagonal coefficient $c_*$ is therefore
zero by Proposition~\ref{prop:universal}. Its point Hilbert factors
and fixed Euler factors are trivial. Formula~\eqref{eq:insertion}
gives
\[
\mu_\alpha=(\alpha H)(-w_1+w_2+x_1-x_2)
=(\alpha H)(-1+x_1-x_2).
\]
Since $m_T=p$ and $m_H=0$, its full pairing is
\begin{align*}
&[x_1^p x_2^0]\,
\mathcal R(x_1,x_2)(x_1+x_2)^p
(\alpha H)^2(-1+x_1-x_2)^2\\
&\hspace{35mm}=(\alpha H)^2\mathcal R(0,0).
\end{align*}
Here all denominators in $\mathcal R$ have nonzero constant term.
Setting $x_2=0$ and extracting $x_1^p$ therefore uses only the constant
term of the factors multiplying $x_1^p$.
We compute this constant exactly. Since there are no point factors,
\eqref{eq:moving-ratio} reduces at $(x_1,x_2)=(0,0)$ to the scalar
weight powers. The $i=2,j=1$ weight is $1$, and the $i=1,j=2$
weight is $-1$, with
\[
P_{12}=T+K,\qquad B_{12}=T-H+K.
\]
Thus
\begin{align*}
\mathcal R(0,0)
&=(-1)^{\chi(T-H+K)-\chi(T+K)}\\
&=(-1)^{-TH+(H^2-KH)/2}
=(-1)^{-TH}.
\end{align*}
The second equality follows by expanding Riemann--Roch, and the last
uses $H^2=KH$. Moreover
\[
TH=(dK-H)H=(d-1)(k+1)
=\begin{cases}6,&k=1,\\2(k+1),&2\leq k\leq5,\end{cases}
\]
which is even. Hence every such term has
\begin{equation}\label{nr:moving-constant}
\mathcal R(0,0)=1.
\end{equation}
Let the sum below run over the effective ordered line decompositions
occurring in the fixed locus with $\alpha H\ne0$, retaining distinct
torsion twists as distinct terms. This is a finite sum, as the fixed
locus has only finitely many component data. It is nonempty by
Lemma~\ref{nr:effective-pair}. Summing the two kinds of contributions
therefore gives
\begin{equation}\label{nr:positive-coefficient}
0=c\alpha^2+\sum_{\alpha H\ne0}(\alpha H)^2,
\qquad
c=-\frac{\displaystyle\sum_{\alpha H\ne0}(\alpha H)^2}{\alpha^2}>0.
\end{equation}
In this equality $c$ is the coefficient already chosen in the Chow
relation: the second group of terms has $c_*=0$, and the first group
accounts for its entire sum. The strict positivity follows from
$\alpha^2<0$ and the nonempty sum of positive real squares.
Corollary~\ref{rel:correspondence} now gives
the rational degree isomorphism on $X$. Lemma~\ref{lem:birational}
passes it to $X_0$, proving the Proposition.
\end{proof}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//nonrank.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//paper.tex ====
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\hypersetup{
pdftitle={Bloch's conjecture for surfaces with pg=0},
pdfauthor={OpenAI},
pdfsubject={The integral Albanese map for complex surfaces with pg=0},
pdfkeywords={Bloch conjecture, zero-cycles, Quot schemes, virtual localization, Hilbert schemes}
}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\setlist[enumerate]{topsep=4pt,itemsep=2pt,parsep=0pt}
\title{Bloch's conjecture for surfaces with \(p_g=0\)}
\author{OpenAI}
\date{September 24, 2026}
\usepackage{accsupp}
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\begin{document}
\maketitle
\begin{abstract}
We prove Bloch's conjecture for smooth connected projective complex
surfaces with \(p_g=0\): the Albanese homomorphism on integral
degree-zero zero-cycles is an isomorphism.
\end{abstract}
\input{sections/introduction}
\input{sections/reductions}
\input{sections/derived}
\input{sections/fixed}
\input{sections/relation}
\input{sections/nonrank}
\input{sections/rankone}
\input{sections/conclusion}
\input{bibliography}
\end{document}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//paper.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//proof-flow.tex ====
\begin{figure}[ht]
\centering
\small
\begin{tabular}{c}
\fbox{\begin{minipage}{0.86\linewidth}\centering
Two marked surface factors in a determinant-fixed Quot calculation\\
Sections~\ref{sec:derived}--\ref{sec:relation}
\end{minipage}}\\[3pt]
$\Downarrow$\\[3pt]
$0=c[\Delta_X]+\Gamma$ in $\CH^2(X\times X)_{\mathbb Q}$,
with $\Gamma$ a sum of external products\\[8pt]
\begin{tabular}{c@{\hspace{1.4em}}c}
\fbox{\begin{minipage}{0.39\linewidth}\centering
$K_{X_0}^2\leq 8$\\
Lattice class and at most three blowups\\
$\alpha^2<0$,\quad $0=c\alpha^2+\sum(\alpha H)^2$\\
A nonempty sum forces $c>0$\\
Section~\ref{sec:nonrank}
\end{minipage}}
&
\fbox{\begin{minipage}{0.39\linewidth}\centering
$K_{X_0}^2=9$\\
Cartwright--Steger comparison\\
$c_{\mathrm{out}}=s c_{\mathrm{mid}}$,\quad $c_{\mathrm{mid}}=12$\\
$c=12(s+\tau_{X_0})>0$\\
Section~\ref{sec:rankone}
\end{minipage}}
\end{tabular}\\[3pt]
$\Downarrow$\\[3pt]
$\CH_0(X_0)^0\otimes\mathbb Q=0$\\[3pt]
$\Downarrow\quad\text{Roitman's torsion theorem and }q=0$\\[3pt]
$\deg:\CH_0(S)\xrightarrow{\sim}\mathbb Z$
\end{tabular}
\caption{The general-type proof, after passage to the minimal model
$X_0$ of $S$. Each branch chooses Quot data and proves that the
resulting diagonal coefficient is nonzero. In the left branch,
the displayed squares come from the external-product terms with
$\alpha H\ne0$; those terms have zero diagonal coefficient.
In the right branch, the outer and middle coefficients are defined
by the same numerical rule on the target and auxiliary surfaces;
$-s$ is the canonical-divisor virtual coefficient on the auxiliary
surface, with $s>0$, and $\tau_{X_0}$ counts the nontrivial torsion
line bundles. Birational invariance returns the conclusion to $S$.}
\label{fig:proof-flow}
\end{figure}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//proof-flow.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//rankone.tex ====
\section{The case \texorpdfstring{\(K^2=9\)}{K squared equals 9}}
\label{sec:rankone}
We now treat a minimal surface with $p_g=q=0$ and $K^2=9$, whose
intersection lattice has rank one. The argument will reduce its diagonal
coefficient to two numerical divisor types. These types occur with
nonnegative multiplicities on the given surface. On an auxiliary surface
with the same Chern numbers, the canonical-divisor type instead has a
negative virtual multiplicity. Comparing the two relations will leave
just one coefficient to evaluate, on a Hilbert scheme of length two.
Throughout this Section, $c_*$ means the coefficient chosen by the fixed
recursion of Proposition~\ref{prop:universal}. Its numerical selection
rule is the same on both surfaces; uniqueness of a diagonal decomposition
on either surface is not required.
\begin{proposition}\label{prop:rankone}
Let $X$ be a smooth minimal projective complex surface of general type
with $p_g(X)=q(X)=0$ and $K_X^2=9$. Take
\begin{equation}\label{r1:data}
L_1=L_2=0,\qquad D=3K_X,\qquad n=20,\qquad p=7.
\end{equation}
The coefficient $c$ in the Chow relation of
Theorem~\ref{thm:chow-relation} is strictly positive.
\end{proposition}
\subsection{Effective line classes and their multiplicities}
Write $K=K_X$. The reductions in Section~\ref{sec:reductions} give
$b_2(X)=1$ and $\int_X c_2(T_X)=3$. The exponential sequence, using
$H^1(X,\mathcal O_X)=H^2(X,\mathcal O_X)=0$, identifies
$\Pic(X)$ with $H^2(X,\mathbb Z)$. Its quotient by torsion has a
positive unimodular intersection pairing of rank one. Choose its
generator $G$ on the ray of an ample divisor. Then
\[
G^2=1,\qquad K\equiv 3G,
\]
where $\equiv$ denotes numerical equivalence. Indeed $K^2=9$ gives the
coefficient $\pm3$, and the nef and big canonical class has positive
intersection with an ample divisor. Every irreducible curve has a
positive coefficient on the same ray: its intersection with an ample
divisor is positive. Consequently $K$ has positive intersection with
every curve, and $K^2>0$. The Nakai--Moishezon criterion proves that
$K$ is ample.
For the data \eqref{r1:data}, Equation~\eqref{eq:vdim} gives
\[
v=2\cdot20-K(3K)=13>9=p+2.
\]
Write $H\equiv aG$ and $T\equiv(9-a)G$. Effectivity of both lines
implies $0\leq a\leq9$. Their virtual divisor dimensions and total
point length, by Equation~\eqref{eq:length}, are
\begin{equation}\label{r1:numerical-classes}
m_H=\frac{a(a-3)}2,\qquad
m_T=\frac{(9-a)(6-a)}2,\qquad
n_1+n_2=20-9a+a^2.
\end{equation}
Here and below an effective numerical class can represent several
distinct line bundles; their contributions must all be counted.
The complete numerical list is
\[
\begin{array}{c|rrrrrrrrrr}
a&0&1&2&3&4&5&6&7&8&9\\ \hline
m_H&0&-1&-1&0&2&5&9&14&20&27\\
m_T&27&20&14&9&5&2&0&-1&-1&0\\
n_1+n_2&20&12&6&2&0&0&2&6&12&20
\end{array}
\]
Proposition~\ref{prop:fixed-pg0} eliminates $a=1,2,7,8$ because one
of the two virtual divisor dimensions is negative. At $a=4,5$ the
point lengths are both zero, so Proposition~\ref{prop:universal} gives
$c_*=0$. Only the outer numerical type $(3K,0)$ and the middle
numerical type $(2K,K)$, with both orders included, can contribute to
$c$.
A numerically trivial effective line bundle is trivial: a nonzero
effective divisor has positive intersection with an ample divisor,
whereas a section with empty zero divisor trivializes its line bundle.
Hence the outer type consists of the single unordered pair $(3K,0)$.
The lines numerically equivalent to $K$ are precisely $K+\epsilon$,
where $\epsilon$ runs over $\Pic(X)_{\mathrm{tors}}$. This torsion
group is finite, since $H^2(X,\mathbb Z)$ is finitely generated. The
untwisted line $K$ has no section. For $\epsilon\neq0$, the preceding
observation gives $H^0(X,-\epsilon)=0$, and therefore
\[
h^2(X,K+\epsilon)=0,\qquad
\chi(K+\epsilon)=1,\qquad
h^0(X,K+\epsilon)=1+h^1(X,K+\epsilon)>0.
\]
Every such line is effective. Its complement $2K-\epsilon$ is also
effective: $(2K-\epsilon)-K$ is ample, so Kodaira vanishing and
Riemann--Roch give $h^0(X,2K-\epsilon)=10$. The same argument gives
$h^0(X,3K)=28$. Thus all the required high lines occur.
Define
\begin{equation}\label{r1:torsion-count}
\tau_X=\#\bigl(\Pic(X)_{\mathrm{tors}}\setminus\{0\}\bigr).
\end{equation}
The notation $\tau_X$ keeps this count distinct from the equivariant
parameter $t$. Let $c_{\mathrm{out}}$ be the sum of the universal
coefficients for the two orders of the numerical pair $(3K,0)$,
including every distribution $n_1+n_2=20$. Let
$c_{\mathrm{mid}}$ be the analogous sum for the two orders of
$(2K,K)$ and all distributions $n_1+n_2=2$. These numbers are
defined by Equation~\eqref{rel:fiber-contribution} and
Proposition~\ref{prop:universal}, including when the indicated precise
line bundles are not effective on $X$. Their numerical inputs are
$K^2=9$, $\int_Xc_2(T_X)=3$, $\chi(\mathcal O_X)=1$, and
\eqref{r1:data}.
For each nontrivial $\epsilon$, the two orders of
$(2K-\epsilon,K+\epsilon)$ have exactly these middle numerical inputs.
Each unordered pair is counted once by $\epsilon$, since its two
members have different numerical classes. The linear systems need
not have the same actual dimension as $\epsilon$ varies; their
$H^1$ obstruction factors have already been incorporated in the
coefficient extraction of Proposition~\ref{prop:fixed-pg0}.
It follows that the total coefficient on $X$ is
\begin{equation}\label{r1:target-coefficient}
c=c_{\mathrm{out}}+\tau_Xc_{\mathrm{mid}}.
\end{equation}
The outer sum involves Hilbert schemes of total length $20$, whereas the
middle sum has total length $2$. We will use the auxiliary relation to
express $c_{\mathrm{out}}$ as a positive multiple of $c_{\mathrm{mid}}$,
avoiding the length-$20$ calculation. Positivity will then follow from
the explicit length-two computation.
\subsection{Comparison on the Cartwright--Steger surface}
Let $Y$ be the Cartwright--Steger surface. Its existence as a smooth
projective compact two-ball quotient follows from the torsion-free
lattice in \cite[Theorem~2]{CKY}. Its invariants are
\begin{equation}\label{r1:auxiliary-invariants}
K_Y^2=9,\qquad \int_Yc_2(T_Y)=3,\qquad
\chi(\mathcal O_Y)=1,\qquad q(Y)=p_g(Y)=1;
\end{equation}
see \cite[Main Theorem~(a) and Lemma~3]{CKY}.
The Bergman metric on the ball descends to $Y$, and its negative
Ricci form gives a positive curvature form on $K_Y$. Kodaira's
positivity criterion therefore makes $K_Y$ ample. Thus $Y$ satisfies
all the hypotheses of the auxiliary construction of
Proposition~\ref{prop:fixed-auxiliary} with $D=3K_Y$.
Choose a nonzero $\omega\in H^0(Y,K_Y)$ and set
$\alpha=[\omega+\overline\omega]\in H^2(Y,\mathbb R)$. Holomorphic
top forms are closed, and
\begin{equation}\label{r1:test-class}
\alpha^2:=\int_Y\alpha\smile\alpha
=2\int_Y\omega\wedge\overline\omega>0.
\end{equation}
The inequality follows in local holomorphic coordinates from positivity
of the associated real volume form. Hodge type gives
$\alpha\cdot P=0$ for every divisor class $P$ on $Y$.
\begin{lemma}\label{r1:auxiliary-comparison}
For the common universal coefficients defined above, one has
\begin{equation}\label{r1:comparison}
c_{\mathrm{out}}-s c_{\mathrm{mid}}=0,
\qquad s>0,
\end{equation}
where $-s$ is the canonical-divisor coefficient in
Equation~\eqref{eq:canonical-cycle} for $Y$.
\end{lemma}
\begin{proof}
Apply the construction on $Y$ with the data \eqref{r1:data}.
Proposition~\ref{prop:fixed-auxiliary} first replaces the low divisor
virtual class by the empty-divisor point with coefficient $1$, or by
the canonical-divisor point with coefficient $-s$. The compatible virtual
pullback along the high morphism is its smooth pullback multiplied by
the high point-obstruction Euler class. Thus the surviving terms are computed on
\[
|3K_Y|\times Y^{[n_1]}\times Y^{[n_2]}
\quad\text{or}\quad
|2K_Y|\times Y^{[n_1]}\times Y^{[n_2]},
\]
with total point lengths $20$ and $2$, respectively, and with both fixed
point-obstruction Euler factors retained. The high linear systems have
dimensions $27$ and $9$. The low divisor is fixed, so its hyperplane
variable is zero. Both orders occur, and the canonical multiplier is
$-s$ in each order.
The projection formula restricts all operational insertions to these
actual fibers. Derived base change likewise restricts the moving
complexes, giving exactly Equation~\eqref{eq:moving-ratio}.
Consequently the remaining formulas are those defining
$c_{\mathrm{out}}$ and $c_{\mathrm{mid}}$, with their indicated
multipliers. This restriction precedes the use of universality: no
integration over a varying Picard component is being compared by
numerical invariants.
After that restriction, the equality of the numerical data in
\eqref{r1:auxiliary-invariants} with those for $X$ permits
Proposition~\ref{prop:universal} to identify the chosen coefficients.
The resulting Chow relation on $Y^2$ has diagonal coefficient
$c_{\mathrm{out}}-s c_{\mathrm{mid}}$. Pair it with
$\alpha\boxtimes\alpha$. Every external term vanishes: bidegree
allows only a divisor on each factor to pair nontrivially, and each
such divisor is orthogonal to $\alpha$. The diagonal pairs to
$\alpha^2$. Equation~\eqref{r1:test-class} now gives
\eqref{r1:comparison}.
\end{proof}
\subsection{The middle coefficient}
\begin{lemma}\label{lem:middle-coefficient}
For the data \eqref{r1:data}, with numerical inputs
$K^2=9$ and $\int c_2=3$, the two-order middle coefficient is
\[
c_{\mathrm{mid}}=12.
\]
\end{lemma}
\begin{proof}
Compute on $Y$ with the class $\alpha$ in \eqref{r1:test-class},
omitting the overall canonical multiplier $-s$. We compute the order
$T=2K_Y$, $H=K_Y$ first, and verify equality with the reverse order
at the end. For this order,
\[
n_1+n_2=2,\qquad m_T=9,\qquad m_H=0.
\]
The three possible point-length distributions are $(2,0)$, $(1,1)$,
and $(0,2)$. Taking the coefficient of $x_2^0$ means setting
$x_2=0$: every moving weight in
Equation~\eqref{eq:moving-ratio} has nonzero constant term $1$ or
$-1$ after $t=1/2$, so all its inverse Euler factors expand with
nonnegative powers of $x_1,x_2$. This remains true for virtual
bundles, by expanding their Chern classes at the nonzero weight.
If $n_2>0$, the low fixed factor at $x_2=0$ is
\[
e\bigl((\mathcal O_Y^{[n_2]})^\vee\bigr)=0.
\]
Indeed the constant function $1$ is a nowhere-zero section of the
rank-$n_2$ tautological bundle, so its top Chern class, and hence
that of its dual, vanishes. Equivalently its Euler polynomial is
divisible by $x_2$; there is no negative power of $x_2$ in the other
factors to cancel it. Therefore only $(n_1,n_2)=(2,0)$ contributes.
Put
\[
U=(-K_Y)^{[2]},\qquad N=\mathcal O_Y^{[2]},\qquad x=x_1,
\qquad
\mu=\int_Y[Z_1]\alpha\in H^2(Y^{[2]},\mathbb R).
\]
All the other terms of Equation~\eqref{eq:insertion} integrate to
zero against $\alpha$, by its orthogonality to divisors and by
degree. Consequently $\mu$ is the class used in the pairing
formula \eqref{rel:pairing}.
For this order, the line data in the moving ratio are
\[
P_{12}=2K_Y,\qquad P_{21}=K_Y,\qquad
B_{12}=K_Y,\qquad B_{21}=-K_Y.
\]
Riemann--Roch gives
$\chi(K_Y)=1$ and $\chi(2K_Y)=\chi(-K_Y)=10$. The scalar part of
the moving ratio is therefore
\begin{equation}\label{r1:scalar-ratio}
\frac{(x-1)^1}{(x-1)^{10}}
\frac{(1-x)^{10}}{1^1}=-(1-x).
\end{equation}
Since the second ideal is $\mathcal O_Y$, its two point-complex
terms are
\begin{equation}\label{r1:point-complexes}
E_{12}(K_Y)=N^\vee,\qquad E_{21}(-K_Y)=U
\quad\text{in }K(Y^{[2]}).
\end{equation}
For the first identity, apply $\RHom(-,K_Y)$ to the point-ideal
sequence: the difference
$[\RGamma(K_Y)]-[\RHom(\mathcal I_1,K_Y)]$ is
$[\RHom(\mathcal O_{Z_1},K_Y)]=[N^\vee[-2]]=[N^\vee]$.
For the second, apply derived sections to that sequence tensored
by $-K_Y$. The even shift in the first identity contributes no
minus sign in $K$-theory.
For any rank-two bundle $F$ one has
$e_{-u}(F^\vee)=e_u(F)$. Hence the moving point numerator
$e_{x-1}(U^\vee)$ cancels the denominator $e_{1-x}(U)$, and the
remaining denominator $e_{x-1}(N^\vee)$ equals $e_{1-x}(N)$.
Including the high fixed Euler factor $e_x(U^\vee)$ and the
insertion $x^7$, the pairing for this order is
\begin{equation}\label{r1:middle-integral}
I=-[x^2]\int_{Y^{[2]}}
\mu^2(1-x)\frac{e_x(U^\vee)}{e_{1-x}(N)}.
\end{equation}
Here is the required homogeneous expansion. Set
$u_i=c_i(U)$ and $\nu_i=c_i(N)$. Then
\[
e_x(U^\vee)=x^2-u_1x+u_2,\qquad
e_{1-x}(N)=(1-x)^2+\nu_1(1-x)+\nu_2,
\]
and, through degree two in the Chern classes on $Y^{[2]}$,
\[
\frac{1-x}{e_{1-x}(N)}
=\frac1{1-x}-\frac{\nu_1}{(1-x)^2}
+\frac{\nu_1^2-\nu_2}{(1-x)^3}
+\text{terms of Chern degree at least three}.
\]
The class $\mu^2$ has real cohomological degree four and $Y^{[2]}$
has complex dimension four. Thus only Chern degree two in the
remaining factor can contribute to the integral. Multiplication
by $x^2-u_1x+u_2$ and extraction of $x^2$ in that degree gives
$u_2+2u_1\nu_1+\nu_1^2-\nu_2$. Equation~\eqref{r1:middle-integral}
therefore becomes
\begin{equation}\label{r1:chern-integral}
I=-\int_{Y^{[2]}}\mu^2
\bigl(c_2(U)+2c_1(U)c_1(N)+c_1(N)^2-c_2(N)\bigr).
\end{equation}
We have reduced the middle coefficient to four tautological Chern
terms in \eqref{r1:chern-integral}. To evaluate them, consider the
consecutive nested Hilbert scheme
\[
\widetilde{Y^2}=Y^{[1,2]}=\operatorname{Bl}_{\Delta_Y}(Y^2),
\qquad
\pi:\widetilde{Y^2}\longrightarrow Y^2,
\qquad
\psi:\widetilde{Y^2}\longrightarrow Y^{[2]}.
\]
The first surface coordinate records the length-one subscheme and
the second the residual point. The morphism $\psi$ is proper and
generically of degree two, since either point of a reduced length-two
subscheme can be selected as its length-one subscheme. The
projectivization description and the tautological sequence are those
of \cite[\S1, equations~(3)--(6), and Lemma~2.1]{EGL}, specialized
to length one. Write $E$ for the exceptional divisor and
$p_i=\operatorname{pr}_i\circ\pi$. The tautological quotient line
of the projectivized diagonal ideal is $\mathcal O(-E)$.
Consequently, for any line bundle $P$ on $Y$, the tautological
sequence is
\begin{equation}\label{r1:tautological-sequence}
0\longrightarrow p_2^*P\otimes\mathcal O(-E)
\longrightarrow\psi^*(P^{[2]})
\longrightarrow p_1^*P\longrightarrow0.
\end{equation}
It follows either from the universal point-ideal sequence, by
tensoring with $P$ and pushing along the finite universal subscheme,
or directly from the quotient recording the residual point.
Use subscripts for pullbacks by $p_i$, and set $P=-K_Y$. The
length-two incidence cycle pulls back to the sum of the two point
graphs, so
\begin{equation}\label{r1:blowup-classes}
\begin{gathered}
\psi^*\mu=\alpha_1+\alpha_2,\qquad
\psi^*c_1(N)=-E,\qquad \psi^*c_2(N)=0,\\
\psi^*c_1(U)=P_1+P_2-E,\qquad
\psi^*c_2(U)=P_1P_2-P_1E.
\end{gathered}
\end{equation}
The incidence assertion also follows by taking the codimension-two
Chern character of the universal point-ideal sequence: its
$\mathcal O(-E)$ twist does not alter the graph's fundamental
codimension-two term. The bundle identities in
\eqref{r1:blowup-classes} follow at once from
\eqref{r1:tautological-sequence}, including the sign of $E$.
For completeness, the exceptional-divisor pushforwards needed here are
\begin{equation}\label{r1:exceptional-push}
\pi_*E=0,\qquad \pi_*(E^2)=-[\Delta_Y].
\end{equation}
The first vanishes because $E$ maps to a smaller-dimensional cycle.
For the second, $E$ is a $\mathbb P^1$-bundle over $\Delta_Y$ and
$\mathcal O(E)|_E=\mathcal O_E(-1)$, whose degree on each fiber is
$-1$. This proves the sign directly.
The bracket in Equation~\eqref{r1:chern-integral} now pulls back to
\[
P_1P_2-(3P_1+2P_2)E+3E^2.
\]
All terms linear in $E$ integrate to zero by
\eqref{r1:exceptional-push} and the projection formula. The remaining
term involving $P$ satisfies
\[
\int_{Y^2}(\alpha_1+\alpha_2)^2P_1P_2
=2(\alpha\cdot P)^2=0;
\]
the two other terms in the square exceed the dimension of one
surface factor. Since $\psi_*[\widetilde{Y^2}]=2[Y^{[2]}]$, the
factor of two from this cover must also be included. Thus
\begin{align*}
I
&=-\frac12\int_{\widetilde{Y^2}}
(\alpha_1+\alpha_2)^2\,3E^2\\
&=\frac32\int_{\Delta_Y}(\alpha_1+\alpha_2)^2
=\frac32\int_Y(2\alpha)^2
=6\alpha^2.
\end{align*}
The external terms pair to zero, so Equation~\eqref{rel:pairing}
and $\alpha^2\neq0$ show that this order has universal diagonal
coefficient $6$.
Finally exchange the two orders. Since $L_1=L_2=0$, exchanging
the two summands is the same as replacing the equivariant parameter
$t$ by $-t$ before evaluation. The ordinary codimension-two output
component has parameter power
\[
p+2-v=7+2-13=-4.
\]
This exponent is even, so the exchange preserves its pairing at
$t=1/2$. The second order therefore also has coefficient $6$.
Both orders and all three point-length distributions have now been
accounted for, and $c_{\mathrm{mid}}=6+6=12$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:rankone}]
Combine Equations~\eqref{r1:target-coefficient} and
\eqref{r1:comparison} with Lemma~\ref{lem:middle-coefficient}. On
the original surface $X$ they give
\[
c=(s+\tau_X)c_{\mathrm{mid}}=12(s+\tau_X)>0,
\]
because $s>0$ and $\tau_X\geq0$.
\end{proof}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//rankone.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//reductions.tex ====
\section{Birational reductions}\label{sec:reductions}
We reduce the zero-cycle theorem to minimal surfaces of general type
and record the numerical range that determines the two branches of
the proof. We also retain the freedom to blow up points, which will
be used in Section~\ref{sec:nonrank}.
\begin{lemma}\label{lem:birational}
Let \(\pi:\widetilde X\to X\) be the blowup of a smooth projective
surface at a point. Then \(p_g\) and \(q\) are unchanged, and
\[
\pi_*:\CH_0(\widetilde X)\xrightarrow{\sim}\CH_0(X)
\]
is a degree-preserving isomorphism with integral coefficients.
Consequently these assertions are invariant under birational maps
between smooth projective surfaces.
\end{lemma}
\begin{proof}
The identities \(\pi_*\mathcal O_{\widetilde X}=\mathcal O_X\)
and \(R^1\pi_*\mathcal O_{\widetilde X}=0\) give invariance of
\(q\). Pullback identifies holomorphic two-forms: a form on
\(\widetilde X\) descends away from the blown-up point and extends
across that point on the smooth surface. This proves invariance of
\(p_g\). The integral blowup formula for Chow groups
\cite[Proposition~6.7(b), (e)]{Fulton} gives, in codimension two,
\[
\CH^2(\widetilde X)
\simeq\CH^2(X)\oplus\CH^1(\operatorname{Spec}\mathbb C)
=\CH^2(X).
\]
The first summand is pulled back from \(X\), and pushforward is its
inverse. Proper pushforward preserves degree. Resolving a birational
map of smooth projective surfaces by point blowups gives the last
assertion. See also \cite[Section~4, first reduction]{BlochRoitman}.
\end{proof}
\begin{proposition}\label{prop:reduction}
To prove Theorem~\ref{thm:bloch} with rational coefficients, it is
enough to prove it for smooth minimal surfaces \(X_0\) of general
type with \(p_g=q=0\). For such a surface, putting
\(k_0=K_{X_0}^2\), one has
\begin{equation}\label{eq:reduction-invariants}
1\le k_0\le9,\qquad
\chi(\mathcal O_{X_0})=1,\qquad
\int_{X_0}c_2(X_0)=12-k_0,\qquad b_2(X_0)=10-k_0.
\end{equation}
It is also enough to prove the assertion on any chosen sequence of
point blowups of each such \(X_0\).
\end{proposition}
\begin{proof}
For Kodaira dimension less than two, the Theorem of Bloch, Kas, and
Lieberman gives zero Albanese kernel
\cite[p.~135]{BKL}. Since \(q=0\), this is the full degree-zero
Chow group. For a surface of general type, pass to a smooth minimal
model and apply Lemma~\ref{lem:birational}.
On the minimal model, \(K_{X_0}\) is nef and big, so \(k_0>0\).
The vanishing of \(p_g\) and \(q\) gives
\(\chi(\mathcal O_{X_0})=1\). Noether's formula gives
\(\int c_2=12-k_0\). Hodge theory and Poincar\'e duality give
\(b_1=b_3=0\), and hence \(\int c_2=2+b_2\). Therefore
\(b_2=10-k_0\). An ample divisor has a nonzero cohomology class,
so \(b_2\ge1\), proving \(k_0\le9\). The final assertion is
another application of Lemma~\ref{lem:birational}.
\end{proof}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//reductions.tex ====
==== BEGIN Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//relation.tex ====
\section{A Chow relation and its universal diagonal coefficient}
\label{sec:relation}
Our goal is the relation \(0=c[\Delta_X]+\Gamma\) of
\eqref{eq:intro-diagonal}, with \(\Gamma\) a sum of external products.
We first obtain a vanishing by equivariant degree, then decompose each
fixed contribution into a diagonal term and external products, retaining
both surface factors.
Throughout this Section Chow groups have rational coefficients. We use
the equivariant perfect obstruction theory of Proposition~\ref{prop:quot-pot},
its fixed theory in Proposition~\ref{prop:fixed-pot}, and the fixed virtual
classes computed in Propositions~\ref{prop:fixed-pg0}
and~\ref{prop:fixed-auxiliary}. Both settings have
$\chi(\mathcal O_X)=1$. The torus parameter is initially the indeterminate
$t$, of equivariant codimension one; the weights on $L_1,L_2$ are $t,-t$.
Only after localization will we set $t=1/2$, obtaining
$w_1=1/2,w_2=-1/2$.
\subsection{The vanishing supplied by equivariant degree}
Fix $p\geq 0$ such that $v>p+2$, where
$v=2n+(A-K)D$ is the virtual dimension in~\eqref{eq:vdim}.
Let $Q$ also denote the perfect universal quotient on $B\times X$.
For the point $o\in X$ used to normalize the determinant, define
\[
h=c_1^{\mathbb G_m}\bigl(\mathbf L i_o^*Q\bigr)
\quad\text{in }\CH^1_{\mathbb G_m}(B),
\qquad i_o:B\longrightarrow B\times X.
\]
The derived restriction in this definition is essential to make it an
operation on the perfect universal family, even where the quotient is
not locally free. Let $q:B\times X\times X\to X\times X$ be projection,
and let the subscripts $(1),(2)$ mean pullback from the two copies of
$B\times X$. Consider the nonlocalized equivariant class
\begin{equation}
\Xi(t)=q_*\left(
h^p\ch^{\mathbb G_m}_2(Q)_{(1)}
\ch^{\mathbb G_m}_2(Q)_{(2)}
\cap\bigl([B]^{\vir}\times[X\times X]\bigr)\right).
\label{rel:equivariant-push}
\end{equation}
All insertions are Chern-class operations of perfect complexes, and $q$
is proper. The total equivariant codimension of $\Xi(t)$ is $p+4-v$.
Since the torus acts trivially on $X\times X$,
\[
\CH^*_{\mathbb G_m}(X\times X)
=\CH^*(X\times X)[t].
\]
Consequently the coefficient of ordinary codimension two in the
localized class would have to be accompanied by
\begin{equation}
t^{p+2-v}.
\label{rel:parameter-power}
\end{equation}
Its exponent is negative, whereas~\eqref{rel:equivariant-push} has no
negative powers of $t$. This component is therefore zero. Virtual
localization~\cite{GP} expresses the same class as a sum of fixed
pushforwards divided by the moving Euler classes. Thus their ordinary
codimension-two components sum to zero, and this remains true after
$t=1/2$. Each individual fixed pushforward has the same total
equivariant codimension $p+4-v$: its ordinary codimension-two part
is therefore $t^{p+2-v}$ times a class independent of $t$.
Consequently changing $t$ to $-t$ multiplies each such contribution
by $(-1)^{p+2-v}$. This termwise statement will allow us to compare
the two orders of a fixed decomposition in Section~\ref{sec:rankone}.
\subsection{A fixed contribution, including its signs}
We first write an individual fiber contribution without the multiplier
from the divisor virtual class. Fix ordered lines
$D_1=T,D_2=H$ with $T+H=D$, and point lengths $n_i\geq0$ satisfying
\eqref{eq:length}. Put
\[
M=X^{[n_1]}\times X^{[n_2]},\qquad
m_i=m_{D_i}=\frac{D_i(D_i-K)}2.
\]
The variables $x_i$ are the hyperplane classes on the divisor linear
systems; they can also be regarded as formal variables in the formula
below. Write $\mathcal I_i$ for the universal ideals pulled to
$M\times X$, and $Z_i$ for their universal subschemes. For a line $P$
let $P^{[n_i]}$ be its rank-$n_i$ tautological bundle.
For a vector bundle $E$ of rank $r$, our Euler convention is
\[
\eu_u(E)=\sum_{j=0}^r c_j(E)u^{r-j}.
\]
For a virtual bundle it is extended multiplicatively whenever $u$ has
nonzero constant equivariant part. Equivalently,
$\eu_u(E)=u^{\rk E}c_{1/u}(E)$, expanded in positive Chow degree.
The fixed point-obstruction factors are
\begin{equation}
\mathcal E(x)=\prod_{i=1}^2
\eu_{x_i}\left(\bigl((K-D_i)^{[n_i]}\bigr)^\vee\right).
\label{rel:fixed-euler}
\end{equation}
Indeed $F_i=\bigl((K-D_i)^{[n_i]}\bigr)^\vee\otimes\mathcal O(1_i)$
on the indicated product or fiber, so~\eqref{rel:fixed-euler} is
$\prod_i\eu(F_i)$.
Define perfect $K$-classes on $M$ by
\[
\begin{split}
P_{ij}&=L_j-L_i+D_i,\\
B_{ij}&=L_j-L_i+D_i-D_j,\\
E_{ij}(P)&=[\RGamma(P)]-
[\RHom(\mathcal I_i,\mathcal I_j(P))],
\qquad i\ne j.
\end{split}
\]
Here and below relative $\RHom$ includes derived pushforward along $X$.
The class $E_{ij}(P)$ is the standard Ext $K$-class of Carlsson--Okounkov
\cite[Section~1.2, equation~(2)]{CarlssonOkounkov}. The moving ratio below
is derived here; their vertex-operator evaluation theorem is not used.
The inverse Euler class of the moving virtual normal bundle is
\begin{equation}
\mathcal R(x)=\prod_{i\ne j}
\frac{(w_j-w_i+x_i-x_j)^{\chi(B_{ij})}}
{(w_j-w_i+x_i)^{\chi(P_{ij})}}
\frac{\eu_{w_j-w_i+x_i}
\left(\bigl((K-P_{ij})^{[n_i]}\bigr)^\vee\right)}
{\eu_{w_j-w_i+x_i-x_j}\bigl(E_{ij}(B_{ij})\bigr)}.
\label{eq:moving-ratio}
\end{equation}
We derive the signs explicitly. On the fiber under consideration,
\[
S'_i=L_i(-D_i)\mathcal I_i\otimes\mathcal O(-1_i).
\]
The ideal sequence and relative Serre duality give
\[
[\RHom(\mathcal I_i,P)]
=[\RGamma(P)]-
\left[\bigl((K-P)^{[n_i]}\bigr)^\vee\right].
\]
The duality shift is $[-2]$, so it contributes with a positive sign
before subtraction by the ideal sequence. Put $u=w_j-w_i$. The moving
class for the ordered pair $(i,j)$, namely
$\RHom(S'_i,L_j)-\RHom(S'_i,S'_j)$, is therefore
\begin{align*}
&[\RGamma(P_{ij})]_{u+x_i}
-\left[\bigl((K-P_{ij})^{[n_i]}\bigr)^\vee\right]_{u+x_i}\\
&\hspace{8mm}-[\RGamma(B_{ij})]_{u+x_i-x_j}
+[E_{ij}(B_{ij})]_{u+x_i-x_j}.
\end{align*}
The subscript denotes the specified weight and line twist. A constant
complex of vector spaces contributes the indicated weight raised to
its Euler characteristic. Taking the inverse Euler class of this
four-term expression proves~\eqref{eq:moving-ratio}; in particular
$E_{ij}$ occurs in the denominator. No individual cohomology dimension
of $P_{ij}$ or $B_{ij}$ remains in the result.
For the Chern-character calculation, temporarily read $w_1,w_2$ as the
degree-one equivariant classes $t,-t$: take the equivariant
codimension-two component before specializing $t=1/2$.
The restriction of the quotient Chern character to a marked surface
factor then follows from
\[
\begin{aligned}
\ch(Q_i)&=e^{L_i+w_i}
\left(1-e^{-D_i-x_i}\ch(\mathcal I_i)\right),\\
\ch(\mathcal I_i)
&=1-[Z_i]+\text{terms of codimension at least three}.
\end{aligned}
\]
Its first Chern class is $D_i+x_i$. The determinant of the universal
point ideal is trivial, so derived restriction to $o$ adds no incidence
term to this first Chern class. Line classes from $X$ restrict to zero
in $\CH^1(B)$, and the quotient has rank zero. Thus $h$ restricts to
$x_1+x_2$. In particular, fixing the determinant in the Picard scheme
does not make $h$ zero. On this fixed product the determinant line of
the universal quotient is
$D\boxtimes\mathcal O(1_1+1_2)$, up to a constant line.
Its based normalization removes the factor from the parameter space,
whereas restriction of the unnormalized line to $o$ retains that factor.
This is exactly the distinction in Equation~\eqref{der:derived-fiber}.
The second Chern character on $M\times X$ is
\begin{equation}
J=\sum_{i=1}^2
\left([Z_i]+(L_i+w_i)(D_i+x_i)-\frac{(D_i+x_i)^2}{2}\right).
\label{eq:insertion}
\end{equation}
Let $\pi:M\times X\times X\to X\times X$ be projection. The
ordinary codimension-two fiber contribution is the following class,
where capping with $[M\times X\times X]$ in the pushforward is implicit:
\begin{equation}
\begin{split}
C_*(T,H;n_1,n_2)
&=\Bigl[\,[x_1^{m_T}x_2^{m_H}]\\
&\qquad\pi_*\bigl(\mathcal E(x)\mathcal R(x)(x_1+x_2)^p
J_{(1)}J_{(2)}\bigr)\Bigr]_2.
\end{split}
\label{rel:fiber-contribution}
\end{equation}
The outer subscript means ordinary codimension two on $X\times X$.
All denominators are expanded at $x_1=x_2=0$ and in positive Chow
degree. Their constant weights are $\pm1$, so these expansions are
well defined. Only finitely many terms can contribute to the specified
$x$-coefficient and Chow degree. A negative value of either $m_i$ means
that the contribution is zero.
For $p_g=q=0$, Proposition~\ref{prop:fixed-pg0} inserts
$x_i^{h^1(D_i)}\eu(F_i)$ on $|D_i|$.
Since $h^2(D_i)=0$ for effective $D_i$ and
$\dim|D_i|=m_i+h^1(D_i)$, projective integration is precisely the
coefficient extraction in~\eqref{rel:fiber-contribution}.
For the auxiliary setting, Proposition~\ref{prop:fixed-auxiliary}
first restricts to a fiber over the empty or canonical low divisor.
The low variable is $x_b=0$, its $m_b$ is zero, and the high projective
factor has dimension $m_a$. The same formula applies, with multiplier
$1$ or $-s$, respectively. This restriction precedes every calculation
with the insertions; there is no remaining integration over a Picard
parameter in~\eqref{rel:fiber-contribution}.
\subsection{Universality with both output factors retained}
Let $R_X\subset\CH^*(X)$ denote the graded subring generated by
$K,c_2(T_X),L_1,L_2,T,H$ and the identity. We shall show that
\eqref{rel:fiber-contribution} lies in
\[
\mathbb Q\cdot[\Delta_X]+\sum_{a+b=2}R_X^a\boxtimes R_X^b.
\]
We need an equality of Chow classes and a consistent numerical choice
of the coefficient of $\Delta_X$. We therefore give the recursion
with the two output factors left unintegrated.
The reason for retaining these factors is already visible on $X^3$.
Let $p_{12}:X^3\to X^2$ retain the first two factors, let $p_i$ be
the individual projections, and write $\Delta_{ij}$ for the locus
where coordinates $i$ and $j$ agree. For
$\gamma,\delta\in R_X$ homogeneous with total codimension two,
\begin{equation}\label{rel:three-slot-model}
\begin{aligned}
p_{12*}\bigl([\Delta_{13}]\cdot[\Delta_{23}]\bigr)
&=[\Delta_X],\\
p_{12*}\bigl([\Delta_{13}]\cdot p_3^*\gamma\cdot p_2^*\delta\bigr)
&=\gamma\boxtimes\delta.
\end{aligned}
\end{equation}
In the first identity both retained coordinates equal the third;
the small diagonal maps isomorphically to $\Delta_X$.
In the second, only the first coordinate is tied to the third;
the second remains independent. These are Chow identities, before
any integration over the outputs. The Hilbert-scheme recursion below
produces products of precisely such equality diagonals, together
with line and tangent Chern classes. Their intersections will give
the same two alternatives, including the excess
factors when an equality is repeated.
\begin{lemma}[Consecutive Hilbert-scheme recursion]
\label{rel:hilbert-recursion}
Let $r>0$, and let $N_r=X^{[r-1,r]}$ be the scheme of nested
subschemes $Z'\subset Z$ of lengths $r-1,r$. Write
\[
\psi:N_r\longrightarrow X^{[r]},\quad
\phi:N_r\longrightarrow X^{[r-1]},\quad
\rho:N_r\longrightarrow X
\]
for the two forgetful maps and the residual-point map. Then $N_r$ is
smooth, projective, irreducible of dimension $2r$;
$\psi_*[N_r]=r[X^{[r]}]$; and, with quotient-line convention,
\[
\sigma=(\phi,\rho):N_r
=\mathbb P(\mathcal I_{r-1})
\longrightarrow X^{[r-1]}\times X.
\]
Put $\mathcal L=\mathcal O_{N_r}(1)$ and $\zeta=c_1(\mathcal L)$.
If $j:N_r\hookrightarrow N_r\times X$ is the graph of $\rho$, then
\begin{equation}
0\longrightarrow(\psi\times\id_X)^*\mathcal I_r
\longrightarrow(\phi\times\id_X)^*\mathcal I_{r-1}
\longrightarrow j_*\mathcal L\longrightarrow0.
\label{rel:universal-ideal-sequence}
\end{equation}
For a two-term vector bundle resolution
$0\to U_1\to U_0\to\mathcal I_{r-1}\to0$ on
$X^{[r-1]}\times X$, one has, for every $k\geq0$,
\begin{equation}
\sigma_*(\zeta^k\cap[N_r])
=\left[\frac{c(U_1^\vee)}{c(U_0^\vee)}\right]_k
\cap[X^{[r-1]}\times X].
\label{rel:projective-pushdown}
\end{equation}
All these identities remain valid after adjoining other Hilbert or
surface factors.
\end{lemma}
\begin{proof}
The geometric description and universal sequence are the consecutive
nested-scheme construction of~\cite[Section~1, equations~(3)--(6)]{EGL}.
A quotient $\mathcal I_{Z'}\twoheadrightarrow\mathbb C_x$ determines
the extra length-one subscheme and conversely. In families its quotient
line is $\mathcal L$, giving the projectivization and
\eqref{rel:universal-ideal-sequence}. The smoothness and dimension of
the consecutive nested scheme are the surface Hilbert-scheme facts
recorded there. Over the open locus of $r$ distinct points, forgetting
which point is residual has degree $r$, proving the pushforward
formula for $\psi$. The universal ideals are flat over their Hilbert
bases, so their ordinary pullbacks in this sequence agree with their
derived pullbacks.
Here is also the pushdown calculation, to fix its convention and sign.
The universal ideal has projective dimension at most one: it is flat
over the smooth Hilbert base and its surface fibers are torsion-free
ideals. A sufficiently negative sum of ample line bundles surjects
onto it; the kernel is locally free, by the fiberwise projective
dimension bound and the local criterion of flatness. This gives the
displayed global resolution. Write $a=\rk U_1$, so $\rk U_0=a+1$,
and let $p:\mathbb P(U_0)\to X^{[r-1]}\times X$.
The inclusion $N_r\subset\mathbb P(U_0)$ is the zero locus of the
section of $p^*U_1^\vee\otimes\mathcal O(1)$ induced by $U_1\to U_0$.
Its codimension is $a$, because both $N_r$ and the base of $\sigma$
have dimension $2r$. The ambient space is smooth, so the section is
regular and its top Chern class represents $[N_r]$.
For $\xi=c_1(\mathcal O_{\mathbb P(U_0)}(1))$ the quotient convention
gives
$p_*(\xi^{a+k})=[c(U_0^\vee)^{-1}]_k$ for $k\geq0$, and zero for
smaller exponents. Hence the left side of
\eqref{rel:projective-pushdown} equals
\[
\begin{split}
p_*\bigl(\xi^k c_a(p^*U_1^\vee\otimes\mathcal O(1))\bigr)
&=\sum_{b=0}^{\min(a,k)}c_b(U_1^\vee)
[c(U_0^\vee)^{-1}]_{k-b}\\
&=\left[\frac{c(U_1^\vee)}{c(U_0^\vee)}\right]_k.
\end{split}
\]
This is also~\cite[Section~1, Lemma~1.1]{EGL}. In particular, for
$r=2$ the map $\sigma$ is the blowup of $X\times X$ along its diagonal,
$\zeta=-E$ for its exceptional divisor, and the formula gives
$\sigma_*(\zeta^2)=-[\Delta_X]$. Flat base change and the projection
formula give all the assertions with extra factors.
\end{proof}
\begin{lemma}[Reduction to decorated equality diagonals]
\label{rel:marked-reduction}
Fix a finite list of line bundles on $X$. Consider a finite polynomial
of Chern characters of universal point ideals and their derived duals,
together with Chern classes of these lines and of $T_X$ pulled from
the surface slots and equality diagonals between those slots, on
\[
X^{[n_1]}\times X^{[n_2]}\times X^\ell,
\qquad \ell\geq2.
\]
Keep two specified surface factors as outputs and push forward over
the Hilbert factors and all other surface factors. The resulting
Chow class is a finite rational linear combination of pushforwards of
products of equality diagonals and line and tangent classes on powers
of $X$. The coefficients and the list of such products can be chosen
by one recursion independent of $X$ and of the chosen line bundles.
\end{lemma}
\begin{proof}
We induct on $n_1+n_2$. When this sum is positive, choose a positive
length $r$ in a fixed order of the Hilbert factors and pull back along
$\psi$ of Lemma~\ref{rel:hilbert-recursion}, with all other factors
unchanged. All spaces used here are smooth, so Chow pullbacks are
defined. Projection formula and $\psi_*[N_r]=r[X^{[r]}]$ express the
original pushforward as $1/r$ times the pushforward of this pullback.
In every surface slot, replace the pulled-back ideal by
\eqref{rel:universal-ideal-sequence} in $K$-theory. Its extra term is
the graph of the new residual point, with the line twist $\mathcal L$.
For that graph, Grothendieck--Riemann--Roch gives
\begin{equation}
\ch(j_*\mathcal L)
=j_*\bigl(e^{\zeta}\rho^*\td(T_X)^{-1}\bigr).
\label{rel:graph-grr}
\end{equation}
Indeed the graph is a regular embedding of codimension two with
normal bundle $\rho^*T_X$. It is the pullback of the diagonal under
$(\rho,\id_X):N_r\times X\to X\times X$. This square is Tor
independent: the two local equations of the diagonal cut out the graph
regularly, in the same codimension. Thus the graph term in
\eqref{rel:graph-grr} is a polynomial in $\zeta$, the equality diagonal
between the new point slot and the slot in use, and tangent classes.
For a perfect complex, the degree-$k$ Chern character of its derived
dual is $(-1)^k\ch_k$; derived duals therefore introduce no new kind
of class.
After this substitution the integrand is a polynomial in $\zeta$
whose coefficients pull back from
$X^{[r-1]}\times X$ and the unchanged factors. Apply
\eqref{rel:projective-pushdown} to each power of $\zeta$.
Its right side is a universal polynomial in the Chern classes of
$\mathcal I_{r-1}$, equivalently in its Chern characters with rational
coefficients. The Hilbert length has decreased by one; the added
surface factor records the residual point. The two output factors
have not been pushed forward at any step. Repeating the construction
ends with Hilbert schemes of length zero, whose ideals are the
structure sheaf. Only surface factors, their diagonals, and the
specified decorations remain. The choices of Hilbert-factor order,
slot order, and polynomial expansion order can be fixed once for all
inputs. This proves the assertion. We have adapted the geometric
recursion underlying the integral formula of~\cite[Section~3,
Proposition~3.1]{EGL}: here the two output factors remain unintegrated
throughout a sequence of Chow identities.
\end{proof}
The recursion has eliminated the Hilbert factors without integrating
the outputs. We now apply it to the localization integrand and use
the remaining diagonal intersections to select its coefficient of
\(\Delta_X\) by a rule independent of the surface.
\begin{proposition}[Chow universality of a fiber contribution]
\label{prop:universal}
The class in~\eqref{rel:fiber-contribution} admits a decomposition
\begin{equation}
C_*(T,H;n_1,n_2)=c_*[\Delta_X]+\Gamma_*,
\qquad
\Gamma_*\in\sum_{a+b=2}R_X^a\boxtimes R_X^b.
\label{rel:universal-decomposition}
\end{equation}
One can choose $c_*$ by a fixed universal rule. Besides the integer
parameters and the chosen weights, this rule depends only on
\begin{equation}
\int_X c_2(T_X),\qquad
\int_X UV\quad
(U,V\in\{K,L_1,L_2,T,H\}).
\label{rel:numerical-data}
\end{equation}
It gives $c_*=0$ when $n_1=n_2=0$. The same rule applies on every
smooth projective surface with the same data and $\chi(\mathcal O_X)=1$,
whether or not the lines $T,H$ are effective; effectivity is needed
only for realizing the formula as a fixed contribution.
\end{proposition}
\begin{proof}
We first put the integrand into the form required by
Lemma~\ref{rel:marked-reduction}. For the surface projection
$\pi_X:M\times X\to M$, Grothendieck--Riemann--Roch gives
\begin{align}
\ch\bigl(\RHom(\mathcal I_i,\mathcal I_j(P))\bigr)
&=\pi_{X*}\bigl(\ch(\mathcal I_i^\vee)
\ch(\mathcal I_j)e^P\td(T_X)\bigr),
\label{rel:ext-grr}\\
\ch(P^{[n_i]})
&=\pi_{X*}\bigl((1-\ch(\mathcal I_i))e^P\td(T_X)\bigr).
\label{rel:tautological-grr}
\end{align}
The dual in~\eqref{rel:ext-grr} is the derived dual. These identities
hold in Chow, since the ideals are perfect and projection is smooth
and proper. The constant complex $\RGamma(P)$ has $K$-class
$\chi(P)[\mathcal O_M]$, with
$\chi(P)=1+P(P-K)/2$. The Chern classes in the Euler factors are
universal polynomials in the resulting Chern characters. The
incidence insertion is $[Z_i]=-\ch_2(\mathcal I_i)$.
Thus, after the finite truncation required by
\eqref{rel:fiber-contribution}, every term is a polynomial in the
classes of the preceding Lemma and in pushforwards of such classes.
Products of surface pushforwards introduce only additional surface
slots. More precisely, if $a,b$ are classes on $M\times X$, the
projection formula and flat base change identify
$\pi_{X*}(a)\pi_{X*}(b)$ with the pushforward of
$a_{(1)}b_{(2)}$ from $M\times X\times X$. The same statement holds
over the base $M$ together with the two marked factors and can be
iterated for any finite product. Apply
Lemma~\ref{rel:marked-reduction} to the resulting expressions.
It remains to perform the surface pushforwards. This reduction will
both prove the Chow decomposition and specify the coefficient to be
compared across surfaces. Represent a monomial in equality diagonals
by a graph $\mathcal G$: its vertices are the surface slots and each
factor $[\Delta_{uv}]$ is an edge between slots $u,v$, with repeated
factors recorded as repeated edges. For each connected component $C$,
let $v_C,e_C$ count its vertices and edges. Let
\[
\iota:\prod_{C\text{ component of }\mathcal G}X
\longrightarrow X^{\{\text{vertices of }\mathcal G\}}
\]
assign the same coordinate to all vertices in a component, and let
$\operatorname{pr}_C$ project its domain to the factor indexed by $C$.
Then
\begin{equation}\label{rel:graph-intersection}
\prod_{uv\text{ edge of }\mathcal G}[\Delta_{uv}]
=\iota_*\left(
\prod_{C\text{ component of }\mathcal G}
\bigl(\operatorname{pr}_C^*c_2(T_X)\bigr)^{e_C-v_C+1}
\right).
\end{equation}
Indeed, a spanning tree in $C$ imposes $v_C-1$ transverse equalities,
giving its small diagonal. Each of the remaining $e_C-v_C+1$ edges
imposes an equality already satisfied. The diagonal
self-intersection formula contributes $c_2(T_X)$ for each such edge.
The exponent is zero for an isolated vertex as well as for a tree.
By projection formula, every line or tangent decoration is then
restricted to the copy of $X$ belonging to its component.
There are three possibilities for the subsequent pushforward:
\begin{enumerate}[label=\textup{(\roman*)}]
\item A component containing neither output is integrated over $X$.
It contributes zero unless its decoration has codimension two;
otherwise it contributes the intersection number of that decoration.
\item A component containing exactly one output maps isomorphically
to that output and leaves its decoration there. When the outputs
belong to different components, their decorations therefore give an
external product of total codimension two, multiplied by the
intersection numbers from components of type~\textup{(i)}.
\item A component containing both outputs maps to their diagonal.
It contributes $\Delta_{X*}(\eta)$, multiplied by the same kind of
intersection numbers. Since $\Delta_X$ already has codimension two,
the codimension-two part of the answer uses only
$\eta\in\CH^0(X)=\mathbb Q[X]$. Thus this case gives a scalar
multiple of $[\Delta_X]$.
\end{enumerate}
Fix the recursion and edge-processing orders once for all inputs.
Carry out this reduction on the formal decorated graphs, replacing
components of type~\textup{(i)} by formal intersection numbers.
Define $c_*$ as the sum of the scalar coefficients in
case~\textup{(iii)}, and assign the terms with separated outputs to
$\Gamma_*$. Only after this assignment do we evaluate the
intersection numbers and Chow classes on $X$.
Equation~\eqref{rel:graph-intersection} and the three cases prove
\eqref{rel:universal-decomposition} in Chow and specify the promised
choice of its coefficient.
Every codimension-two decoration integrated over a surface is a
linear combination of $c_2(T_X)$ and products of the lines in
\eqref{rel:numerical-data}, since $c_1(T_X)=-K$.
The scalar Euler exponents are determined by the same data through
Riemann--Roch. Hence the rule gives the same $c_*$ on any two
surfaces with those numerical data. We do not require the resulting
diagonal decomposition to be unique on an individual surface:
relations between its diagonal and external products are never
used to select $c_*$.
Finally, when both point lengths are zero, $M$ is a point,
$\mathcal E=1$, and $E_{ij}=0$. There is no incidence term in $J$.
The two copies of $J$ involve only classes on their separate output
factors and the scalar variables $x_i,w_i$. The graph procedure has
no edge joining the outputs, so it selects $c_*=0$. None of this
formal calculation requires sections of $T$ or $H$, proving the last
assertion as well.
\end{proof}
All diagonal coefficients below use this fixed formal choice.
In particular, numerical comparison transports the coefficient of
the explicit fiber formula~\eqref{rel:fiber-contribution}; it does
not transport the geometry or multiplicity of the divisor component
from which that formula arose.
\subsection{The relation, cohomological tests, and its action}
The fixed contributions now have Chow decompositions with consistently
chosen diagonal coefficients. Summing them gives the required relation;
cohomological pairings will then test its coefficient, and its
correspondence action will control zero-cycles.
\begin{theorem}[Chow relation]
\label{thm:chow-relation}
For either surface setting, and any $p\geq0$ with $v>p+2$, sum the
classes~\eqref{rel:fiber-contribution} with the following multiplicities:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $p_g=q=0$, sum over the effective ordered line pairs $T+H=D$
and nonnegative point lengths satisfying~\eqref{eq:length}, with
$m_T,m_H\geq0$. Each fixed part has multiplier $1$.
\item In the auxiliary setting, $D=3K$. Sum over the ordered pairs
$(3K,0),(0,3K)$ with multiplier $1$, and $(2K,K),(K,2K)$ with
multiplier $-s$, with the admissible point lengths. Here $s>0$ is
the canonical-divisor coefficient of~\eqref{eq:canonical-cycle}.
\end{enumerate}
All sums are finite. Writing $c$ for the same weighted sum of the
coefficients chosen in Proposition~\ref{prop:universal}, one obtains
\begin{equation}
0=c[\Delta_X]+\Gamma
\qquad\text{in }\CH^2(X\times X),
\label{rel:chow-identity}
\end{equation}
where $\Gamma$ is a sum of external products of total codimension two.
\end{theorem}
\begin{proof}
The finite-type fixed locus has only finitely many discrete
decompositions and Hilbert polynomials, as in Section~\ref{sec:fixed}.
Proposition~\ref{prop:fixed-pg0} gives the first list and discards
negative $m_i$. Proposition~\ref{prop:fixed-auxiliary} gives the second
list, its fiber restrictions, and its multiplicities. The moving
factor and insertions were computed in~\eqref{eq:moving-ratio}
and~\eqref{eq:insertion}, so these are exactly the contributions to the
ordinary codimension-two part of localization of
\eqref{rel:equivariant-push}. That part vanishes by
\eqref{rel:parameter-power}. Substitute
\eqref{rel:universal-decomposition} for each contribution to obtain
\eqref{rel:chow-identity}.
\end{proof}
For the subsequent nonvanishing arguments, let
$\alpha\in H^2(X,\mathbb R)$ and write
$\alpha U=\int_X\alpha\smile U$ and
$\alpha^2=\int_X\alpha\smile\alpha$.
Recall $\pi_X:M\times X\to M$ and set
\[
\mu_i(\alpha)=\pi_{X*}([Z_i]\smile\alpha),\qquad
\mu_\alpha=\pi_{X*}(J\smile\alpha).
\]
Here $[Z_i]$ and $J$ denote their cycle classes; $\alpha$ need not be
algebraic. Expansion of~\eqref{eq:insertion} gives the useful formula
\begin{equation}
\mu_\alpha=\sum_{i=1}^2
\left(\mu_i(\alpha)+(w_i-x_i)(\alpha D_i)
+x_i(\alpha L_i)\right).
\label{rel:mu-formula}
\end{equation}
Integrating over the two output factors first and using projection
formula shows that
\begin{multline}
\int_{X\times X}C_*(T,H;n_1,n_2)
\smile(\alpha\boxtimes\alpha)\\
= [x_1^{m_T}x_2^{m_H}]\int_M
\mathcal E(x)\mathcal R(x)(x_1+x_2)^p\mu_\alpha^2.
\label{rel:pairing}
\end{multline}
The integral takes the top ordinary cohomological degree on $M$.
It automatically selects precisely the ordinary codimension-two
part on the left. In particular, no algebraicity assumption on
$\alpha$ is used.
If $\alpha$ is orthogonal to $K,L_1,L_2,T,H$, then
\begin{equation}
\int_{X\times X}C_*(T,H;n_1,n_2)
\smile(\alpha\boxtimes\alpha)
=c_*\alpha^2.
\label{rel:orthogonal-test}
\end{equation}
Indeed, an external product of codimensions $(0,2)$ or $(2,0)$ pairs
to zero by degree, and one of codimensions $(1,1)$ pairs to the
product of two intersections with $\alpha$. Those intersections
vanish because the codimension-one part of $R_X$ is spanned by the
listed line classes. The diagonal pairs to $\alpha^2$.
This is a test for the chosen coefficient of a relation already
proved in Chow.
\begin{corollary}
\label{rel:correspondence}
If $p_g=q=0$ and the coefficient $c$ in
\eqref{rel:chow-identity} is nonzero, then
$\CH_0(X)^0\otimes\mathbb Q=0$.
\end{corollary}
\begin{proof}
For $Z\in\CH^2(X\times X)$, use its correspondence action
\[
Z_*(z)=\operatorname{pr}_{2*}
\bigl(\operatorname{pr}_1^*z\cdot Z\bigr).
\]
The diagonal acts as the identity. If $Z=a\boxtimes b$ has total
codimension two and $a$ has positive codimension, then
$z\cdot a=0$ on the surface for every zero-cycle $z$, by dimension.
If $a$ has codimension zero, connectedness of $X$ gives
$a=\lambda[X]$ and $Z_*(z)=\lambda\deg(z)b$.
Every summand of $\Gamma$ therefore annihilates degree-zero
zero-cycles. Acting by~\eqref{rel:chow-identity} gives $cz=0$ in
$\CH_0(X)\otimes\mathbb Q$. Since $c\ne0$, this implies $z=0$.
\end{proof}
==== END Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0//relation.tex ====
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