64c3814
==== TABLE OF CONTENTS ====
P-equals-W-in-composite-rank-for-fixed-determinant//01-introduction.tex
P-equals-W-in-composite-rank-for-fixed-determinant//02-lefschetz.tex
P-equals-W-in-composite-rank-for-fixed-determinant//03-betti.tex
P-equals-W-in-composite-rank-for-fixed-determinant//04-diagram.tex
P-equals-W-in-composite-rank-for-fixed-determinant//05-logarithmic.tex
P-equals-W-in-composite-rank-for-fixed-determinant//06-translation.tex
P-equals-W-in-composite-rank-for-fixed-determinant//07-specialization.tex
P-equals-W-in-composite-rank-for-fixed-determinant//08-commutator.tex
P-equals-W-in-composite-rank-for-fixed-determinant//09-conclusion.tex
P-equals-W-in-composite-rank-for-fixed-determinant//main.tex
P-equals-W-in-composite-rank-for-fixed-determinant//preamble.tex
P-equals-W-in-composite-rank-for-fixed-determinant//references.bib
===========================
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//01-introduction.tex ====
\section{Introduction}
Nonabelian Hodge theory identifies two moduli spaces whose cohomology carries very different geometric filtrations. On the Higgs-bundle side, the Hitchin map defines the perverse Leray filtration. On the character-variety side, the mixed Hodge structure defines the weight filtration. The $P=W$ conjecture asserts that these filtrations coincide after doubling the indices of the first.
The geometric setting comes from Hitchin's moduli spaces and integrable systems \cite{Hitchin1987SelfDuality,Hitchin1987Integrable}. The analytic correspondence rests on the foundational work of Hitchin, Donaldson, Corlette, and Simpson \cite{Hitchin1987SelfDuality,Donaldson1987,Corlette1988,Simpson1992}. We use its twisted, fixed-determinant form, including the action by torsion line bundles, as described by Hausel and Thaddeus \cite[Sections~2 and~5]{HT2003}.
Let $C$ be a smooth projective complex curve of genus $g\ge2$. Fix an integer $n\ge2$, an integer $d$ coprime to $n$, and a line bundle $L\in\Pic^d(C)$. Write
\[
X=M_{\mathrm{Dol}}^L(\SL_n)
\]
for the moduli space of stable rank-$n$ Higgs bundles $(E,\theta)$ with $\det E\simeq L$ and $\tr\theta=0$. Here stability means that every proper nonzero $\theta$-invariant subbundle has slope smaller than $d/n$. Its Hitchin map is
\[
h:X\longrightarrow A=\bigoplus_{i=2}^nH^0(C,K_C^i),
\]
given by the coefficients of the characteristic polynomial. Put
\[
R=(n^2-1)(g-1),\qquad b=\frac{n(n-1)}2,
\qquad \zeta=\exp(2\pi\sqrt{-1}d/n).
\]
The corresponding twisted character variety is
\begin{equation}\label{intro:betti}
X^B=\left\{(A_1,B_1,\ldots,A_g,B_g)\in\SL_n(\C)^{2g}:
\prod_{j=1}^g[A_j,B_j]=\zeta I_n\right\}\mathbin{/\mkern-6mu/}\SL_n(\C),
\end{equation}
where the group acts by simultaneous conjugation and $[A,B]=ABA^{-1}B^{-1}$. Coprimality makes both spaces smooth and makes every representation in \eqref{intro:betti} irreducible. Nonabelian Hodge theory gives a diffeomorphism between them \cite{HT2003,Simpson1997}.
We use rational cohomology unless complex coefficients are indicated. The Hitchin map is proper, $\dim X=2R$, $\dim A=R$, and
\[
\dim(X\times_A X)-\dim X=R.
\]
Thus the standard $P=W$ normalization is
\begin{equation}\label{intro:P}
P_kH^m(X)=\im\left\{
H^{m-R}\!\left(A,{}^p\tau_{\le k}Rh_*\Q_X[R]\right)
\longrightarrow H^m(X)\right\}.
\end{equation}
Here ${}^p\tau$ denotes truncation for the middle perverse $t$-structure. We recall the dimension and shift conventions in Section~\ref{sec:lefschetz}. Transport $P$ to $H^*(X^B)$ by nonabelian Hodge theory, and let $W$ denote Deligne's weight filtration there \cite{Deligne1971,Deligne1974}.
\begin{theorem}\label{thm:main}
For every smooth projective complex curve $C$ of genus $g\ge2$, every composite integer $n\ge4$, every integer $d$ with $\gcd(n,d)=1$, and every $L\in\Pic^d(C)$, one has
\[
P_kH^m(X^B,\Q)=W_{2k}H^m(X^B,\Q)
=W_{2k+1}H^m(X^B,\Q)
\]
for all $m,k\ge0$.
\end{theorem}
The assertion concerns the entire cohomology. The group
$\Gamma=\Pic^0(C)[n]$ acts on $X$ by tensor product. Its averaging projector
\[
\operatorname{av}_{\Gamma}=\frac1{|\Gamma|}\sum_{\gamma\in\Gamma}\gamma^*
\]
has image $H^*(X,\Q)^\Gamma$ and kernel the \emph{variant cohomology}
$H^*_{\mathrm{var}}(X,\Q)$. Thus, after complexification, the theorem includes every nontrivial $\Gamma$-character, not only the invariant part. Combining Theorem~\ref{thm:main} with the established prime-rank cases \cite[Theorem~0.3]{MS2024} resolves the smooth coprime fixed-determinant, trace-free $P=W$ conjecture in every rank $n\ge2$.
\subsection{History and the fixed-determinant problem}
The weight symmetries behind $P=W$ arose from the arithmetic study of character varieties by Hausel and Rodriguez-Villegas \cite{HRV2008}, extended to generic puncture data by Hausel, Letellier, and Rodriguez-Villegas \cite{HLRV2011}. De Cataldo, Hausel, and Migliorini formulated the filtration comparison and established it in rank two \cite{dCHM2012}. Their theorem relates the topology of an algebraic integrable system to the mixed Hodge theory of an affine character variety: relative hard Lefschetz on the Higgs side explains curious hard Lefschetz on the Betti side.
Tautological classes have been central to the higher-rank problem. For $\GL_n$, generation by these classes was established in rank two by Hausel and Thaddeus \cite{HTGenerators2004} and in arbitrary rank by Markman \cite[Theorem~7]{Markman2002}; Shende determined their Betti weights \cite{Shende2017}. De Cataldo, Maulik, and Shen proved $P=W$ for $\GL_n$ in genus two and obtained the required perversity bounds for tautological classes in arbitrary genus \cite[Theorems~0.2 and~0.4--0.6]{dCMSJAMS2022}.
Maulik and Shen proved the coprime $\GL_n$ conjecture in arbitrary rank and genus \cite{MS2024}. Further proofs use the Hecke-algebra action of Hausel, Mellit, Minets, and Schiffmann \cite{HMMS2025} and the Fourier-transform method of Maulik, Shen, and Yin \cite[Theorem~0.4]{MSY2025}.
For fixed determinant, the quotient by $\Gamma$ detects the invariant cohomology, and the projective tautological generation theorem applies to this quotient \cite[Theorem~2.1(i)]{MS2024}. It does not generate the variant part. De Cataldo, Maulik, and Shen determined the two filtrations on the variant part in prime rank and reduced $\SL_p$ to $\GL_p$ \cite{dCMS2022}; together with the general $\GL_p$ theorem, this gives the prime-rank cases cited above. In arbitrary rank, Maulik and Shen constructed endoscopic correspondences compatible with the perverse filtrations \cite[Theorem~5.4]{MSEndoscopy2021}, without identifying the full filtrations in composite rank. Corollary~\ref{cor:endoscopic-weights} combines our all-rank equality with cover-side $\GL_r$ $P=W$ to give the exact Betti-weight image equality asked in \cite[Question~5.5]{MSEndoscopy2021}, for the canonical-twist operator transported by nonabelian Hodge theory.
Our argument constructs an operator on the full cohomology, before taking $\Gamma$-invariants. Its algebraic pattern follows the common $\mathfrak{sl}_2$ structure and flag-correspondence methods of Hausel, Mellit, Minets, and Schiffmann \cite[Proposition~7.9 and Section~8]{HMMS2025}. The fixed-determinant construction requires two further geometric steps. First, we pull Mellit's toric stratifications through a determinant-root cover and prove integral local constancy and ordinary base change. The inputs include the relative product stratification over ordered eigenvalues, including collisions \cite[Section~7.4, Theorem~7.5, and proof of Proposition~8.7]{Mellit2025}. This yields curious hard Lefschetz on the full fixed-determinant cohomology, rather than only its invariant part.
Second, we use a finite diagram of algebraic fibers to give weight zero to the cochains of a topological moving-point base. Shende used this weight convention for simplicial bases and tautological classes \cite[proof of the main Theorem]{Shende2017}. Here the construction also carries variations and proper Gysin maps, using mixed Hodge modules and their coherent enhancement \cite{SaitoExtension1990,TubachStacks2025}. Its finite weight-graded monodromy supplies the lowering operator. These diagram weights are distinct from the usual weights of an algebraic moving-point curve; that distinction is needed for integration over the curve to have weight shift zero.
\subsection{The operator behind the comparison}
The universal projective bundle on $C\times X$ has local vector-bundle lifts $\mathcal E$. The following characteristic class is independent of the lift and descends to $C\times X$; integration over $C$ gives
\[
e=\int_C\left(\ch_2(\mathcal E)-\frac{c_1(\mathcal E)^2}{2n}\right)
=\frac1{2n}\int_C\ch_2(\End\mathcal E)\in H^2(X).
\]
We also write $e$ for cup product by this class. Section~\ref{sec:lefschetz} proves that $H^2(X)=\Q e$ and that $e$ satisfies relative hard Lefschetz, with center $R$. Section~\ref{sec:betti} proves curious hard Lefschetz for the same class and the half-weight grading, again with center $R$.
It remains to construct one degree-$-2$ operator $f$ such that
\begin{equation}\label{intro:operator-goal}
fP_k\subset P_{k-2},\qquad
fW_j\subset W_{j-4},\qquad [[f,e],e]=-2e.
\end{equation}
An elementary Lefschetz-module argument then identifies both filtrations as sums of eigenspaces of the same operator $[e,f]$. It also handles extensions between filtration grades.
To construct $f$, allow one logarithmic pole, move it over a finite cover $S\to C$, and choose a full flag at the pole. The logarithmic $\lambda$-connection parameter interpolates between Higgs bundles at $\lambda=0$ and connections at $\lambda=1$. The underlying moduli constructions and semistable reduction come from \cite{dCFHLogHodge,FHZ2025}; the restriction argument uses the semiprojective geometry of \cite{HRV2015}. We verify the fixed determinant, residue, and flag conditions, as well as compatibility with the full nonabelian-Hodge identification. Write $X_\lambda$ for the fixed-pole scalar-residue moduli space, with $X_0=X$. At scalar residue, the flag space fits into a correspondence
\[
X_\lambda\xleftarrow{\ \pr\ }S\times X_\lambda
\xleftarrow{\ \pi\ }Z_\lambda
\xrightarrow{\ \iota\ }Y_{\lambda,0}.
\]
Here $Y_{\lambda,0}$ permits a flag-preserving residue with prescribed scalar diagonal, while $Z_\lambda$ requires the residue itself to be scalar. We first deform the ordered residues to distinct values and shift two residue eigenline lattices in opposite directions. Restriction isomorphisms from the total logarithmic Hodge family transport the resulting translation to a degree-zero operator $T$ on $H^*(Y_{\lambda,0})$. On the connection side, $T$ is eigenvalue monodromy. A power has unipotent action, and its logarithm $N$ lowers the diagram weights by two.
At $\lambda=0$, the resulting operator, before multiplication by a nonzero scalar, is
\[
f'=\pr_*\pi_*\iota^*N^2\iota_*\pi^*\pr^*.
\]
The flag shifts cancel. The square $N^2$ lowers weights by four, and integration over $S$ has weight shift zero in the diagram construction. On the Higgs side, nearby cycles realize this same operator as a morphism
$Rh_*\Q_X\to Rh_*\Q_X[-2]$, giving the perverse bound. A lattice Riemann--Roch calculation then gives the double commutator in \eqref{intro:operator-goal} after explicit rescaling.
The proof keeps three comparison statements separate: ordinary base change for the Betti family, restriction isomorphisms for the logarithmic Hodge family, and specialization over the Hitchin base. Each is proved with its own hypotheses. Section~\ref{sec:lefschetz} gives the Lefschetz comparison criterion. Section~\ref{sec:betti} proves ordinary base change and curious hard Lefschetz for the full fixed-determinant Betti family. Section~\ref{sec:diagram} constructs the diagram weights, and Section~\ref{sec:logarithmic} establishes the logarithmic restriction isomorphisms and full-cohomology comparison. Section~\ref{sec:translation} constructs the common operator and proves its weight bound; Section~\ref{sec:specialization} proves its perverse bound by specialization. Section~\ref{sec:commutator} computes the double commutator, and Section~\ref{sec:conclusion} assembles these results to prove Theorem~\ref{thm:main}.
==== END P-equals-W-in-composite-rank-for-fixed-determinant//01-introduction.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//02-lefschetz.tex ====
\section{Lefschetz operators and the comparison criterion}
\label{sec:lefschetz}
For the proof of Theorem~\ref{thm:main}, assume that $n\ge4$ is composite.
We first fix the perverse normalization and identify a common candidate for
the two Lefschetz operators. We then give the linear-algebra argument that
will compare the filtrations once a common lowering operator has been
constructed. Throughout this section, cohomology has rational coefficients.
\subsection{The normalized perverse filtration}
Recall that $X$ parametrizes stable, trace-free Higgs bundles of rank $n$
and determinant $L$ on $C$, and that $h:X\to A$ is its Hitchin map.
The standard geometry of this system gives a proper surjective morphism
from a smooth quasi-projective variety, together with a holomorphic
symplectic form of weight one for Higgs-field scaling;
see \cite[Section~1.1]{dCMS2022}.
\begin{lemma}\label{norm:dimensions}
Set $R=(n^2-1)(g-1)$. Then $\dim X=2R$, $\dim A=R$, and every fiber of
$h$ has dimension $R$. In particular,
\[
r:=\dim(X\times_A X)-\dim X=R.
\]
\end{lemma}
\begin{proof}
Riemann--Roch gives
\[
\dim A=\sum_{i=2}^n h^0(C,K_C^i)
=\sum_{i=2}^n(2i-1)(g-1)=R.
\]
The trace-free Higgs deformation complex has vanishing zeroth and second
hypercohomology, by stability and Serre duality. Its Euler characteristic
is $-2R$, so the tangent space has dimension $2R$ at every point.
We recall why the fiber bound holds also over the singular Hitchin values.
Higgs-field scaling makes $X$ semiprojective: properness of $h$ extends
each scaling orbit at zero, since its Hitchin value extends there, and
the fixed locus is closed in the proper fiber $h^{-1}(0)$.
For the Bia\l{}ynicki--Birula decomposition \cite{BialynickiBirula1973}
in this semiprojective setting, see \cite[Section~1.2]{HRV2015}.
At a fixed point, the symplectic form pairs a tangent weight $w$ with
weight $1-w$. The weights are integers, so the nonpositive tangent
weights account for exactly half the tangent dimension. Consequently
each repelling stratum, including its fixed component, has dimension
$R$.
The union of these strata is $h^{-1}(0)$. Indeed, a point with a limit
at infinity must have zero Hitchin value, because every Hitchin coordinate
has positive scaling weight. Conversely, an orbit in $h^{-1}(0)$ has a
limit at infinity because that fiber is proper. Thus
$\dim h^{-1}(0)=R$. Pulling $h$ back along the weighted scaling curve
from any $a\in A$ to zero, upper semicontinuity of fiber dimension for a
proper morphism gives $\dim h^{-1}(a)\le R$. The dimension inequality
for a morphism from a smooth $2R$-dimensional variety to an
$R$-dimensional variety gives the reverse inequality on each nonempty
fiber. Surjectivity therefore proves the asserted equidimensionality.
It follows that $\dim(X\times_A X)=R+2R=3R$.
\end{proof}
Put
\[
K=Rh_*\mathbb Q_X[R].
\]
We normalize the perverse filtration by
\begin{equation}\label{norm:perverse-filtration}
P_kH^m(X)=\operatorname{Im}\left\{
H^{m-R}\bigl(A,{}^p\tau_{\le k}K\bigr)
\longrightarrow H^m(X)\right\}.
\end{equation}
This is the normalization of \cite[Section~1.1]{MS2024}.
The fiber bound implies that $Rh_*\mathbb Q_X[2R]$ has perverse
cohomology only in degrees $[-R,R]$: the upper bound follows from the
support criterion, and the lower bound follows by Verdier duality.
Indeed its ordinary stalk cohomology vanishes in positive degrees,
and every support has dimension at most $R$.
After shifting by $[-R]$, the perverse degrees of $K$ lie in $[0,2R]$.
In particular,
\begin{equation}\label{norm:perverse-range}
P_{-1}H^*(X)=0,\qquad P_{2R}H^*(X)=H^*(X).
\end{equation}
We will use the following shift convention repeatedly. A morphism
$K\to K[s]$ induces a map of cohomological degree $s$ sending $P_k$ into
$P_{k+s}$, since
\begin{equation}\label{norm:shift-rule}
{}^p\tau_{\le k}(K[s])
=({}^p\tau_{\le k+s}K)[s].
\end{equation}
Thus an actual morphism $Rh_*\mathbb Q_X\to Rh_*\mathbb Q_X[-2]$
will supply precisely the lowering bound needed below.
\subsection{The degree-two class}
The universal projective bundle on $C\times X$ determines the characteristic class
\[
\widetilde{\operatorname{ch}}_2(E)
=\operatorname{ch}_2(E)-\frac{c_1(E)^2}{2n}
=\frac{1}{2n}\operatorname{ch}_2(\operatorname{End}E).
\]
The last expression also defines it when a universal vector bundle has
not been chosen. Define
\begin{equation}\label{norm:e}
e=\int_C\widetilde{\operatorname{ch}}_2(E)\in H^2(X).
\end{equation}
We also write $e$ for cup product by this class.
\begin{lemma}\label{norm:low-cohomology}
For the composite ranks $n\ge4$ under consideration,
\[
H^0(X)=\mathbb Q,\qquad H^1(X)=0,\qquad H^2(X)=\mathbb Q e,
\qquad e\ne0.
\]
Every ample class on $X$ is a nonzero scalar multiple of $e$.
\end{lemma}
\begin{proof}
The finite group $\Gamma=\operatorname{Pic}^0(C)[n]$ acts on $X$ by
tensor product. Averaging over $\Gamma$ decomposes rational cohomology
into its invariant summand and its complementary variant summand.
Let $p$ be the smallest prime divisor of $n$, and put
\[
c=n(n-n/p)(g-1).
\]
Proposition~1.4 of \cite{dCMS2022}, which applies to arbitrary rank,
gives
\[
H^i_{\mathrm{var}}(X)=P_{i-c}H^i_{\mathrm{var}}(X).
\]
Here $c\ge n^2/2\ge8$, so \eqref{norm:perverse-range} makes the
variant summand vanish in degrees $0$, $1$, and $2$.
The invariant summand is the rational cohomology of the corresponding
$\mathrm{PGL}_n$ moduli space. Markman's generation theorem \cite[Theorem~7]{Markman2002}, in the
projective-bundle formulation of \cite[Theorem~2.1(i)]{MS2024}, states
that this algebra is generated by the K\"unneth components of the
projective Chern characters with indices $j\ge2$. These components
have degrees $2j$, $2j-1$, and $2j-2$ on the moduli space. There are
therefore no degree-one generators and only one possible degree-two
generator, namely a nonzero normalization of the expression defining
$e$. This proves $H^0(X)=\mathbb Q$, $H^1(X)=0$, and
$H^2(X)=\mathbb Q e$, with nonvanishing still to be checked.
Choose an ample line bundle on $X$. By Lemma~\ref{norm:dimensions},
a Hitchin fiber is projective of positive dimension, and hence contains
an integral projective curve. The ample bundle has positive degree on
that curve, so its first Chern class is nonzero in $H^2(X)$. The
one-dimensional upper bound just proved now gives $e\ne0$ and the
final assertion.
\end{proof}
\begin{proposition}\label{norm:relative-lefschetz}
Cup product by $e$ sends $P_kH^m(X)$ into $P_{k+2}H^{m+2}(X)$.
For every $i\ge0$ and every $m$, it induces an isomorphism
\begin{equation}\label{norm:relative-hl}
e^i:\operatorname{Gr}^P_{R-i}H^m(X)
\xrightarrow{\ \sim\ }
\operatorname{Gr}^P_{R+i}H^{m+2i}(X).
\end{equation}
\end{proposition}
\begin{proof}
The filtration shift follows from \eqref{norm:shift-rule} applied to
the cup-product morphism. Choose an ample class on $X$, which is
also relatively ample for the proper morphism $h$, making $h$
projective. The decomposition theorem and relative hard Lefschetz
\cite[Theorems~6.2.5 and~6.2.10]{BBD1982} give \eqref{norm:relative-hl} for that
class: the perverse degrees of $Rh_*\mathbb Q_X[2R]$ have center zero,
and the shift to $K$ moves the center to $R$. The decomposition theorem
also makes the perverse cohomology spectral sequence degenerate, so
these sheaf-theoretic isomorphisms give the displayed isomorphisms on
associated-graded cohomology. Lemma~\ref{norm:low-cohomology} replaces
the ample class by a nonzero scalar multiple of $e$, without changing
invertibility.
\end{proof}
\subsection{A common lowering operator determines the filtration}
For a finite graded vector space $V=\bigoplus_k V_k$ and a degree-two
operator $E$, say that $E$ is \emph{Lefschetz with center $r$} if
\[
E^i:V_{r-i}\xrightarrow{\sim}V_{r+i}
\quad\text{for every }i\ge0.
\]
Here $r$ is an integer, and grades outside the finite support are zero.
Proposition~\ref{norm:relative-lefschetz} gives this property for the
total perverse associated graded, with center $R$. The next statement
explains the role of the lowering operator in the proof.
The use of an $\mathfrak{sl}_2$-operator to determine the perverse
filtration is also central to \cite[Propositions~7.9 and~8.2--8.3]{HMMS2025}.
We prove the precise two-filtration criterion needed here, including
the passage from associated gradeds to the filtrations themselves.
\begin{proposition}[Comparison by a common lowering operator]
\label{alg:comparison}
Let $H$ be a finite-dimensional vector space over a field of
characteristic zero, with finite, exhaustive, separated increasing
integer-indexed filtrations $F_\bullet$ and $G_\bullet$.
Suppose that $e,f\in\operatorname{End}(H)$ satisfy, for
$A_\bullet=F_\bullet$ and $A_\bullet=G_\bullet$,
\[
e(A_k)\subseteq A_{k+2},\qquad f(A_k)\subseteq A_{k-2}.
\]
Assume that the induced raiser on each associated graded is Lefschetz
with the same center $r$, and that
\begin{equation}\label{alg:double-commutator}
[[f,e],e]=-2e,
\qquad [u,v]=uv-vu.
\end{equation}
Then $F_k=G_k$ for every integer $k$.
If $H$ has a further grading respected by both filtrations, and $e,f$
have degrees $2,-2$ for that grading, the equality holds degree by
degree.
\end{proposition}
\begin{proof}
Fix either associated graded, and denote its raiser by $E$.
The Lefschetz decomposition expresses it as a sum of primitive strings
\[
v,Ev,\ldots,E^\ell v,\qquad \deg v=r-\ell.
\]
On each string set
\[
F_0(E^jv)=j(\ell-j+1)E^{j-1}v.
\]
If $H_0$ acts by $k-r$ on grade $k$, direct calculation gives
\[
[E,F_0]=H_0,\qquad [H_0,E]=2E,\qquad [H_0,F_0]=-2F_0.
\]
Thus $(E,H_0,F_0)$ is an $\mathfrak{sl}_2$-triple, and
$[[F_0,E],E]=-2E$.
We claim that a degree $-2$ endomorphism satisfying this last identity
must equal $F_0$. On the endomorphism space, the adjoint
$\mathfrak{sl}_2$-action identifies grading degree $-2$ with
$H_0$-weight $-2$. Decompose this finite-dimensional representation
into irreducibles. In any irreducible summand containing weight $-2$,
two raising steps map that weight space nontrivially to weight $2$.
Consequently
\[
(\operatorname{ad}E)^2:
\operatorname{End}(\operatorname{Gr}^A H)_{-2}
\longrightarrow
\operatorname{End}(\operatorname{Gr}^A H)_{2}
\]
is injective. Applying this to $\operatorname{Gr}^A(f)-F_0$ and
\eqref{alg:double-commutator} proves the claim.
It follows that the single operator $h_0=[e,f]$ on $H$ preserves both
filtrations and acts on their grade $k$ by $k-r$. Choose integers
$a\le b$ containing all their nonzero grades. For either filtration,
\[
(h_0-(k-r))A_k\subseteq A_{k-1}.
\]
Applying these factors successively from $k=b$ down to $k=a$ gives
\[
\prod_{k=a}^b(h_0-(k-r))=0.
\]
The polynomial has distinct roots, so $h_0$ is semisimple.
Every $h_0$-stable subspace is therefore the direct sum of its
intersections with the eigenspaces.
If a nonzero eigenvector first belongs to $A_j$, its image in
$\operatorname{Gr}^A_jH$ shows that its eigenvalue is $j-r$.
Since each $A_k$ is $h_0$-stable, this proves
\begin{equation}\label{alg:eigenspace-filtration}
A_k=\bigoplus_{j\le k}\ker\bigl(h_0-(j-r)\bigr).
\end{equation}
The right side is independent of the choice of filtration, proving
$F_k=G_k$. In the additionally graded setting, $h_0$ has degree zero,
so its eigenspaces and \eqref{alg:eigenspace-filtration} decompose
degree by degree.
\end{proof}
We also record a cancellation fact for the Betti Lefschetz argument.
\begin{lemma}\label{alg:tensor}
Let $V,W$ be nonzero finite integer-graded vector spaces over a field
of characteristic zero, and let $E_V,E_W$ have degree two.
If $E_V\otimes1+1\otimes E_W$ is Lefschetz with center $r$ on
$V\otimes W$, then $E_V$ and $E_W$ are Lefschetz with centers
$r_V,r_W$, respectively, where $r_V+r_W=r$.
\end{lemma}
\begin{proof}
Choose homogeneous Jordan-string decompositions for the two operators,
which are nilpotent because the gradings have finite support.
The tensor product of any pair of strings is a graded invariant direct
summand, with invariant complement, for the sum operator. Each
Lefschetz isomorphism is block diagonal with respect to these summands,
so it is an isomorphism on each of them.
For a string whose lowest and highest grades are $p$ and $q$, call
$(p+q)/2$ its center. The average of the lowest and highest grades of
a tensor product of two strings is the sum of their centers. Its
Lefschetz symmetry forces that sum to equal $r$, for every pair of
strings. Fixing one string in either factor shows that all strings
in the other factor have the same center. A Jordan string is
Lefschetz about its own center, which proves the conclusion.
\end{proof}
We will apply Proposition~\ref{alg:comparison} to total cohomology,
with $F_k=P_k$ and $G_k=W_{2k}$ under nonabelian Hodge theory.
Section~\ref{sec:betti} proves that $e$ is also Lefschetz for the
half-weight grading, with center $R$, and that all odd weight-graded
pieces vanish. The remaining geometric construction supplies one
operator $f:H^m(X)\to H^{m-2}(X)$ satisfying both lowering bounds
and \eqref{alg:double-commutator}. The proposition then identifies
the two filtrations in every cohomological degree; the separate
even-weight assertion gives $W_{2k}=W_{2k+1}$.
==== END P-equals-W-in-composite-rank-for-fixed-determinant//02-lefschetz.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//03-betti.tex ====
\section{Curious hard Lefschetz on the full character variety}
\label{sec:betti}
We now establish the Betti Lefschetz statement needed for the comparison
criterion. The point requiring care is the full cohomology of the
fixed-determinant space. We obtain it from a finite cover of a
$\mathrm{GL}_n$ family, retaining every component of every lifted stratum.
For a finite-dimensional vector space graded by half weights, a class of
cohomological degree two and weight four acts with grading degree two.
We say it satisfies \emph{Lefschetz with center $c$} if its $i$th power
is an isomorphism from grade $c-i$ to grade $c+i$, in the corresponding
cohomological degrees, for every $i\geq0$.
A mixed Hodge structure is \emph{Hodge--Tate} if its odd weight-graded
pieces vanish and its weight-$2r$ piece has only Hodge type $(r,r)$.
\begin{theorem}\label{betti:lefschetz}
The mixed Hodge structure on $H^*(X^{\mathrm B},\mathbb Q)$ is Hodge--Tate.
Under the nonabelian-Hodge identification, the class $e$ has nonzero image
in $\operatorname{Gr}^W_4H^2(X^{\mathrm B},\mathbb Q)$, and
\[
e^i:\operatorname{Gr}^W_{2R-2i}H^m(X^{\mathrm B},\mathbb Q)
\xrightarrow{\ \sim\ }
\operatorname{Gr}^W_{2R+2i}H^{m+2i}(X^{\mathrm B},\mathbb Q)
\qquad(i\geq0).
\]
In particular, $W_{2k}=W_{2k+1}$ on this cohomology. These assertions
concern the entire cohomology, including all nontrivial
$\Gamma$-character summands after complexification.
\end{theorem}
\subsection{The ordered family and its scalar fiber}
Let
\[
V=\left\{(t_1,\ldots,t_n)\in(\mathbb C^*)^n:
\prod_i t_i=1,\quad
\prod_{i\in I}t_i\ne1\text{ for }\varnothing\ne I\subsetneq\{1,\ldots,n\}
\right\},
\qquad q=(\zeta,\ldots,\zeta).
\]
The space $V$ is a connected smooth open subset of a torus, and $q\in V$
since $\zeta$ has order $n$. Write $\nu=\prod_{j=1}^g[A_j,B_j]$.
The fiber $Y^{\mathrm B}_t$ parametrizes tuples
$(A_1,B_1,\ldots,A_g,B_g)\in\mathrm{SL}_n^{2g}$ together with a full flag
preserved by $\nu$, on whose ordered graded lines $\nu$ acts by
$t_1,\ldots,t_n$, modulo simultaneous conjugation. Repeated eigenvalues
are allowed.
Every such tuple is irreducible. Indeed, on a common invariant proper
subspace the determinant of $\nu$ is one, while its eigenvalues form a
proper nonempty submultiset of the $t_i$. This contradicts the definition
of $V$. Thus conjugation gives geometric quotients by a free
$\mathrm{PGL}_n$ action. They may be constructed by the usual
\'etale-local principal-bundle charts on irreducible tuples; with a full
flag, these charts can also be obtained by completing its first line to
a basis using words in the matrices.
\begin{lemma}\label{betti:geometry}
The morphism $Y^{\mathrm B}\to V$ is smooth of relative dimension
$2R+2b$. There is a correspondence
\[
X^{\mathrm B}\xleftarrow{\ \pi\ }Z^{\mathrm B}
\xrightarrow{\ \iota\ }Y^{\mathrm B}_q,
\]
where $Z^{\mathrm B}$ imposes $\nu=\zeta I$ and $\pi$ is the full flag
bundle. The map $\iota$ is a regular embedding of codimension $b$ with
normal bundle $T_\pi^*$. The cohomological operators
\[
\mathsf A=\pi_*\iota^*,\qquad
\mathsf B=\iota_*\pi^*
\]
have degrees $-2b$ and $2b$, respectively, and satisfy
\begin{equation}\label{betti:split}
\mathsf A\mathsf B=(-1)^b n!\,\mathrm{id}.
\end{equation}
Here and below a covariant star denotes the Gysin map.
\end{lemma}
\begin{proof}
The commutator-product map is a submersion at every irreducible tuple.
We give the differential check. For a pair $a,b'$ set
$c=ab'a^{-1}b'^{-1}$. Its right-translated differential has terms
\[
(1-\operatorname{Ad}_{ab'a^{-1}})u,\qquad
(\operatorname{Ad}_a-\operatorname{Ad}_c)v.
\]
Using the trace pairing, an annihilating vector $z$ is fixed by
$ab'a^{-1}$ and satisfies
$\operatorname{Ad}_{a^{-1}}z=\operatorname{Ad}_{b'}z$.
It is therefore fixed by $ab'$, hence by $a$ and $b'$.
For a product of commutators one applies this argument to the testing
vector conjugated by each prefix. Since the vector is fixed by the
current commutator, it stays unchanged from one pair to the next.
It consequently centralizes every matrix. Irreducibility forces a
scalar, and the trace-free condition forces zero.
The space of a matrix in a flag Borel, with its flag, is smooth over
its ordered diagonal torus, with fibers of dimension $2b$.
Pulling this space back along the submersion and then taking the free
quotient gives smoothness. The relative dimension is
\[
2g(n^2-1)+2b-2(n^2-1)=2R+2b.
\]
At $q$, the section $\zeta^{-1}\nu-I$ takes values in the nilradical
bundle over the flag space. Its zero locus is $Z^{\mathrm B}$, and
submersivity proves that this section is regular. The trace pairing
identifies the nilradical with the dual of the flag tangent space, so
its associated bundle is $T_\pi^*$. Self-intersection and the projection
formula now give
\[
\mathsf A\mathsf B(\alpha)
=\pi_*\bigl(c_b(T_\pi^*)\pi^*\alpha\bigr)
=(-1)^b\chi(\mathrm{Fl}_n)\alpha
=(-1)^b n!\alpha.
\]
All constructions descend through $\mathrm{PGL}_n$ and apply to full
cohomology.
\end{proof}
\subsection{Lifting the toric stratifications}
Set $D_1=(\mathbb C^*)^{2g}$ and
$\widehat Y^{\mathrm B}=Y^{\mathrm B}\times D_1$.
Multiplying each of the $2g$ matrices by its corresponding scalar gives
\begin{equation}\label{betti:cover}
\widehat Y^{\mathrm B}\longrightarrow Y^{\mathrm B}_{\mathrm{GL}},
\end{equation}
where the target is the same ordered family with matrices in
$\mathrm{GL}_n$. This is the pullback of the $n$th-power map on the
$2g$ determinant coordinates, hence is finite \'etale of degree
$n^{2g}$. Its source is the indicated product: dividing by the chosen
determinant roots gives matrices in $\mathrm{SL}_n$, without changing
commutators or flags.
We use Mellit's toric stratifications of the $\mathrm{GL}_n$ family
\cite{Mellit2025}. To identify the family exactly, take two punctures
with first monodromy $\zeta^{-1}I$ and last monodromy
$\zeta\nu^{-1}$. The latter has ordered eigenvalues
$(\zeta t_i^{-1})_i$. The product relation is
$\nu\zeta^{-1}I\cdot\zeta\nu^{-1}=I$, and the genericity condition of
\cite[Definition~4.9]{Mellit2025} is precisely the subproduct condition
in the definition of $V$. Its identity last monodromy corresponds to
$q$. The ordering condition in that reference is automatic for the
scalar first monodromy and for the distinct-eigenvalue last monodromy.
We recall the two features of the stratifications which will be used.
First, at a parameter with distinct $t_i$, an auxiliary rank-$b$ vector
bundle over $Y^{\mathrm B}_{\mathrm{GL},t}$ has a finite stratification
refined into vector bundles over products of tori and affine spaces.
The restriction of
Mellit's degree-two class gives compact-support curious Lefschetz on
every final stratum, with the same middle weight. If
$\delta=\dim Y^{\mathrm B}_{\mathrm{GL},t}$, that middle \emph{actual}
weight is $\delta+2b$; removal of the vector-bundle shift gives
$\delta$ \cite[Theorem~6.14, Section~7.4, and Theorem~7.5]{Mellit2025}.
Second, over all of $V$, an auxiliary affine bundle of rank $b$ has a
finite stratification whose refinements are vector bundles over
\[
\mathbb A^a\times(\mathbb C^*)^s\times V
\]
with structural map the projection to $V$
\cite[proof of Proposition~8.7]{Mellit2025}.
These assertions are about the final torus--affine refinements, not
about arbitrary braid strata occurring earlier in their construction.
Here is the additional finite-cover argument needed for
\eqref{betti:cover}.
\begin{lemma}\label{betti:finite-cover}
Pulling the preceding auxiliary bundles and stratifications back along
\eqref{betti:cover} has the following consequences.
\begin{enumerate}
\item Every fiber of $\widehat Y^{\mathrm B}\to V$ and of
$Y^{\mathrm B}\to V$ has Hodge--Tate cohomology. For
$\widehat Y^{\mathrm B}$ at a parameter with distinct eigenvalues, the pulled-back
class satisfies curious hard Lefschetz on ordinary cohomology with
half-weight center $\delta/2=R+b+g$.
\item The cohomology sheaves of both ordinary and compact-support
integral direct images over $V$ are locally constant and finitely
generated. Ordinary direct image commutes with restriction to every
fiber. The same assertions hold for $Y^{\mathrm B}\to V$.
\end{enumerate}
\end{lemma}
\begin{proof}
Consider first a final stratum $F=(\mathbb C^*)^s\times\mathbb A^a$,
or a vector bundle over such a stratum. Its homotopy type is that of
a torus. Each connected component $\widetilde F$ of its finite \'etale
preimage corresponds to a finite-index subgroup of $\mathbb Z^s$.
The pullback
\[
H^*(F,\mathbb Q)\longrightarrow H^*(\widetilde F,\mathbb Q)
\]
is therefore an isomorphism: both rings are the exterior algebras of
the rational duals of these lattices. Since the map is algebraic, it
is an isomorphism of mixed Hodge structures. Both spaces are smooth
of the same dimension, and finite-degree trace together with
Poincar\'e duality gives the same assertion for compact supports.
These isomorphisms commute with cup product by the pulled-back class.
We apply this argument to each connected component separately.
The cohomology of a disconnected preimage is their direct sum, and
all components have the same Lefschetz center. Preimages of the closed
filtration of the original stratification still give a finite closed
filtration. In the compact-support localization sequences, strictness
of mixed Hodge morphisms \cite[Theorem~2.3.5]{Deligne1971} permits
passage to weight-gradeds. Comparing
these exact sequences by a fixed power of the common class and using
the five lemma passes the Lefschetz isomorphisms from the strata to
the entire auxiliary space. The same sequences show that its
cohomology is Hodge--Tate.
The pulled-back auxiliary bundle has the same rank $b$. Its Thom
isomorphism shifts cohomological degree and actual weight by $2b$.
Thus the compact-support middle weight drops from $\delta+2b$ to
$\delta$. Smooth Poincar\'e duality sends weight $w$ to
$2\delta-w$, so ordinary cohomology has the same middle weight
$\delta$, or center $\delta/2$ in half weights. The relative product
stratification proves the Hodge--Tate assertion on every fiber,
including repeated eigenvalues, by the same componentwise argument
and the affine-bundle Thom isomorphism.
For integral local constancy, use the relative stratification and
choose a small contractible analytic open set $U\subset V$.
A finite cover of $F\times U$ is pulled back from a finite cover of
$F$, since $\pi_1(F\times U)=\pi_1(F)$. The same statement holds for
the total space of a vector bundle over $F\times U$, which is homotopy
equivalent to its base. Topologically the bundle may be identified,
in the $U$ direction, with the pullback of its restriction over one
point. Equivalently, one may use the integral Thom isomorphism for
its complex orientation. It follows that the compact-support
cohomology sheaves of each covered stratum are locally constant
finitely generated abelian groups on $U$. Localization triangles
pass this conclusion through the finite stratification: kernels,
cokernels, and extensions of locally constant sheaves are locally
constant. The auxiliary affine-bundle trace removes its shift.
Write $f$ for the covered smooth family of relative dimension
$\delta$. The complex $K=Rf_!\mathbb Z$ is locally perfect: its
cohomology is bounded and finitely generated, and $\mathbb Z$ has
finite global dimension. Smooth Verdier duality gives
\[
Rf_*\mathbb Z\simeq
R\mathcal Hom(K,\mathbb Z_V)[-2\delta].
\]
The right side is locally constant and perfect, so taking its stalk
commutes with this duality. Compact-support base change and the
fiberwise cup--trace pairing identify its stalk with
$R\Gamma(f^{-1}(t),\mathbb Z)$. This identification is precisely the
natural ordinary base-change morphism: both are adjoint to the cup
product followed by the smooth orientation trace, whose formation
commutes with compact-support base change. This proves the ordinary
assertion, without using a nonproper smooth base-change principle.
Finally, write $p:Y^{\mathrm B}\times D_1\to Y^{\mathrm B}$ for the
projection and $s$ for its section at the identity of $D_1$. For ordinary
direct images, $s^*p^*=\mathrm{id}$ gives the required summand. For
compact supports use the Gysin map of the closed section, followed by
integration over $D_1$: these maps have degrees $4g$ and $-4g$ and
compose to the identity. Equivalently, the integral compact-support
K\"unneth complex of $D_1$ has its top class as a split summand. Thus
both local-constancy assertions pass to $Y^{\mathrm B}$, and natural
ordinary base change passes through the first splitting. The ordinary
splitting is algebraic on each fiber, so it also proves the asserted
Hodge--Tate property for $Y^{\mathrm B}$.
\end{proof}
\subsection{The scalar summand and the determinant factor}
We have now obtained Lefschetz on a regular-semisimple fiber of the
cover and cohomological local constancy over the entire ordered base.
We next pass to $q$ and then remove the flag and determinant factors.
For a smooth algebraic family whose ordinary direct images are lisse (that is, have locally constant
cohomology sheaves) and satisfy fiberwise base change, mixed Hodge module direct image
gives admissible graded-polarizable variations of mixed Hodge
structure on these local systems. In particular, parallel transport
preserves the weight filtration; cup products are parallel as well.
The construction and its diagram version are given in
Proposition~\ref{diag:variation}. Apply this fact to the family in
Lemma~\ref{betti:finite-cover}. Along a path in the connected space
$V$, transport a Lefschetz class from a distinct-eigenvalue fiber to
$q$. We obtain
\[
\omega\in\operatorname{Gr}^W_4
H^2(\widehat Y^{\mathrm B}_q,\mathbb C)
\]
with Lefschetz center $R+b+g$. A choice of path suffices; no assertion
that this class is invariant under all monodromy is needed.
Take the product of the correspondence in
Lemma~\ref{betti:geometry} with $D_1$, writing
$\widehat X^{\mathrm B}=X^{\mathrm B}\times D_1$ and
$\widehat Z^{\mathrm B}=Z^{\mathrm B}\times D_1$.
The flag bundle has rational algebraic Leray--Hirsch classes. More
explicitly, the ratios of the ordered graded lines descend for the
universal projective bundle, and their first Chern classes restrict
to generators of the flag cohomology over $\mathbb Q$. Indeed,
differences of Chern roots generate rationally because the sum of the
roots vanishes on the flag fiber. Polynomial representatives of a
fiber basis therefore give the required global classes.
The additional degree-two classes have weight two. Consequently
\[
\pi^*:\operatorname{Gr}^W_4H^2(\widehat X^{\mathrm B},\mathbb C)
\xrightarrow{\ \sim\ }
\operatorname{Gr}^W_4H^2(\widehat Z^{\mathrm B},\mathbb C).
\]
There is a unique class $\xi$ in the space on the left such that
$\iota^*\omega=\pi^*\xi$ on the weight-graded.
By the projection formula, $\mathsf A$ and $\mathsf B$ intertwine
multiplication by $\omega$ and $\xi$ on weight-gradeds. Their actual
weight shifts are $-2b$ and $2b$. Equation~\eqref{betti:split} exhibits
the image of $\mathsf B$ as a graded stable summand, with stable
complement $\ker\mathsf A$. Each Lefschetz isomorphism for $\omega$
is block diagonal for this decomposition and thus restricts to an
isomorphism on the summand. The half-weight shift of $\mathsf B$ is
$b$, so $\xi$ has center $R+g$ on
$H^*(\widehat X^{\mathrm B},\mathbb C)$. This correspondence also
makes its cohomology a summand of a Tate twist of the Hodge--Tate
cohomology upstairs, hence Hodge--Tate.
It remains to remove $D_1$.
By Lemma~\ref{norm:low-cohomology} and the given nonabelian-Hodge
diffeomorphism, $H^0(X^{\mathrm B})=\mathbb Q$ and
$H^1(X^{\mathrm B})=0$. K\"unneth therefore gives
\[
\xi=\xi_X+\xi_D,
\qquad
\xi_X\in\operatorname{Gr}^W_4H^2(X^{\mathrm B},\mathbb C),
\quad
\xi_D\in H^2(D_1,\mathbb C).
\]
The tensor-factor Lefschetz lemma, Lemma~\ref{alg:tensor}, shows that
each factor has a common Lefschetz center and that the two centers
add to $R+g$. The cohomology of $D_1$ is an exterior algebra on $2g$
generators of cohomological degree one and weight two. Its smallest
and largest half-weight grades are $0$ and $2g$, so its center is
$g$. Thus $\xi_X$ has center $R$. Projection and a section also
show that $X^{\mathrm B}$ itself has Hodge--Tate cohomology.
Since $H^0(X^{\mathrm B})\ne0$ and $R>0$, this Lefschetz operator
$\xi_X$ is nonzero. Lemma~\ref{norm:low-cohomology} gives
$H^2(X^{\mathrm B},\mathbb Q)=\mathbb Q e$ as a vector space, so the
image of $e$ in $\operatorname{Gr}^W_4$ is nonzero and is a nonzero
scalar multiple of $\xi_X$. Here $e\in W_4H^2$ because
$X^{\mathrm B}$ is smooth. Rescaling a Lefschetz operator preserves
all its isomorphisms. They hold over $\mathbb Q$, since the rational
maps in question become isomorphisms over $\mathbb C$. This proves
Theorem~\ref{betti:lefschetz}. No step took invariants under the
finite group $\Gamma$.
==== END P-equals-W-in-composite-rank-for-fixed-determinant//03-betti.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//04-diagram.tex ====
\section{Weights for flat bundles of algebraic families}
\label{sec:diagram}
Consider first a product $S\times X^{\mathrm B}$, with $S$ a closed
oriented smooth surface. The moving-point construction in the proof overview
requires integration over $S$ to lower cohomological degree by two while
preserving weights. On this product, give the K\"unneth factor
$H^a(S,\mathbb Q)$ pure weight zero and Hodge type $(0,0)$ in every degree,
and retain Deligne's mixed Hodge structure on $H^*(X^{\mathrm B},\mathbb Q)$.
Integration over $S$ then acts on the first factor without a Tate twist.
Even when $S$ is an algebraic curve, this convention differs from its
ordinary Deligne weights.
The moving-puncture flag spaces need not be products. We therefore
extend this convention to bundles whose transition maps are locally
constant algebraic automorphisms of a fiber family. A finite diagram of
algebraic fibers will give the mixed Hodge structures together with their
compatibility with parallel transport and proper Gysin maps. Section~\ref{sec:logarithmic}
will construct the required bundles and identify the scalar unflagged
bundle as an actual flat product. The simplicial mixed-Hodge formalism
goes back to Deligne \cite[Section~8.3]{Deligne1974}; Shende uses weight-zero
simplicial cochains to compute weights of tautological classes on
character varieties \cite[proof of the main Theorem]{Shende2017}.
Here we supply the relative construction and the Gysin compatibility
needed for the moving-point correspondence.
\subsection{The finite diagram}
Let $V$ be a smooth connected complex algebraic variety and let
$p:H\to V$ be an algebraic morphism of finite type. We assume that
$Rp_*\mathbb Z_H$ has bounded, locally constant, finitely generated
cohomology sheaves and that its natural ordinary base-change maps
\begin{equation}
\label{diag:basechange}
i_t^*Rp_*\mathbb Z_H\longrightarrow
R\Gamma(H_t,\mathbb Z)
\end{equation}
are isomorphisms for all $t\in V$, where $i_t:\{t\}\hookrightarrow V$.
These are hypotheses about ordinary direct image; smoothness alone would
not imply them for a nonproper morphism.
Let $S$ be a compact smooth manifold. A \emph{flat bundle of the family
$H\to V$ over $S$} means a space $\mathscr H$ with a map to $S\times V$
and local identifications
\[
\mathscr H|_{U_i\times V}\simeq U_i\times H
\]
whose transition maps have the form
$(s,h)\mapsto(s,g_{ij}(h))$, with $g_{ij}$ an algebraic automorphism
over $V$, locally constant in $s$. The maps $g_{ij}$ satisfy the cocycle
identity. Write $\mathscr H_t$ for the total space over $S$ at $t\in V$.
Thus $\mathscr H_t$ includes the parameter space $S$; it need not be an
algebraic variety.
A map of flat bundles will mean one which, in simultaneous flat
trivializations, is the identity on $S$ times an algebraic map of the
fiber families.
Choose a finite good cover $\mathcal U=\{U_i\}$ refining a flat
trivializing cover. Every nonempty intersection $U_I=\bigcap_{i\in I}U_i$
is contractible. Choose a trivialization on each $U_I$. Inclusion of
intersections then gives a finite diagram of copies
$p_I:H_I\to V$ and algebraic isomorphisms over $V$.
The cocycle identity makes this a diagram of maps of varieties, before
passing to cohomology. Its index category is the poset of nonempty
intersections, ordered in the direction of cohomological restriction.
We use the enhanced bounded derived category of mixed Hodge modules,
whose six operations and rational realization are compatible with
diagrams \cite[Section~2.1, Theorem~2.1.1, Remark~2.1.2, and
Proposition~4.2.6]{TubachStacks2025}.
For an algebraic variety $T$, let $\mathbb Q_T^H$ denote the constant
mixed-Hodge-module complex with underlying complex $\mathbb Q_T$,
without a dimension shift. Define
\begin{equation}
\label{diag:complex}
K_{\mathscr H}
=\mathop{\mathrm{holim}}_I Rp_{I*}\mathbb Q_{H_I}^H
\quad\text{in }D^b\mathrm{MHM}(V).
\end{equation}
Here the homotopy limit means the derived limit, retaining the
transition maps and their compatibilities. The index poset has a
finite-dimensional nerve, so this is a finite limit and remains
bounded. In particular, \eqref{diag:complex} uses the full derived
diagram, not only the cohomology groups of its terms.
\begin{proposition}[Diagram weights]
\label{diag:variation}
Under the preceding hypotheses, $H^j(\mathscr H_t,\mathbb Q)$ carries
a graded-polarizable mixed Hodge structure, functorial for maps of flat
bundles that are algebraic in flat trivializations. As $t$ varies,
these structures form an admissible graded-polarizable variation of
mixed Hodge structures, with an integral underlying local system up
to torsion. They are computed by $H^j(i_t^*K_{\mathscr H})$.
If $H^j(H_t,\mathbb Q)$ is Hodge--Tate for every $j$ and $t$, then
$H^j(\mathscr H_t,\mathbb Q)$ is Hodge--Tate as well. We call its
weight filtration the \emph{diagram weight filtration}.
\end{proposition}
\begin{proof}
First we identify the underlying ordinary complex. Open-cover
descent computes the ordinary direct image of $\mathscr H\to V$
from the spaces $U_I\times H_I$. Projection to $H_I$ induces a
quasi-isomorphism on direct images over $V$: over any open subset
of $V$, its fiber in this product is the contractible space $U_I$.
These projections commute with the transition maps. Their derived
limit therefore identifies the rational realization of
$K_{\mathscr H}$ with the ordinary direct image of $\mathscr H\to V$.
The same argument works with integral coefficients.
A finite limit of bounded complexes with locally constant cohomology
again has locally constant cohomology. Its integral cohomology
sheaves have finite type. Point pullback commutes with finite limits,
and the termwise base-change maps are isomorphisms by
\eqref{diag:basechange}. The corresponding maps in mixed Hodge
modules are isomorphisms because rational realization is conservative.
It follows that $i_t^*K_{\mathscr H}$ computes the derived diagram of
the algebraic fibers $H_{I,t}$ and that its underlying cohomology is
$H^*(\mathscr H_t,\mathbb Q)$.
Put $v=\dim V$. Since the ordinary cohomology sheaves of
$K_{\mathscr H}$ are locally constant, its perverse cohomology
${}^pH^{j+v}(K_{\mathscr H})$ is a smooth mixed Hodge module.
Smooth mixed Hodge modules, after the dimension shift, are admissible
graded-polarizable variations
\cite[Section~3.2, Equation~(3.2.1)]{SaitoExtension1990}.
Applying point pullback to the perverse truncations shows that the
fiber of this variation is $H^j(i_t^*K_{\mathscr H})$: a smooth
perverse object has only one nonzero fiber degree, namely $-v$.
Finally, mixed Hodge structures whose weight gradeds have only Hodge
types $(r,r)$ are closed under subobjects, quotients, and extensions.
The long exact sequence of a cone thus shows that complexes with
Hodge--Tate cohomology are closed under finite limits. Applying this
observation to the fiber diagram proves the last assertion.
Algebraic maps of the diagrams give the stated functoriality.
A common good refinement compares different choices of cover and
trivializations: its comparison maps are isomorphisms after ordinary
descent, hence also in mixed Hodge modules by conservativity.
\end{proof}
The construction also preserves the cup product. Each algebraic
direct image of the constant complex has its functorial cup product,
and the transition pullbacks preserve it. The derived limit inherits
this product. Under the open-cover comparison in the proof of
Proposition~\ref{diag:variation}, it is the ordinary cup product on
$H^*(\mathscr H_t,\mathbb Q)$. Thus parallel transport in $V$ preserves
both the cohomology ring and its weight filtration.
\subsection{Products and proper Gysin maps}
We next identify the maps that preserve these weights.
\begin{proposition}[Weight shifts]
\label{diag:gysin}
The diagram weight filtrations have the following properties.
\begin{enumerate}
\item For the product $S\times M$, with $M$ a complex algebraic
variety, the ordinary K\"unneth isomorphism identifies its mixed Hodge
structure with that on
\[
H^*(S,\mathbb Q)\otimes H^*(M,\mathbb Q),
\]
where every $H^a(S,\mathbb Q)$ is pure of weight zero and type $(0,0)$,
and $H^*(M,\mathbb Q)$ has its Deligne mixed Hodge structure.
If $S$ is oriented and closed of real dimension $d_S$, projection
pullback and integration over $S$ preserve weight; the latter has
cohomological degree $-d_S$.
\item Pullbacks by maps of flat bundles preserve weight.
\item Suppose $u:\mathscr H_t\to\mathscr H'_t$ is a proper map of
flat bundles with smooth equidimensional algebraic fibers. Put
$c=\dim_{\mathbb C}H'_t-\dim_{\mathbb C}H_t$. With the complex
orientations on the fibers, its ordinary Gysin map satisfies
\[
u_*:W_aH^j(\mathscr H_t)\longrightarrow
W_{a+2c}H^{j+2c}(\mathscr H'_t).
\]
If $S$ is oriented, this is the Gysin map for the induced orientations
on the total spaces.
\end{enumerate}
\end{proposition}
\begin{proof}
For a product, the diagram is constant. Its derived limit is
\[
C^*(N\mathcal U,\mathbb Q(0))\otimes R\Gamma(M,\mathbb Q_M^H),
\]
where $N\mathcal U$ is the finite nerve of the good cover. The
first complex has a copy of $\mathbb Q(0)$ for each simplex, with
its ordinary simplicial differential. All its cohomology therefore
has weight zero. The external-product comparison on the cover
gives exactly the ordinary K\"unneth map. Evaluation on a simplicial
fundamental cycle defines
\[
C^*(N\mathcal U,\mathbb Q(0))\longrightarrow
\mathbb Q(0)[-d_S].
\]
This is a morphism of complexes of mixed Hodge structures and
realizes ordinary integration over $S$. It has no Tate twist.
Projection pullback is induced by the unit of the same cochain
complex. This proves the first assertion; functoriality of algebraic
pullbacks proves the second.
For the third assertion, fix $t$ and choose simultaneous
trivializations of the map. On each algebraic fiber map
$u_{I,t}:H_{I,t}\to H'_{I,t}$, smooth
dualizing complexes and the proper counit give the Gysin morphism
\begin{equation}
\label{diag:trace}
Ru_{I,t*}\mathbb Q^H_{H_{I,t}}\longrightarrow
\mathbb Q^H_{H'_{I,t}}[2c](c).
\end{equation}
Its degree and Tate twist give the displayed weight shift.
Complex orientations are preserved by the algebraic transition
isomorphisms, and the proper counit is natural. Consequently
\eqref{diag:trace} gives a coherent map of the finite diagrams and
hence of their derived limits.
It remains to identify its ordinary realization. On
$U_I\times H_{I,t}$ the map is the identity on $U_I$ times $u_{I,t}$.
The orientation trace
for this product is the fiber trace, tensored with the identity on
$U_I$. It is compatible with restriction to intersections and with
the transition isomorphisms. Open-cover descent therefore identifies
the realized map with the ordinary topological Gysin map of the
total spaces. In particular, the construction gives the actual
Gysin map, not merely a map with the same source and target.
\end{proof}
\subsection{Finite monodromy on weight gradeds}
The diagram construction has now supplied a variation on the
cohomology of the total spaces. Hodge--Tate fibers and the integral
structure put a strong restriction on its monodromy.
\begin{lemma}
\label{diag:finite}
Assume the Hodge--Tate hypothesis of
Proposition~\ref{diag:variation}. The combined monodromy action
on all $\operatorname{Gr}^W_aH^j(\mathscr H_t,\mathbb Q)$ has finite
image. For any loop with cohomology monodromy $T$, there is an integer
$M>0$ such that $T^M$ is unipotent and
\begin{equation}
\label{diag:log}
N=\frac{1}{M}\log(T^M)
\quad\text{satisfies}\quad N(W_a)\subset W_{a-2}.
\end{equation}
The logarithm is a finite polynomial. On the total cohomology ring,
$N$ is a derivation of cohomological degree zero.
\end{lemma}
\begin{proof}
Each nonzero weight graded is a polarizable variation of pure type
$(r,r)$ and weight $2r$. After the conventional sign, its flat
polarization is positive definite on the underlying real vector
space. Let $\mathcal L$ be the torsion-free quotient of the integral
cohomology local system. Intersecting $\mathcal L$ with the rational
weight sub-local systems and taking successive quotients gives an
invariant full lattice in each rational graded piece. These
intersections are saturated because the subspaces are rational.
The automorphisms of a lattice preserving a positive definite form
constitute a finite group: in a fixed lattice basis, every column of
such an automorphism has a prescribed bounded norm and hence lies
in a finite set. Thus each graded monodromy image is finite. Only
finitely many cohomological degrees and weights occur, which proves
the combined assertion.
Choose $M$ killing the action of the specified loop on all weight
gradeds. Then
\[
(T^M-1)W_a\subset W_{a-1}=W_{a-2}
\]
whenever $a$ is even; the identical conclusion for odd $a$ follows
because successive odd and even steps coincide. The operator
$T^M-1$ is nilpotent, since the weight filtration is finite.
Its finite logarithm has the same lowering bound.
Parallel transport preserves cup products. For every integer
$k\geq0$, the polynomial operator $\exp(kMN)=T^{Mk}$ is therefore
a ring automorphism. For cohomology classes $\alpha,\beta$, the
identity
\[
\exp(kMN)(\alpha\beta)
=\exp(kMN)(\alpha)\exp(kMN)(\beta)
\]
is an equality of polynomials in $k$. Comparing linear coefficients
proves $N(\alpha\beta)=N(\alpha)\beta+\alpha N(\beta)$.
Monodromy preserves cohomological degree, so $N$ does as well.
\end{proof}
To apply these results, we will verify locally constant algebraic
transition maps for the family with a moving marked point, together
with simultaneous such transitions for its flag correspondence.
We will also identify the scalar unflagged bundle as a flat product.
Once those inputs are established, Proposition~\ref{diag:gysin}
controls its push-pull maps, and Lemma~\ref{diag:finite} supplies the
weight-lowering logarithm on the same ordinary cohomology groups.
==== END P-equals-W-in-composite-rank-for-fixed-determinant//04-diagram.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//05-logarithmic.tex ====
\section{Moving logarithmic poles and cohomology comparisons}
\label{sec:logarithmic}
We now construct a single family through which the Higgs and Betti
correspondences can be compared. The point carrying the full flag must move
over a compact curve. Its motion changes the determinant, so the first step
is to compensate for that change by a line bundle. The resulting comparison
will apply to the whole fixed-determinant cohomology.
\subsection{The determinant correction and the scalar correspondence}
Choose a point $x_0\in C$. For the construction we first take
$L=\mathcal O_C(d x_0)$; we return to an arbitrary determinant at the end of
this section. The line bundle $\mathcal O_C(d x)$ has a canonical
logarithmic connection $d_{d x}$: its canonical meromorphic section is
horizontal. Its residue at $x$ is $-d$. Multiplying a connection by
$\lambda$ produces a $\lambda$-connection, whose Leibniz rule is
$D(fs)=\lambda\,df\otimes s+fD(s)$.
Pull back the finite \emph{\'etale} map
$[n]:\operatorname{Pic}^0(C)\to\operatorname{Pic}^0(C)$ by
\[
C\longrightarrow\operatorname{Pic}^0(C),\qquad
x\longmapsto\mathcal O_C(d x-d x_0),
\]
and choose a connected component $S$ of this pullback. Then $S$ is a smooth
projective connected curve, and its map $x:S\to C$ is finite \emph{\'etale}
and surjective. Write $\Delta\subset C\times S$ for the graph of $x$.
A Poincar\'e bundle gives a line bundle $J$ on $C\times S$ such that
\begin{equation}
J^{\otimes n}\simeq
\mathcal O_{C\times S}\bigl(d\Delta-d(x_0\times S)\bigr)
\otimes\operatorname{pr}_S^*B
\label{log:root}
\end{equation}
for some line bundle $B$ on $S$. The possible base factor is the only
ambiguity in this equality of families.
A line pulled back from $S$ has a canonical relative connection along $C$.
Equip the right side of \eqref{log:root} with the relative logarithmic
connection given by its canonical meromorphic section and this connection
on the base factor. Dividing its local connection forms by $n$ defines
a unique compatible connection on $J$. This division glues: the
transition equation for $J^{\otimes n}$ is $n$ times the equation for $J$.
Let $D_{J,\lambda}$ be the resulting relative $\lambda$-connection. Its
residues are $-\lambda d/n$ along $\Delta$ and $+\lambda d/n$ along
$x_0\times S$.
Put
\[
\mathfrak t_0=\{(u_1,\ldots,u_n)\in\mathbb A^n:
\textstyle\sum_i u_i=0\},\qquad
\Lambda=\mathbb A^1_\lambda\times\mathfrak t_0.
\]
Let $\mathcal X\to\mathbb A^1_\lambda$ parametrize stable logarithmic
$\lambda$-connections on rank-$n$ bundles, with determinant
$(\mathcal O_C(d x_0),\lambda d_{d x_0})$ and residue
$-\lambda(d/n)I$ at $x_0$. Stability here and below is slope stability for
subbundles preserved by the operator. At $\lambda=0$ the scalar residue
is zero, so the fiber $X_0$ is the ordinary trace-free Higgs moduli space $X$.
Let $\mathcal Y\to S\times\Lambda$ parametrize the following data:
\begin{itemize}
\item a stable logarithmic $\lambda$-connection $(E,D)$ with its only
possible pole at $x(s)$ and determinant
$(\mathcal O_C(d x(s)),\lambda d_{d x(s)})$;
\item a full flag of $E_{x(s)}$, preserved by the residue, on whose
ordered graded lines $Q_i$ the residue acts by $-\lambda d/n+u_i$.
\end{itemize}
The flag does not enter the stability condition. We write $Y_{\lambda,u}$
for the fiber over $(\lambda,u)\in\Lambda$, \emph{retaining all of $S$}.
Thus $Y_{\lambda,u}$ is itself a family over the moving-point curve.
At $u=0$, let $\mathcal Z$ be the closed locus where the residue is scalar,
and let $\mathcal O$ be the same scalar-residue family with the flag
forgotten. The correspondence over $\mathbb A^1_\lambda$ is shown
fiberwise in Figure~\ref{log:scalar-correspondence}.
\begin{figure}[htbp]
\centering
\begin{tikzcd}[column sep=large]
X_\lambda & O_\lambda\simeq S\times X_\lambda
\arrow[l,"\operatorname{pr}"'] & Z_\lambda
\arrow[l,"\pi"'] \arrow[r,hook,"\iota"] & Y_{\lambda,0}
\end{tikzcd}
\caption{The scalar correspondence. All moving-pole spaces retain the
curve $S$; $\operatorname{pr}$ has fiber $S$, $\pi$ is a
flag projection of relative dimension $b$, and $\iota$ has codimension
$b$. The residue on $Y_{\lambda,0}$ may have a strictly flag-lowering
nilpotent part in addition to its prescribed scalar part; $Z_\lambda$
requires that nilpotent part to vanish. The derived morphisms and their cohomological and diagram-weight shifts
are recorded in Table~\ref{spec:shift-table}.}
\label{log:scalar-correspondence}
\end{figure}
Here $\pi$ forgets the flag and $\iota$ is the scalar-residue inclusion.
The first map comes from the following product identification.
\begin{lemma}
\label{log:scalar-product}
Tensoring by $(J,D_{J,\lambda})$ gives an algebraic isomorphism
$\mathcal O\simeq S\times\mathcal X$ over $S\times\mathbb A^1$.
At $\lambda=0$ it preserves the ordinary Hitchin coefficients.
\end{lemma}
\begin{proof}
By \eqref{log:root}, tensoring fixed-pole data by $J$ changes its
determinant to $\mathcal O_C(d x(s))$, up to a base line, and changes its
determinant operator to the prescribed one. At $x_0$, the residue
$+\lambda d/n$ of $J$ cancels the residue $-\lambda d/n$ of the original
rank-$n$ connection.
This cancellation also holds where the moving and fixed sections meet.
In a coordinate $z$ with moving section $z=a(s)$, the scalar polar part
contributed by $J$ is
\[
-\frac{\lambda d}{n}\frac{dz}{z-a(s)}
+\frac{\lambda d}{n}\frac{dz}{z}.
\]
Adding the fixed-pole form leaves only the first term. The identity is an
identity of relative meromorphic forms over the whole local base, including
$a(s)=0$. Thus the tensor product has a single logarithmic pole at the
moving point. The inverse construction tensors by $J^{-1}$.
Base-line twists do not change the induced map to coarse moduli, and
$\deg J_s=0$ preserves stability. When $\lambda=0$, the Higgs field of
$J$ is zero, so the characteristic polynomial is unchanged.
\end{proof}
The relative quasiprojective moduli of logarithmic $\lambda$-connections
exist by the standard construction; we use the formulation in
\cite[Section~2.2]{dCFHLogHodge}. Fixed determinant is obtained by taking
the inverse image of its rank-one section. Adding a residue-invariant
full flag is projective over the unflagged moduli; its ordered residue
eigenvalues are then regular functions on the flag parameter space.
The prescribed-residue equations are closed.
We will use universal bundles only \emph{\'etale} locally. Coprimality
makes semistable pairs stable, with scalar automorphisms. Their stack is
a scalar gerbe over its coarse space, or a $\mu_n$-gerbe after choosing a
determinant isomorphism; compare
\cite[Lemma~5.3 and Remark~5.4]{dCFHLogHodge}.
The universal projective bundle, $\operatorname{End}E$, and the line
ratios $Q_i\otimes Q_j^{-1}$ have trivial scalar action and descend.
Individual $Q_i$ need not descend, and no construction below requires them
to do so.
\subsection{Smooth families and actual restriction isomorphisms}
Our next objective is to identify the cohomology of all fibers by
restriction from the total space. Smoothness alone would not suffice,
because these moduli spaces are not proper. The scaling action supplies
the additional compactness property that is needed.
\begin{proposition}
\label{log:families}
The morphism $\mathcal Y\to S\times\Lambda$ is smooth of relative
dimension $2R+2b$, and $\mathcal X\to\mathbb A^1$ is smooth of relative
dimension $2R$. Their total spaces are semiprojective for scaling the
operators, which scales $\lambda$ and every $u_i$ with weight one.
The same holds for the inverse image in $\mathcal Y$ of every linear
subspace of $\Lambda$ through zero. In
Figure~\ref{log:scalar-correspondence}, $\pi$ is a smooth proper flag
projection of relative dimension $b$, and $\iota$ is a regular embedding
of codimension $b$ with normal bundle $T_\pi^*$.
\end{proposition}
\begin{proof}
We first establish zero limits for scaling. Given an unflagged object,
keep its bundle fixed and scale its operator. This extends the orbit to
the stack of all logarithmic $\lambda$-connections, without imposing
semistability on the special fiber. Semistable reduction
\cite[Proposition~4.48]{FHZ2025} replaces this DVR family by a semistable
one after a finite extension, without changing its generic fiber.
Its hypotheses hold for $\mathrm{GL}_n$ over $\mathbb C$.
The determinant of the replacement and the prescribed determinant agree
generically, hence agree over the DVR by separatedness of the rank-one
moduli space. A generic flag extends because the flag variety of the
extended bundle at the pole is proper. Residue preservation and the
\emph{ordered} diagonal residue equations are closed, and therefore
persist at the special fiber. Coprimality makes the limiting semistable
pair stable. The same reasoning applies to the scalar-residue condition.
A finite extension of the DVR is enough for a limit in the original
coarse space. Indeed, embed this quasiprojective space in projective
space as a locally closed subvariety. The original orbit has a unique
projective limit. The limit after finite base change is that point and
belongs to the coarse moduli space, so the original orbit extends there.
This proves the required zero-limit property.
We next check smoothness over $\lambda=u=0$. At such a point put
\[
\mathcal F=\operatorname{ParEnd}_0(E),\qquad
\mathcal G=\mathcal F^\vee\otimes K_C,
\]
where $\mathcal F$ is the sheaf of trace-free endomorphisms preserving
the flag at $x(s)$. Trace pairing identifies $\mathcal G$ with the
trace-free logarithmic Higgs fields having strictly flag-lowering
residue. The relative deformation complex is
\begin{equation}
\mathcal F\xrightarrow{[\theta,-]}\mathcal G.
\label{log:deformation}
\end{equation}
Here is the lifting argument, including the parameter directions.
For a small Artin extension with prescribed lifts of $s,\lambda,u$, the
bundle, determinant isomorphism and flag lift because the obstruction
lies in $H^2(C,\mathcal F)=0$. In local frames respecting the lifted
flag, the operator lifts with its prescribed determinant and diagonal
residues: these are affine-linear conditions, and trace splits in
characteristic zero. Its gluing obstruction lies in
$H^1(C,\mathcal G)$ tensored with the square-zero ideal. Changing the
bundle and flag gluing changes this obstruction by the image of
$H^1(C,\mathcal F)$ under \eqref{log:deformation}.
This map on $H^1$ is surjective. Its Serre dual is, up to sign, the map
on $H^0$ in \eqref{log:deformation}; its kernel consists of trace-free
automorphisms of the stable Higgs pair preserving the flag, hence is
zero. Thus every such lifting problem has a solution. Since
\[
\chi(\mathcal F)=-(n^2-1)(g-1)-b=-R-b,
\]
the relative tangent dimension is $-2\chi(\mathcal F)=2R+2b$.
For the scalar unflagged family, replace $\mathcal F$ by
$\operatorname{End}_0(E)$ and obtain dimension $2R$ in the same way.
Finite scalar stabilizers act trivially on these deformations, so the
calculation also proves smoothness of the coarse spaces.
The nonsmooth locus of the morphism is closed and scaling invariant.
If it contained a point, it would contain that point's zero limit,
contradicting the calculation over $\lambda=u=0$. This proves smoothness
everywhere.
A scaling-fixed point has $\lambda=u=0$. Its logarithmic Hitchin
coefficients vanish, because their scaling weights are positive.
The logarithmic Hitchin map is proper; its base has coefficients in
$H^0(C,K_C(x(s))^{\otimes i})$ as in
\cite[Section~2.4, Equation~(4)]{dCFHLogHodge}.
The fixed locus is closed in its nilpotent fiber after adding flags
and imposing determinant and residue conditions. Since flags and $S$
are proper, that fixed locus is proper. Together with the zero limits,
this proves semiprojectivity. The inverse image of a linear subspace
through zero is smooth by base change and inherits both properties.
Finally $\pi$ is the full flag bundle of the universal projective bundle.
On the $u=0$ family, subtracting the prescribed scalar from the residue
gives a section of the bundle of strictly flag-lowering matrices.
Its zero scheme is $\mathcal Z$. The smoothness and dimensions just
proved show that it is a regular zero scheme of codimension $b$.
Trace pairing identifies this residue bundle along $\mathcal Z$ with
the cotangent bundle of the flag fibers, giving $N_\iota\simeq T_\pi^*$.
\end{proof}
\begin{corollary}
\label{log:restriction}
For every $(\lambda,u)\in\Lambda$, restriction induces an isomorphism
\begin{equation}
\rho_{\lambda,u}:H^*(\mathcal Y,\mathbb Q)
\xrightarrow{\ \sim\ }H^*(Y_{\lambda,u},\mathbb Q).
\label{log:rho}
\end{equation}
Restriction to the inverse image of any line through zero in $\Lambda$
is also an isomorphism, as is restriction from that inverse image to any
of its fibers. The analogous restriction isomorphisms hold for all
four families in Figure~\ref{log:scalar-correspondence} over the
$\lambda$-line. They commute with pullbacks and with the proper Gysin
maps in that correspondence, including integration in the $S$ factor.
\end{corollary}
\begin{proof}
Recall that the core of a smooth semiprojective variety is the union
of the sets whose scaling orbits have limits as the parameter tends
to infinity. Its inclusion induces a cohomology isomorphism
\cite[Theorem~1.3.1]{HRV2015}. Because every coordinate on $\Lambda$
has positive weight, a point of $\mathcal Y$ with such a limit must
lie over zero. Hence $\mathcal Y$ and its inverse image
$\mathcal Y_\ell$ of any line $\ell$ through zero have the same core.
Their restrictions to the core commute, so
$H^*(\mathcal Y)\to H^*(\mathcal Y_\ell)$ is an isomorphism.
The map $\mathcal Y_\ell\to\ell$ is smooth and surjective. For
surjectivity, its image is a scaling-invariant open subset containing
zero, and therefore contains the whole affine line. Apply
\cite[Corollary~1.3.3]{HRV2015}: for a smooth surjection from a smooth
semiprojective variety to the affine line, equivariant for a positive
base weight, restriction to each fiber is an isomorphism. This proves
\eqref{log:rho} and its line version. The same proof applies to
$\mathcal X$, $\mathcal O$ and $\mathcal Z$.
Pullback compatibility is functorial. For Gysin maps, the families
are smooth over the line, the scalar embedding is transverse to fiber
restriction, and the flag and product projections are smooth proper.
The transverse base-change identities for Gysin maps give the claimed
compatibilities.
\end{proof}
\subsection{Riemann--Hilbert with a moving puncture}
Let $V$, $q$ and $Y^{\rm B}\to V$ be the ordered eigenvalue family from
Section~\ref{sec:betti}. The exponent domain we need is
\begin{equation}
U=\left\{u\in\mathfrak t_0^{\rm an}:
t(u)=\bigl(\zeta e^{-2\pi\sqrt{-1}u_i}\bigr)_i\in V,\quad
u_i-u_j\notin\mathbb Z\setminus\{0\}\right\}.
\label{log:nonresonance}
\end{equation}
Notice that zero differences are allowed. In particular $0\in U$,
which is essential for the scalar correspondence.
There is a flat bundle over $S$ with algebraic fiber family
$Y^{\rm B}\to V$. To define it, use local trivializations of the
punctured surfaces $C\setminus\{x(s)\}$ and record a full flag of local
subsystems near the puncture. We choose the peripheral-loop convention
in which its monodromy is the product of commutators used to define
$Y^{\rm B}$. Changes of surface markings act on the handle matrices
by group words and on the peripheral flag by the corresponding
conjugating word. They are algebraic maps over $V$.
These transition maps are locally constant and satisfy the cocycle
condition as algebraic maps on the coarse spaces. To check this
precisely, let $\delta$ be the product of commutators in the free
fundamental group of the punctured surface. A change of marking
gives an outer automorphism represented by $\varphi$, with
$\varphi(\delta)=w\delta w^{-1}$. Word evaluation transforms a
representation to $\rho\varphi$ and its flag by $\rho(w)$.
Changing $\varphi$ by an inner automorphism gives simultaneous
conjugation. Changing $w$ multiplies it on the right by an element
centralizing $\delta$. The cyclically reduced word $\delta$ is not
a proper power: each positive handle generator occurs just once.
Its centralizer in the free group is therefore $\langle\delta\rangle$.
Thus this last ambiguity is a power of peripheral monodromy and
preserves every invariant flag, including those with repeated
eigenvalues or nonsemisimple monodromy. The resulting maps compose
strictly after passage to the geometric quotient. Consequently no
choice of eigenlines or higher homotopy of transitions is needed.
The scalar flag correspondence from
Section~\ref{sec:betti} gives a corresponding diagram of flat bundles
over $S$.
\begin{proposition}
\label{log:rh}
Over $U$, Riemann--Hilbert identifies $Y_{1,u}$ with the total space
over $S$ of this flat bundle at parameter $t(u)$. These identifications
are homeomorphisms of families. At $u=0$ they identify the scalar
correspondence of Figure~\ref{log:scalar-correspondence} at $\lambda=1$ with
the flat Betti scalar correspondence, compatibly with its pullbacks
and Gysin maps.
\end{proposition}
\begin{proof}
The local exponents of a connection are $r_i=-d/n+u_i$.
For the regular-singular correspondence and logarithmic lattices, including
Manin's extension theorem and Katz's nonresonance corollary, see
Deligne \cite[II, Proposition~5.4, Remark~5.5(ii),
Corollary~5.6, and Theorem~5.9]{Deligne1970}. We spell out the
nonresonant local construction to track the ordered flags and dependence
on the moving parameters.
In a disc coordinate $z$, write a logarithmic connection as
$d+A(z)\,dz/z$, with $A(0)=A_0$. The regular-singular reduction
to $d+A_0\,dz/z$ by a gauge equal to the identity at zero is obtained
recursively using the operators $k+\operatorname{ad}(A_0)$, $k\geq1$.
Their eigenvalues are $k+r_i-r_j$, so they are invertible under
\eqref{log:nonresonance}, even when $A_0$ has repeated eigenvalues or
a nonzero nilpotent part. On a small parameter neighborhood their
inverse norms are $O(1/k)$ for large $k$. The usual power-series
majorant for the analytic coefficients therefore gives convergence
on a common smaller disc and continuous dependence on parameters.
In this constant-residue model a residue-invariant flag gives a
monodromy-invariant flag. Conversely, the prescribed exponents select
one logarithm for each monodromy eigenvalue: equal exponential values
have equal selected exponents, by \eqref{log:nonresonance}. Functional
calculus on each generalized eigenspace then gives a logarithm of
monodromy preserving its invariant flag. This constructs the inverse
logarithmic extension with the specified ordered residues. Coordinate
and logarithm changes in the constant-residue model act by functions
of the residue, and hence preserve the flag; the construction is
intrinsic. In rank one the same construction gives exactly
$(\mathcal O_C(d x(s)),d_{d x(s)})$, so the determinant is the required
one. The subproduct condition defining $V$ makes the representation
irreducible, and thus ensures stability.
Holonomy along local moving generators, together with this local
normal form, makes the forward maps continuous in the bundle and
in $s,u$. Regard the Betti family as pulled back to $U$ via $t$;
there is no assertion that $t$ is globally injective. The forward
map is a continuous bijection between manifolds of equal real
dimension, by Proposition~\ref{log:families} and the Betti
smoothness. Invariance of domain gives its inverse continuity.
For a fixed point of $S$ it is holomorphic on the moduli fibers,
so it preserves their complex orientations.
At $u=0$, scalar residues correspond to scalar monodromy, and
forgetting or retaining the flag commutes with this construction.
The proper maps in the correspondence have their usual complex
orientations in the algebraic fibers and the same orientation on
$S$. Consequently their topological Gysin maps agree under these
homeomorphisms.
\end{proof}
\begin{lemma}
\label{log:scalar-flat-product}
The product identification in Lemma~\ref{log:scalar-product}, at
$\lambda=1$, identifies the scalar unflagged Betti family with
$S\times X^{\rm B}$ as a flat bundle with algebraic fiber. Its
projection to $X^{\rm B}$ corresponds to $\operatorname{pr}$.
\end{lemma}
\begin{proof}
Away from the fixed and moving poles, the nth power of the connection
on $J$ is the canonically trivial connection. Its holonomies
therefore lie in $\mu_n$. Locally in $S$, take a disc containing
$x_0$ and the moving point, and choose the handle generators outside
that disc. Their line holonomies depend continuously on $s$ and
take values in the finite group $\mu_n$, so are locally constant.
Tensoring multiplies the handle matrices by these constant scalars,
an algebraic map on the representation variety. The argument also
applies in collision neighborhoods. Thus the algebraic product
identification of the scalar connection family is compatible with
the locally constant algebraic Betti transitions.
\end{proof}
We have now obtained the actual flat diagrams to which
Section~\ref{sec:diagram} will be applied. In particular, when that
construction gives weight zero to the cochains of $S$, its product
and integration statements concern the same projection
$\operatorname{pr}$ as the logarithmic correspondence.
\subsection{The nonabelian-Hodge comparison on full cohomology}
The restriction isomorphisms for $\mathcal X$ and Riemann--Hilbert give
\begin{equation}
H^*(X^{\rm B})\xrightarrow{\ \sim\ }H^*(X_1)
\xleftarrow{\ \sim\ }H^*(\mathcal X)
\xrightarrow{\ \sim\ }H^*(X).
\label{log:comparison}
\end{equation}
We must check that this comparison transports the weight filtration
specified by nonabelian Hodge theory, including its nontrivial
$\Gamma$-character summands. Agreement only after taking
$\Gamma$-invariants would not suffice.
\begin{lemma}
\label{log:finite-ambiguity}
Let a finite group $G$ act algebraically on a smooth connected complex
variety $B$, and let $q:B\to B/G$ be the quotient. Suppose
$g:A\to B$ is a homeomorphism and $f:A\to B$ is continuous with
$qf=qg$. Then $f=\gamma g$ for a single $\gamma\in G$.
\end{lemma}
\begin{proof}
Divide out the kernel of the action. Smoothness and connectedness
make $B$ irreducible. Removing the fixed loci of the nonidentity
elements gives a dense connected Zariski open $B^\circ$ on which
the action is free. For $a\in g^{-1}(B^\circ)$ there is a unique
$\gamma(a)$ with $f(a)=\gamma(a)g(a)$. The free finite quotient
is a covering there, so $\gamma(a)$ is locally constant. It is
constant because this open set is connected. Continuity extends
the equality over its closure, which is all of $A$.
\end{proof}
\begin{proposition}
\label{log:nah}
The comparison \eqref{log:comparison} transports the Deligne weight
filtration on the whole $H^*(X^{\rm B},\mathbb Q)$ in the same way as
the nonabelian-Hodge identification. The conclusion is unchanged
when $\mathcal O_C(d x_0)$ is replaced by an arbitrary
$L\in\operatorname{Pic}^d(C)$ and the Higgs spaces are identified by
tensoring with a degree-zero nth root of their determinant ratio.
\end{proposition}
\begin{proof}
The full fixed-determinant nonabelian-Hodge diffeomorphism, including
its $\Gamma$-equivariance, is part of the twisted construction in
\cite[Sections~2 and~5]{HT2003}. We compare its cohomology map with
the particular logarithmic family above.
For $(E,\theta)\in X$, choose a harmonic metric $H$ normalized to a
fixed metric on its determinant. Let $\partial_H$ denote the
$(1,0)$ part of its Chern connection. For real $0\leq\tau\leq1$
consider the holomorphic structure and logarithmic $\tau$-operator
\begin{equation}
\bar\partial_E+\tau\theta^{\dagger_H},\qquad
\tau\partial_H+\theta+
\frac{\tau}{n}
\bigl(d^{1,0}_{d x_0}-\partial_{\det H}\bigr)I.
\label{log:harmonic-path}
\end{equation}
Its projectivization is the usual harmonic preferred section
\cite[Section~4]{Simpson1997}. We verify the scalar adjustment.
The trace of $\theta$ and of $\theta^{\dagger_H}$ is zero, so the
determinant holomorphic structure stays fixed and the determinant
operator is exactly $\tau d_{d x_0}$. The trace-free
holomorphicity equation follows from
$\bar\partial_E\theta=0$, $\partial_H\theta^{\dagger_H}=0$ and
the projective Hitchin equation. The scalar equation is that of
the prescribed logarithmic determinant connection. The only
nonregular term is the scalar logarithmic term at $x_0$.
For $\tau\ne0$, divide the operator by $\tau$.
Its puncture monodromy is $\zeta I$. A proper invariant subspace
of rank $r$ would have $\zeta^r=1$ by taking determinants of the
commutator relation, contrary to $\gcd(n,d)=1$. Thus these
connections are irreducible and stable. At $\tau=0$ we recover
the original Higgs pair. Hence \eqref{log:harmonic-path} lies in
$\mathcal X$ throughout.
This construction is a continuous homotopy of maps $X\to\mathcal X$.
One may work in local families of stable Higgs pairs. The normalized
harmonic metrics vary smoothly: their trace-free metric linearization
is self-adjoint elliptic, and its kernel consists of infinitesimal
symmetries, which stability kills. This is also the parameter
dependence used for preferred sections in
\cite[Section~4]{Simpson1997}. To pass to the analytic topology of
coarse moduli, choose a sufficiently positive twist near any one
parameter so the underlying bundles are generated and have vanishing
$H^1$. The kernels of the corresponding Dolbeault operators then
give continuously varying bases of sections. Their evaluations
give continuous quotient presentations; the logarithmic operators,
viewed as morphisms from the first jet bundle with fixed symbol,
also have continuous coefficients in such presentations. Passing
to the stable quotient gives the asserted continuity, independently
of the local choices.
At $\tau=0$ the homotopy is the inclusion $X\hookrightarrow\mathcal X$.
At $\tau=1$ it factors through $X_1$, and then through
Riemann--Hilbert to a map $F:X\to X^{\rm B}$.
The pullback $F^*$ is therefore precisely
\eqref{log:comparison}, because both restrictions from
$\mathcal X$ are isomorphisms. Projectivizing $F$ gives the
usual projective nonabelian-Hodge map.
Let $\Phi:X\to X^{\rm B}$ be the fixed-determinant
nonabelian-Hodge diffeomorphism. The maps $F$ and $\Phi$ agree
after projectivization. Two determinant-one lifts of the same
projective representation differ on each handle generator by an
nth root of unity; thus their classes differ by the finite group
of determinant-preserving characters, identified with $\Gamma$.
The low-degree calculation of Section~\ref{sec:lefschetz} gives
$H^0(X^{\rm B})=\mathbb Q$, so $X^{\rm B}$ is connected; it is
smooth in the coprime setting. Lemma~\ref{log:finite-ambiguity}
therefore gives $F=\gamma\Phi$ for one $\gamma\in\Gamma$.
Its action on $X^{\rm B}$ is algebraic and preserves Deligne
weights. Consequently $F^*$ and $\Phi^*$ transport the weight
filtration identically on every cohomology group. No averaging
or projection to invariants occurs.
Finally choose $M\in\operatorname{Pic}^0(C)$ with
$M^{\otimes n}\simeq L\otimes\mathcal O_C(-d x_0)$.
Tensoring by $M$ identifies the two Higgs moduli spaces
algebraically over the same Hitchin base. Their projective
Higgs bundles, and therefore their projective harmonic
correspondences, are unchanged. Applying the same finite-character
argument to the two fixed-determinant nonabelian-Hodge maps shows
that this identification also preserves the transported weight
filtration. It preserves the perverse filtration because it is
an isomorphism over the Hitchin base.
\end{proof}
==== END P-equals-W-in-composite-rank-for-fixed-determinant//05-logarithmic.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//06-translation.tex ====
\section{A common cohomological operator}\label{sec:translation}
We now use the logarithmic families to turn an elementary modification of Higgs bundles into monodromy of Betti spaces. All cohomology identifications in this section are the restriction maps of Corollary~\ref{log:restriction}. This specifies a single operator before we estimate its effect on either filtration.
\subsection{Modification of two eigenlines}
Choose $v\in\sqrt{-1}\mathbb R^n$ with sum zero, distinct entries, and no vanishing proper nonempty partial sum. Such vectors exist because the excluded conditions form finitely many proper real hyperplanes. Set
\[
\ell=(1,-1,0,\ldots,0).
\]
At a logarithmic operator with distinct residue eigenvalues, a positive
elementary modification allows a simple pole in a chosen residue eigenline;
a negative one takes the kernel of the projection to that eigenline at $x$.
With the convention that a positive shift replaces a local generator $s$
by $z^{-1}s$, the $\lambda$-Leibniz rule changes its residue eigenvalue by
$-\lambda$. A negative shift changes it by $+\lambda$.
We perform the shifts specified by $\ell$ successively: first the positive
modification on the first labeled eigenline, then, after reselecting the
labeled eigenspaces of the new residue, the negative modification on the
second labeled eigenline. The ordered eigenspaces of the final residue
define the final flag.
\begin{lemma}\label{trans:translation}
The two modifications define a continuous family of isomorphisms
\begin{equation}\label{trans:path}
F_a:Y_{a,v}\longrightarrow Y_{a,v-a\ell},\qquad 0\le a\le1.
\end{equation}
At $a=0$ they define an algebraic automorphism of $Y_{0,v}$. More generally, the same modification defines an automorphism over the punctured Higgs line $(\lambda,u)=(0,zv)$, $z\ne0$, preserving the logarithmic Hitchin map. At $a=1$, the Riemann--Hilbert identifications make $F_1$ the identity on the underlying Betti space.
\end{lemma}
\begin{proof}
For each single modification, choose a local frame adapted to its current
residue eigenspaces. The off-diagonal coefficients involving the chosen
eigenline vanish at the pole, so the singly modified lattice remains
preserved by the logarithmic operator. The preceding calculation gives its
new residue eigenvalues. Their imaginary parts remain distinct throughout
\eqref{trans:path}, including after the first step. We can therefore
reselect the residue eigenspaces before performing the second modification.
The two-step composite preserves the determinant and its operator because
the lattice shifts have sum zero. Its inverse performs the opposite
single modifications in reverse order, again reselecting eigenspaces
between steps. These elementary transformations and their inverses vary
algebraically where the successive residue eigenvalues are distinct,
giving the asserted continuous family.
We check stability, which is not supplied by lattice modification alone. On the Higgs line with $z\ne0$, a proper invariant saturated subbundle would have a trace Higgs differential with a single possible simple pole. Its residue would be $z\sum_{i\in I}v_i$ for a proper nonempty subset $I$. This is nonzero, contrary to the residue theorem. The modified Higgs bundles satisfy the same argument. Distinct residue eigenvalues split the local Higgs operator into formal eigenline summands; shifting these summands and shifting them back proves invertibility.
For $a>0$, divide the operator by $a$. A proper invariant monodromy subspace would have determinant monodromy one around the puncture. The imaginary part of the sum of its exponents is $a^{-1}\sum_{i\in I}\operatorname{Im}v_i$, which is nonzero. Thus no such subspace exists. Differences of exponents have nonzero imaginary part, so no nonzero integral resonance occurs. The inverse modifications are valid and remain stable by the same tests.
The characteristic polynomial on the Higgs line is unchanged away from the pole and hence unchanged as a meromorphic polynomial on $C$. Finally, modifications of a connection change only its logarithmic lattice at the puncture. At $a=1$ the exponent changes are integers, so the meromorphic flat bundle and the ordered monodromy eigenlines are unchanged. This is precisely the endpoint identification asserted in the lemma.
\end{proof}
\subsection{Translation as eigenvalue monodromy}
Write $T=F_0$ and transport its pullback to the total family:
\begin{equation}\label{trans:global-T}
\mathsf T=\rho_{0,v}^{-1}T^*\rho_{0,v}
\quad\text{on }H^*(\mathcal Y,\Q).
\end{equation}
To compare this operator with the endpoint $F_1$, we need constancy along
the two families in \eqref{trans:path}. Their fibers are nonproper; the
restriction isomorphisms, rather than properness, supply the following
criterion.
\begin{lemma}
\label{log:compact-cycles}
Let $I$ be a connected real interval and let $p:M\to I$ and
$q:N\to I$ be smooth submersions, with finite-dimensional rational
cohomology on their fibers. Suppose there are spaces $A,B$ and
continuous maps $a:M\to A$, $b:N\to B$ whose restrictions induce
isomorphisms
\[
\alpha_t:H^*(A,\mathbb Q)\xrightarrow{\sim}H^*(M_t,\mathbb Q),
\qquad
\beta_t:H^*(B,\mathbb Q)\xrightarrow{\sim}H^*(N_t,\mathbb Q)
\]
for every $t\in I$. If $F:M\to N$ is continuous over $I$, then
$\alpha_t^{-1}F_t^*\beta_t:H^*(B,\mathbb Q)\to H^*(A,\mathbb Q)$
is independent of $t$.
\end{lemma}
\begin{proof}
Fix a degree and a parameter $t_0$. Choose finitely many compact
singular cycles forming a rational homology basis of $M_{t_0}$.
A lift of the interval vector field near the compact union of their
carriers has a flow for a common short time. It transports these
cycles to nearby fibers; the resulting cycles sweep homologies in
$M$. Their images in $A$ are therefore homologous, so their
pairings with the restrictions of a fixed basis of $H^*(A)$ are
constant. The restriction isomorphisms show that the transported
cycles remain a basis of the fiber homology.
Applying the continuous map $bF$ to these swept cycles gives
homologies in $B$. Hence their pairings with every fixed class
of $H^*(B)$ are also constant. These pairings are the matrix
coefficients of $\alpha_t^{-1}F_t^*\beta_t$, relative to the
fixed bases. The operator is thus locally constant, including
one-sided neighborhoods of interval endpoints, and therefore
constant on $I$.
\end{proof}
Apply the lemma with $I=[0,1]$ and with $M,N$ the pullbacks of
$\mathcal Y\to\Lambda$ along
\[
a\longmapsto(a,v),\qquad a\longmapsto(a,v-a\ell),
\]
respectively. These are smooth submersions by
Proposition~\ref{log:families}, and \eqref{trans:path} gives the continuous
map $F:M\to N$. Take both ambient spaces in the lemma to be
$\mathcal Y$, with the natural maps from the two pullbacks. Their fiber
restrictions induce the isomorphisms of
Corollary~\ref{log:restriction}. The constancy conclusion therefore gives
\begin{equation}\label{trans:endpoint}
\mathsf T=\rho_{1,v}^{-1}F_1^*\rho_{1,v-\ell}.
\end{equation}
In particular, this identification does not require the nonproper smooth families to be globally locally trivial.
Recall the eigenvalue map
\[
t(u)=(\zeta e^{-2\pi\sqrt{-1}u_i})_{i=1}^n.
\]
The path $u=v-a\ell$, $0\le a\le1$, maps to a loop in $V$. Every forbidden proper subproduct has modulus different from one because the corresponding partial sum of $v$ has nonzero imaginary part. The same condition excludes resonance along the entire path. By Proposition~\ref{log:rh}, this is a path in the actual moving-puncture Betti family. Restrictions of classes from $\mathcal Y$ are flat sections along this path: the Betti direct images satisfy ordinary base change and are locally constant. Since $F_1$ is the endpoint identity, \eqref{trans:endpoint} identifies $\mathsf T$ with the pullback convention for its eigenvalue monodromy.
\begin{lemma}\label{trans:logarithm}
There is an integer $M>0$ such that
\begin{equation}\label{trans:N}
N=\frac1M\log(\mathsf T^M)
\end{equation}
is a well-defined nilpotent derivation of $H^*(\mathcal Y,\Q)$. Let
$N_{\lambda,u}=\rho_{\lambda,u}N\rho_{\lambda,u}^{-1}$.
On $H^*(Y_{1,0},\Q)$, equipped with the diagram weights of Section~\ref{sec:diagram}, one has
\[
N_{1,0}W_j\subset W_{j-2}.
\]
\end{lemma}
\begin{proof}
The full-flag Betti family has integral lisse direct images, ordinary base change, and Hodge--Tate fibers by Section~\ref{sec:betti}. The moving-puncture transition maps are algebraic and locally constant by Proposition~\ref{log:rh}. The finite-monodromy statement of Section~\ref{sec:diagram} therefore applies to this very family. Choose $M$ that kills the monodromy action on every weight-graded piece of its total cohomology. There are finitely many such pieces. Then $\mathsf T^M-1$ is nilpotent, and \eqref{trans:N} is a finite logarithmic sum. The weights are even, so $N$ lowers them by at least two at $(1,v)$.
The path $(1,av)$, $0\le a\le1$, stays in the nonresonant inverse image of $V$. At $a=0$, the scalar point belongs to $V$ because $\zeta$ has order $n$. For $a>0$, the imaginary-part argument applies again. Parallel transport preserves the diagram weights and coincides with the restriction identifications. This gives the stated estimate at $(1,0)$.
Lemma~\ref{diag:finite} makes the logarithm a derivation on the moving-puncture Betti cohomology. The restriction identifications are ring isomorphisms, so this property transports to $H^*(\mathcal Y,\Q)$.
\end{proof}
\subsection{Flag push-pull and the weight bound}
For the scalar correspondence at $\lambda$, put
\[
\mathsf A_\lambda=\pi_{\lambda,*}\iota_\lambda^*,
\qquad
\mathsf B_\lambda=\iota_{\lambda,*}\pi_\lambda^*.
\]
Its unflagged moving-pole space is $S\times X_\lambda$, with projection $\pr_\lambda$. We define
\begin{equation}\label{trans:fprime}
f'=\pr_{0,*}\,\mathsf A_0 N_{0,0}^2\mathsf B_0\,\pr_0^*
:H^m(X,\Q)\longrightarrow H^{m-2}(X,\Q).
\end{equation}
The flag embedding and projection have opposite degree shifts, $2b$ and
$-2b$. Integration over the compact curve $S$ supplies the remaining
degree $-2$. Table~\ref{spec:shift-table} records the individual derived morphisms
and their degree and weight shifts. Section~\ref{sec:specialization}
derives the perverse bound from their composite over the Hitchin base.
We first prove the weight bound.
\begin{proposition}\label{trans:weights}
Under nonabelian Hodge theory, the operator \eqref{trans:fprime} satisfies
\[
f'W_jH^m(X^B,\Q)\subset W_{j-4}H^{m-2}(X^B,\Q).
\]
\end{proposition}
\begin{proof}
The scalar correspondence and its Gysin maps commute with the restriction identifications of Corollary~\ref{log:restriction}. Thus \eqref{trans:fprime} transports to the same expression at $\lambda=1$. By Proposition~\ref{log:nah}, these identifications transport the ordinary Betti weight filtration on the full cohomology of $X^B$. The possible difference from a chosen nonabelian-Hodge identification is one algebraic $\Gamma$-action, which preserves every weight subspace, including those in variant cohomology.
At $\lambda=1$, the scalar Betti space is the flat algebraic product $S\times X^B$ of Lemma~\ref{log:scalar-flat-product}. Its diagram mixed Hodge structure assigns weight zero to the cochains of $S$ and the ordinary Deligne weights to $H^*(X^B)$. Hence both $\pr_1^*$ and $\pr_{1,*}$ preserve weights. The flag maps are maps of the same flat transition diagrams. Their ordinary topological Gysin maps are the diagram Gysin maps, with weight shifts $2b$ for $\mathsf B_1$ and $-2b$ for $\mathsf A_1$. These shifts cancel, while $N_{1,0}^2$ lowers weight by four by Lemma~\ref{trans:logarithm}.
\end{proof}
We have constructed the operator and proved its weight bound. The remaining tasks are to realize it over the Hitchin base, which gives the perverse bound, and to compute its double commutator with $e$.
==== END P-equals-W-in-composite-rank-for-fixed-determinant//06-translation.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//07-specialization.tex ====
\section{Specialization over a Higgs line}\label{sec:specialization}
We now prove the perverse-filtration bound for the operator $f'$ in
\eqref{trans:fprime}. The translation is defined over a logarithmic Hitchin
base away from zero on a Higgs line. Nearby cycles extend its action to the
zero fiber as a sheaf endomorphism. A final integration over the moving-point
curve supplies the shift by $-2$. The essential comparison with ordinary
cohomology uses the restriction isomorphisms of
Corollary~\ref{log:restriction}; it does not require the parameter morphism
to be proper.
\subsection{The proper morphism and its relative translation}
Keep the vector $v$ and the translation from Section~\ref{sec:translation}.
Let $p_C:C\times S\to S$ be the projection and let $\Delta\subset C\times S$
be the graph of $x:S\to C$. The vector bundle of trace-free logarithmic
Hitchin coefficients is
\[
\mathcal E=
\bigoplus_{i=2}^n
p_{C,*}\bigl(K_{C\times S/S}(\Delta)^{\otimes i}\bigr).
\]
We use the same notation for its total space. These direct images are locally
free and commute with base change: each twisting line on a fiber has degree
$i(2g-1)>2g-2$, so its first cohomology vanishes. The inclusion of ordinary
coefficient spaces defines a closed vector-subbundle embedding
\[
j:S\times A\hookrightarrow\mathcal E.
\]
Indeed both the ordinary spaces and the logarithmic spaces commute with base
change, and their quotient has constant rank.
Let $D=\mathbb A^1_z$, let $D^*=D\setminus\{0\}$, and define
\[
\mathcal Y_v=\mathcal Y\times_{\Lambda}D,
\qquad z\longmapsto(0,zv).
\]
Thus its fiber at $z$ is $Y_{0,zv}$, including the moving-point coordinate in
$S$. Characteristic coefficients and the parameter define
\begin{equation}\label{spec:hitchin-line}
g_v:\mathcal Y_v\longrightarrow\mathcal E\times D.
\end{equation}
Write $g_0:Y_{0,0}\to\mathcal E$ for its zero fiber.
\begin{lemma}\label{spec:geometry}
The morphism $g_v$ is proper, and $\mathcal Y_v\to D$ is smooth. The
translation defines an automorphism $T_v$ of $\mathcal Y_v|_{D^*}$ over
$\mathcal E\times D^*$. Scaling gives an isomorphism
\[
\mathcal Y_v|_{D^*}\simeq Y_{0,v}\times D^*
\]
over $D^*$, and restriction from $H^*(\mathcal Y_v)$ to either $Y_{0,0}$ or
$Y_{0,v}$ is an isomorphism.
\end{lemma}
\begin{proof}
Smoothness and the restriction assertions follow from
Proposition~\ref{log:families} and Corollary~\ref{log:restriction}. For properness,
start with the relative logarithmic Higgs Hitchin morphism over $S$.
Imposing determinant $\mathcal O_C(dx(s))$ and trace zero gives a closed
subspace. Adding a residue-invariant full flag is projective over that
subspace. Finally the requirement that the ordered diagonal residues equal
$zv_i$ is closed after adjoining $z$. The properness of the logarithmic
Hitchin morphism from Section~\ref{sec:logarithmic} therefore implies
properness of \eqref{spec:hitchin-line}. Coprimality ensures that the
semistable limits in this argument remain stable.
For $z\ne0$ the residual eigenvalues $zv_i$ are distinct, so the elementary
modifications defining $T_v$ and their inverses are algebraic in $z$.
They remain in the stable moduli by the trace-residue argument of
Section~\ref{sec:translation}. A modified Higgs field agrees with the
original meromorphic Higgs field away from the pole. Their characteristic
coefficients are consequently equal as sections of
$K_C(x(s))^{\otimes i}$, which proves that $T_v$ is over $\mathcal E$.
Multiplying the Higgs field by $z$ identifies the fiber at $1$ with that at
$z$; division by $z$ is its inverse. This gives the claimed product over
$D^*$. The product need not fix the Hitchin coefficient in $\mathcal E$;
only the translation must do so.
\end{proof}
\subsection{Specialization and ordinary cohomology}
Here is the comparison needed for the noncompact spaces in
Lemma~\ref{spec:geometry}. Throughout, nearby cycles are unshifted;
we use the nearby-cycle and proper-base-change formalism recalled in
\cite[Sections~5.5 and~5.8]{dCM2009}.
\begin{lemma}\label{spec:specialization}
Let $Y\to D=\mathbb A^1$ be a smooth morphism of complex algebraic varieties,
and let $g:Y\to B\times D$ be a proper morphism commuting with the maps to
$D$. Suppose that $Y|_{D^*}$ is
isomorphic to $Y_1\times D^*$ over $D^*$ and that the actual restriction maps
\[
r_i:H^*(Y,\mathbb Q)\longrightarrow H^*(Y_i,\mathbb Q),
\qquad i=0,1,
\]
are isomorphisms. An automorphism $T$ of $Y|_{D^*}$ over $B\times D^*$
then induces an endomorphism $\tau_0$ of $Rg_{0,*}\mathbb Q_{Y_0}$ whose
cohomological action is
\[
H^*(\tau_0)=r_0r_1^{-1}T_1^*r_1r_0^{-1}.
\]
Neither $B$ nor $Y_i$ is required to be proper.
\end{lemma}
\begin{proof}
Smoothness gives the canonical isomorphism
$\mathbb Q_{Y_0}\simeq\psi\mathbb Q_Y$.
Compatibility of nearby cycles with proper direct image gives
\[
Rg_{0,*}\mathbb Q_{Y_0}
\simeq Rg_{0,*}\psi\mathbb Q_Y
\simeq\psi(Rg_*\mathbb Q_Y).
\]
Apply nearby cycles to the pullback endomorphism induced by $T$ on the
punctured direct image, and use these isomorphisms to define $\tau_0$.
We verify its action on global cohomology without interchanging nearby
cycles with a nonproper direct image to a point. Put $K=Rg_*\mathbb Q_Y$
on $B\times D$. Let $\widetilde D^*\to D^*$ be the universal cover and write
\[
\widetilde j_B:B\times\widetilde D^*\longrightarrow B\times D,
\qquad i_B:B\times\{0\}\hookrightarrow B\times D,
\qquad \widetilde K=\widetilde j_B^*K.
\]
Set $\widetilde Y^*=Y|_{D^*}\times_{D^*}\widetilde D^*$.
Proper base change identifies $\widetilde K$ with the direct image of the
constant sheaf from $\widetilde Y^*$. The nearby-cycle formula downstairs
is $\psi K=i_B^*R\widetilde j_{B,*}\widetilde K$. Restriction of this
complex gives a natural morphism
\begin{equation}\label{spec:comparison-map}
\begin{split}
R\Gamma(\widetilde Y^*,\mathbb Q)
&\simeq R\Gamma(B\times D,R\widetilde j_{B,*}\widetilde K)\\
&\longrightarrow R\Gamma(B,\psi K)
\simeq R\Gamma(Y_0,\mathbb Q).
\end{split}
\end{equation}
The last isomorphism uses proper nearby-cycle compatibility and smoothness
of $Y\to D$. The composite of \eqref{spec:comparison-map} with restriction
from $R\Gamma(Y,\mathbb Q)$ is ordinary restriction to $Y_0$.
Indeed the specialization unit for $K$ corresponds, under proper direct
image, to the unit $\mathbb Q_{Y_0}\to\psi\mathbb Q_Y$, which is the
canonical smooth isomorphism used above.
Choose a lift of $1\in D^*$. The assumed product and the contractibility of
$\widetilde D^*$ identify $H^*(\widetilde Y^*)$ with $H^*(Y_1)$ by actual
restriction to that lift. Pullback from $H^*(Y)$ therefore becomes $r_1$
and is an isomorphism. Since its composite with
\eqref{spec:comparison-map} is $r_0$, the latter map is also an isomorphism
on cohomology.
The comparison was constructed on $B\times D$ so that it is equivariant
for the required operator: because $T$ is over $B\times D^*$, it induces
an endomorphism of $K|_{B\times D^*}$ and hence of
$\widetilde K$, $R\widetilde j_{B,*}\widetilde K$, and $\psi K$.
Every arrow in \eqref{spec:comparison-map} respects these actions.
Restricting at the chosen lift identifies the action on its domain with
$T_1^*$, and its action on the codomain is $H^*(\tau_0)$.
Thus the displayed formula for $H^*(\tau_0)$ follows. No extension of $T$
to an automorphism of $Y$ is asserted or needed.
\end{proof}
Apply Lemma~\ref{spec:specialization} to \eqref{spec:hitchin-line}. The
resulting endomorphism $\tau_0$ of $Rg_{0,*}\mathbb Q_{Y_{0,0}}$ acts on
cohomology as the translation transported to $Y_{0,0}$ by
Corollary~\ref{log:restriction}. In particular its action is the operator
whose logarithm is $N_{0,0}$ in Section~\ref{sec:translation}.
A finite polynomial suffices to realize this logarithm over the Hitchin
base. Choose $M$ as in Lemma~\ref{trans:logarithm}, and choose an integer
$r\ge1$ for which $(\mathsf T^M-1)^r=0$ on total cohomology. Put
\[
p(t)=\frac1M\sum_{a=1}^{r-1}
\frac{(-1)^{a+1}}{a}(t^M-1)^a.
\]
The polynomial $p(\tau_0)$ is a legitimate endomorphism of
$Rg_{0,*}\mathbb Q_{Y_{0,0}}$, and its action on cohomology is $N_{0,0}$.
No unipotence assertion about $\tau_0$ in the derived category is needed.
\subsection{The derived shift}
We have now realized the middle operator in $f'$ over the Hitchin base.
We next include the flag correspondence and the moving-point integration,
keeping every shift explicit.
\begin{proposition}\label{spec:perverse-lowering}
The operator $f'$ of \eqref{trans:fprime} is induced by a morphism
\begin{equation}\label{spec:derived-lowering}
Rh_*\mathbb Q_X\longrightarrow Rh_*\mathbb Q_X[-2]
\end{equation}
in the constructible derived category on $A$. Consequently
\[
f'(P_kH^m(X,\mathbb Q))\subseteq P_{k-2}H^{m-2}(X,\mathbb Q)
\]
for all integers $m,k$.
\end{proposition}
\begin{proof}
Let $O_0=S\times X$ using the scalar product identification of
Lemma~\ref{log:scalar-product}, and write $q=\operatorname{id}_S\times h$.
Both maps in the scalar flag correspondence of
Figure~\ref{log:scalar-correspondence} at $\lambda=0$ are over $\mathcal E$.
The flag projection has relative complex dimension $b$, and the scalar
inclusion has complex codimension $b$. Thus their pullbacks and proper
Gysin maps give, with
\[
\begin{gathered}
F_O=j_*Rq_*\mathbb Q_{O_0},\qquad
F_Y=Rg_{0,*}\mathbb Q_{Y_{0,0}},\\
F_Z=R(g_0\circ\iota)_*\mathbb Q_{Z_0},
\end{gathered}
\]
morphisms
\[
\mathsf B_0:F_O\longrightarrow F_Y[2b],
\qquad
\mathsf A_0:F_Y\longrightarrow F_O[-2b].
\]
Composing these with the appropriate shifts of $p(\tau_0)^2$ gives an
endomorphism of $F_O$ with zero net shift. Its cohomological action is
$\mathsf A_0N_{0,0}^2\mathsf B_0$.
Since direct image by the closed embedding $j$ is fully faithful, this is
an endomorphism of $Rq_*\mathbb Q_{O_0}$ over $S\times A$.
Let $p_S:S\times A\to A$ be the projection, and write
$K=Rh_*\mathbb Q_X$. The product identification gives
\[
Rp_{S,*}Rq_*\mathbb Q_{O_0}
\simeq K\otimes R\Gamma(S,\mathbb Q).
\]
Push the preceding endomorphism to $A$, precompose with the projection unit
$K\to K\otimes R\Gamma(S,\mathbb Q)$, and postcompose with the orientation
trace
\[
K\otimes R\Gamma(S,\mathbb Q)\longrightarrow K[-2].
\]
This constructs \eqref{spec:derived-lowering}. On ordinary cohomology the
unit is $\operatorname{pr}_0^*$, the trace is
$\operatorname{pr}_{0,*}$, and the middle map is the flag composite already
identified. Hence the resulting operator is exactly $f'$.
For completeness, set $K_{\mathrm{norm}}=K[R]$, the normalization used in
the statement of the theorem. Functoriality of perverse truncation applied
to \eqref{spec:derived-lowering} gives
\[
{}^p\tau_{\le k}K_{\mathrm{norm}}
\longrightarrow{}^p\tau_{\le k}(K_{\mathrm{norm}}[-2])
=({}^p\tau_{\le k-2}K_{\mathrm{norm}})[-2].
\]
Taking hypercohomology in degree $m-R$ and then its image in ordinary
cohomology proves the stated inclusion. This uses neither a splitting nor
multiplicativity of the perverse filtration.
\end{proof}
We finish by recording the derived morphisms and their shifts. Retain the
complexes $F_O,F_Z,F_Y$ and $K$ from the proof, and put
$K_S=K\otimes R\Gamma(S,\mathbb Q)$.
The first panel of Table~\ref{spec:shift-table} consists of morphisms on
$\mathcal E$. Compose the flag maps and $p(\tau_0)^2$ there to obtain an
endomorphism of $F_O$, descend it through $j$, and then apply $Rp_{S,*}$.
This produces the degree-zero endomorphism of $K_S$ in the second panel,
whose base is $A$. The unit and trace then give $K\to K[-2]$; it is this
complete morphism that yields the normalized perverse bound above.
No perverse exactness of $Rp_{S,*}$ is used.
The weight columns refer to the cohomological actions after transport
to $\lambda=1$ and use the diagram weights of Section~\ref{sec:diagram}.
In particular $p(\tau_0)$ acts as $N_{1,0}$ after this comparison, while
the unit and trace preserve weight because the cochains of $S$ have
weight zero.
\begin{table}[htbp]
\centering
\small
\begin{tabular}{llrr}
\toprule
\multicolumn{4}{c}{Morphisms on $\mathcal E$}\\
\midrule
Operation & Morphism & Degree & $\Delta W$\\
\midrule
$\pi_0^*$ & $F_O\to F_Z$ & $0$ & $0$\\
$\iota_{0,*}$ & $F_Z\to F_Y[2b]$ & $2b$ & $2b$\\
$p(\tau_0)$ & $F_Y\to F_Y$ & $0$ & $-2$\\
$p(\tau_0)^2$ & $F_Y\to F_Y$ & $0$ & $-4$\\
$\iota_0^*$ & $F_Y\to F_Z$ & $0$ & $0$\\
$\pi_{0,*}$ & $F_Z\to F_O[-2b]$ & $-2b$ & $-2b$\\
Flag composite & $F_O\to F_O$ & $0$ & $-4$\\
\midrule
\multicolumn{4}{c}{Morphisms on $A$}\\
\midrule
Operation & Morphism & Degree & $\Delta W$\\
\midrule
Projection unit & $K\to K_S$ & $0$ & $0$\\
Pushed composite & $K_S\to K_S$ & $0$ & $-4$\\
Orientation trace & $K_S\to K[-2]$ & $-2$ & $0$\\
Complete $f'$ & $K\to K[-2]$ & $-2$ & $-4$\\
\bottomrule
\end{tabular}
\caption{Derived degrees and diagram-weight bounds for the correspondence
maps. A weight bound $s$ means that $W_k$ maps into $W_{k+s}$ after
transport to the Betti side. The row for
$p(\tau_0)$ records $N$ separately; the flag composite uses its square.}
\label{spec:shift-table}
\end{table}
==== END P-equals-W-in-composite-rank-for-fixed-determinant//07-specialization.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//08-commutator.tex ====
\section{The translation class and the double commutator}\label{sec:commutator}
The preceding sections give the two filtration bounds for the same operator
$f'$. We now compute its double commutator with $e$. The calculation has two
parts: an elementary modification differentiates the universal degree-two
class into a difference of flag classes, and flag integration converts the
square of that difference back into $e$.
\subsection{The characteristic class under translation}
The universal projective bundle on $C\times\mathcal Y$ defines a universal
endomorphism bundle. Let
\[
e_{\mathcal Y}
=\frac1{2n}\int_C\operatorname{ch}_2(\operatorname{End}E)
\in H^2(\mathcal Y,\mathbb Q).
\]
Individual ordered flag lines $Q_i$ can be taken on local universal-bundle
charts; their ratios $Q_i\otimes Q_j^{-1}$ descend to the coarse moduli
space. We use the global classes
\[
y=c_1(Q_1\otimes Q_2^{-1}),\qquad
t_C=x^*c_1(T_C),
\]
where the second is pulled back from $S$. We retain the same notation for
the restrictions of these classes to a fiber $Y_{\lambda,u}$.
\begin{lemma}\label{calc:translation-formula}
For the translation $T:Y_{0,v}\to Y_{0,v}$ associated with
$\ell=(1,-1,0,\ldots,0)$ and every integer $k\ge0$,
\begin{equation}\label{calc:translation-polynomial}
T^{k*}e_{\mathcal Y}=e_{\mathcal Y}+ky+k^2t_C.
\end{equation}
Consequently the logarithm from Section~\ref{sec:translation} satisfies
\begin{equation}\label{calc:logarithm}
N(e_{\mathcal Y})=y
\end{equation}
on $H^*(\mathcal Y,\mathbb Q)$ and, by restriction, on every fiber.
\end{lemma}
\begin{proof}
Work first on $Y_{0,v}$. Denote by $i:\Delta\hookrightarrow C\times Y_{0,v}$
the graph of the moving point. Its normal line is the pullback of $x^*T_C$,
so
\[
\mathcal O(\Delta)|_\Delta\simeq x^*T_C,
\qquad c_1(N_\Delta)=t_C.
\]
The residue eigenvalues $v_i$ are distinct. Consequently the Higgs operator
has uniquely labeled eigenline summands on the formal neighborhood of
$\Delta$. This follows by lifting its simple residual eigenvalues and
their idempotent projectors to every finite order. Multiplying a local
frame of the logarithmic canonical line by a unit does not change those
projectors. Thus the formal summands are intrinsic, and any finite number
of their jets suffices for an elementary modification.
The $k$-fold translation replaces the $i$th formal line lattice by its twist
by $k\ell_i\Delta$. On the $(i,j)$ block of $\operatorname{End}E$, the
lattice shift is therefore
\[
m_{ij}=k(\ell_i-\ell_j).
\]
If a line block has restriction $F$ to $\Delta$, its K-theory change under
an integer shift $m$ is
\begin{equation}\label{calc:signed-jets}
[\text{new block}]-[\text{old block}]
=\sum_{a=1}^{m}[i_*(F\otimes N_\Delta^{\otimes a})].
\end{equation}
For $m<0$, the right side means the negative of the sum over $m<a\le0$;
for $m=0$ it is zero. For positive $m$, the formula follows by successive
quotients of the lattices at pole orders $1,\ldots,m$. For negative $m$,
apply the same quotient sequence between the smaller lattice and the
original one. In particular $m=-1$ contributes $-[i_*F]$, as it must for
the kernel elementary transformation.
The resulting identity for $\operatorname{End}E$ is global in K-theory,
before applying a characteristic class. Indeed the intrinsic formal
idempotents lie in the descended endomorphism algebra, so its $(i,j)$
blocks are global formal line bundles. Identify the original and modified
endomorphism bundles away from $\Delta$, and choose a common sufficiently
small lattice inside both. Their quotients by this lattice are supported
on a finite infinitesimal neighborhood of $\Delta$. The formal block
projectors and the filtration by pole order give global finite
filtrations of these quotients. Their associated graded terms are
graph pushforwards of the descended line bundles
$Q_i\otimes Q_j^{-1}\otimes N_\Delta^{\otimes a}$. Subtracting the two
quotient classes cancels the common pole orders and leaves exactly the
signed sums in \eqref{calc:signed-jets}. Additivity for these filtrations
therefore gives the identity in the Grothendieck group of coherent sheaves
on $C\times Y_{0,v}$; the common lattice also cancels.
Grothendieck--Riemann--Roch for the graph \cite[Section~15.2]{Fulton1998} gives
\[
\operatorname{ch}(i_*F)
=i_*\bigl(\operatorname{ch}(F)\operatorname{td}(N_\Delta)^{-1}\bigr).
\]
Hence the contribution of a summand in \eqref{calc:signed-jets} to
$\int_C\operatorname{ch}_2$ is
\[
c_1(F)+(a-\tfrac12)t_C.
\]
This also exhibits the sign of the normal-bundle correction: the normal
line is the tangent line, and its inverse Todd class starts with
$1-t_C/2$.
For the signed sums just specified, the elementary identities
\[
\sum_{a=1}^{m}1=m,
\qquad
\sum_{a=1}^{m}(a-\tfrac12)=\frac{m^2}{2}
\]
hold for every integer $m$.
Write $w_i=c_1(Q_i)$ on a universal-bundle chart. Only $w_i-w_j$ and
expressions invariant under common translation of the $w_i$ will occur.
Summing the contributions for $F=Q_i\otimes Q_j^{-1}$ gives
\[
T^{k*}e_{\mathcal Y}-e_{\mathcal Y}
=\frac1{2n}\sum_{i,j}
\left(k(\ell_i-\ell_j)(w_i-w_j)
+\frac{k^2}{2}(\ell_i-\ell_j)^2t_C\right).
\]
Since $\sum_i\ell_i=0$,
\[
\sum_{i,j}(\ell_i-\ell_j)(w_i-w_j)=2n\sum_i\ell_iw_i,
\qquad
\sum_{i,j}(\ell_i-\ell_j)^2=2n\sum_i\ell_i^2.
\]
For the chosen cocharacter these are $2ny$ and $4n$, respectively, proving
\eqref{calc:translation-polynomial}. Since the lattice identity was global
on the coarse moduli space before applying Grothendieck--Riemann--Roch,
this is a global cohomological identity there.
Let $M$ be the integer used to define $N$. Replace $k$ by $Mq$ in
\eqref{calc:translation-polynomial}. Since $\mathsf T^{Mq}=\exp(MqN)$,
both sides are polynomial functions of the nonnegative integer $q$ with
values in a finite-dimensional cohomology group. They agree as
polynomials. Their linear coefficients are respectively $MN(e_{\mathcal
Y})$ and $My$, giving \eqref{calc:logarithm} on $Y_{0,v}$.
Corollary~\ref{log:restriction} transports this identity to
$H^*(\mathcal Y)$ and then to all its fibers: the three classes in
\eqref{calc:translation-polynomial} are restrictions of the global classes
just defined.
\end{proof}
The quadratic term in \eqref{calc:translation-polynomial} is consistent
with the behavior of the flag lines themselves. Restricting a shifted
lattice to $\Delta$ gives
\[
T^{k*}y=y+2kt_C,
\qquad N(y)=2t_C,
\qquad N(t_C)=0.
\]
In particular one must not assume $N(y)=0$ in the commutator calculation.
\subsection{The flag integral}
We record explicitly the permutation identity that evaluates the flag
correspondence. It applies to projective bundles because every polynomial
used below is invariant under simultaneous translation of the roots.
\begin{lemma}\label{calc:flag-sum}
Let $\pi:\operatorname{Fl}(V)\to B$ be the full-flag bundle of a rank-$n$
complex vector bundle, and let $w_1,\ldots,w_n$ be its formal Chern roots.
For a polynomial $F$ in the first Chern classes of the ordered flag lines,
\[
\pi_*\bigl(F(Q_\bullet)c_b(T_\pi)\bigr)
=\sum_{\sigma\in\mathfrak S_n}
F(w_{\sigma(1)},\ldots,w_{\sigma(n)}).
\]
If $\sum_i\ell_i=0$, then
\begin{equation}\label{calc:permutation}
\sum_{\sigma\in\mathfrak S_n}
\left(\sum_i\ell_iw_{\sigma(i)}\right)^2
=n(n-2)!\|\ell\|^2
\left(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\right).
\end{equation}
Both identities, when their expressions are invariant under common
translation of the roots, hold for the associated flag bundle of a
principal $\mathrm{PGL}_n$-bundle with rational coefficients.
\end{lemma}
\begin{proof}
The projective-bundle pushforward formula is a sum over the roots, with
denominator the product of the other-root differences. Iterating it along
the full-flag tower gives the sum over ordered roots, with denominator the
tangent Euler class at that ordering. Multiplication by $c_b(T_\pi)$
cancels this denominator, giving the first identity. One may perform this
calculation after passing to a splitting space and inverting root
differences; the resulting equality is a polynomial identity in the
universal Chern classes, so it holds without the inversion. The
projective-bundle pushforward is characterized by its Chern-polynomial
relation and the normalization that the top power of the hyperplane class
integrates to one; using the complex tangent Euler class fixes the
orientation signs in the displayed cancellation.
For \eqref{calc:permutation}, a fixed root occurs $(n-1)!$ times at a fixed
position, and a fixed ordered pair of distinct roots occurs $(n-2)!$ times
at two prescribed distinct positions. Since
$\sum_{i\ne j}\ell_i\ell_j=-\|\ell\|^2$, expansion of the square gives
\[
\begin{split}
&(n-1)!\|\ell\|^2\sum_iw_i^2
-(n-2)!\|\ell\|^2\sum_{i\ne j}w_iw_j\\
&\hspace{15mm}
=n(n-2)!\|\ell\|^2
\left(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\right).
\end{split}
\]
Finally, rational pullback from $B\mathrm{PGL}_n$ to $B\mathrm{GL}_n$ is
injective: on a maximal torus it is the inclusion of symmetric polynomials
in the root differences into symmetric polynomials in all roots.
Universal characteristic-class identities invariant under common root
translation can therefore be checked after this pullback, proving the
last assertion.
\end{proof}
\subsection{The common lowering operator}
\begin{proposition}\label{calc:double-commutator}
Let $\delta=\deg(x:S\to C)$. The operator $f'$ of
\eqref{trans:fprime} satisfies
\begin{equation}\label{calc:scalar}
[[f',e],e]=8(-1)^b n(n-2)!\delta\,e.
\end{equation}
Consequently
\begin{equation}\label{calc:f}
f=\frac{(-1)^{b+1}}{4n(n-2)!\delta}\,f'
\end{equation}
has cohomological degree $-2$, lowers perversity by two and Betti weights
by four, and satisfies $[[f,e],e]=-2e$.
\end{proposition}
\begin{proof}
Work on the scalar correspondence at $\lambda=0$. The restriction of
$e_{\mathcal Y}$ along $\iota:Z_0\hookrightarrow Y_{0,0}$ equals
$\pi^*\operatorname{pr}_0^*e$: the universal projective characteristic
class is unchanged by the line twist $J$ in the scalar product
identification. Projection formulas therefore show that the maps outside
$N_{0,0}^2$ in $f'$ intertwine multiplication by the corresponding classes
$e$ and $e_{\mathcal Y}$.
For a degree-zero derivation $N$ and an even class $a$, write $m_a$ for
multiplication by $a$. Then $[N,m_a]=m_{N(a)}$, and direct expansion gives
\[
[[N^2,m_a],m_a]=2m_{N(a)^2}.
\]
Indeed the first commutator is
$m_{N^2(a)}+2m_{N(a)}N$; the first term commutes with $m_a$, while the
second contributes $2m_{N(a)^2}$. Thus no vanishing assertion about
$N^2(a)$ is needed. By Lemma~\ref{calc:translation-formula},
\begin{equation}\label{calc:middle-commutator}
[[f',e],e]
=2\operatorname{pr}_{0,*}\mathsf A_0
(y^2\cup-)\mathsf B_0\operatorname{pr}_0^*.
\end{equation}
The normal bundle of $\iota$ is $T_\pi^*$. Self-intersection therefore
replaces $\iota^*\iota_*$ in \eqref{calc:middle-commutator} by
\[
c_b(T_\pi^*)\cup-=(-1)^b c_b(T_\pi)\cup-.
\]
For a class pulled back from $X$, the projection formula and
Lemma~\ref{calc:flag-sum} reduce the flag integral to the symmetric class
\[
\begin{split}
\sum_{\sigma\in\mathfrak S_n}
(w_{\sigma(1)}-w_{\sigma(2)})^2
&=2n(n-2)!
\left(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\right)\\
&=4n(n-2)!\,\widetilde{\operatorname{ch}}_2(E_{x(s)}).
\end{split}
\]
Here $E_{x(s)}$ denotes the universal projective data restricted to the
graph of $x$; the normalized expression is well defined without an honest
universal vector bundle.
Under $O_0\simeq S\times X$, this characteristic class is the pullback of
$\widetilde{\operatorname{ch}}_2(E)$ from $C\times X$ along
$x\times\operatorname{id}_X$. Tensoring by $J$ contributes nothing to it.
Integration along $S$ multiplies integration along $C$ by $\delta$:
\[
\operatorname{pr}_{0,*}
\widetilde{\operatorname{ch}}_2(E_{x(s)})=\delta e.
\]
This follows directly from the degree formula $x_*[S]=\delta[C]$, or from
the $C$-degree-two term in the K\"unneth decomposition. The other curve
degrees integrate to zero.
Combining this equality, the factor $2$ in
\eqref{calc:middle-commutator}, and the self-intersection sign proves
\eqref{calc:scalar}.
The integer $\delta$ is positive because $x$ is a nonempty finite
\'{e}tale cover of connected projective curves. Thus
\eqref{calc:f} is a nonzero rational rescaling and gives the stated double
commutator. The degree of $f'$ is $-2$, since the two flag shifts cancel
and integration over $S$ has degree $-2$. The filtration bounds are
Propositions~\ref{spec:perverse-lowering} and~\ref{trans:weights}; they are unchanged by the rescaling.
\end{proof}
==== END P-equals-W-in-composite-rank-for-fixed-determinant//08-commutator.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//09-conclusion.tex ====
\section{Equality of the filtrations}\label{sec:conclusion}
We assemble the preceding results on the same vector space $H^*(X,\Q)$, identified with $H^*(X^B,\Q)$ by nonabelian Hodge theory.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
First take $L=\mathcal O_C(dx_0)$. Proposition~\ref{norm:relative-lefschetz} gives hard Lefschetz for cup product by $e$ on $\Gr^P H^*(X)$, with grading shift two and center $R$. Section~\ref{sec:betti} gives the same statement on $\Gr^W H^*(X^B)$ when actual weight $2k$ is assigned grade $k$. It also proves that all odd weight-graded pieces vanish.
Let $\delta=\deg(S\to C)>0$ and set
\[
f=\frac{(-1)^{b+1}}{4n(n-2)!\,\delta}\,f'.
\]
Propositions~\ref{trans:weights} and~\ref{spec:perverse-lowering} show that $f$ lowers the half-weight filtration and the perverse filtration by two. The calculation in Section~\ref{sec:commutator} gives $[[f,e],e]=-2e$. Apply Proposition~\ref{alg:comparison} to $P_k$ and $W_{2k}$. It follows that
$P_kH^m(X)=W_{2k}H^m(X^B)$ in every cohomological degree. The vanishing of odd weight-graded pieces gives $W_{2k}=W_{2k+1}$.
For arbitrary $L\in\Pic^d(C)$, choose $M\in\Pic^0(C)$ with
$M^n\simeq L\otimes\mathcal O_C(-dx_0)$. Tensoring by $M$ identifies the Higgs spaces over the Hitchin base. Proposition~\ref{log:nah} proves that this identification transports the full Betti weight filtration as well. This proves the theorem for every determinant.
\end{proof}
\begin{corollary}
Let $C$ be a smooth projective complex curve of genus $g\ge2$, let
$n\ge2$ and $d$ be coprime integers, and let $L\in\Pic^d(C)$.
For the moduli space $X$ of stable rank-$n$, determinant-$L$, trace-free
Higgs bundles and its corresponding twisted $\SL_n$ character variety
$X^B$, nonabelian Hodge theory identifies
\[
P_k H^m(X,\Q)=W_{2k}H^m(X^B,\Q)=W_{2k+1}H^m(X^B,\Q)
\qquad\text{for all integers }m,k.
\]
The equality holds on full rational cohomology, including its
$\Pic^0(C)[n]$-variant part.
\end{corollary}
\begin{proof}
The composite ranks are Theorem~\ref{thm:main}; the prime ranks are \cite[Theorem~0.3]{MS2024}.
\end{proof}
\begin{corollary}[Endoscopic weight compatibility]\label{cor:endoscopic-weights}
Keep $C,n,d,L$ as in the preceding corollary. Put $\Gamma=\Pic^0(C)[n]$,
let $\gamma\in\Gamma$ have order $m$, and let $\kappa\in\widehat{\Gamma}$
be the character $\kappa(\gamma')=\langle\gamma,\gamma'\rangle_\Gamma$
defined by the Weil pairing in \cite[Section~1.3]{MSEndoscopy2021}. Let
$\pi:C'\to C$ be the associated cyclic \'etale cover, set $r=n/m$, and put
\[
d_\gamma=\operatorname{codim}_{\mathcal A^{K_C}}\mathcal A_\gamma^{K_C},
\]
where $\mathcal A_\gamma^{K_C}$ is their endoscopic Hitchin base.
For $D=K_C$ and $D'=\pi^*K_C=K_{C'}$, let $\mathfrak p_\kappa^B$ denote
their nonabelian-Hodge-transported operator in
\cite[Section~5.4]{MSEndoscopy2021}. Write
$\widetilde{\mathcal M}_{r,d}^B(C')$ for the full degree-$d$ twisted
$\GL_r$ character variety and $\mathcal M_{n,L}^B(C)=X^B$ for the
fixed-determinant twisted $\SL_n$ character variety. On complexified
cohomology, for all integers $i,k\ge0$,
\begin{multline*}
\mathfrak p_\kappa^B\!\left(W_{2k}H^i(\widetilde{\mathcal M}_{r,d}^B(C'),\C)\right)\\
=W_{2k+2d_\gamma}H^{i+2d_\gamma}(\mathcal M_{n,L}^B(C),\C)_\kappa,
\end{multline*}
where the subscript $\kappa$ denotes the pullback-action summand on which
$\gamma'^*\alpha=\kappa(\gamma')\alpha$ for every $\gamma'\in\Gamma$.
\end{corollary}
\begin{proof}
Since $r\mid n$, one has $\gcd(r,d)=1$, and \'etale Riemann--Hurwitz gives
$g(C')=m(g-1)+1\ge2$. The all-rank $\GL_r$ theorem
\cite[Theorem~0.2]{MS2024} therefore identifies $W_{2k}$ with $P_k$ on the
source, including when $r=1$. The preceding corollary identifies
$P_{k+d_\gamma}$ with $W_{2k+2d_\gamma}$ on the target after
complexification and passage to the $\Gamma$-$\kappa$ summand: the
$\Gamma$-action preserves both filtrations, and the target nonabelian-Hodge
identification is $\Gamma$-equivariant. With the common
defect normalization, \cite[Theorem~5.4]{MSEndoscopy2021} gives the exact
image of $P_kH^i$ as $P_{k+d_\gamma}H^{i+2d_\gamma}_\kappa$.
Writing $\eta_s,\eta_t$ for the source and target nonabelian-Hodge
cohomology identifications from Betti to Dolbeault, the definition
$\mathfrak p_\kappa^B=\eta_t^{-1}\mathfrak p_\kappa\eta_s$ then gives the
claim. Its nonzero scalar ambiguity does not affect images.
\end{proof}
The standard implication from $P=W$ to perverse multiplicativity
\cite[Section~0.4.3, equation~(10)]{dCMSJAMS2022} now applies to the full
fixed-determinant cohomology.
\begin{corollary}[Multiplicativity of the perverse filtration]\label{cor:perverse-multiplicativity}
In the setting of the all-rank corollary above, use the normalization
\eqref{intro:P}. For all $i,j\ge0$ and $a,b\in\Z$,
\[
P_aH^i(X,\Q)\smile P_bH^j(X,\Q)
\subseteq P_{a+b}H^{i+j}(X,\Q).
\]
Put $\Gamma=\Pic^0(C)[n]$ and
$P_aH^i(X,\C)=P_aH^i(X,\Q)\otimes_{\Q}\C$. For the pullback-action
isotypic parts indexed by $\chi,\psi\in\widehat\Gamma
=\operatorname{Hom}(\Gamma,\C^\times)$, one also has
\[
\bigl(P_aH^i(X,\C)\bigr)_\chi\smile
\bigl(P_bH^j(X,\C)\bigr)_\psi
\subseteq\bigl(P_{a+b}H^{i+j}(X,\C)\bigr)_{\chi\psi}.
\]
This includes trivial and distinct characters, with the same normalized
$P$ on every character summand.
\end{corollary}
\begin{proof}
The nonabelian-Hodge diffeomorphism induces a graded ring isomorphism
$\eta:H^*(X^B,\Q)\to H^*(X,\Q)$ \cite[Sections~2 and~5]{HT2003}.
Cup product is a morphism of mixed Hodge structures
\cite[Corollaire~(8.2.11)]{Deligne1974}, so it sends
$W_{2a}H^i(X^B,\Q)\smile W_{2b}H^j(X^B,\Q)$ into
$W_{2(a+b)}H^{i+j}(X^B,\Q)$. Transport by $\eta$ and the all-rank
$P=W$ equality give the first assertion. The filtration is
$\Gamma$-stable, and
$\gamma^*(\alpha\smile\beta)=\gamma^*\alpha\smile\gamma^*\beta$
multiplies the two eigencharacters, giving the second assertion.
\end{proof}
==== END P-equals-W-in-composite-rank-for-fixed-determinant//09-conclusion.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//main.tex ====
\pdfinfoomitdate=1
\pdftrailerid{}
\pdfsuppressptexinfo=15
\documentclass[11pt]{article}
\input{preamble}
\title{$P=W$ in composite rank for fixed determinant}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the $P=W$ conjecture on the full rational cohomology of
fixed-determinant, trace-free Higgs moduli spaces in composite rank and
coprime degree, for smooth projective complex curves of genus at least two.
Together with the established prime-rank cases, this gives the equality in
every coprime rank.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-lefschetz}
\input{sections/03-betti}
\input{sections/04-diagram}
\input{sections/05-logarithmic}
\input{sections/06-translation}
\input{sections/07-specialization}
\input{sections/08-commutator}
\input{sections/09-conclusion}
{\small
\bibliographystyle{alpha}
\bibliography{references}
}
\end{document}
==== END P-equals-W-in-composite-rank-for-fixed-determinant//main.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//preamble.tex ====
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\hypersetup{pdftitle={P=W in composite rank for fixed determinant},pdfauthor={OpenAI}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
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\newtheorem{remark}[theorem]{Remark}
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\newcommand{\C}{\mathbb C}
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==== END P-equals-W-in-composite-rank-for-fixed-determinant//preamble.tex ====
==== BEGIN P-equals-W-in-composite-rank-for-fixed-determinant//references.bib ====
@article{MS2024,
author = {Maulik, Davesh and Shen, Junliang},
title = {The {$P=W$} conjecture for {$\mathrm{GL}_n$}},
journal = {Annals of Mathematics}, volume = {200}, number = {2},
year = {2024}, pages = {529--556}, doi = {10.4007/annals.2024.200.2.3}
}
@article{dCMS2022,
author = {de Cataldo, Mark Andrea A. and Maulik, Davesh and Shen, Junliang},
title = {On the {$P=W$} conjecture for {$\mathrm{SL}_n$}},
journal = {Selecta Mathematica}, volume = {28}, year = {2022},
pages = {90}, doi = {10.1007/s00029-022-00803-0}
}
@article{MSEndoscopy2021,
author = {Maulik, Davesh and Shen, Junliang},
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==== END P-equals-W-in-composite-rank-for-fixed-determinant//references.bib ====