==== TABLE OF CONTENTS ====
The-Campana-Peternell-conjecture-in-dimension-six//glyph-unicode.tex
The-Campana-Peternell-conjecture-in-dimension-six//main.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/01-introduction.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/02-minimal-curves.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/02-p1-cohomology.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/03-pseudoindex-four.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/04-characteristic-classes.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/05-arithmetic.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/06-grassmannian.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/07-conclusion.tex
The-Campana-Peternell-conjecture-in-dimension-six//sections/08-certificate.tex
The-Campana-Peternell-conjecture-in-dimension-six//references.bib
===========================

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==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//main.tex ====
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\title{The Campana--Peternell conjecture\protect\\ in dimension six}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that every smooth connected complex projective Fano sixfold with nef
tangent bundle is rational homogeneous, resolving the Campana--Peternell
conjecture in complex dimension six.
As a consequence, a connected compact K\"ahler manifold $X$ with nef
holomorphic tangent bundle and
$\dim_{\mathbb C}X-\widetilde q(X)\leq6$ has ordinary universal cover
$F\times\mathbb C^{\widetilde q(X)}$, where $F$ is a rational homogeneous
manifold and $\widetilde q(X)$ is the maximal irregularity of a connected
finite \'{e}tale cover.
\end{abstract}
\tableofcontents
\clearpage
\input{sections/01-introduction}
\input{sections/02-minimal-curves}
\input{sections/02-p1-cohomology}
\input{sections/03-pseudoindex-four}
\input{sections/04-characteristic-classes}
\input{sections/05-arithmetic}
\input{sections/06-grassmannian}
\input{sections/07-conclusion}
\appendix
\input{sections/08-certificate}
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\end{document}
==== END The-Campana-Peternell-conjecture-in-dimension-six//main.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/01-introduction.tex ====
\section{Introduction}

A smooth connected complex projective variety $X$ is \emph{Fano} if
its anticanonical line bundle $\omega_X^{-1}$ is ample. A vector bundle
$E$ is \emph{nef} if the tautological line bundle
$\mathcal O_{\mathbb P(E)}(1)$ has nonnegative degree on every curve.
We use the quotient convention
$\mathbb P(E)=\operatorname{Proj}\operatorname{Sym}E$.
A variety is \emph{rational homogeneous} if it is isomorphic to $G/P$,
where $G$ is a connected semisimple complex algebraic group and $P$ is
a parabolic subgroup.

The tangent bundle of a rational homogeneous variety is generated by
global sections and is therefore nef. Mori's characterization of
projective space by ampleness of its tangent bundle
\cite{Mori1979} makes the corresponding question for nef tangent
bundles natural. The Campana--Peternell conjecture asks whether every
smooth complex Fano variety with nef tangent bundle is rational
homogeneous \cite{CampanaPeternell1991}. Thus it predicts that a
numerical positivity condition determines a variety with a transitive
algebraic group action. We prove the conjecture in dimension six.

\begin{theorem}\label{thm:main}
Every smooth connected complex projective Fano variety of dimension six
with nef tangent bundle is rational homogeneous.
\end{theorem}

The Picard number $\rho(X)$ is the rank of the group of divisor classes
modulo numerical equivalence, and the \emph{pseudoindex} is the least
anticanonical degree of a rational curve on $X$.
These two invariants organize much of the progress on the conjecture.
Campana and Peternell established it in dimensions at most three and
for fourfolds of Picard number greater than one
\cite{CampanaPeternell1991,CampanaPeternell1993}.
The remaining fourfold case was settled through Mok's and Hwang's
study of the tangent directions of minimal rational curves
\cite{Mok2002,Hwang2006}. Watanabe proved the fivefold case of Picard
number greater than one \cite[Theorem~1.2]{Watanabe2014}, and
Kanemitsu completed the fivefold classification
\cite[Theorem~0.2]{Kanemitsu2017}.

Kanemitsu also proved that a smooth complex Fano $n$-fold with nef
tangent bundle is rational homogeneous whenever $\rho(X)>n-5$
\cite[Theorem~0.2]{Kanemitsu2016}. For a sixfold, this covers every
$\rho(X)\geq2$, leaving only Picard number one. In that range,
standard characterization results reduce the problem to pseudoindices
four and five. Watanabe treated the pseudoindex-four sixfold case
\cite[Theorem~2.4]{Watanabe2021}. We include an independent intersection
argument for it; the principal new step is pseudoindex five, where a
characteristic-number constraint leads to the Grassmannian.

The difficulty is to pass from positivity of $T_X$ to a specific
projective model for the tangent directions of minimal rational curves.
Nefness gives strong smoothness and deformation properties, but these
properties do not by themselves identify the threefold of directions
that occurs in pseudoindex five. We extract additional information by
using the universal family in two ways: as a family of rational curves
over their parameter space, and as a family of threefolds over $X$.
The compatibility of these two maps yields a characteristic-number
identity strong enough to determine the projective variety of tangent
directions.

For a covering family of rational curves of least anticanonical degree,
the closure of the tangent directions to its members smooth at a general
point is called the \emph{variety of minimal rational tangents} (VMRT).
Characteristic classes and the geometry of minimal curves already
interact in Mok's work on one-dimensional varieties of minimal rational
tangents \cite{Mok2002}. Under an additional fourth-Betti-number
hypothesis, his argument uses the two rational-curve fibrations of the
universal family and characteristic-class inequalities for a rank-two
bundle on the parameter space to constrain the tangent image.
Hwang removed that hypothesis
\cite[Section~4]{Hwang2006}; the resulting classification is stated in
\cite[Theorem~15]{Mok2008}. A related structural approach studies
elementary Mori contractions, the contractions of extremal rays:
if all of these contractions of a Fano manifold are smooth
$\mathbb P^1$-fibrations, the manifold is a complete flag variety
\cite[Theorem~1.2]{OcchettaSolaCondeWatanabeWisniewski2017}.
These earlier results show how multiple curve fibrations can constrain
both characteristic classes and homogeneous structure.
The calculation here concerns a three-dimensional tangent image. It
uses the two fibrations to produce explicit linear intersection
identities, then combines them with the product identities furnished
by the projection formula.

\subsection{The argument}

Assume $\rho(X)=1$. Standard characterization theorems
leave pseudoindices four and five; the other possibilities are excluded
or give projective space $\mathbb P^6$ and the smooth quadric $Q^6$.
The proof is organized around the universal family of rational curves
of least degree. Its evaluation map and its projection to the parameter
space are both smooth. Along every fibre of the latter map, the
relative tangent bundle of evaluation is a sum of copies of
$\mathcal O_{\mathbb P^1}(-1)$.

For pseudoindex four, Kanemitsu's factorization theorem reduces the
problem to a variety admitting two fibrations. We rule out the possible
fibres by Hard Lefschetz and intersection theory. The last case is an
odd-dimensional quadric bundle: its orthogonal description forces a
vanishing characteristic number, whereas the second fibration would
make an integral divisor have nonintegral degree.

For pseudoindex five and a general point $x\in X$, the evaluation fibre
$F$ is a smooth threefold parametrizing minimal rational curves marked at
$x$. Sending a marked
curve to its tangent direction at $x$ gives a finite morphism
$F\to\mathbb P(T_x^*X)\simeq\mathbb P^5$. Its image is the variety of
minimal rational tangents, and the pullback $L$ of $\mathcal O(1)$ is
ample. At this point the degree of the map and the projective type of
its image are both undetermined. The central numerical step is
\[
 32-20u+3u^2=0,
 \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}.
\]
We derive this identity from the cohomology of the universal family,
Deligne's fixed-part theory \cite{Deligne1971}, and
Grothendieck--Riemann--Roch \cite{BorelSerre1958}. All resulting characteristic-number
constraints are written explicitly. A finite rational calculation then
extracts the quadratic relation.

This method separates two useful ingredients. The two fibrations give
linear relations among intersection numbers; integration along evaluation
also makes selected numbers factor as products. Multiplying these
relations forces a nonlinear constraint on the polarized fibre.
The proof explains both ingredients and gives an exact certificate for
the finite elimination.

Finally, this image spans $\mathbb P^5$. The sectional-genus
formula excludes $u=4$ and forces $L^3=3$ when $u=8/3$.
The variety of minimal rational tangents is consequently the Segre
threefold $\mathbb P^1\times\mathbb P^2\subset\mathbb P^5$.
The VMRT method of Hwang and Mok
\cite[Propositions~1.2.1--1.2.2]{HwangMok1999} supplies the
nondegeneracy criterion, and Mok's recognition theorem
\cite{Mok2008Recognition} identifies $X$ with
$\operatorname{Gr}(2,5)$, the Grassmannian of two-dimensional subspaces
of $\mathbb C^5$.

\subsection{Consequences of the classification}

The classification also has a consequence for compact K\"ahler manifolds
that need not be projective. A differential-geometric antecedent is
Mok's uniformization theorem for compact K\"ahler manifolds with
nonnegative holomorphic bisectional curvature \cite{Mok1988}.
For nef tangent bundles, the structure theorem of
Demailly--Peternell--Schneider relates the compact K\"ahler problem to
the Fano conjecture through the fibres of the Albanese map
\cite[Main Theorem and Theorem~3.14]{DemaillyPeternellSchneider1994}.
To state the consequence precisely, a holomorphic vector bundle
is \emph{nef in the analytic sense} when its tautological line bundle
admits smooth Hermitian metrics whose curvature is bounded below by
$-\varepsilon\omega$ for every $\varepsilon>0$, for a fixed Hermitian
form $\omega$ on its projectivization. This agrees with the preceding
numerical definition in the projective setting
\cite[Section~1.A and Definitions~1.2 and~1.9]{DemaillyPeternellSchneider1994}.
For a connected compact K\"ahler manifold $X$ with nef tangent bundle, put
\[
 \widetilde q(X)=
 \max\bigl\{h^1(X',\mathcal O_{X'}):
       X'\longrightarrow X\text{ is a connected finite \'{e}tale cover}\bigr\}.
\]
This maximum exists and is at most $\dim_{\mathbb C}X$
by the Demailly--Peternell--Schneider Albanese theorem
\cite[Proposition~3.9 and Theorem~3.14]{DemaillyPeternellSchneider1994}.

\begin{corollary}[Albanese fibres and uniformization]\label{cor:nef-tangent-uniformization}
Let $X$ be a connected compact K\"ahler manifold whose holomorphic
tangent bundle is nef in the analytic sense. Suppose
\[
 0\leq d=\dim_{\mathbb C}X-\widetilde q(X)\leq6.
\]
Then there is a connected finite \'{e}tale cover $X'\to X$ with the
following properties.
\begin{enumerate}
\item The Albanese map
      $\alpha:X'\to A=\operatorname{Alb}(X')$ is a smooth proper
      surjection with connected fibres, where $A$ is a complex torus of
      dimension $\widetilde q(X)$. Every fibre is rational homogeneous
      of dimension $d$, with a point allowed when $d=0$.
\item There is one such fibre $F$ for which $\alpha$ is a holomorphically
      locally trivial $F$-bundle. The ordinary universal cover of $X$
      is biholomorphic to
      \[
       \widetilde X\simeq F\times\mathbb C^{\widetilde q(X)}.
      \]
      It is therefore biholomorphic to a quasi-projective variety and
      is holomorphically convex. Its Remmert reduction is the projection
      \[
       F\times\mathbb C^{\widetilde q(X)}
       \longrightarrow\mathbb C^{\widetilde q(X)}.
      \]
\end{enumerate}
In particular, these conclusions hold whenever $\dim_{\mathbb C}X\leq6$.
\end{corollary}

The first conclusion applies the Fano classification fibrewise to the
Demailly--Peternell--Schneider fibration. The second also uses local
deformation rigidity of rational homogeneous manifolds, the
Fischer--Grauert theorem, and the universal-cover splitting theorem of
Claudon--H\"oring--Koll\'ar. The torus $A$ may be nonprojective, and the
finite-cover bundle may remain twisted: the product is asserted only
for the ordinary universal cover. We prove the corollary after the
sixfold theorem in Section~\ref{sec:conclusion}.

The Fano classification also gives a positive K\"ahler--Einstein metric
on every Fano manifold of positive dimension at most six with nef tangent bundle;
Corollary~\ref{cor:positive-ke} records the normalization of this metric.

\paragraph{Organization.}
Section~\ref{sec:minimal-curves} establishes the common geometry,
including the rational-curve fibration calculation in
Section~\ref{subsec:p1-cohomology}.
Section~\ref{sec:pseudoindex-four} settles pseudoindex four.
Sections~\ref{sec:characteristic-classes} and~\ref{sec:arithmetic}
derive the quadratic relation. Section~\ref{sec:grassmannian}
recognizes the Grassmannian, and Section~\ref{sec:conclusion}
combines the cases with Kanemitsu's Picard-number theorem and derives
Corollary~\ref{cor:nef-tangent-uniformization}.
Appendix~\ref{app:certificate} includes the complete executable
certificate for the finite calculation.
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/01-introduction.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/02-minimal-curves.tex ====
\section{Minimal rational curves}\label{sec:minimal-curves}

We now assume that $X$ has Picard number one.  The first task is to
construct a smooth universal family of rational curves and determine its
relative tangent bundle along every member.  These facts will provide the
common geometric input for the two remaining pseudoindices.

All cohomology groups in the intersection calculations have rational
coefficients.  A class in $H^{2i}$ has \emph{weight} $i$.  For a smooth
morphism $a$ we write $T_a=\ker(da)$ and $-K_a=c_1(T_a)$.
Kodaira vanishing gives $H^i(X,\mathcal O_X)=0$ for $i>0$.
The exponential sequence and the Lefschetz $(1,1)$ theorem therefore give
\begin{equation}\label{eq:picard-cohomology}
  H^2(X,\mathbb Q)=\mathbb Q\,c_1(T_X).
\end{equation}
We also use that smooth complex Fano manifolds are rationally connected
and simply connected; see
\cite[Theorems~2.11 and~5.3.2]{CampanaPeternell1999}.

The \emph{pseudoindex} of $X$ is the positive integer
\[
 d=\min\{-K_X\cdot B:B\subset X\text{ is an irreducible rational curve}\}.
\]
Choose an irreducible component $V$ of the normalized space of rational
curves whose members have degree $d$.  Let
\begin{equation}\label{eq:universal-family}
 \begin{tikzcd}[column sep=large]
 U \arrow[r,"e"] \arrow[d,"\pi"'] & X\\
 V &
 \end{tikzcd}
\end{equation}
be its universal family and evaluation map.  Thus a fibre of $\pi$ is the
normalization of the corresponding curve, and its map to $X$ is
birational onto that curve.

\begin{proposition}\label{prop:minimal-family}
Let $X$ be a smooth complex Fano manifold of dimension $n$ with nef tangent
bundle, and let $V$ be a component of rational curves of minimal
anticanonical degree $d$.  Then $U$ and $V$ are smooth projective varieties,
$\dim V=n+d-3$, the map $\pi:U\to V$ is a smooth $\mathbb P^1$-fibration,
and $e:U\to X$ is smooth and surjective with connected fibres of dimension
$d-2$.  For every fibre $C$ of $\pi$,
\begin{equation}\label{eq:relative-tangent-splitting}
 T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus(d-2)}.
\end{equation}
\end{proposition}

\begin{proof}
The properties of $U,V,\pi,e$ are the minimal-family theorem recalled in
\cite[Proposition~1.6]{Kanemitsu2017}; that statement applies in arbitrary
dimension.  We recall why its hypotheses hold here.  For every
normalization $\varphi:\mathbb P^1\to X$, nefness implies a splitting
\[
 \varphi^*T_X=\bigoplus_{i=1}^n\mathcal O(a_i),\qquad a_i\geq0.
\]
In particular, each such map is \emph{free}, meaning that its pulled-back
tangent bundle is generated by global sections.  Minimality of $d$ rules
out a degeneration into a sum of rational curves: each nonzero component
would already have anticanonical degree at least $d$.  This is the
properness condition for the minimal family.  Freeness makes the spaces
of maps and the evaluation smooth.  Finally, the finite map in the Stein
factorization of a smooth proper evaluation is \'{e}tale; simple
connectedness of $X$ makes it an isomorphism, giving connected fibres.

It remains to prove \eqref{eq:relative-tangent-splitting}; here it is
essential to allow \emph{every} member of $V$.  Fix a fibre
$C\simeq\mathbb P^1$ and set $\varphi=e|_C$.  Let $H$ be the corresponding
component of $\operatorname{Hom}_{\mathrm{bir}}(\mathbb P^1,X)$.
Since all its maps are free, $H$ is smooth, and
\[
 T_{[\varphi]}H=H^0(\mathbb P^1,\varphi^*T_X).
\]
The construction of the normalized rational-curve space gives a
principal $\operatorname{PGL}_2$-bundle $H\to V$ and a compatible map
\[
 \lambda:\mathbb P^1\times H\longrightarrow U,
 \qquad e\circ\lambda(t,f)=f(t);
\]
see \cite[Sections~2.1--2.2]{BrionKannan2021}.
The normalization on the map space introduces no change because $H$ is
smooth.  Over $[\varphi]$, the map $\lambda$ identifies $\mathbb P^1$ with
$C$.

For $\xi\in T_{[\varphi]}H$, differentiating $\lambda$ in the constant
parameter direction $(0,\xi)$ produces a section of $T_U|_C$.
Its image under $de$ is precisely the section of $\varphi^*T_X$
represented by $\xi$.  Consequently
\begin{equation}\label{eq:sections-evaluation-surjective}
 H^0(C,T_U|_C)\longrightarrow H^0(C,\varphi^*T_X)
 \quad\text{is surjective}.
\end{equation}
The tangent sequence of $\pi$, restricted to $C$, is
\[
 0\longrightarrow\mathcal O(2)\longrightarrow T_U|_C
 \longrightarrow\mathcal O^{\oplus(n+d-3)}\longrightarrow0.
\]
It gives $h^0(T_U|_C)=n+d$ and $H^1(T_U|_C)=0$.
The nonnegative splitting of $\varphi^*T_X$ likewise gives
$h^0(\varphi^*T_X)=n+d$.  Thus the map in
\eqref{eq:sections-evaluation-surjective} is an isomorphism.
The cohomology sequence of
\[
 0\longrightarrow T_e|_C\longrightarrow T_U|_C
 \longrightarrow\varphi^*T_X\longrightarrow0
\]
now yields $H^0(C,T_e|_C)=H^1(C,T_e|_C)=0$.
Every line-bundle summand of $T_e|_C$ therefore has degree $-1$, proving
the assertion.  No immersion assumption on $\varphi$ was used.
\end{proof}

We can now isolate the cases that require further work.

\begin{proposition}\label{prop:pseudoindex-reduction}
Let $X$ be a smooth complex Fano sixfold with $\rho(X)=1$ and nef tangent
bundle.  Then either $X\simeq\mathbb P^6$, $X\simeq Q^6$, or its
pseudoindex is $4$ or $5$.
\end{proposition}

\begin{proof}
For a nonconstant map $\varphi:\mathbb P^1\to X$, the differential
$\mathcal O(2)\to\varphi^*T_X$ is nonzero.  At least one summand of the
nonnegative splitting of $\varphi^*T_X$ therefore has degree at least
$2$, so $d\geq2$.
If $d=2$, Proposition~\ref{prop:minimal-family} makes $e$ a smooth proper
map of relative dimension zero with connected fibres, hence an
isomorphism.  The map $\pi$ would then give a $\mathbb P^1$-fibration
$X\to V$ with $\dim V=5$.  The pullback of an ample divisor on $V$ is
nonzero and nef but has degree zero on the fibres.  This contradicts
$\rho(X)=1$.

For a general point $x\in X$, the \emph{variety of minimal rational
tangents} is the closure in $\mathbb P(T_x^*X)$ of the tangent lines at
$x$ to members of $V$ smooth at $x$.  The marked family $e^{-1}(x)$ has
dimension $d-2$, and the finite tangent-map theorem gives the same
dimension for this variety; see \cite[Theorem~3.4]{Kebekus2002}.
When $d=3$, it is therefore one-dimensional.  The Mok--Hwang classification
\cite[Theorem~15]{Mok2008} gives only $\mathbb P^2$, $Q^3$, and the
five-dimensional homogeneous contact manifold of type $G_2$; see also
\cite[Section~1]{Kanemitsu2017}.
None of the three resulting manifolds has dimension six.

If $d\geq6$, every rational curve $B\subset X$ satisfies
$-K_X\cdot B\geq\dim X$.  The numerical characterization of
Dedieu--H\"oring \cite[Theorem~1.3]{DedieuHoring2017} gives
$X\simeq\mathbb P^6$ or $Q^6$.
Thus the only cases left are $d=4$ and $d=5$.
\end{proof}
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/02-minimal-curves.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/02-p1-cohomology.tex ====
\subsection{Cohomology of a rational-curve fibration}
\label{subsec:p1-cohomology}

The two remaining cases use integration along a smooth
$\mathbb P^1$-fibration. The following lemma expresses this operation
through an involution and a product rule, without requiring the
fibration to be the projectivization of a vector bundle.

\begin{lemma}\label{lem:p1-involution}
Let $p:B\to D$ be a smooth projective morphism of smooth connected
projective varieties whose fibres are isomorphic to $\mathbb P^1$.
Set $c=c_1(T_p)$ and $\delta=p^*p_*$.
Every class on $B$ is uniquely $a+cb$, with $a,b$ pulled back from $D$,
and $c^2$ is pulled back from $D$. Consequently
\[
 \overline{a+cb}=a-cb
\]
defines a ring involution. On even cohomology,
\begin{align}
 \delta(a+cb)&=2b,&
 \bar\alpha&=\alpha-c\delta\alpha,\label{eq:p1-involution}\\
 \delta(\alpha\gamma)
   &=(\delta\alpha)\gamma+\bar\alpha\,\delta\gamma.
   \label{eq:p1-product}
\end{align}
For a class $h$ of weight one with fibre degree one, both $2h-c$ and
$\alpha-h\delta\alpha$ are pulled back from $D$.
If $\alpha$ has top degree on $B$, then
$\int_B\bar\alpha=-\int_B\alpha$.
\end{lemma}

\begin{proof}
The restrictions of $1$ and $c/2$ form a basis of the cohomology of every
fibre. Leray--Hirsch therefore gives the asserted unique decomposition,
and fibre integration gives $p_*c=2$ and $p_*p^*a=0$.
The higher direct images of $\mathcal O_B$ are $p_*\mathcal O_B=\mathcal
O_D$ and $R^jp_*\mathcal O_B=0$ for $j>0$. Relative
Grothendieck--Riemann--Roch~\cite{BorelSerre1958} consequently gives
\[
 p_*\operatorname{td}(T_p)=1.
\]
Its component of weight one on $D$ is $p_*c^2/12=0$.
Writing $c^2=a+cb$ now gives $2b=0$, so $c^2$ is pulled back. This proves
that negating $c$ preserves products. The first two identities follow
from the decomposition, and multiplication of two such decompositions
proves~\eqref{eq:p1-product}.

Write $h=a_0+c/2$. Then $2h-c=2a_0$, while
$\alpha-h\delta\alpha=a-2a_0b$ for $\alpha=a+cb$.
Finally, in top degree the summand pulled back from $D$ vanishes by
dimension. The involution negates the remaining summand.
\end{proof}

In particular, if $\bar h=\ell-h$ and $\ell=2h-c$, repeated use of
\eqref{eq:p1-product} gives the polynomial identities
\begin{equation}\label{eq:p1-powers}
 \delta(h^a)=\sum_{k=0}^{a-1}h^k(\ell-h)^{a-1-k},\qquad
 \delta(c^a)=(1-(-1)^a)c^{a-1}\quad(a\ge1).
\end{equation}
These formulas are obtained by induction, so do not require division
by a cohomology class.
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/02-p1-cohomology.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/03-pseudoindex-four.tex ====
\section{Pseudoindex four}\label{sec:pseudoindex-four}

Watanabe treats this case in \cite[Theorem~2.4]{Watanabe2021}.
We give an intersection-theoretic argument using the factorization of the
universal family established by Kanemitsu. His theorem gives either
rational homogeneity or a factorization of the evaluation map; we exclude
the latter possibility. The relative-degree calculation below reproves
Kanemitsu's Lemma~4.3, and the exclusion of projective-plane fibres
adapts the Betti-number argument in his proof of Theorem~0.2
\cite{Kanemitsu2017}. In dimension six, the remaining quadric-bundle
case is excluded by the orthogonal-bundle calculation given here.

Let $X$ be a smooth connected complex projective Fano sixfold with nef
tangent bundle, Picard number one and pseudoindex four. Let
$\pi:U\to V$ and $e:U\to X$ be the universal family and evaluation map
of a minimal rational component. By Proposition~\ref{prop:minimal-family}, we have
\[
 T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus2}
 \qquad(C\text{ a }\pi\text{-fibre}).
\]
\begin{proposition}\label{prop:pseudoindex-four}
A smooth complex Fano sixfold with nef tangent bundle, Picard number one
and pseudoindex four is rational homogeneous.
\end{proposition}

\begin{proof}
Kanemitsu's Theorem~3.3 and Proposition~4.1
\cite{Kanemitsu2017} apply in arbitrary dimension; the latter point is
explicit in his Remark~4.2.
They show that either $X$ is rational homogeneous, or there is a diagram
\[
\begin{tikzcd}[column sep=large,row sep=large]
 U \arrow[r,"f"] \arrow[d,"\pi"'] &
 W \arrow[r,"g"] \arrow[d,"q"] & X\\
 V \arrow[r] & Y &
\end{tikzcd}
\]
with $e=g\circ f$, with $f,g,\pi$ smooth $\mathbb P^1$-fibrations,
and with $q$ a smooth elementary Mori contraction, that is, a Mori
contraction of relative Picard number one. We exclude the
second possibility. Here $\dim W=7$. Applied to $f$ and $\pi$,
\cite[Remark~2.3]{Kanemitsu2017} says that all fibres of $q$ are isomorphic to
one member of the list
\[
 \mathbb P^1,\quad\mathbb P^2,\quad\mathbb P^3,\quad
 Q^3,\quad Q^5,\quad K(G_2).
\]
Here $Q^r$ is a smooth quadric of dimension $r$, and $K(G_2)$ is the
five-dimensional rational homogeneous contact manifold of type $G_2$.
In particular, $\dim Y\ge2$.

\smallskip
\noindent\emph{The two relative degrees.}
Write
\[
 H=\frac{c_1(T_X)}4,\qquad h=g^*H,\qquad c=c_1(T_g).
\]
The classes $h,c$ form a basis of $H^2(W,\mathbb Q)$: a smooth
$\mathbb P^1$-fibration has the rational Leray--Hirsch decomposition,
and $H^2(X,\mathbb Q)=\mathbb QH$. These classes are algebraic, so
the same basis describes $N^1(W)_{\mathbb Q}$, the rational divisor
classes modulo numerical equivalence.
Fix a $\pi$-fibre $C$. The map $e|_C$ is birational to its image,
so $f|_C$ is also birational to its image and $h\cdot f(C)=1$.
Set
\[
 a=c_1(T_f)\cdot C,\qquad b=c\cdot f(C).
\]
The relative tangent sequence gives $a+b=-2$. Moreover,
$T_f|_C$ is a line subbundle of $\mathcal O(-1)^{\oplus2}$, whence
$a\le-1$. Both $a$ and $b$ are integers.

We show that $b<0$. The ray contracted by $q$ is generated by $f(C)$.
Since $\rho(W)=2$ and $q$ is elementary, $\rho(Y)=1$. Thus a
positive multiple of $c-bh$ is the pullback
of an ample rational class on $Y$. The sign is positive because
$c-bh$ has degree two on a $g$-fibre, whose numerical class is not
in the contracted ray. By Lemma~\ref{lem:p1-involution},
\[
 g_*c=2,\qquad g_*c^2=0,
\]
and consequently
\[
 g_*(c-bh)^2=-4bH.
\]
Intersecting with any complete-intersection curve in $X$ proves
$b\le0$.

If $b=0$, the square of $c$ has zero integral over $g^{-1}(D)$
for every smooth curve $D\subset X$. Each $g$-fibre has positive
degree against $q^*A$, for an ample class $A$ on $Y$, and hence maps
onto a curve under $q$. Since $q$ is surjective and $\dim Y\ge2$,
two such image curves are distinct. Choose a smooth irreducible
complete-intersection curve $D\subset X$ through the corresponding
two points of $X$. The inverse image $g^{-1}(D)$ is a smooth
irreducible surface, and its image under $q$ contains both of those
distinct curves. Thus this image has dimension two. It follows that
$(q^*A)^2\cdot[g^{-1}(D)]>0$, a contradiction. We have proved
\[
 a=b=-1.
\]
Since $-K_f\cdot C=a=-1$, Kanemitsu's Remark~2.3 now reduces the
possible fibres of $q$ to
$\mathbb P^2$, $Q^3$, and $K(G_2)$.

\smallskip
\noindent\emph{The projective-plane fibres.}
Suppose that $q$ has fibres $\mathbb P^2$, so that $\dim Y=5$.
Relative ample classes and their powers give Leray--Hirsch bases for
both $g$ and $q$. Writing $b_i(Z)=\dim H^i(Z,\mathbb Q)$, we obtain
$b_2(Y)=1$, and Poincar\'e duality
on $Y$ gives
\[
\begin{split}
 b_4(X)+1&=b_4(W)=b_4(Y)+2,\\
 b_6(X)+b_4(X)&=b_6(W)=2b_4(Y)+1.
\end{split}
\]
Hence $b_6(X)=b_4(X)-1$. This contradicts the injectivity of
cup-product by an ample class from $H^4(X,\mathbb Q)$ to
$H^6(X,\mathbb Q)$, which follows from Hard Lefschetz.

\smallskip
\noindent\emph{The remaining fibres.}
We may therefore assume that $\dim Y\le4$. Put
\[
 z=\frac{h+c}{2},\qquad m_i=\int_W h^{7-i}z^i\quad(0\le i\le7).
\]
Since $z\cdot f(C)=0$, the class $z$ is pulled back from
$H^2(Y,\mathbb Q)$. Thus $z^5=0$, and
\[
 m_0=m_5=m_6=m_7=0,\qquad m_1=\int_X H^6>0.
\]
The rational-cohomology involution of $g$ fixes classes pulled back
from $X$ and sends $c$ to $-c$. It sends $z$ to $h-z$ and acts by
$-1$ on top cohomology. For $i=2,4,6$, therefore,
\[
 \sum_{j=0}^i(-1)^j\binom ij m_j=-m_i.
\]
In order, these identities give
\[
 m_2=m_1,\qquad m_1-2m_3+m_4=0,
 \qquad9m_1-20m_3+15m_4=0.
\]
Consequently
\[
 m_1=m_2=5m_4,\qquad m_3=3m_4.
\]
In particular $m_4>0$, so $\dim Y\ge4$. Thus $\dim Y=4$ and
every fibre of $q$ is $Q^3$.

It remains to show that this quadric fibration is incompatible with
$g$. Since $-K_W=4h+c$ has degree three on $f(C)$, write
\[
 -K_q=3h-\lambda z.
\]
The coefficient $\lambda$ is integral: a $g$-fibre has respective
degrees zero and one against $h$ and $z$, and $-K_q$ is an integral
divisor class. We will determine $\lambda$ from the moments above
using the quadric-bundle identity
\[
 q_*(-K_q)^4=0.
\]
To prove this identity, first apply Fischer--Grauert
\cite{FischerGrauert1965}: the smooth projective morphism $q$, all of
whose fibres are $Q^3$, is analytically locally trivial. Its structure
group is $\operatorname{Aut}(Q^3)=\operatorname{PGO}_5(\mathbb C)$.
In odd rank this group is isomorphic to $\operatorname{SO}_5$.
Explicitly, fix a nondegenerate symmetric $5\times5$ matrix $J$ defining
the quadric. If $A$ represents a projective automorphism and
$A^{\mathsf T}JA=\mu J$, then
\[
 \widetilde A=\frac{\mu^2}{\det A}A
\]
has determinant one and preserves $J$. This expression is unchanged
when $A$ is multiplied by a scalar; it is the unique special
orthogonal lift and respects products. The transition functions thus
define a rank-five holomorphic vector bundle $E$ equipped with a
nondegenerate symmetric form with values in $\mathcal O_Y$ and
trivial determinant. GAGA algebraizes this bundle and its form
\cite[Theorems~2--3 and Propositions~15, 18]{Serre1956}.
The resulting analytic identification of $W$ with the relative quadric
is algebraic as well.
Accordingly, in the quotient convention for projectivization,
$W\subset\mathbb P_Y(E)$ is the quadric of class $2\zeta$, where
$\zeta=c_1(\mathcal O_{\mathbb P(E)}(1))$. Self-duality identifies
the line and quotient descriptions of this quadric. If
$p:\mathbb P_Y(E)\to Y$ is the projection, adjunction gives
\[
 -K_q=3\zeta|_W,
 \qquad q_*(-K_q)^4=2\cdot3^4p_*\zeta^5
 =2\cdot3^4c_1(E)=0.
\]

Now the projection formula and $z^5=0$ give
\[
 0=\int_W(-K_q)^4z^3=3^4m_3-4\cdot3^3\lambda m_4.
\]
Since $m_3=3m_4$ and $m_4>0$, this forces $\lambda=9/4$,
contrary to integrality. This excludes the factorization case and
proves the proposition.
\end{proof}
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/03-pseudoindex-four.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/04-characteristic-classes.tex ====
\section{Characteristic-class constraints in pseudoindex five}
\label{sec:characteristic-classes}

We now assume that $X$ is a Fano sixfold with nef tangent bundle,
$\rho(X)=1$, and pseudoindex five. Let
\[
 \pi:U\longrightarrow V,\qquad e:U\longrightarrow X
\]
be its smooth universal family. Thus $\dim U=9$, $\dim V=8$,
the fibres of $\pi$ are projective lines, and $e$ has relative dimension
three. The preceding geometric argument gives
$T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus3}$ for every
$\pi$-fibre $C$. Our goal is to constrain the polarization of an
evaluation fibre by combining the two fibrations.

Put
\begin{equation}\label{eq:d5-generators}
 h_X=\frac{c_1(T_X)}5,\qquad h=e^*h_X,\qquad
 c=c_1(T_\pi),\qquad \ell=2h-c,\qquad E=T_e.
\end{equation}
The class $h$ has degree one on a $\pi$-fibre. Lemma~\ref{lem:p1-involution}
therefore shows that $\ell$ comes from $V$ and that $\bar h=\ell-h$.

\subsection{The polarized evaluation fibre}

\begin{lemma}\label{lem:marked-tangent}
For a general point $x\in X$, the fibre $F=e^{-1}(x)$ is a smooth
irreducible projective threefold, and the tangent direction at the marking
defines a finite morphism
\[
 \tau_x\colon F\longrightarrow\mathbb P(T_x^*X).
\]
Its image $\mathcal C_x$ is the variety of minimal rational tangents of
$V$ at $x$.  The line bundle $L=\tau_x^*\mathcal O(1)$ is ample and
generated by global sections, with $c_1(L)=\ell|_F$.
\end{lemma}

\begin{proof}
Smoothness, projectivity and connectedness follow from the established
properties of $e$.  A smooth connected variety is irreducible.
Let $V_x\subset V$ be the reduced locus of curves passing through $x$.
The restriction $\pi|_F\colon F\to V_x$ is proper and has finite fibres:
each curve has only finitely many points on its normalization mapping to
$x$.  It is therefore finite.  Kebekus's Theorem~3.3(1)
\cite{Kebekus2002}, applied with the ample line bundle
$\omega_X^{-1}$, says that only finitely many members of $V_x$ can be
singular at a general $x$, and that even these curves are immersed at
their markings.  Since $\dim V_x=3$, a general member is smooth at $x$,
so $\pi|_F$ is birational.  As $F$ is normal, it is the normalization of
$V_x$.  The hypotheses of Kebekus's Theorem~3.4 are satisfied: $V$ is an
irreducible covering family and $V_x$ is proper.  That theorem gives the
finite tangent morphism on $F$.

With our quotient convention for projectivization,
$\mathbb P(T_x^*X)$ parametrizes lines in $T_xX$.  The differential of
$e$ is nonzero on $T_\pi$ at every marking over $x$, and identifies
$T_\pi|_F$ with the pullback of the tautological subline
$\mathcal O(-1)\subset T_xX\otimes\mathcal O$.  Consequently
$c_1(L)=-c|_F=\ell|_F$.  Pullback by a finite morphism preserves
ampleness, and the pullbacks of hyperplane sections generate $L$.
\end{proof}

\subsection{Classes descending to the parameter space}

Define $r,s,t$ to be the elementary symmetric polynomials in the three
formal roots obtained by adding $h$ to the Chern roots of $E$.
Equivalently,
\begin{align*}
 r&=c_1(E)+3h,\\
 s&=c_2(E)+2hc_1(E)+3h^2,\\
 t&=c_3(E)+hc_2(E)+h^2c_1(E)+h^3.
\end{align*}
This definition uses only rational cohomology; it does not assert that
$h$ is the first Chern class of a line bundle.

\begin{lemma}\label{lem:shifted-descent}
The classes $r,s,t$ are pulled back from $V$.
\end{lemma}
\begin{proof}
There is an actual line bundle
\[
 \mathcal A=e^*\omega_X^{-1}\otimes(T_\pi^\vee)^{\otimes2},
 \qquad c_1(\mathcal A)=5h-2c=h+2\ell.
\]
Its degree on every $\pi$-fibre is $5-4=1$.
Thus $E\otimes\mathcal A$ restricts to the trivial rank-three bundle
on each fibre. Cohomology and base change show that
$\mathcal B=\pi_*(E\otimes\mathcal A)$ is locally free of rank three.
The evaluation map $\pi^*\mathcal B\to E\otimes\mathcal A$ is an
isomorphism on every fibre, and hence is an isomorphism.
The formal roots defining $r,s,t$ are consequently the roots of
$\pi^*\mathcal B$ minus $2\ell$. Their elementary symmetric
polynomials are pulled back from $V$.
\end{proof}

For the fibre $F$ and polarization $L$ of
Lemma~\ref{lem:marked-tangent}, the definitions and $h|_F=0$ give
\begin{equation}\label{eq:fibre-polarization}
 c_1(L)=\ell|_F,
 \qquad r|_F=c_1(T_F).
\end{equation}
We shall constrain the ratio
\begin{equation}\label{eq:ratio-u}
 u=\frac{\int_F c_1(T_F)c_1(L)^2}{\int_F c_1(L)^3}.
\end{equation}
Its denominator is positive.
The intersection identities below will force a quadratic equation for
$u$. In Section~\ref{sec:grassmannian}, this equation and the
sectional-genus formula will determine the degree of $L$.

\subsection{Two direct-image identities}

We first record why the Hodge-theoretic direct images appearing in
Riemann--Roch contribute no positive-degree Chern character.

\begin{lemma}\label{lem:constant-hodge}
Let $f:Z\to B$ be a smooth projective morphism between smooth connected
complex projective varieties, and suppose that $B$ is simply connected.
For all $p,j\ge0$, the vector bundle $R^jf_*\Omega^p_{Z/B}$ is
holomorphically trivial. In particular its positive-degree rational
Chern character vanishes.
\end{lemma}
\begin{proof}
The local systems $R^kf_*\mathbb Q$ are constant. The global invariant
cycle theorem gives a surjection
\[
 H^k(Z,\mathbb Q)\longrightarrow H^0(B,R^kf_*\mathbb Q)
   \simeq H^k(Z_b,\mathbb Q)
\]
for every $b\in B$; see Deligne~\cite[Theorem~4.1.1(ii)]{Deligne1971}.
Restriction is a morphism of pure Hodge structures. Therefore each
Hodge component of $H^k(Z_b,\mathbb C)$ is spanned by restrictions of
classes of the same type on $Z$. A class on $Z$ restricts to a flat
section of the local system, and a class of fixed Hodge type keeps that
type on every fibre. Thus the Hodge decomposition, and hence the Hodge
filtration, is constant in the flat trivialization. This is also the
fixed-part statement of~\cite[Corollary~4.1.2]{Deligne1971}.
The relative Hodge--de Rham degeneration identifies
$R^jf_*\Omega^p_{Z/B}$ with the degree-$p$ graded bundle of the Hodge
filtration on $R^{p+j}f_*\mathbb C\otimes\mathcal O_B$; see
\cite[Section~2(a)--(c)]{Griffiths1970}. Since that
filtration is constant, each graded bundle is trivial.
\end{proof}

The base $X$ is simply connected, so the lemma applies to $e$.
Relative Grothendieck--Riemann--Roch
\cite[\S7, p.~113]{BorelSerre1958} for $\mathcal O_U$ and
$\Omega^1_{U/X}=E^\vee$ gives
\begin{equation}\label{eq:grr-vanishings}
 e_*\operatorname{td}_i(E)=0,
 \qquad
 e_*[\operatorname{td}(E)\operatorname{ch}(E^\vee)]_i=0
 \quad(i>3).
\end{equation}
Indeed these are the components of weight $i-3>0$ of
\[
 \operatorname{ch}\!\left(\sum_j(-1)^jR^je_*\mathcal O_U\right)
 \quad\text{and}\quad
 \operatorname{ch}\!\left(\sum_j(-1)^jR^je_*\Omega^1_{U/X}\right),
\]
respectively.

These two choices capture the equations from all relative holomorphic
forms. Indeed, for a rank-three bundle and $p=0,1$,
$\bigwedge^{3-p}E^\vee=\det(E^\vee)\otimes\bigwedge^pE$ and
$\operatorname{td}(E^\vee)=e^{-c_1(E)}\operatorname{td}(E)$.
Replacing the Chern roots by their negatives therefore gives
\[
 [\operatorname{td}(E)\operatorname{ch}(\textstyle\bigwedge^{3-p}E^\vee)]_i
 =(-1)^i
 [\operatorname{td}(E)\operatorname{ch}(\textstyle\bigwedge^pE^\vee)]_i.
\]
Thus $\Omega^2_{U/X}$ and $\Omega^3_{U/X}$ repeat, up to sign in
each weight, the equations from $\Omega^1_{U/X}$ and $\mathcal O_U$.

\subsection{Explicit polynomial representatives}

We now turn the tangent-sequence and Riemann--Roch identities into
linear relations among weight-nine intersection numbers on $U$. The
projection formula will then identify selected normalized numbers as
products involving $u$.
Set
\[
 x_i=e^*(i!\operatorname{ch}_i(T_X)),\qquad i\ge1.
\]
Then $x_1=5h$ and $x_i=0$ for $i\ge7$, by dimension. Work in the
polynomial ring
\[
 R=\mathbb Q[h,x_2,x_3,x_4,x_5,x_6,\ell,r,s,t],
\]
with respective weights $(1,2,3,4,5,6,1,1,2,3)$.
Evaluation of these generators as cohomology classes defines a graded
homomorphism $R\to H^{2*}(U,\mathbb Q)$.

Our unknowns are the integrals of weight-nine monomials in $R$.
We obtain linear relations among them from three sources: classes
descending to the eight-dimensional space $V$, classes pulled back
from the six-dimensional space $X$ and their $\pi$-integrals, and the
two Riemann--Roch vanishings. The tangent sequences make the
$\pi$-integrals explicit. Recall that
$\delta=\pi^*\pi_*$ lowers weight by one. The recurrences below
express these operations as polynomials, before we collect the
resulting relations into a matrix.

Let $p_j$ be the Newton power sum of the three formal roots with
elementary symmetric functions $r,s,t$, and put $p_0=3$.
For complete specificity, if $s_1=r,s_2=s,s_3=t$, Newton's identities
read
\begin{equation}\label{eq:newton-recursion}
 p_i=\sum_{j=1}^{\min(3,i)}(-1)^{j+1}s_j
 \begin{cases}
 i,&j=i,\\
 p_{i-j},&j<i.
 \end{cases}
\end{equation}
Define
\begin{equation}\label{eq:unshifted-powers}
 q_i=\sum_{j=0}^i\binom ijp_j(-h)^{i-j}.
\end{equation}
These represent $i!\operatorname{ch}_i(E)$, with $q_0=3$.
The two tangent exact sequences on $U$ give, for every $i\ge1$,
\[
 q_i+x_i=c^i+\pi^*(i!\operatorname{ch}_i(T_V)).
\]
Applying $\delta=\pi^*\pi_*$ and using~\eqref{eq:p1-powers} gives
$\delta x_i=D_i$, where
\begin{equation}\label{eq:D-polynomials}
\begin{split}
 D_i={}&(1-(-1)^i)(2h-\ell)^{i-1}\\
 &-\sum_{j=0}^{i-1}\binom ijp_j(-1)^{i-j}
      \sum_{k=0}^{i-j-1}h^k(\ell-h)^{i-j-1-k}.
\end{split}
\end{equation}
In particular $D_1=5$, consistently with $x_1=5h$, and $D_i$ evaluates
to zero for $i\ge7$. This calculation uses the polynomial product
rule for $\delta$, so remains valid even when the evaluated classes
are zero divisors.

For a monomial $m$ in the six base generators
$(h,x_2,x_3,x_4,x_5,x_6)$ define $\Delta m$ recursively by
\begin{equation}\label{eq:Delta-recursion}
 \Delta1=0,\qquad\Delta h=1,\qquad\Delta x_i=D_i\ (2\le i\le6),
 \qquad
 \Delta(gn)=(\Delta g)n+(g-c\Delta g)\Delta n.
\end{equation}
In the last formula, remove the first generator $g$ dividing $m=gn$
in the displayed order. By~\eqref{eq:p1-product}, this recursively
specified polynomial evaluates to $\delta m$. No independence of
different formal recursion orders is needed.

Finally, write $\theta_i$ for a polynomial representing
$\operatorname{td}_i(E)$. Through weight nine, put
\begin{equation}\label{eq:todd-recursion}
 \theta_0=1,\qquad
 \theta_i=\frac1i\sum_{j=1}^i\beta_jq_j\theta_{i-j},\qquad
 (\beta_1,\beta_2,\beta_4,\beta_6,\beta_8)
 =\left(\frac12,-\frac1{12},\frac1{720},-\frac1{30240},
         \frac1{1209600}\right),
\end{equation}
with the remaining $\beta_j$ zero. To obtain this formula, apply the
logarithmic derivative to the product of the Todd factors:
\[
 z\frac{d}{dz}\log\frac{z}{1-e^{-z}}
  =1-\frac{z}{e^z-1}
  =\frac z2-\frac{z^2}{12}+\frac{z^4}{720}
       -\frac{z^6}{30240}+\frac{z^8}{1209600}+O(z^{10}).
\]
Since $\operatorname{ch}(E^\vee)=\sum_{j\ge0}(-1)^jq_j/j!$, the second
polynomial in~\eqref{eq:grr-vanishings} is represented by
\begin{equation}\label{eq:todd-dual}
 \psi_i=\sum_{j=0}^i\frac{(-1)^j}{j!}q_j\theta_{i-j}.
\end{equation}

\subsection{The linear constraints and their geometric meaning}

Let $\mathcal B_k$ be the weight-$k$ monomials in
$(h,x_2,x_3,x_4,x_5,x_6)$, let $\mathcal I_k$ be those in
$(\ell,r,s,t)$, and let $\mathcal T_k$ be all weight-$k$ monomials in
$R$. A negative subscript means the empty list. Monomials are ordered
recursively: for an ordered generator list ending in a variable $g$ of
weight $a$, list $g^j$ times the monomials of weight $k-aj$ in the
preceding variables, in increasing order of $j$.

Form a rational matrix $A$ whose columns are indexed by $\mathcal T_9$.
Its rows are coefficient vectors of the following polynomials. We
group them according to their geometric source.

First, for a base monomial $m$, both $m-h\delta m$ and $\delta m$
descend to $V$ by Lemma~\ref{lem:p1-involution}, as does $\ell$.
The classes $r,s,t$ descend by Lemma~\ref{lem:shifted-descent}.
Products of these classes of
weight nine vanish because $\dim V=8$. This gives the rows
\begin{alignat}{3}
 &w(m-h\Delta m),&&\quad m\in\mathcal B_i,\quad
 w\in\mathcal I_{9-i},&&\quad0\le i\le6;\label{eq:matrix-rows-a}\\
 &w\Delta m,&&\quad m\in\mathcal B_i,\quad
 w\in\mathcal I_{10-i},&&\quad0\le i\le6.\label{eq:matrix-rows-b}
\end{alignat}
Second, a base monomial $m$ of weight greater than six vanishes,
as does $\delta m$. Also $x_i=0$ for $i\ge7$, so
$D_i=\delta x_i$ evaluates to zero. Their multiples give
\begin{alignat}{3}
 &wm,&&\quad m\in\mathcal B_i,\quad
 w\in\mathcal T_{9-i},&&\quad7\le i\le10;\label{eq:matrix-rows-c}\\
 &w\Delta m,&&\quad m\in\mathcal B_i,\quad
 w\in\mathcal T_{10-i},&&\quad7\le i\le10;\label{eq:matrix-rows-d}\\
 &wD_i,&&\quad w\in\mathcal T_{10-i},&&\quad7\le i\le10.\label{eq:matrix-rows-e}
\end{alignat}
Finally, the Riemann--Roch identities~\eqref{eq:grr-vanishings} and
the projection formula give zero integrals for
\begin{alignat}{3}
 &w\theta_i,\quad w\psi_i,&&\quad w\in\mathcal B_{9-i},&&\quad4\le i\le9.
 \label{eq:matrix-rows-f}
\end{alignat}
Zero rows may be omitted. Let
\[
 v=\left(\int_U m\right)_{m\in\mathcal T_9}.
\]
The preceding vanishings prove $Av=0$. The first five row families
come from zero cohomology classes; the final family only requires
the stated integrals to vanish.

The constraints are linear in the intersection numbers on $U$. To
recover a useful condition on $u$, we next use the product structure
forced by integration along $e$.

We use invariant monomials of weights three and four because their
pushforwards along $e$ lie in $H^0(X,\mathbb Q)$ and
$H^2(X,\mathbb Q)=\mathbb Qh_X$, respectively. Each pushforward
is determined by one scalar. These two cohomology groups
therefore supply scalar product identities.
Write the ordered lists $M=\mathcal B_6$ and
$G=(\mathcal I_3,\mathcal I_4)$ explicitly as
\begin{align*}
 M={}&(h^6,h^4x_2,h^2x_2^2,x_2^3,h^3x_3,hx_2x_3,x_3^2,
                 h^2x_4,x_2x_4,hx_5,x_6),\\
 G={}&(\ell^3,\ell^2r,\ell r^2,r^3,\ell s,rs,t,\ell^4,
       \ell^3r,\ell^2r^2,\ell r^3,r^4,\ell^2s,\ell rs,r^2s,s^2,
       \ell t,rt).
\end{align*}
Indices begin at zero, and $w_j$ denotes the weight of $G_j$.
Put $D=\int_F c_1(L)^3>0$ and define
\begin{equation}\label{eq:normalized-products}
 \xi_i=\frac{\int_X M_i}{\int_Xh_X^6},\qquad
 e_*G_j=D\eta_jh_X^{w_j-3}.
\end{equation}
The first expression uses the corresponding base classes on $X$.
The second defines a scalar because $w_j-3$ is zero or one and
$H^2(X,\mathbb Q)=\mathbb Qh_X$. We have
$\xi_0=\eta_0=1$ and $\eta_1=u$.

Use the index set
\[
 J=\{0,1,2,4,5,6,7,8,9,10,12,13,15,16,17\}
\]
for the selected invariant monomials. This subcollection suffices for
the elimination in Section~\ref{sec:arithmetic} and reduces its size;
no vanishing is asserted for the omitted monomials $r^3$, $r^4$,
and $r^2s$. Let
\[
 C_1=\{(i,(j)):0\le i\le10,\ j\in J,
                  \ w_j=3\ \text{or}\ h\mid M_i\}.
\]
Order $C_1$ by increasing $i$ and then increasing $j$.
For each such pair the monomial $M_i/h^{w_j-3}$ is defined, and the
projection formula gives
\begin{equation}\label{eq:product-factorization}
 \int_U G_j\frac{M_i}{h^{w_j-3}}
   =D\int_Xh_X^6\,\xi_i\eta_j.
\end{equation}
The common factor on the right is $\int_U h^6\ell^3>0$.
We now retain all linear relations that $Av=0$ induces on these
selected coordinates. The following construction expresses them as
a matrix $N$.
If $K$ is any matrix whose columns form a rational basis of $\ker A$,
let $P$ consist of its rows indexed, with repetitions if necessary, by
the monomials on the left of~\eqref{eq:product-factorization}.
Let $N$ be the nonzero rows in the reduced row echelon form of a matrix
whose rows form a basis of $\ker(P^{\mathsf T})$. Thus $N$ is determined
by the ordered coordinates, independently of the chosen basis $K$.
Then
\begin{equation}\label{eq:N-products}
 N(\xi_i\eta_j)_{(i,(j))\in C_1}=0.
\end{equation}
Indeed $v\in\ker A$, so its selected coordinates lie in the column
space of $P$, and division by the single positive scalar in
\eqref{eq:product-factorization} gives the displayed equation.
This completes the geometric derivation of the finite constraints.
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/04-characteristic-classes.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/05-arithmetic.tex ====
\section{The numerical constraint}
\label{sec:arithmetic}

The geometric argument has produced the bilinear equations
\eqref{eq:N-products}. We now prove that they allow only two values of
$u$. This section is a finite calculation over $\mathbb Q$: its input
is the explicitly defined matrix $A$ and its selected coordinates
$C_1$, and its output is a quadratic equation for $\eta_1$.

\begin{lemma}\label{lem:finite-ratio}
Let $A$, $C_1$ and $N$ be the rational matrices and index set defined
in Section~\ref{sec:characteristic-classes}. Suppose that
$\xi\in\mathbb Q^{11}$ and $\eta\in\mathbb Q^{18}$ satisfy
\[
 \xi_0=\eta_0=1,
 \qquad N(\xi_i\eta_j)_{(i,(j))\in C_1}=0.
\]
Then
\begin{equation}\label{eq:ratio-quadratic}
 32-20\eta_1+3\eta_1^2=0.
\end{equation}
\end{lemma}

We give the finite certificate, including the distinction between
the modular rank bound and the rational identities needed to finish
the proof. The exact recurrences and elimination procedure are
implemented in Appendix~\ref{app:certificate}.

\subsection{Multiplying the constraints}

Write $y_{i,j}=\xi_i\eta_j$ for the selected bilinear coordinates.
Multiplying each equation of $Ny=0$ by $\eta_a\eta_b$, with
$a,b\in J$, gives linear equations in the cubic products
\[
 v_{i,\operatorname{sort}(j,a,b)}=\xi_i\eta_j\eta_a\eta_b,
 \qquad(i,(j))\in C_1.
\]
Here $\operatorname{sort}$ arranges the three indices in nondecreasing
order, identifying equal products.
We will impose those linear equations without requiring an arbitrary
solution $v$ to have this product form. In the resulting larger
solution space, we prove that
\begin{equation}\label{eq:target-functional}
 32v_{0,(0,0,0)}-20v_{0,(0,0,1)}+3v_{0,(0,1,1)}=0.
\end{equation}
For the product vector, $\xi_0=\eta_0=1$ makes the left side
$32-20\eta_1+3\eta_1^2$. Thus a linear statement on cubic
coordinates will prove the desired quadratic relation. The following
spaces make this relaxation precise; the intermediate quadratic
space also makes its computation smaller.

For $n=2,3$, form $C_n$ from $C_{n-1}$ by appending an index $j\in J$
to its tuple and sorting. Repeated coordinates are identified.
Equivalently, $(i,z)\in C_n$ means that $z$ is a size-$n$ multiset
on $J$, and either $h\mid M_i$ or $z$ contains an index of weight
three. Let $\mathcal K_1=\ker N$. Recursively define
$\mathcal K_n\subseteq\mathbb Q^{C_n}$ by requiring, for every
$j\in J$, that the slice
\[
 (v_{i,\operatorname{sort}(z,j)})_{(i,z)\in C_{n-1}}
 \quad\text{belong to }\mathcal K_{n-1}.
\]
Thus $\mathcal K_3$ is the kernel of a single rational matrix:
for each unordered pair $a,b\in J$, allowing $a=b$, impose $N$ on the slice
\[
 \bigl(v_{i,\operatorname{sort}(j,a,b)}\bigr)_{(i,(j))\in C_1}.
\]
There are $\binom{16}{2}=120$ such pairs. In particular,
\begin{equation}\label{eq:geometric-cubic-column}
 v_{i,z}=\xi_i\prod_{j\in z}\eta_j
\end{equation}
belongs to $\mathcal K_3$: every double slice is a scalar multiple
of the given vector in $\ker N$.

Seven monomials $M_i$ contain $h$, and six of the fifteen indices
in $J$ have weight three. Therefore
\[
 |C_n|=7\binom{14+n}{n}
       +4\left(\binom{14+n}{n}-\binom{8+n}{n}\right),
\]
which gives $|C_1|=129$, $|C_2|=1140$ and $|C_3|=6820$.

\subsection{An upper bound from finite-field elimination}

The coefficient recurrences in Section~\ref{sec:characteristic-classes}
give $1785$ nonzero rows and $581$ columns for $A$. Exact elimination
gives a matrix $K$ of size $581\times65$ with $AK=0$.
The selected-row matrix $P$ has size $129\times65$. Choose $N$ in
reduced row echelon form. The following checks give, in particular,
complete bases rather than merely lists of relations:
\begin{center}
\begin{tabular}{c|c|c}
matrix & size & rank modulo $10007$\\\hline
$A$ & $1785\times581$ & $516$\\
$K$ & $581\times65$ & $65$\\
$P$ & $129\times65$ & $60$\\
$N$ & $69\times129$ & $69$
\end{tabular}
\end{center}
The identities $AK=0$ and $NP=0$ hold over $\mathbb Q$.
Every denominator in these matrices is prime to $10007$.
Consequently the first two ranks prove that $K$ is a basis of
$\ker A$, and the last two prove that $N$ is the full left
annihilator of $P$.
Write $\mathcal K_n(\mathbb F_{10007})$ for the spaces defined
by the same slice equations using the reduction of $N$ modulo
$10007$.

Here is the elimination procedure used for the larger spaces.
Suppose that the columns of $Z$ are a basis for the solutions on a union
$S$ of slices, and the columns of $B$ are a basis for the solutions on the
next slice $T$. On the overlap, the two parameter vectors $a,b$
must satisfy
\[
 Z_{S\cap T}a-B_{S\cap T}b=0.
\]
If the columns of $\binom{V_1}{V_2}$ form the kernel of this matrix,
then the columns of
\[
 \begin{pmatrix} ZV_1\\ B_{T\setminus S}V_2\end{pmatrix}
\]
parametrize all solutions on $S\cup T$. This parametrization is
injective: zero on the union is zero on each piece, and the old
parametrizations were injective. Appending a fixed index to a
multiset is itself injective, so there are no hidden identifications
within an individual slice. Ordinary Gaussian elimination and
successive application of this construction give
\begin{equation}\label{eq:finite-field-dimensions}
 \dim\mathcal K_1(\mathbb F_{10007})=60,
 \qquad\dim\mathcal K_2(\mathbb F_{10007})=163,
 \qquad\dim\mathcal K_3(\mathbb F_{10007})=6.
\end{equation}

For the rational conclusion, use the single double-slice matrix
defining $\mathcal K_3$. It has $69\cdot120=8280$ rows and $6820$
columns, and its denominators are units modulo $10007$. Its modular
rank is $6814$, so its rational rank is at least $6814$. Thus
\begin{equation}\label{eq:rational-dimension-upper}
 \dim_{\mathbb Q}\mathcal K_3\le6.
\end{equation}
This argument does not assume that intermediate rational kernel
bases specialize to bases.

\subsection{Six rational columns}

It remains to exhibit six independent rational solutions and show that
the functional giving the desired quadratic vanishes on their span.
These columns are witnesses for the linear system: no assertion that
they arise from varieties is needed. Three have product form, and
three are specified by their small supports. The relaxed system
allows both kinds without imposing the original normalization or
requiring a product representation.
Define the following rational vectors:
\begin{align*}
 B(v)={}&\bigl(1,7-2v/5,45-4v,275-30v,11,65-2v,\\
             &\hspace{29mm}85,17,95-2v,25,35\bigr),\\
 B'={}&(1,11/5,5,11,-1,-1,10,-3,-1,0,14),\\
 H^a={}&\bigl(1,4,16,64,6,24,4,3,12,\\
       &\hspace{29mm}48,192,768,17,68,272,96,9,36\bigr),\\
 H^b={}&\tfrac13\bigl(3,8,21,54,9,24,6,14,37,\\
       &\hspace{29mm}97,252,648,37,97,252,96,16,42\bigr).
\end{align*}
For $(B,H)=(B(0),H^a),(B(1),H^a),(B',H^b)$, let
\[
 v(B,H)_{i,z}=B_i\prod_{j\in z}H_j.
\]
Direct rational substitution gives
$N(B_iH_j)_{C_1}=0$, so these three columns lie in $\mathcal K_3$.
Three further columns $s_1,s_2,s_3$ have the following entries and
are zero elsewhere:
\begin{center}
\begin{tabular}{c|c|rrr}
$i$ & $z$ & $s_1$ & $s_2$ & $s_3$\\\hline
$5$ & $(15,15,15)$ & $3$ & $0$ & $0$\\
$7$ & $(15,15,15)$ & $1$ & $0$ & $0$\\
$9$ & $(15,15,15)$ & $-5$ & $0$ & $0$\\
$7$ & $(10,10,15)$ & $0$ & $2$ & $0$\\
$7$ & $(10,10,10)$ & $0$ & $4$ & $1$\\
$9$ & $(10,10,10)$ & $0$ & $25$ & $0$
\end{tabular}
\end{center}
Applying $N$ to each of their $120$ double slices gives zero over
$\mathbb Q$. These are exact rational equalities, separate from
the finite-field calculation.

The six columns are independent. Indeed, restrict them to the rows
\begin{gather*}
 (0,(0,0,0)),\quad (0,(0,0,1)),\quad (1,(0,0,0)),\\
 (5,(15,15,15)),\quad (7,(10,10,15)),\quad (7,(10,10,10)).
\end{gather*}
The upper-right $3\times3$ block is zero. The upper-left block has
determinant $-8/15$, and the lower-right block is
$\left(\begin{smallmatrix}3&0&0\\0&2&0\\0&4&1\end{smallmatrix}\right)$,
with determinant $6$. The full minor is $-16/5$. Together with
\eqref{eq:rational-dimension-upper}, this proves that the six
displayed columns form a rational basis of $\mathcal K_3$.

\begin{proof}[Proof of Lemma~\ref{lem:finite-ratio}]
The functional in~\eqref{eq:target-functional} vanishes on the six
basis columns. On a displayed product column its value
is $32-20H_1+3H_1^2$, with $H_1=4$ or $8/3$; the sparse columns
are zero at all three coordinates. Apply it to
\eqref{eq:geometric-cubic-column} and use $\xi_0=\eta_0=1$.
\end{proof}

Returning to the evaluation fibre, \eqref{eq:N-products} verifies
the hypothesis of the lemma and $\eta_1=u$. Hence
\[
 (u-4)(3u-8)=0.
\]
It remains to identify which value is compatible with the tangent
map and to recognize the resulting variety of minimal rational
tangents.
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/05-arithmetic.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/06-grassmannian.tex ====
\section{Recognizing the Grassmannian}\label{sec:grassmannian}

We finish the pseudoindex-five case by interpreting the numerical constraint
on the fibres of the evaluation map.  Throughout this section, $X$ is a
smooth complex Fano sixfold with nef tangent bundle, Picard number one and
pseudoindex five.  We use its minimal rational component $V$, universal
family $\pi\colon U\to V$, and evaluation $e\colon U\to X$.
Thus $e$ is smooth with connected three-dimensional fibres. For a general
point $x\in X$, let $F=e^{-1}(x)$ and let $L$ be the tangent-map
polarization of Lemma~\ref{lem:marked-tangent}.
Our goal is to recover $X$ from the projective geometry of the tangent
directions to its minimal rational curves.
For this polarization, Lemma~\ref{lem:finite-ratio} gives
\begin{equation}\label{eq:vmrt-ratio-input}
 32-20u+3u^2=0,
 \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}.
\end{equation}
We first show that the tangent image spans $\mathbb P^5$. Its degree
bound and the sectional-genus formula will then determine $L^3$ and
the degree of the tangent map. Finally, normality of the image will
let us use the smoothness of $F$ to select the Segre threefold among
the varieties of minimal degree.

\begin{lemma}\label{lem:vmrt-nondegenerate}
For a general point $x\in X$, the threefold
$\mathcal C_x\subset\mathbb P(T_x^*X)\simeq\mathbb P^5$ is linearly
nondegenerate.
\end{lemma}

\begin{proof}
Write $\widehat{\mathcal C}_x\subset T_xX$ for its affine cone and
$W_x=\operatorname{Span}(\widehat{\mathcal C}_x)$.  Suppose that $W_x$
is proper.  Since $\mathcal C_x$ has dimension three,
$4\leq\dim W_x\leq5$.  On a dense open subset of $X$, these spans
define the distribution $W\subset T_X$.

We use two results of Hwang and Mok
\cite[Propositions~1.2.1 and~1.2.2]{HwangMok1999}.  For an irreducible
variety of minimal rational tangents on a uniruled projective manifold
with $b_2=1$, a proper spanned distribution cannot be integrable.  On
the other hand, it is integrable if the bivectors
\begin{equation}\label{eq:tangent-bivectors}
 \alpha\wedge v,\qquad
 \alpha\in\widehat{\mathcal C}_x\setminus\{0\}
 \text{ a smooth point},\quad
 v\in T_\alpha\widehat{\mathcal C}_x,
\end{equation}
span $\bigwedge^2W_x$ at general $x$.  These results apply here:
$b_2(X)=1$ was established earlier, and $\mathcal C_x$ is irreducible
as the image of $F$.

We verify the spanning condition directly, without assuming that
$\mathcal C_x$ is smooth.  Otherwise a nonzero alternating form
$\omega\in\bigwedge^2W_x^*$ annihilates all bivectors
\eqref{eq:tangent-bivectors}.  The one-form $\omega(\alpha,d\alpha)$
vanishes on the smooth cone locus.  Its exterior derivative is the
restriction of $2\omega$, so every four-dimensional tangent space
$T_\alpha\widehat{\mathcal C}_x$ is isotropic.  An alternating form
of rank $2r$ on an $m$-dimensional space has isotropic subspaces of
dimension at most $m-r$.  Since $m\leq5$ and $r\geq1$, we must have
\[
 4\leq m-r\leq4,
\]
and hence $m=5$ and $r=1$.

Projection to the two-dimensional symplectic quotient
$W_x/\ker\omega$ has differential of rank at most one on the smooth
cone locus.  In characteristic zero, the Zariski closure of the image therefore
has dimension at most one.  It is an irreducible cone, hence is either
the origin or a line.  But $\widehat{\mathcal C}_x$ spans $W_x$, so
its projection spans the two-dimensional quotient, a contradiction.
Thus the bivectors do span $\bigwedge^2W_x$, making $W$ integrable
and contradicting the other Hwang--Mok result.
\end{proof}

\begin{proposition}\label{prop:pseudoindex-five-recognition}
Every smooth complex Fano sixfold with nef tangent bundle, Picard number
one and pseudoindex five is isomorphic to $\operatorname{Gr}(2,5)$.
\end{proposition}

\begin{proof}
The geometric constraints~\eqref{eq:N-products} and
Lemma~\ref{lem:finite-ratio} give $u=4$ or $u=8/3$. We first determine
the degree of $L$.  By Bertini, two general members of $|L|$ meet in a smooth
curve $B$.  Both successive divisors are connected:
for a smooth projective variety $Z$ of dimension at least two and an
ample divisor $D$, Kodaira vanishing and Serre duality give
$H^1(Z,\mathcal O_Z(-D))=0$, and the divisor exact sequence then gives
$H^0(D,\mathcal O_D)=\mathbb C$.  Applying this first to $F$ and then
to its smooth ample surface section proves that $B$ is connected.
Adjunction gives
\begin{equation}\label{eq:vmrt-sectional-genus}
 g(B)=1+\frac{(K_F+2L)\cdot L^2}{2}
     =1+\left(1-\frac u2\right)L^3.
\end{equation}
Here and below $L^3$ denotes its top intersection number.

By Lemma~\ref{lem:vmrt-nondegenerate}, $\mathcal C_x$ is a
nondegenerate threefold in $\mathbb P^5$, so its degree is at least
three.  The degree formula for the finite morphism $\tau_x$ yields
\[
 L^3=\deg(\tau_x)\deg(\mathcal C_x)\geq3.
\]
If $u=4$, Equation~\eqref{eq:vmrt-sectional-genus} gives
$g(B)=1-L^3<0$.  Thus $u=8/3$, and the same equation gives
$g(B)=1-L^3/3\geq0$.  It follows that
\[
 L^3=3,\qquad\deg(\tau_x)=1,\qquad\deg(\mathcal C_x)=3.
\]

The del Pezzo--Bertini classification of varieties of minimal degree
\cite[Theorem~0.1]{EisenbudGreenHulekPopescu2006} now makes
$\mathcal C_x$ a rational normal scroll, possibly a cone.  Indeed,
linear spaces and quadrics have smaller degree, while a cone over the
Veronese surface has degree four.  A three-dimensional scroll of
degree three has parameters $a_1,a_2,a_3\geq0$ with
$a_1+a_2+a_3=3$.

To turn the finite degree-one tangent map into an isomorphism, we need
normality of its target. These scrolls are normal, including the cone
cases, as follows. Their homogeneous coordinate rings are generated
by the monomials
\[
 z_i s_0^j s_1^{a_i-j},\qquad
 1\leq i\leq3,\quad0\leq j\leq a_i.
\]
These monomials generate exactly the invariant subring of
$\mathbb C[z_1,z_2,z_3,s_0,s_1]$ for the multiplicative-group action
with weights $(-a_1,-a_2,-a_3,1,1)$, graded by total degree in the
$z_i$. In this grading every displayed generator has degree one.
To see that they generate the invariant ring, distribute the
$s_0$ and $s_1$ exponents of an invariant monomial among its $z_i$
factors, each of which requires exactly $a_i$ such exponents.  An
invariant subring of a normal domain is normal: an element of its
fraction field integral over it belongs to the original domain and
remains invariant.  Thus the scroll is normal.

The finite birational morphism $\tau_x\colon F\to\mathcal C_x$ is
therefore an isomorphism.  In particular, $\mathcal C_x$ is smooth.
If some $a_i$ vanished, the scroll would be a nontrivial cone over a
nonlinear variety and hence singular along its vertex.  Consequently
all $a_i$ are positive, so $(a_1,a_2,a_3)=(1,1,1)$.  We have proved
the projective equivalence
\[
 \mathcal C_x\simeq
 \mathbb P^1\times\mathbb P^2\hookrightarrow\mathbb P^5,
 \qquad\text{embedded by }\mathcal O(1,1).
\]

This is the variety of minimal rational tangents of
$S=\operatorname{Gr}(2,5)$.  At a two-plane $A\subset\mathbb C^5$,
$T_AS=\operatorname{Hom}(A,\mathbb C^5/A)$, and tangent directions
to lines are exactly the rank-one maps.  Their projectivization is
the displayed Segre variety.  For example, in graph coordinates the
Pl\"ucker coordinates consist of the constant coordinate $1$, the entries of a map, and its
$2\times2$ minors; a direction determines a line precisely when the
quadratic minors vanish.

Finally, Mok's recognition theorem
\cite[Main Theorem, \S2.1]{Mok2008Recognition} states that a Fano
manifold of Picard number one whose general variety of minimal
rational tangents is projectively equivalent to that of an irreducible
compact Hermitian symmetric space is isomorphic to that space.
The model $S=\operatorname{Gr}(2,5)$ has rank two, and we have
verified the required projective equivalence for the minimal component
$V$.  The theorem gives $X\simeq S$.
\end{proof}
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/06-grassmannian.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/07-conclusion.tex ====
\section{The sixfold theorem and its consequences}\label{sec:conclusion}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $X$ satisfy its hypotheses. Since $X$ is projective and positive
dimensional, an ample divisor has nonzero numerical class, so
$\rho(X)\geq1$. If $\rho(X)\geq2$, then
$\rho(X)>\dim X-5$. Kanemitsu's Theorem~0.2
\cite{Kanemitsu2016} applies and gives rational homogeneity.

Suppose now that $\rho(X)=1$.
By Proposition~\ref{prop:pseudoindex-reduction}, either $X$ is
$\mathbb P^6$ or $Q^6$, or its pseudoindex is four or five.
The first two varieties are rational homogeneous.
Proposition~\ref{prop:pseudoindex-four} settles pseudoindex four.
In pseudoindex five, the characteristic-class identities of
Section~\ref{sec:characteristic-classes} and
Lemma~\ref{lem:finite-ratio} give $32-20u+3u^2=0$.
Proposition~\ref{prop:pseudoindex-five-recognition} therefore gives
$X\simeq\operatorname{Gr}(2,5)$, which is rational homogeneous.
These cases exhaust the possibilities.
\end{proof}

\subsection{A K\"ahler--Einstein consequence}

\begin{corollary}\label{cor:positive-ke}
Let $X$ be a smooth connected complex projective Fano variety with nef
tangent bundle and $1\leq\dim_{\mathbb C}X\leq6$.
Then $X$ admits a positive K\"ahler--Einstein metric. More precisely,
with the Ricci-form convention
\[
 \rho(\omega)=-i\partial\bar\partial\log\det(g_{a\bar b}),
\]
there is a K\"ahler form $\omega$ such that
\[
 \rho(\omega)=\omega,\qquad [\omega]=2\pi c_1(-K_X).
\]
\end{corollary}

\begin{proof}
In dimension six, Theorem~\ref{thm:main} makes $X$ rational homogeneous.
For $1\leq n=\dim_{\mathbb C}X\leq5$, its Picard number is at least
one and hence greater than $n-5$, so Kanemitsu's Theorem~0.2
\cite{Kanemitsu2016} gives the same conclusion. Thus the
lower-dimensional cases already follow from the known classification.
The homogeneous metric construction of Alekseevskii and Perelomov
\cite{AlekseevskiiPerelomov1986}, recalled in
\cite[\S1, p.~285]{Demailly2018AlmostKE}, supplies a positive
K\"ahler--Einstein metric on $X$.
Write its K\"ahler form as $\sigma$, with
$\rho(\sigma)=\lambda\sigma$ and $\lambda>0$. Constant rescaling
does not change the Ricci form, so $\omega=\lambda\sigma$ satisfies
$\rho(\omega)=\omega$. Since the Ricci form represents
$2\pi c_1(-K_X)$ in this convention, the asserted class follows.
\end{proof}

\subsection{Albanese fibres and the ordinary universal cover}

We now combine the classification with the analytic structure theorems
for compact K\"ahler manifolds with nef tangent bundle.

\begin{proof}[Proof of Corollary~\ref{cor:nef-tangent-uniformization}]
If $\dim_{\mathbb C}X=0$, connectedness makes $X$ a point and every
assertion follows. Assume henceforth that $\dim_{\mathbb C}X>0$.
Every finite \'{e}tale cover remains compact K\"ahler, and its tangent
bundle is analytically nef by Proposition~3.4 of
Demailly--Peternell--Schneider.
Their Proposition~3.9
\cite{DemaillyPeternellSchneider1994}, applied to every connected finite
\'{e}tale cover, makes its Albanese map surjective; hence its
irregularity is at most $\dim_{\mathbb C}X$. These irregularities are
nonnegative integers and include that of $X$, so their maximum
$q=\widetilde q(X)$ is attained.
Choose a connected finite \'{e}tale cover $X'\to X$ attaining it.
The Main Theorem and Theorem~3.14 of
Demailly--Peternell--Schneider, together with their Proposition~3.9,
give a smooth proper Albanese surjection
\[
 \alpha:X'\longrightarrow A=\operatorname{Alb}(X')
\]
with connected fibres, where $A$ is a complex torus of dimension $q$.
Every positive-dimensional fibre is a smooth projective Fano manifold
with nef tangent bundle and has dimension $d=\dim_{\mathbb C}X-q$.
Analytic nefness agrees with numerical nefness on these projective
fibres, so the algebraic classification applies.

If $d=6$, Theorem~\ref{thm:main} makes every fibre rational homogeneous.
For $1\leq d\leq5$, a projective fibre $F_a$ has
$\rho(F_a)\geq1>d-5$, and Kanemitsu's Theorem~0.2
\cite{Kanemitsu2016} gives the same conclusion. This is precisely the
established lower-dimensional range recalled in the introduction.
If $d=0$, each connected fibre is a point. This proves the first
part of the corollary.

Suppose first that $d>0$. Every rational homogeneous manifold $G/P$ satisfies
\[
 H^1(G/P,T_{G/P})=0
\]
and is locally rigid under complex deformations
\cite[Introduction and Section~3.7]{BienBrion1996}.
Applying this at each fibre of the proper
holomorphic submersion $\alpha$ shows that the fibre's biholomorphism
type is locally constant on $A$. Since $A$ is connected, all fibres
are biholomorphic to one fixed rational homogeneous manifold $F$.
The analytic Fischer--Grauert theorem
\cite{FischerGrauert1965}, in the form recalled in
\cite[Theorem~1.1]{Poczobut2025}, now makes $\alpha$ holomorphically
locally trivial. If $d=0$, the proper submersion with one-point fibres
is a biholomorphism, so this conclusion also holds with $F$ a point.

The universal cover of the complex torus $A$ is $\mathbb C^q$, which
is Stein and contractible even when $A$ is not projective.
Theorem~4.4 of Claudon--H\"oring--Koll\'ar
\cite{ClaudonHoringKollar2013} applies to the locally trivial proper
fibration $\alpha$ and gives
\[
 \widetilde {X'}\simeq\widetilde F\times\mathbb C^q.
\]
Here the tildes denote ordinary universal covers. A rational homogeneous
manifold $F$ is simply connected: write it using the simply connected
semisimple cover of its acting group; the corresponding parabolic
subgroup is connected, and the homotopy exact sequence of that
homogeneous fibration gives $\pi_1(F)=0$. This is also true for a point.
Thus $\widetilde F=F$. The connected finite \'{e}tale cover $X'\to X$
has the same ordinary universal cover as $X$, proving
$\widetilde X\simeq F\times\mathbb C^q$.

Since $F$ is projective, $F\times\mathbb A^q$ is quasi-projective and
its analytification is $F\times\mathbb C^q$. The projection of this
product to $\mathbb C^q$ is proper, surjective, and has connected compact
fibres. For every open set $U\subseteq\mathbb C^q$, a holomorphic
function on $F\times U$ is constant on each fibre and hence descends
holomorphically to $U$ by restriction to $\{f_0\}\times U$ for any
fixed $f_0\in F$. Thus the
projection has direct image of its structure sheaf equal to
$\mathcal O_{\mathbb C^q}$. Since the target is Stein, this projection
is the Remmert reduction, and the product is holomorphically convex.
The argument includes $q=0$, when the target is a point.
\end{proof}
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/07-conclusion.tex ====

==== BEGIN The-Campana-Peternell-conjecture-in-dimension-six//sections/08-certificate.tex ====
\section{Exact arithmetic certificate}
\label{app:certificate}

The following complete Python program verifies the finite calculations
in Section~\ref{sec:arithmetic}. It requires SymPy and NumPy. Run
\texttt{python3 verification/verify.py} with assertions enabled to perform
the checks described below. The monomial order, coefficient recurrences
and projection are exactly those defined in
Section~\ref{sec:characteristic-classes}.

The calculation has three logically separate parts. First, exact
rational identities $AK=0$ and $NP=0$, together with the four modular
ranks in Section~\ref{sec:arithmetic}, certify the kernel and
annihilator bases. Reduction of a rational entry explicitly inverts
its denominator modulo $10007$, so a noninvertible denominator would
stop the calculation. Second, the slice-merging procedure computes
the finite-field spaces and checks their dimensions. The final basis
is also tested directly against every double-slice constraint.
Third, rational substitution checks the six displayed columns, their
nonzero minor $-16/5$, and the final linear functional.

SymPy performs the polynomial and rational operations exactly.
The finite-field arrays use signed 64-bit integers. After each
reduction, their entries lie between $0$ and $10006$. The inner
dimension of every matrix product is at most $6820$, since a basis
has no more columns than the number of coordinates of its space.
Every accumulated dot product is therefore at most
\[
 6820\cdot10006^2=682818645520<2^{63}-1.
\]
The products and subtractions in a pivot step have smaller absolute
values. Thus no arithmetic operation can wrap around before modular
reduction.

\lstinputlisting[language=Python]{../verification/verify.py}
==== END The-Campana-Peternell-conjecture-in-dimension-six//sections/08-certificate.tex ====

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