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% P = W in composite rank for fixed determinant
% A rewritten, self-contained proof.
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\title{$P=W$ in composite rank for fixed determinant}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We prove the $P=W$ conjecture for 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.
Combined with the known prime-rank cases, this gives the equality in every
coprime rank.
\end{abstract}

% ============================================================
\section*{Overview: what is being proved, and why it is not obvious}
% ============================================================

\paragraph{The two spaces.} Fix a smooth projective complex curve $C$ of
genus $g\ge 2$. Fix a rank $n\ge 2$, a degree $d$ coprime to $n$, and a
line bundle $L$ of degree $d$ on $C$. Two very different spaces sit over $C$.

The \emph{Higgs moduli space} $X$ parametrizes stable pairs $(E,\theta)$:
$E$ is a rank-$n$ holomorphic vector bundle with $\det E\simeq L$; $\theta$
is a \emph{Higgs field}, a traceless map $E\to E\otimes K_C$ (where $K_C$ is
the canonical line bundle). The \emph{character variety} $X^B$ parametrizes
flat connections, equivalently representations of the fundamental group of
$C$ with product of commutators $\zeta I_n$, where
$\zeta=\exp(2\pi\sqrt{-1}\,d/n)$. Nonabelian Hodge theory says $X$ and $X^B$
are diffeomorphic, so they have the same cohomology.

\paragraph{Two filtrations.} On $H^*(X)$ the Hitchin map $h:X\to A$ (recording
the coefficients of $\det(\theta-t\,I)$) defines the \emph{perverse Leray
filtration} $P_k$. On $H^*(X^B)$ Deligne's theory of mixed Hodge structures
defines the \emph{weight filtration} $W_j$. These come from completely
different geometry. The $P=W$ conjecture says they coincide after doubling
the perverse index:
\[
  P_kH^m(X)\;=\;W_{2k}H^m(X^B)\;=\;W_{2k+1}H^m(X^B).
\]

\paragraph{Why it is hard.} In the \emph{unfixed} determinant ($\GL_n$) case
this is known in every rank \cite{MS2024}. The fixed-determinant case splits
into the part invariant under the finite group $\Gamma=\Pic^0(C)[n]$ (acting
by tensor product), which reduces to $\GL_n$, and the \emph{variant} part,
which does not. Our proof does not throw away the variant part: it constructs
a single operator on the whole of $H^*(X)$ before taking any invariants.

\paragraph{The strategy in one paragraph.} Both filtrations are controlled by
a degree-two raising operator $e$: cup product with a fixed class in $H^2$.
For each filtration, $e$ satisfies a hard Lefschetz theorem with the same
center $R=(n^2-1)(g-1)$. A classical lemma (proved in
\Cref{sec:lefschetz}) says: if one degree-$(-2)$ operator $f$ lowers \emph{both}
filtrations by two and satisfies the $\mathfrak{sl}_2$ relation
$[[f,e],e]=-2e$, then the two filtrations are equal, because both are rebuilt
from the eigenspaces of the same operator $[e,f]$. So the whole proof reduces
to producing such an $f$. That is done by a geometric construction: insert a
marked point with a logarithmic pole, move it over a finite cover of $C$,
read off monodromy, take a logarithm $N$, and set $f\sim \pr_*\pi_*\iota^*N^2
\iota_*\pi^*\pr^*$.

\paragraph{Where each ingredient is proved.}
\begin{itemize}[leftmargin=1.4em]
\item \Cref{sec:lefschetz}: normalizations, the class $e$, the abstract
      criterion (a common lowering operator determines the filtration).
\item \Cref{sec:betti}: hard Lefschetz for $e$ on the Betti side, on the
      \emph{whole} cohomology.
\item \Cref{sec:diagram}: a bookkeeping device giving weight zero to the
      cochains of the moving-point curve, so that integrating over it does
      not shift weight.
\item \Cref{sec:logarithmic}: the moving logarithmic pole, restriction
      isomorphisms, and compatibility with nonabelian Hodge theory on the
      whole cohomology.
\item \Cref{sec:translation}: the elementary modification, and its
      logarithm $N$.
\item \Cref{sec:specialization}: the perverse bound, via nearby cycles.
\item \Cref{sec:commutator}: the double commutator $[[f',e],e]$.
\item \Cref{sec:conclusion}: assembly.
\end{itemize}

\paragraph{Standing conventions.} Cohomology is with rational coefficients
unless $\C$ is written. $R=(n^2-1)(g-1)$ and $b=\tfrac{n(n-1)}2$ are fixed
throughout. The group $\Gamma=\Pic^0(C)[n]$ acts on $X$ by tensor product;
its averaging projector splits $H^*(X)$ into invariants and the variant part.
We prove the theorem for the composite integers $n\ge4$; the prime cases are
\cite[Theorem~0.3]{MS2024}, and the two together cover every $n\ge2$.

% ============================================================
\section{Lefschetz operators and the comparison criterion}
\label{sec:lefschetz}
% ============================================================

We fix once and for all the perverse normalization, exhibit the common
degree-two class $e$, and prove the abstract linear-algebra lemma that turns
"one common lowering operator" into "equal filtrations." From now on we
assume $n\ge4$ is composite; we say so again when coprimality is used.

\subsection{The normalized perverse filtration}

\begin{lemma}\label{norm:dimensions}
With $R=(n^2-1)(g-1)$ one has $\dim X=2R$, $\dim A=R$, every fiber of $h$
has dimension $R$, and $\dim(X\times_A X)-\dim X=R$.
\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 $H^0$ and $H^2$ by
stability and Serre duality, and Euler characteristic $-2R$; so each tangent
space of $X$ has dimension $2R$.

For the fibers, use Higgs-field scaling: $X$ is semiprojective, so every
scaling orbit closable at $\infty$ has a limit, and the limit lies in the
proper fiber $h^{-1}(0)$ (all Hitchin coordinates have positive scaling
weight). At a fixed point the symplectic form pairs a tangent weight $w$ with
$1-w$; since weights are integers, the nonpositive part has exactly half the
tangent dimension, so every repelling stratum has dimension $R$. These strata
fill $h^{-1}(0)$, hence $\dim h^{-1}(0)=R$. Pulling $h$ back along the scaling
curve toward $0$ and using upper semicontinuity of fiber dimension for the
proper map $h$ gives $\dim h^{-1}(a)\le R$ for every $a$; the dimension
inequality for a map from a smooth $2R$-fold onto an $R$-fold gives
$\dim h^{-1}(a)\ge R$ on every nonempty fiber. Thus every fiber is
$R$-dimensional and $\dim(X\times_A X)=R+2R=3R$.
\end{proof}

Put $K=Rh_*\Q_X[R]$ and define the normalized perverse filtration by
\begin{equation}\label{norm:perverse-filtration}
 P_kH^m(X)=\im\!\left(
 H^{m-R}\bigl(A,{}^p\tau_{\le k}K\bigr)\to H^m(X)\right).
\end{equation}
The fiber bound forces the perverse cohomology of $Rh_*\Q_X[2R]$ to live in
degrees $[-R,R]$ (support criterion gives the upper bound, Verdier duality
the lower); after the shift by $[-R]$, $K$ has perverse degrees in $[0,2R]$,
so
\begin{equation}\label{norm:perverse-range}
 P_{-1}H^*(X)=0,\qquad P_{2R}H^*(X)=H^*(X).
\end{equation}
We use the shift rule for a morphism $K\to K[s]$:
\begin{equation}\label{norm:shift-rule}
 {}^p\tau_{\le k}(K[s])=({}^p\tau_{\le k+s}K)[s],
\end{equation}
so an actual morphism $Rh_*\Q_X\to Rh_*\Q_X[-2]$ lowers perversity by two.

\subsection{The degree-two class}

The universal projective bundle on $C\times X$ determines the class
\[
 \widetilde{\ch}_2(E)=\ch_2(E)-\frac{c_1(E)^2}{2n}
 =\frac1{2n}\ch_2(\End E),
\]
the last expression being defined even without a chosen universal vector
bundle. Define
\begin{equation}\label{norm:e}
 e=\int_C\widetilde{\ch}_2(E)\in H^2(X),
\end{equation}
and write $e$ also for cup product with it.

\begin{lemma}\label{norm:low-cohomology}
For composite $n\ge4$:
$H^0(X)=\Q$, $H^1(X)=0$, $H^2(X)=\Q e$, and $e\ne0$.
Every ample class on $X$ is a nonzero multiple of $e$.
\end{lemma}

\begin{proof}
The group $\Gamma=\Pic^0(C)[n]$ acts; averaging splits $H^*$ into invariants
and the variant part $H^*_{\mathrm{var}}$. Let $p$ be the smallest prime
divisor of $n$ and $c=n(n-n/p)(g-1)\ge8$. \cite[Proposition~1.4]{dCMS2022},
valid in any rank, gives $H^i_{\mathrm{var}}=P_{i-c}H^i_{\mathrm{var}}$, and
\eqref{norm:perverse-range} kills the variant part in degrees $0,1,2$.

The invariant part is the cohomology of the $\PGL_n$ moduli space. Markman's
generation theorem \cite[Theorem~7]{Markman2002}, in the projective-bundle form
of \cite[Theorem~2.1(i)]{MS2024}, says this algebra is generated by the
K\"unneth components of projective Chern characters of index $j\ge2$, of
degrees $2j,\,2j-1,\,2j-2$. So there is no degree-one generator and a single
possible degree-two generator, a nonzero normalization of $e$. Hence
$H^0=\Q$, $H^1=0$, $H^2=\Q e$ up to nonvanishing.

A Hitchin fiber is projective of positive dimension, so it contains a curve;
an ample class has positive degree on it and is nonzero in $H^2$, forcing
$e\ne0$ and the last claim.
\end{proof}

\begin{proposition}\label{norm:relative-lefschetz}
Cup product by $e$ maps $P_kH^m(X)\to P_{k+2}H^{m+2}(X)$, and for every
$i\ge0$ it induces an isomorphism
\begin{equation}\label{norm:relative-hl}
 e^i:\Gr^P_{R-i}H^m(X)\xrightarrow{\sim}\Gr^P_{R+i}H^{m+2i}(X).
\end{equation}
\end{proposition}

\begin{proof}
The shift follows from \eqref{norm:shift-rule}. Choose an ample class, also
relatively ample for the projective $h$. The decomposition theorem and
relative hard Lefschetz \cite[Theorems~6.2.5, 6.2.10]{BBD1982} give
\eqref{norm:relative-hl} for that class (the perverse degrees of
$Rh_*\Q_X[2R]$ are centered at $0$, and passing to $K$ centers them at $R$).
The decomposition theorem makes the perverse spectral sequence degenerate,
so sheaf-level isomorphisms give the associated-graded isomorphisms. Replace
the ample class by its nonzero multiple $e$.
\end{proof}

\subsection{A common lowering operator determines the filtration}

For a finite graded $V=\bigoplus_kV_k$ and a degree-two $E$, say $E$ is
\emph{Lefschetz with center $r$} if $E^i:V_{r-i}\xrightarrow{\sim}V_{r+i}$
for all $i\ge0$. \Cref{norm:relative-lefschetz} says the raiser on $\Gr^P$ is
Lefschetz with center $R$.

\begin{proposition}[Comparison by a common lowering operator]
\label{alg:comparison}
Let $H$ be finite-dimensional over a characteristic-zero field, with finite
exhaustive separated increasing integer filtrations $F_\bullet,G_\bullet$.
Suppose $e,f\in\End(H)$ satisfy, for $A_\bullet$ equal to $F_\bullet$ or
$G_\bullet$,
\[
 e(A_k)\subseteq A_{k+2},\quad f(A_k)\subseteq A_{k-2},
\]
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 all $k$. If $H$ carries a further grading respected by both
filtrations with $e,f$ of degrees $2,-2$, the equality is degreewise.
\end{proposition}

\begin{proof}
Fix one associated graded with raiser $E$. The Lefschetz decomposition writes
$E$ on primitive strings $v,Ev,\dots,E^\ell v$ with $\deg v=r-\ell$. Put
\[
 F_0(E^jv)=j(\ell-j+1)E^{j-1}v,\qquad
 H_0=k-r\ \text{on grade }k.
\]
Direct computation gives $[E,F_0]=H_0$, $[H_0,E]=2E$, $[H_0,F_0]=-2F_0$, so
$(E,H_0,F_0)$ is an $\mathfrak{sl}_2$-triple and $[[F_0,E],E]=-2E$.

A degree-$(-2)$ endomorphism obeying this identity must equal $F_0$: on
$\End(\Gr^A H)$ the adjoint action labels degree $-2$ by $H_0$-weight $-2$,
and in any irreducible summand two raising steps inject the weight-$(-2)$
space into weight $2$. Thus $(\mathrm{ad}\,E)^2$ is injective on
degree-$(-2)$ endomorphisms, and applying it to $\Gr^A(f)-F_0$ plus
\eqref{alg:double-commutator} gives $\Gr^A(f)=F_0$.

Hence $h_0=[e,f]$ preserves both filtrations and acts by $k-r$ on grade $k$.
Then $(h_0-(k-r))A_k\subseteq A_{k-1}$, and multiplying these factors from
$k=b$ down to $k=a$ over the range of nonzero grades gives
$\prod_{k=a}^b(h_0-(k-r))=0$. The roots are distinct, so $h_0$ is semisimple
and every $h_0$-stable subspace splits along eigenspaces. An eigenvector first
entering $A_j$ has eigenvalue $j-r$, so
\begin{equation}\label{alg:eigenspace-filtration}
 A_k=\bigoplus_{j\le k}\ker\bigl(h_0-(j-r)\bigr),
\end{equation}
whose right side does not depend on the filtration. In the graded setting
$h_0$ has degree zero, so \eqref{alg:eigenspace-filtration} is degreewise.
\end{proof}

\begin{lemma}\label{alg:tensor}
Let $V,W$ be nonzero finite integer-graded spaces over a characteristic-zero
field, $E_V,E_W$ of degree two. If $E_V\otimes1+1\otimes E_W$ is Lefschetz
with center $r$ on $V\otimes W$, then $E_V,E_W$ are Lefschetz with centers
$r_V,r_W$ and $r_V+r_W=r$.
\end{lemma}

\begin{proof}
Decompose into homogeneous Jordan strings (nilpotent, finite support). Each
tensor of two strings is a graded invariant direct summand with invariant
complement for the sum operator, and each Lefschetz isomorphism is block
diagonal, hence an isomorphism on each summand. For a string with extreme
grades $p,q$ call $(p+q)/2$ its center; the center of a tensor of two strings
is the sum of centers, and Lefschetz symmetry forces that sum to be $r$ for all
pairs. Fixing one factor's string forces all strings in the other factor to
share a center, and a Jordan string is Lefschetz about its own center.
\end{proof}

We will apply \Cref{alg:comparison} to total cohomology with $F_k=P_k$ and
$G_k=W_{2k}$, once \Cref{sec:betti} proves $e$ is Betti-Lefschetz with center
$R$ and all odd weight-graded pieces vanish, and \Cref{sec:translation,%
sec:specialization,sec:commutator} produce the one operator $f$ with both
lowering bounds and \eqref{alg:double-commutator}.

% ============================================================
\section{Curious hard Lefschetz on the full character variety}
\label{sec:betti}
% ============================================================

Here we prove that $e$ is a Lefschetz operator for the weight filtration on
the \emph{whole} cohomology of $X^B$, including every nontrivial
$\Gamma$-character summand. The trick is to compute first on an ordered family
over a torus of eigenvalues, using a finite cover to reach the $\GL_n$ toric
stratifications, and only at the very end specialize the eigenvalue parameter
to the scalar point and strip off flag and determinant factors.

Recall 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 $H^*(X^{\mathrm B},\Q)$ is Hodge--Tate. Under the
nonabelian-Hodge identification, $e$ has nonzero image in
$\Gr^W_4H^2(X^{\mathrm B},\Q)$, and for all $i\ge0$
\[
 e^i:\Gr^W_{2R-2i}H^m(X^{\mathrm B},\Q)\xrightarrow{\sim}
      \Gr^W_{2R+2i}H^{m+2i}(X^{\mathrm B},\Q).
\]
In particular $W_{2k}=W_{2k+1}$ on this cohomology, including all
$\Gamma$-character summands after complexification.
\end{theorem}

\subsection{The ordered family and its scalar fiber}

Let
\[
 V=\Bigl\{(t_1,\dots,t_n)\in(\C^*)^n:\ \textstyle\prod_i t_i=1,\
 \prod_{i\in I}t_i\ne1\ \text{for }\varnothing\ne I\subsetneq\{1,\dots,n\}\Bigr\},
 \qquad q=(\zeta,\dots,\zeta).
\]
Then $V$ 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 $(A_1,B_1,\dots,A_g,B_g)\in\SL_n^{2g}$
together with a full flag preserved by $\nu$, on whose ordered graded lines
$\nu$ acts by $t_1,\dots,t_n$, modulo simultaneous conjugation; repeated
eigenvalues are allowed.

Every such tuple is irreducible: on a proper invariant subspace,
$\det\nu=1$ but the eigenvalues are a proper nonempty submultiset of the
$t_i$, contradicting the definition of $V$. So conjugation gives geometric
quotients by a free $\PGL_n$ action (built from \'etale-local charts on
irreducible tuples; a full flag lets one complete the first line to a basis by
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;
$\iota$ is a regular embedding of codimension $b$ with normal bundle
$T_\pi^*$. The operators $\mathsf A=\pi_*\iota^*$ and $\mathsf B=\iota_*\pi^*$
have degrees $-2b$ and $2b$ and satisfy
\begin{equation}\label{betti:split}
 \mathsf A\mathsf B=(-1)^b n!\,\id.
\end{equation}
(Covariant star denotes Gysin.)
\end{lemma}

\begin{proof}
The commutator product is a submersion at every irreducible tuple. For a pair
$a,b'$ set $c=ab'a^{-1}b'^{-1}$; the right-translated differential has terms
$(1-\mathrm{Ad}_{ab'a^{-1}})u$ and $(\mathrm{Ad}_a-\mathrm{Ad}_c)v$. An
annihilating vector $z$ (trace pairing) is fixed by $ab'a^{-1}$ and satisfies
$\mathrm{Ad}_{a^{-1}}z=\mathrm{Ad}_{b'}z$; iterating over prefixes shows $z$
centralizes all matrices, so irreducibility makes it scalar and tracelessness
makes it zero.

Matrices in a Borel, with their flags, form a smooth space over the ordered
diagonal torus with $2b$-dimensional fibers; pulling back along the submersion
and quotienting gives smoothness with relative dimension
$2g(n^2-1)+2b-2(n^2-1)=2R+2b$. At $q$, the section $\zeta^{-1}\nu-I$ lands in
the nilradical bundle over the flag space, with zero locus $Z^{\mathrm B}$;
submersivity makes it regular, and the trace pairing identifies the
nilradical with $T_\pi^*$. Self-intersection via the projection formula gives
$\mathsf A\mathsf B(\alpha)=\pi_*(c_b(T_\pi^*)\pi^*\alpha)
=(-1)^b\chi(\Fl_n)\alpha=(-1)^b n!\,\alpha$. Everything descends through
$\PGL_n$ and applies to full cohomology.
\end{proof}

\subsection{Lifting the toric stratifications}

Set $D_1=(\C^*)^{2g}$ and $\widehat Y^{\mathrm B}=Y^{\mathrm B}\times D_1$.
Scaling each of the $2g$ matrices by the corresponding scalar gives a finite
\'etale map of degree $n^{2g}$
\begin{equation}\label{betti:cover}
 \widehat Y^{\mathrm B}\to Y^{\mathrm B}_{\mathrm{GL}},
\end{equation}
the target being the ordered family with matrices in $\GL_n$: the source is a
genuine product because dividing by chosen determinant roots returns matrices
in $\SL_n$ without changing commutators or flags.

We use Mellit's toric stratifications \cite{Mellit2025}. To identify the
family, take two punctures with first monodromy $\zeta^{-1}I$ and last
monodromy $\zeta\nu^{-1}$, whose ordered eigenvalues are $(\zeta t_i^{-1})_i$.
The product relation reads $\nu\zeta^{-1}I\cdot\zeta\nu^{-1}=I$, and the
genericity condition of \cite[Definition~4.9]{Mellit2025} is exactly the
subproduct condition defining $V$; the identity last monodromy is $q$, and the
ordering condition is automatic for a scalar first monodromy and a
distinct-eigenvalue last monodromy.

Two features of these stratifications are 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; Mellit's degree-two
class then gives compact-support curious Lefschetz on every final stratum,
with the same middle actual weight $\delta+2b$ ($\delta=\dim
Y^{\mathrm B}_{\mathrm{GL},t}$), and removing the vector-bundle shift gives
middle weight $\delta$ \cite[Theorem~6.14, Section~7.4, Theorem~7.5]{Mellit2025}.
Second, over all of $V$, an auxiliary affine bundle of rank $b$ has a finite
stratification refined into vector bundles over
$\mathbb A^a\times(\C^*)^s\times V$, projecting to $V$
\cite[proof of Proposition~8.7]{Mellit2025}. These statements concern the
final torus--affine refinements.

\begin{lemma}\label{betti:finite-cover}
Pulling the preceding bundles and stratifications back along
\eqref{betti:cover}:
\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 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, and
ordinary direct image commutes with restriction to every fiber; the same holds
for $Y^{\mathrm B}\to V$.
\end{enumerate}
\end{lemma}

\begin{proof}
Consider a final stratum $F=(\C^*)^s\times\mathbb A^a$, or a vector bundle over
it; its homotopy type is a torus. Each connected component
$\widetilde F$ of a finite \'etale preimage corresponds to a finite-index
subgroup of $\Z^s$, and $H^*(F,\Q)\to H^*(\widetilde F,\Q)$ is an isomorphism
of exterior algebras, hence of mixed Hodge structures (algebraic map). Both
spaces are smooth of the same dimension, and finite trace plus Poincar\'e
duality give the compact-support analogue; these isomorphisms commute with the
class.

Apply componentwise, then use the finite closed filtration and the strictness
of mixed Hodge morphisms \cite[Theorem~2.3.5]{Deligne1971} on compact-support
localization sequences; the five lemma passes the Lefschetz isomorphisms and
Hodge--Tate property to the whole auxiliary space. The Thom isomorphism shifts
degree and weight by $2b$; compact-support middle weight drops from
$\delta+2b$ to $\delta$, and Poincar\'e duality ($w\mapsto2\delta-w$) gives the
same middle weight $\delta$, i.e.\ center $\delta/2$. The relative product
stratification handles repeated eigenvalues the same way.

For integral local constancy, take $U\subset V$ small contractible analytic;
a finite cover of $F\times U$ is pulled back from one of $F$ (since
$\pi_1(F\times U)=\pi_1(F)$), and the same holds for a vector bundle over $F$
(homotopy equivalent to its base; equivalently use the complex-orientation
Thom isomorphism). So each covered stratum has locally constant finitely
generated compact-support sheaves on $U$; localization triangles pass this up
a finite stratification, and the auxiliary affine trace removes the shift.

For the covered smooth family $f$ of relative dimension $\delta$,
$K=Rf_!\Z$ is locally perfect (bounded, finitely generated), so smooth Verdier
duality gives
$Rf_*\Z\simeq R\Hom(K,\Z_V)[-2\delta]$; the right side is locally perfect, so
stalks commute with the duality, and compact-support base change plus the
fiberwise cup--trace pairing identify the stalk with
$R\Gamma(f^{-1}(t),\Z)$, verifying ordinary base change. Passing to
$Y^{\mathrm B}$: the projection $p:Y^{\mathrm B}\times D_1\to Y^{\mathrm B}$
splits off a section for ordinary direct images, and the Gysin map of the
closed section followed by integration over $D_1$ splits it for compact
supports. Thus local constancy and base change descend, and the algebraic
splitting transfers Hodge--Tate-ness to $Y^{\mathrm B}$.
\end{proof}

\subsection{The scalar summand and the determinant factor}

For a smooth algebraic family with lisse ordinary direct images satisfying
fiberwise base change, mixed Hodge module direct image gives admissible
graded-polarizable variations of mixed Hodge structure on the local systems
(constructed in Proposition~\ref{diag:variation}); parallel transport
preserves the weight filtration, and cup products are parallel. Applied to
\Cref{betti:finite-cover}, transport a Lefschetz class from a
distinct-eigenvalue fiber along a path in the connected $V$ to $q$, giving
\[
 \omega\in\Gr^W_4H^2(\widehat Y^{\mathrm B}_q,\C)
\]
with center $R+b+g$. (A choice of path suffices.)

Take the product of \Cref{betti:geometry} with $D_1$, setting
$\widehat X^{\mathrm B}=X^{\mathrm B}\times D_1$,
$\widehat Z^{\mathrm B}=Z^{\mathrm B}\times D_1$. The flag bundle has rational
algebraic Leray--Hirsch classes: ratios of ordered graded lines descend for
the universal projective bundle and their first Chern classes generate the
flag cohomology over $\Q$ (differences of Chern roots generate rationally
because the roots sum to zero on a fiber), and polynomial representatives give
global classes. These extra degree-two classes have weight two, hence
\[
 \pi^*:\Gr^W_4H^2(\widehat X^{\mathrm B},\C)
 \xrightarrow{\sim}\Gr^W_4H^2(\widehat Z^{\mathrm B},\C),
\]
and there is a unique $\xi$ on the left with $\iota^*\omega=\pi^*\xi$ on
weight-gradeds. By the projection formula $\mathsf A,\mathsf B$ intertwine
multiplication by $\omega$ and $\xi$ on weight-gradeds; their actual weight
shifts are $-2b,2b$. Since \eqref{betti:split} exhibits the image of $\mathsf B$
as a graded stable summand with stable complement $\ker\mathsf A$, each
Lefschetz isomorphism for $\omega$ restricts to one on the summand. The
half-weight shift of $\mathsf B$ is $b$, so $\xi$ has center $R+g$ on
$H^*(\widehat X^{\mathrm B},\C)$, whose cohomology is thus a Tate-twisted
summand of Hodge--Tate cohomology, hence Hodge--Tate.

Remove $D_1$: \Cref{norm:low-cohomology} and nonabelian Hodge give
$H^0(X^{\mathrm B})=\Q$, $H^1(X^{\mathrm B})=0$, so K\"unneth gives
$\xi=\xi_X+\xi_D$ with $\xi_X\in\Gr^W_4H^2(X^{\mathrm B},\C)$ and
$\xi_D\in H^2(D_1,\C)$. \Cref{alg:tensor} shows both factors have common
Lefschetz centers summing to $R+g$; $H^*(D_1)$ is an exterior algebra on $2g$
generators of degree one and weight two, with half-weight grades $0$ to $2g$,
hence center $g$. Thus $\xi_X$ has center $R$, and projection/section again
give $X^{\mathrm B}$ Hodge--Tate cohomology.

Since $H^0(X^{\mathrm B})\ne0$ and $R>0$, $\xi_X$ is nonzero; by
\Cref{norm:low-cohomology} $H^2(X^{\mathrm B},\Q)=\Q e$, so the image of $e$ in
$\Gr^W_4$ is nonzero and a nonzero multiple of $\xi_X$ (and $e\in W_4$ since
$X^{\mathrm B}$ is smooth). Rescaling preserves Lefschetz isomorphisms, which
hold over $\Q$ since they become isomorphisms over $\C$. This proves
\Cref{betti:lefschetz}, with no step taking $\Gamma$-invariants.

% ============================================================
\section{Weights for flat bundles of algebraic families}
\label{sec:diagram}
% ============================================================

Consider a product $S\times X^{\mathrm B}$ with $S$ a closed oriented smooth
surface. The moving-point construction needs integration over $S$ to lower
degree by two while preserving weight. So on this product we give the
K\"unneth factor $H^a(S,\Q)$ pure weight zero and type $(0,0)$ in every degree,
and keep Deligne's mixed Hodge structure on $H^*(X^{\mathrm B},\Q)$; then
integration over $S$ acts on the first factor without a Tate twist. Even for
an algebraic curve $S$ this differs from its ordinary Deligne weights. Because
the moving-puncture flag spaces are not products, we extend the convention to
bundles whose transition maps are locally constant algebraic automorphisms of
a fiber family. A finite diagram of algebraic fibers carries the mixed Hodge
structures together with parallel transport and proper Gysin maps. The
simplicial idea goes back to Deligne \cite[Section~8.3]{Deligne1974} and was
used by Shende for tautological classes \cite[proof of the main
Theorem]{Shende2017}; we supply the relative construction and Gysin
compatibility.

\subsection{The finite diagram}

Let $V$ be a smooth connected complex variety and $p:H\to V$ an algebraic
morphism of finite type. Assume the cohomology sheaves of $Rp_*\Z_H$ are
bounded, locally constant, and finitely generated, and that the natural
ordinary base-change maps
\begin{equation}\label{diag:basechange}
 i_t^*Rp_*\Z_H\to R\Gamma(H_t,\Z)
\end{equation}
are isomorphisms for all $t\in V$ (these concern ordinary direct image;
smoothness alone does not give them for a nonproper morphism).

For a compact smooth manifold $S$, a \emph{flat bundle of the family $H\to V$
over $S$} is a space $\mathscr H\to S\times V$ with local identifications
$\mathscr H|_{U_i\times V}\simeq U_i\times H$ whose transition maps are
$(s,h)\mapsto(s,g_{ij}(h))$ with $g_{ij}$ an algebraic automorphism over $V$
locally constant in $s$, satisfying the cocycle identity. Write
$\mathscr H_t$ for the total space over $S$ at $t$ (it includes $S$ and need
not be algebraic). A \emph{map of flat bundles} is the identity on $S$ times an
algebraic map of fiber families in simultaneous flat trivializations.

Choose a finite good cover $\mathcal U=\{U_i\}$ refining a flat trivializing
cover (all $U_I=\bigcap_{i\in I}U_i$ contractible), a trivialization on each
$U_I$; inclusions give a finite diagram of copies $H_I\to V$ and algebraic
isomorphisms over $V$, indexed by the poset of nonempty intersections ordered
toward restriction. Using the enhanced bounded derived category of mixed Hodge
modules with diagram-compatible six operations \cite[Section~2.1,
Theorem~2.1.1, Remark~2.1.2, Proposition~4.2.6]{TubachStacks2025}, and writing
$\Q_T^H$ for the constant mixed Hodge module complex without dimension shift,
define the \emph{derived} diagram
\begin{equation}\label{diag:complex}
 K_{\mathscr H}=\holim_I\,Rp_{I*}\Q_{H_I}^H\quad\text{in }D^b\MHM(V).
\end{equation}
The index poset has finite-dimensional nerve, so this is a finite limit and
stays bounded; \eqref{diag:complex} uses the whole derived diagram, not only
cohomology of its terms.

\begin{proposition}[Diagram weights]\label{diag:variation}
Under these hypotheses, $H^j(\mathscr H_t,\Q)$ carries a graded-polarizable
mixed Hodge structure, functorial for maps of flat bundles algebraic in flat
trivializations, forming as $t$ varies an admissible graded-polarizable
variation with integral local system up to torsion; it is computed by
$H^j(i_t^*K_{\mathscr H})$. If $H^j(H_t,\Q)$ is Hodge--Tate for all $j,t$,
then so is $H^j(\mathscr H_t,\Q)$. We call its weight filtration the
\emph{diagram weight filtration}.
\end{proposition}

\begin{proof}
Open-cover descent computes the ordinary direct image of $\mathscr H\to V$ from
the spaces $U_I\times H_I$; projection to $H_I$ has contractible ($U_I$) fibers
over $V$ and is quasi-isomorphic on direct images, commuting with transitions,
so the derived limit identifies the rational realization of $K_{\mathscr H}$
with the ordinary direct image (same integrally).

A finite limit of bounded complexes with locally constant cohomology has
locally constant finitely generated cohomology; point pullback commutes with
finite limits, and termwise base change is an isomorphism by
\eqref{diag:basechange}, with the corresponding mixed Hodge module maps
isomorphisms by conservativity of rational realization. So $i_t^*K_{\mathscr
H}$ computes the derived diagram of the algebraic fibers $H_{I,t}$ and its
underlying cohomology is $H^*(\mathscr H_t,\Q)$.

For $v=\dim V$, the locally constant ordinary cohomology of $K_{\mathscr H}$
makes ${}^pH^{j+v}(K_{\mathscr H})$ a smooth mixed Hodge module, which after
the dimension shift is an admissible graded-polarizable variation
\cite[Section~3.2, Equation~(3.2.1)]{SaitoExtension1990}; point pullback on
perverse truncations shows the fiber is $H^j(i_t^*K_{\mathscr H})$ (a smooth
perverse object has only fiber degree $-v$).

Finally, mixed Hodge structures with weight gradeds of Hodge type $(r,r)$ only
are closed under subobjects, quotients, and extensions; the cone long exact
sequence then makes complexes with Hodge--Tate cohomology closed under finite
limits, giving the last assertion. Algebraic diagram maps give functoriality,
and a common good refinement compares choices of cover.
\end{proof}

The construction preserves the cup product: each algebraic direct image of the
constant complex has its functorial product preserved by transitions, the
derived limit inherits it, and under the open-cover comparison it is the
ordinary cup product on $H^*(\mathscr H_t,\Q)$. So parallel transport preserves
both the ring and its weight filtration.

\subsection{Products and proper Gysin maps}

\begin{proposition}[Weight shifts]\label{diag:gysin}
The diagram weight filtrations satisfy:
\begin{enumerate}
\item For $S\times M$ with $M$ algebraic, the K\"unneth isomorphism
identifies the structure with that on $H^*(S,\Q)\otimes H^*(M,\Q)$, where
every $H^a(S,\Q)$ is pure of weight zero and type $(0,0)$ and $H^*(M,\Q)$ has
Deligne's structure. If $S$ is closed oriented of real dimension $d_S$,
projection pullback and integration over $S$ preserve weight, the latter with
degree $-d_S$.
\item Pullbacks by maps of flat bundles preserve weight.
\item For a proper map of flat bundles $u:\mathscr H_t\to\mathscr H'_t$ with
smooth equidimensional algebraic fibers, $c=\dim_\C H'_t-\dim_\C H_t$, its
ordinary Gysin map satisfies $u_*:W_aH^j(\mathscr H_t)\to
W_{a+2c}H^{j+2c}(\mathscr H'_t)$. If $S$ is oriented, this is the Gysin map for
the induced orientations.
\end{enumerate}
\end{proposition}

\begin{proof}
For a product the diagram is constant with derived limit
$C^*(N\mathcal U,\Q(0))\otimes R\Gamma(M,\Q_M^H)$, the first factor a copy of
$\Q(0)$ per simplex with simplicial differential, all of weight zero; the
external-product comparison gives the ordinary K\"unneth map. Evaluation on a
simplicial fundamental cycle gives $C^*(N\mathcal U,\Q(0))\to\Q(0)[-d_S]$, a
morphism of mixed Hodge complexes realizing integration over $S$ with no Tate
twist; projection pullback is the unit of the same complex. This gives (1),
and functoriality of algebraic pullbacks gives (2).

For (3), fix $t$ and choose simultaneous trivializations. On each 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*}\Q^H_{H_{I,t}}\to\Q^H_{H'_{I,t}}[2c](c),
\end{equation}
whose degree and Tate twist give the weight shift; complex orientations are
preserved by the algebraic transitions and the counit is natural, so
\eqref{diag:trace} gives a coherent diagram map and hence a map of derived
limits. On $U_I\times H_{I,t}$ the map is $\id_{U_I}\times u_{I,t}$, whose
orientation trace is the fiber trace tensored with $\id_{U_I}$, compatible with
restrictions and transitions; open-cover descent identifies the realization
with the ordinary topological Gysin map.
\end{proof}

\subsection{Finite monodromy on weight gradeds}

\begin{lemma}\label{diag:finite}
Assume the Hodge--Tate hypothesis of \Cref{diag:variation}. The combined
monodromy on all $\Gr^W_aH^j(\mathscr H_t,\Q)$ has finite image. For a loop with
cohomology monodromy $T$ there is $M>0$ with $T^M$ unipotent and
\begin{equation}\label{diag:log}
 N=\tfrac1M\log(T^M)\ \text{satisfying}\ N(W_a)\subset W_{a-2}.
\end{equation}
The logarithm is a finite polynomial, and $N$ is a degree-zero derivation of
the cohomology ring.
\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 real part. Taking the torsion-free quotient of the integral
local system and intersecting with the rational weight sub-local systems gives
an invariant full lattice in each rational graded piece (saturated, as the
subspaces are rational). Automorphisms of a lattice preserving a positive
definite form form a finite group (columns have bounded norm in a fixed basis),
so each graded monodromy image is finite; finitely many degrees and weights
give the combined finiteness.

Choose $M$ killing the specified loop on all weight gradeds; then
$(T^M-1)W_a\subseteq W_{a-1}=W_{a-2}$ for even $a$ (and identically for odd $a$
since consecutive odd/even steps coincide). $T^M-1$ is nilpotent (finite
filtration), and its finite logarithm has the same lowering bound.

Parallel transport preserves products, so $\exp(kMN)=T^{Mk}$ is a ring
automorphism for every $k\ge0$; expanding the polynomial identity in $k$ and
comparing linear coefficients gives the derivation property, and monodromy
preserves degree.
\end{proof}

The next sections verify locally constant algebraic transitions for the family
with a moving marked point and its flag correspondence, and identify the scalar
unflagged bundle as a flat product, so that \Cref{diag:gysin} controls
push-pull and \Cref{diag:finite} supplies the weight-lowering logarithm.

% ============================================================
\section{Moving logarithmic poles and cohomology comparisons}
\label{sec:logarithmic}
% ============================================================

We build one family through which the Higgs and Betti correspondences are
compared. The point carrying the full flag must move over a compact curve. Its
motion changes the determinant, so we first compensate with a line bundle. The
resulting comparison applies to the whole fixed-determinant cohomology.

\subsection{The determinant correction and the scalar correspondence}

Choose $x_0\in C$ and first take $L=\mathcal O_C(d\,x_0)$ (we return to
arbitrary $L$ at the end of the section). The bundle $\mathcal O_C(d\,x)$ has a
canonical logarithmic connection $d_{d x}$ (its canonical meromorphic section
is horizontal), with residue $-d$ at $x$. Multiplying a connection by
$\lambda$ gives a $\lambda$-connection whose Leibniz rule is
$D(fs)=\lambda\,df\otimes s+fD(s)$.

Pull back the finite \'etale map $[n]:\Pic^0(C)\to\Pic^0(C)$ along
$C\to\Pic^0(C)$, $x\mapsto\mathcal O_C(d x-d x_0)$, and choose a connected
component $S$. Then $S$ is a smooth projective connected curve and $x:S\to C$
is finite \'etale and surjective. Let $\Delta\subset C\times S$ be the graph of
$x$. A Poincar\'e bundle gives $J$ on $C\times S$ with
\begin{equation}\label{log:root}
 J^{\otimes n}\simeq\mathcal O_{C\times S}\bigl(d\Delta-d(x_0\times S)\bigr)
 \otimes\pr_S^*B
\end{equation}
for some line bundle $B$ on $S$ (the only ambiguity). A line pulled back from
$S$ has a canonical relative connection along $C$; give the right side the
relative logarithmic connection from its canonical meromorphic section and the
base connection, and divide local forms by $n$ to define a compatible
connection on $J$ (division glues since the transition equation for
$J^{\otimes n}$ is $n$ times that 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 $\mathfrak t_0=\{(u_1,\dots,u_n)\in\mathbb A^n:\sum_i u_i=0\}$ and
$\Lambda=\mathbb A^1_\lambda\times\mathfrak t_0$. 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: slope stability for preserved
subbundles). At $\lambda=0$ the scalar residue vanishes, so $X_0=X$.

Let $\mathcal Y\to S\times\Lambda$ parametrize: a stable logarithmic
$\lambda$-connection $(E,D)$ with its only possible pole at $x(s)$ and
determinant $(\mathcal O_C(d x(s)),\lambda d_{d x(s)})$; and a full flag of
$E_{x(s)}$ preserved by the residue, acting on the ordered graded lines $Q_i$
by $-\lambda d/n+u_i$. The flag does not enter stability. Write $Y_{\lambda,u}$
for the fiber over $(\lambda,u)$, retaining all of $S$, so $Y_{\lambda,u}$ 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
$\mathcal O$ the same family with the flag forgotten. The scalar
correspondence over $\mathbb A^1_\lambda$ is
\begin{equation}\label{log:corr}
 X_\lambda\xleftarrow{\ \pr\ }O_\lambda\simeq S\times X_\lambda
 \xleftarrow{\ \pi\ }Z_\lambda\xrightarrow{\ \iota\ }Y_{\lambda,0},
\end{equation}
where $\pi$ forgets the flag (relative dimension $b$), $\iota$ is the
scalar-residue inclusion (codimension $b$; on $Y_{\lambda,0}$ the residue may
have a strictly flag-lowering nilpotent part, and $Z_\lambda$ requires it
vanishing), and $\pr$ has fiber $S$. The left identification is:

\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 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 the determinant operator as
prescribed; at $x_0$ the residue $+\lambda d/n$ of $J$ cancels the original
$-\lambda d/n$. This cancellation persists where the poles collide: in a
coordinate $z$ with moving section $z=a(s)$ the scalar polar part contributed
by $J$ is $-\frac{\lambda d}{n}\frac{dz}{z-a(s)}+\frac{\lambda d}{n}\frac{dz}{z}$,
and adding the fixed-pole form leaves only the first term, an identity of
relative meromorphic forms including $a(s)=0$. So the product has a single
logarithmic pole at the moving point, and the inverse tensors by $J^{-1}$.
Base-line twists do not change the map to coarse moduli, $\deg J_s=0$ preserves
stability, and at $\lambda=0$ the Higgs field of $J$ is zero, so the
characteristic polynomial is unchanged.
\end{proof}

The relative quasiprojective moduli exist by the standard construction
\cite[Section~2.2]{dCFHLogHodge}; fixing the determinant is the inverse image
of a rank-one section, and adding a residue-invariant full flag is projective
over the unflagged moduli with ordered residue eigenvalues as regular
functions, the prescribed-residue equations being closed. Universal bundles are
used only \'etale-locally: coprimality makes semistable pairs stable with scalar
automorphisms, and the stack is a scalar (or $\mu_n$-) gerbe over its coarse
space \cite[Lemma~5.3, Remark~5.4]{dCFHLogHodge}; the universal projective
bundle, $\End E$, and the ratios $Q_i\otimes Q_j^{-1}$ have trivial scalar
action and descend (individual $Q_i$ need not, and nothing needs them to).

\subsection{Smooth families and restriction isomorphisms}

\begin{proposition}\label{log:families}
The morphisms $\mathcal Y\to S\times\Lambda$ (relative dimension $2R+2b$) and
$\mathcal X\to\mathbb A^1$ (relative dimension $2R$) are smooth, and their
total spaces are semiprojective for scaling the operators (which scales
$\lambda$ and every $u_i$ with weight one), as are the inverse images in
$\mathcal Y$ of linear subspaces of $\Lambda$ through zero. In \eqref{log:corr},
$\pi$ 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}
\emph{Zero limits.} Given an unflagged object, keep the bundle and scale the
operator; semistable reduction \cite[Proposition~4.48]{FHZ2025} (valid for
$\GL_n/\C$) replaces the resulting DVR family by a semistable one after a finite
extension without changing the generic fiber. The replacement's determinant
agrees with the prescribed one generically, hence over the DVR by separatedness
of the rank-one moduli; a generic flag extends by properness of the pole flag
variety; residue preservation and the ordered diagonal residue equations are
closed, so persist; coprimality makes the limiting semistable pair stable. The
scalar-residue condition behaves the same. A finite DVR extension suffices for a
limit in the coarse space (embed it quasiprojectively in projective space: the
original orbit has a unique projective limit, which is the finite-base-change
limit and lies in the coarse space).

\emph{Smoothness over $\lambda=u=0$.} Put $\mathcal F=\mathrm{ParEnd}_0(E)$
(trace-free endomorphisms preserving the pole flag) and
$\mathcal G=\mathcal F^\vee\otimes K_C$ (trace-free logarithmic Higgs fields
with strictly flag-lowering residue, up to trace pairing). The relative
deformation complex is $\mathcal F\xrightarrow{[\theta,-]}\mathcal G$. For an
Artin extension with prescribed lifts of $s,\lambda,u$: bundle, determinant, 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 affine-linear
prescriptions (trace splits in characteristic zero); the gluing obstruction lies
in $H^1(C,\mathcal G)$, changed by the image of $H^1(C,\mathcal F)$ under the
complex. The map on $H^1$ is surjective (its Serre dual is the map on $H^0$
whose kernel is zero by stability). So every lifting problem solves, and with
$\chi(\mathcal F)=-(n^2-1)(g-1)-b=-R-b$ the relative tangent dimension is
$-2\chi(\mathcal F)=2R+2b$. For the scalar unflagged family replace
$\mathcal F$ by $\End_0(E)$ and get $2R$; finite scalar stabilizers act trivially
on these deformations, so coarse spaces are smooth.

The nonsmooth locus is closed and scaling-invariant; if nonempty it would
contain a point's zero limit, contradicting the computation over
$\lambda=u=0$. So the morphism is smooth everywhere.

\emph{Semiprojectivity.} A scaling-fixed point has $\lambda=u=0$ and vanishing
logarithmic Hitchin coefficients (positive weights); the logarithmic Hitchin map
is proper with base $\bigoplus H^0(C,K_C(x(s))^{\otimes i})$
\cite[Section~2.4, Equation~(4)]{dCFHLogHodge}, and the fixed locus is closed in
its nilpotent fiber after adding flags and determinant/residue conditions; with
flags and $S$ proper this fixed locus is proper. With the zero limits this gives
semiprojectivity, inherited by inverse images of linear subspaces through zero.

Finally $\pi$ is the full flag bundle of the universal projective bundle; on
$u=0$, subtracting the prescribed scalar from the residue gives a section of the
bundle of strictly flag-lowering matrices with zero scheme $\mathcal Z$, a
regular zero scheme of codimension $b$ (by the dimensions just proved), and 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 is an isomorphism
\begin{equation}\label{log:rho}
 \rho_{\lambda,u}:H^*(\mathcal Y,\Q)\xrightarrow{\sim}H^*(Y_{\lambda,u},\Q),
\end{equation}
as is restriction to the inverse image of any line through zero in $\Lambda$ and
from it to any fiber; the analogues hold for the four families in
\eqref{log:corr} over the $\lambda$-line, commuting with pullbacks and with the
proper Gysin maps (including integration in $S$).
\end{corollary}

\begin{proof}
The core of a smooth semiprojective variety is the union of sets whose scaling
orbits have limits at $\infty$; its inclusion is a cohomology isomorphism
\cite[Theorem~1.3.1]{HRV2015}. Every $\Lambda$-coordinate has positive weight, so
a $\mathcal Y$-point with such a limit lies over zero; hence $\mathcal Y$ and its
inverse image $\mathcal Y_\ell$ over a line $\ell$ through zero share the same
core, and $H^*(\mathcal Y)\to H^*(\mathcal Y_\ell)$ is an isomorphism. The map
$\mathcal Y_\ell\to\ell$ is smooth and surjective (image a scaling-invariant open
set containing zero), so \cite[Corollary~1.3.3]{HRV2015} makes fiber restriction
an isomorphism, proving \eqref{log:rho} and its line version; the same holds for
$\mathcal X,\mathcal O,\mathcal Z$. 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/product projections are smooth
proper, so transverse base change gives the claimed compatibilities.
\end{proof}

\subsection{Riemann--Hilbert with a moving puncture}

Let $V,q,Y^{\rm B}\to V$ be the ordered eigenvalue family of
\Cref{sec:betti}. Let
\begin{equation}\label{log:nonresonance}
 U=\Bigl\{u\in\mathfrak t_0^{\rm an}:t(u)=(\zeta e^{-2\pi\sqrt{-1}u_i})_i\in V,\
 u_i-u_j\notin\Z\setminus\{0\}\Bigr\}.
\end{equation}
Zero differences are allowed, so $0\in U$. There is a flat bundle over $S$ with
algebraic fiber family $Y^{\rm B}\to V$: use local trivializations of the
punctured surfaces $C\setminus\{x(s)\}$ and record a full flag of local
subsystems near the puncture, with the peripheral loop equal to the commutator
product. Changes of marking act by group words on the handle matrices and by the
conjugating word on the peripheral flag, algebraic over $V$.

\begin{proposition}\label{log:rh}
Over $U$, Riemann--Hilbert identifies $Y_{1,u}$ with the total space over $S$ of
this flat bundle at $t(u)$, as a homeomorphism of families. At $u=0$ it
identifies the scalar correspondence \eqref{log:corr} at $\lambda=1$ with the
flat Betti scalar correspondence, compatibly with pullbacks and Gysin maps.
\end{proposition}

\begin{proof}
The transition maps are locally constant and satisfy the cocycle condition as
algebraic maps on coarse spaces. Let $\delta$ be the commutator-product word; a
change of marking gives an outer automorphism $\varphi$ with
$\varphi(\delta)=w\delta w^{-1}$, acting on representations by $\rho\varphi$ and
on flags by $\rho(w)$. Inner changes give conjugation; changing $w$ multiplies
it by an element centralizing $\delta$, and the cyclically reduced $\delta$ is
not a proper power (each positive handle generator occurs once), so this
centralizer is $\langle\delta\rangle$, a power of peripheral monodromy preserving
every invariant flag, including repeated or nonsemisimple cases. The maps
compose strictly on the geometric quotient, so no eigenline choices are needed.

Local exponents are $r_i=-d/n+u_i$; in a disc write a logarithmic connection as
$d+A(z)\,dz/z$ with $A(0)=A_0$. Regular-singular reduction to $d+A_0\,dz/z$ uses
operators $k+\mathrm{ad}(A_0)$ ($k\ge1$) with eigenvalues $k+r_i-r_j$,
invertible under \eqref{log:nonresonance} even for repeated eigenvalues or a
nonzero nilpotent part; inverse norms $O(1/k)$ give a convergent majorant with
continuous parameter dependence. In the constant-residue model a
residue-invariant flag gives a monodromy-invariant flag, and conversely the
prescribed exponents select one logarithm per eigenvalue (equal exponential
values have equal exponents by \eqref{log:nonresonance}), giving an inverse
logarithmic extension; the construction is intrinsic and, in rank one, yields
$(\mathcal O_C(d x(s)),d_{d x(s)})$, so the determinant is right. The subproduct
condition makes representations irreducible, hence stable.

Holonomy along local moving generators plus the normal form makes forward maps
continuous; they are continuous bijections between equal-dimensional manifolds
(\Cref{log:families} and Betti smoothness), so invariance of domain gives inverse
continuity, and for fixed $s$ they are holomorphic, preserving complex
orientations. At $u=0$ scalar residues correspond to scalar monodromy and flags
match, and the proper maps' algebraic-fiber orientations agree with
topological Gysin.
\end{proof}

\begin{lemma}\label{log:scalar-flat-product}
At $\lambda=1$, the product identification of \Cref{log:scalar-product}
identifies the scalar unflagged Betti family with $S\times X^{\rm B}$ as a flat
bundle with algebraic fiber; its projection to $X^{\rm B}$ is $\pr$.
\end{lemma}

\begin{proof}
Away from the poles the $n$th power of the connection on $J$ is canonically
trivial, so its holonomies lie in $\mu_n$; locally choose the handle generators
outside a disc containing both poles, and their line holonomies are continuous
$\mu_n$-valued, hence locally constant. Tensoring multiplies the handle matrices
by these constants, an algebraic map on the representation variety, also near
collisions. So the algebraic product identification is compatible with the
locally constant algebraic Betti transitions.
\end{proof}

We now have the actual flat diagrams to which \Cref{sec:diagram} applies; in
particular the weight-zero treatment of the cochains of $S$ concerns the same
projection $\pr$ as the logarithmic correspondence.

\subsection{The nonabelian-Hodge comparison on full cohomology}

The restriction isomorphisms for $\mathcal X$ and Riemann--Hilbert give
\begin{equation}\label{log:comparison}
 H^*(X^{\rm B})\xrightarrow{\sim}H^*(X_1)\xleftarrow{\sim}H^*(\mathcal X)
 \xrightarrow{\sim}H^*(X).
\end{equation}
We must check it transports the weight filtration of nonabelian Hodge theory on
the whole cohomology, including nontrivial $\Gamma$-characters.

\begin{lemma}\label{log:finite-ambiguity}
Let a finite group $G$ act algebraically on a smooth connected complex $B$ with
quotient $q:B\to B/G$. If $g:A\to B$ 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. $B$ is irreducible; removing nonidentity fixed loci gives
a dense connected free open $B^\circ$. On $g^{-1}(B^\circ)$ there is a unique
$\gamma(a)$ with $f(a)=\gamma(a)g(a)$, locally constant because $B\to B/G$ is a
covering there, hence constant since $g^{-1}(B^\circ)$ 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 $H^*(X^{\rm B},\Q)$ exactly as the nonabelian-Hodge identification does.
The conclusion is unchanged for arbitrary $L\in\Pic^d(C)$, the Higgs spaces
being identified by tensoring with a degree-zero $n$th root of the determinant
ratio.
\end{proposition}

\begin{proof}
The full twisted fixed-determinant nonabelian-Hodge diffeomorphism, including
its $\Gamma$-equivariance, is in \cite[Sections~2,~5]{HT2003}; we compare its
cohomology map with the particular logarithmic family. For $(E,\theta)\in X$
choose a harmonic metric $H$ normalized to a fixed determinant metric, let
$\partial_H$ be the $(1,0)$ part of its Chern connection, and for
$0\le\tau\le1$ take holomorphic structures and logarithmic $\tau$-operators
\begin{equation}\label{log:harmonic-path}
 \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.
\end{equation}
Its projectivization is the usual harmonic preferred section
\cite[Section~4]{Simpson1997}. The scalar adjustment is correct: traces of
$\theta,\theta^{\dagger_H}$ vanish so the determinant structure is fixed and the
determinant operator is exactly $\tau d_{d x_0}$; the trace-free holomorphy
equation follows from $\bar\partial_E\theta=0$, $\partial_H\theta^{\dagger_H}=0$,
and the projective Hitchin equation; the only nonregular term is the scalar
logarithmic term at $x_0$.

For $\tau\ne0$ divide by $\tau$; the puncture monodromy is $\zeta I$, and a proper
invariant subspace of rank $r$ would have $\zeta^r=1$ by taking determinants of
the commutator relation, against $\gcd(n,d)=1$. So these connections are
irreducible and stable, and \eqref{log:harmonic-path} lies in $\mathcal X$
throughout, recovering the Higgs pair at $\tau=0$.

This is a continuous homotopy $X\to\mathcal X$: normalized harmonic metrics vary
smoothly (trace-free metric linearization self-adjoint elliptic with
stability-killed kernel), matching \cite[Section~4]{Simpson1997}; to pass to the
analytic topology of coarse moduli, twist so bundles are generated with
vanishing $H^1$; Dolbeault kernels give continuously varying bases, whose
evaluations give continuous quotient presentations in which the logarithmic
operators (morphisms from the first jet bundle with fixed symbol) have continuous
coefficients, hence continuity on the stable quotient.

At $\tau=0$ this is $X\hookrightarrow\mathcal X$; at $\tau=1$ it factors through
$X_1$ and then Riemann--Hilbert to $F:X\to X^{\rm B}$, and $F^*$ is exactly
\eqref{log:comparison} since both restrictions from $\mathcal X$ are
isomorphisms. Projectivizing gives the usual projective nonabelian-Hodge map.
Let $\Phi$ be the fixed-determinant nonabelian-Hodge diffeomorphism; $F$ and
$\Phi$ agree after projectivization, and two determinant-one lifts differ per
handle by an $n$th root of unity, so their classes differ by $\Gamma$. Since
$H^0(X^{\rm B})=\Q$ (\Cref{norm:low-cohomology}) makes $X^{\rm B}$ connected and
it is smooth, \Cref{log:finite-ambiguity} gives $F=\gamma\Phi$; the algebraic
$\Gamma$-action preserves Deligne weights, so $F^*$ and $\Phi^*$ transport the
weight filtration identically on every group, with no averaging.

For arbitrary $L$, choose $M\in\Pic^0(C)$ with
$M^{\otimes n}\simeq L\otimes\mathcal O_C(-d x_0)$; tensoring by $M$ identifies
the Higgs spaces algebraically over the same Hitchin base, leaving projective
Higgs bundles and harmonic correspondences unchanged, and the same finite-character
argument shows this identification preserves the transported weight filtration
(and the perverse filtration, as an isomorphism over the Hitchin base).
\end{proof}

% ============================================================
\section{A common cohomological operator}
\label{sec:translation}
% ============================================================

We turn an elementary modification of Higgs bundles into monodromy of Betti
spaces. All cohomology identifications here are the restriction maps of
\Cref{log:restriction}, which pin down one operator before we bound its effect
on either filtration.

\subsection{Modification of two eigenlines}

Choose $v\in\sqrt{-1}\R^n$ with sum zero, distinct entries, and no vanishing
proper nonempty partial sum (generic: the excluded conditions are finitely many
proper real hyperplanes). Set $\ell=(1,-1,0,\dots,0)$. 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
the residue eigenvalue by $-\lambda$; a negative shift changes it by $+\lambda$.
We perform the shifts of $\ell$ 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. 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}\to Y_{a,v-a\ell},\qquad 0\le a\le1.
\end{equation}
At $a=0$ they give an algebraic automorphism of $Y_{0,v}$; more generally they
give 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 a single modification choose a local frame adapted to the current residue
eigenspaces: off-diagonal coefficients involving the chosen eigenline vanish at
the pole, so the modified lattice is preserved by the operator, with the
prescribed new residue eigenvalues. Their imaginary parts stay distinct along
\eqref{trans:path}, even after the first step, so we can reselect before the
second modification. The two-step composite preserves the determinant and its
operator (the shifts sum to zero), and its inverse does the opposite single
modifications in reverse with reselection between; both vary algebraically where
successive residues are distinct.

Stability is not automatic. On the Higgs line with $z\ne0$, a proper invariant
saturated subbundle would have a trace Higgs differential with a single possible
simple pole, whose residue $z\sum_{i\in I}v_i$ (proper nonempty $I$) is nonzero,
against the residue theorem; modified bundles obey the same argument. Distinct
residue eigenvalues split the local operator into formal eigenline summands;
shifting and shifting back proves invertibility.

For $a>0$ divide by $a$: a proper invariant monodromy subspace would have
determinant monodromy one around the puncture, but the imaginary part of the sum
of its exponents is $a^{-1}\sum_{i\in I}\mathrm{Im}\,v_i\ne0$; and differences of
exponents have nonzero imaginary part, so no integral resonance occurs, keeping
inverse modifications valid and stable.

The characteristic polynomial on the Higgs line is unchanged away from the pole,
hence unchanged as a meromorphic polynomial. Modifications change only the
logarithmic lattice at the puncture; at $a=1$ the exponent changes are integers,
so the meromorphic flat bundle and ordered monodromy eigenlines are unchanged,
which is exactly the endpoint identification.
\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 with the endpoint $F_1$ we need constancy along the two families of
\eqref{trans:path}; their fibers are nonproper, so the restriction isomorphisms,
not properness, supply it.

\begin{lemma}\label{log:compact-cycles}
Let $I$ be a connected interval and $p:M\to I$, $q:N\to I$ smooth submersions
with finite-dimensional rational fiber cohomology. Suppose $a:M\to A$ and
$b:N\to B$ restrict to isomorphisms
$\alpha_t:H^*(A,\Q)\xrightarrow{\sim}H^*(M_t,\Q)$ and
$\beta_t:H^*(B,\Q)\xrightarrow{\sim}H^*(N_t,\Q)$ for every $t$. If $F:M\to N$
is continuous over $I$, then $\alpha_t^{-1}F_t^*\beta_t$ is independent of $t$.
\end{lemma}

\begin{proof}
Fix a degree and $t_0$; choose compact cycles giving a rational homology basis of
$M_{t_0}$. A lift of the interval vector field near their compact carriers flows
for a common short time, transporting them to nearby fibers so that they sweep
homologies in $M$. Their images in $A$ are homologous, so pairings with a fixed
basis of $H^*(A)$ are constant, and the restriction isomorphisms keep them a
basis of the fiber homology. Applying $bF$ gives homologies in $B$, so pairings
with any class of $H^*(B)$ are also constant; these are the matrix coefficients
of $\alpha_t^{-1}F_t^*\beta_t$ in fixed bases. Hence locally constant, including
one-sided neighborhoods of endpoints, hence constant on $I$.
\end{proof}

Apply with $I=[0,1]$ and $M,N$ the pullbacks of $\mathcal Y\to\Lambda$ along
$a\mapsto(a,v)$ and $a\mapsto(a,v-a\ell)$ (smooth submersions by
\Cref{log:families}), with $F:M\to N$ from \eqref{trans:path}. Take both ambient
spaces $\mathcal Y$; fiber restrictions induce the isomorphisms of
\Cref{log:restriction}, so constancy gives
\begin{equation}\label{trans:endpoint}
 \mathsf T=\rho_{1,v}^{-1}F_1^*\rho_{1,v-\ell}.
\end{equation}
No global local triviality of the nonproper smooth families is needed.

The path $u=v-a\ell$, $0\le a\le1$, maps under
$t(u)=(\zeta e^{-2\pi\sqrt{-1}u_i})_i$ to a loop in $V$: every forbidden proper
subproduct has modulus different from one (its partial $v$-sum has nonzero
imaginary part), excluding resonance along the whole path. By \Cref{log:rh}
this is a path in the actual moving-puncture Betti family, and restrictions of
$\mathcal Y$-classes are flat sections (Betti direct images are locally constant
with ordinary base change). Since $F_1$ is the endpoint identity,
\eqref{trans:endpoint} identifies $\mathsf T$ with eigenvalue monodromy.

\begin{lemma}\label{trans:logarithm}
There is $M>0$ such that
\begin{equation}\label{trans:N}
 N=\tfrac1M\log(\mathsf T^M)
\end{equation}
is a well-defined nilpotent derivation of $H^*(\mathcal Y,\Q)$. With
$N_{\lambda,u}=\rho_{\lambda,u}N\rho_{\lambda,u}^{-1}$, one has
$N_{1,0}W_j\subset W_{j-2}$ on $H^*(Y_{1,0},\Q)$ with the diagram weights of
\Cref{sec:diagram}.
\end{lemma}

\begin{proof}
The full-flag Betti family has integral lisse direct images, ordinary base change,
and Hodge--Tate fibers by \Cref{sec:betti}, and algebraic locally constant
transitions by \Cref{log:rh}, so \Cref{diag:finite} applies. Choose $M$ killing
the monodromy on every weight-graded piece of the total cohomology (finitely
many). Then $\mathsf T^M-1$ is nilpotent, \eqref{trans:N} is a finite log sum,
and since weights are even $N$ 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 is in $V$ ($\zeta$ has order $n$), and for $a>0$ the
imaginary-part argument applies. Parallel transport preserves diagram weights and
coincides with the restriction identifications, giving the estimate at $(1,0)$.
\Cref{diag:finite} makes $N$ a derivation, and restriction isomorphisms (ring
isomorphisms) transport it to $H^*(\mathcal Y,\Q)$.
\end{proof}

\subsection{Flag push-pull and the weight bound}

For the scalar correspondence at $\lambda$ set
$\mathsf A_\lambda=\pi_{\lambda,*}\iota_\lambda^*$ and
$\mathsf B_\lambda=\iota_{\lambda,*}\pi_\lambda^*$; the unflagged moving-pole
space is $S\times X_\lambda$, with projection $\pr_\lambda$. 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)\to 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 $-2$.
\Cref{sec:specialization} derives the perverse bound from this composite over the
Hitchin base; we first record the weight bound.

\begin{proposition}\label{trans:weights}
Under nonabelian Hodge theory,
$f'W_jH^m(X^B,\Q)\subset W_{j-4}H^{m-2}(X^B,\Q)$.
\end{proposition}

\begin{proof}
The scalar correspondence and its Gysin maps commute with the restriction
isomorphisms of \Cref{log:restriction}, so \eqref{trans:fprime} transports to the
same expression at $\lambda=1$. By \Cref{log:nah} these isomorphisms transport the
Betti weight filtration on the full cohomology of $X^B$; the possible difference
from a chosen nonabelian-Hodge identification is one algebraic $\Gamma$-action,
preserving every weight subspace, including in variant cohomology.

At $\lambda=1$ the scalar Betti space is the flat algebraic product
$S\times X^B$ of \Cref{log:scalar-flat-product}, whose diagram structure gives
weight zero to the cochains of $S$ and the ordinary Deligne weights to
$H^*(X^B)$. So $\pr_1^*$ and $\pr_{1,*}$ preserve weights. The flag maps are maps
of the same flat transition diagrams, and their ordinary topological Gysin maps
are the diagram Gysin maps, shifting weight by $2b$ ($\mathsf B_1$) and $-2b$
($\mathsf A_1$); these cancel, while $N_{1,0}^2$ lowers weight by four by
\Cref{trans:logarithm}.
\end{proof}

It remains to realize $f'$ over the Hitchin base (perverse bound) and compute its
double commutator with $e$.

% ============================================================
\section{Specialization over a Higgs line}
\label{sec:specialization}
% ============================================================

We prove the perverse-filtration bound for $f'$ of \eqref{trans:fprime}. The
translation is defined over a logarithmic Hitchin base away from zero on a Higgs
line; nearby cycles extend it to the zero fiber as a sheaf endomorphism; a final
integration over the moving-point curve supplies the shift $-2$. The comparison
with ordinary cohomology uses the restriction isomorphisms of
\Cref{log:restriction}, not properness of the parameter morphism.

\subsection{The proper morphism and its relative translation}

Keep $v$ and the translation of \Cref{sec:translation}. Let $p_C:C\times S\to S$
be the projection and $\Delta\subset C\times S$ the graph of $x:S\to C$. The
vector bundle of trace-free logarithmic Hitchin coefficients is
\[
 \mathcal E=\bigoplus_{i=2}^n p_{C,*}\bigl(K_{C\times S/S}(\Delta)^{\otimes i}\bigr),
\]
with 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 $H^1$ vanishes). The inclusion of ordinary coefficient spaces
is a closed vector-subbundle embedding $j:S\times A\hookrightarrow\mathcal E$
(both spaces commute with base change, and their quotient has constant rank).

Let $D=\mathbb A^1_z$, $D^*=D\setminus\{0\}$, and
$\mathcal Y_v=\mathcal Y\times_\Lambda D$ with $z\mapsto(0,zv)$; 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\to\mathcal E\times D,
\end{equation}
with zero fiber $g_0:Y_{0,0}\to\mathcal E$.

\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 $Y_{0,0}$ or $Y_{0,v}$ is an
isomorphism.
\end{lemma}

\begin{proof}
Smoothness and the restriction assertions follow from \Cref{log:families} and
\Cref{log:restriction}. For properness, start with the relative logarithmic Higgs
Hitchin morphism over $S$; imposing determinant $\mathcal O_C(dx(s))$ and trace
zero is closed; adding a residue-invariant full flag is projective over that;
and imposing ordered diagonal residues $=zv_i$ is closed after adjoining $z$. So
properness of the logarithmic Hitchin morphism (\Cref{sec:logarithmic}) gives
properness of \eqref{spec:hitchin-line}, and coprimality keeps the semistable
limits stable.

For $z\ne0$ the residues $zv_i$ are distinct, so the elementary modifications
defining $T_v$ and their inverses are algebraic in $z$ and stable (trace-residue
argument of \Cref{sec:translation}). A modified Higgs field agrees with the
original meromorphic one away from the pole, so characteristic coefficients are
equal as sections of $K_C(x(s))^{\otimes i}$, proving $T_v$ is over $\mathcal E$;
multiplying the Higgs field by $z$ identifies fiber $1$ with fiber $z$, whose
inverse is division by $z$, giving the product over $D^*$. The product need not
fix the Hitchin coefficient; only the translation must.
\end{proof}

\subsection{Specialization and ordinary cohomology}

Throughout, nearby cycles are unshifted, with the formalism recalled in
\cite[Sections~5.5, 5.8]{dCM2009}.

\begin{lemma}\label{spec:specialization}
Let $Y\to D=\mathbb A^1$ be smooth, and $g:Y\to B\times D$ proper, commuting with
the maps to $D$. Suppose $Y|_{D^*}\simeq Y_1\times D^*$ over $D^*$ and that the
ordinary restrictions $r_i:H^*(Y,\Q)\to H^*(Y_i,\Q)$, $i=0,1$, are isomorphisms.
An automorphism $T$ of $Y|_{D^*}$ over $B\times D^*$ then induces an endomorphism
$\tau_0$ of $Rg_{0,*}\Q_{Y_0}$ with cohomological action
\[
 H^*(\tau_0)=r_0r_1^{-1}T_1^*r_1r_0^{-1}.
\]
Neither $B$ nor $Y_i$ need be proper.
\end{lemma}

\begin{proof}
Smoothness gives $\Q_{Y_0}\simeq\psi\Q_Y$; proper compatibility of nearby cycles
gives $Rg_{0,*}\Q_{Y_0}\simeq Rg_{0,*}\psi\Q_Y\simeq\psi(Rg_*\Q_Y)$. Apply
nearby cycles to the pullback endomorphism from $T$ on the punctured direct
image to define $\tau_0$.

To read its global action without interchanging nearby cycles with a nonproper
direct image to a point: put $K=Rg_*\Q_Y$ on $B\times D$, let
$\widetilde D^*\to D^*$ be the universal cover,
$\widetilde j_B:B\times\widetilde D^*\to B\times D$, $i_B:B\times\{0\}\to B\times D$,
$\widetilde K=\widetilde j_B^*K$, $\widetilde Y^*=Y|_{D^*}\times_{D^*}\widetilde D^*$.
Proper base change identifies $\widetilde K$ with the direct image from
$\widetilde Y^*$, and downstairs $\psi K=i_B^*R\widetilde j_{B,*}\widetilde K$.
Restriction gives a natural morphism
\begin{equation}\label{spec:comparison-map}
 R\Gamma(\widetilde Y^*,\Q)\simeq R\Gamma(B\times D,R\widetilde j_{B,*}\widetilde K)
 \to R\Gamma(B,\psi K)\simeq R\Gamma(Y_0,\Q),
\end{equation}
the last step using proper nearby cycles and smoothness of $Y\to D$. Composing
\eqref{spec:comparison-map} with restriction from $R\Gamma(Y,\Q)$ is ordinary
restriction to $Y_0$ (the specialization unit for $K$ corresponds under proper
direct image to $\Q_{Y_0}\to\psi\Q_Y$).

Choose a lift of $1\in D^*$; product and contractibility of $\widetilde D^*$
identify $H^*(\widetilde Y^*)$ with $H^*(Y_1)$ by restriction to that lift, so
pullback from $H^*(Y)$ becomes $r_1$, an isomorphism; composed with
\eqref{spec:comparison-map} (which then must also be an isomorphism) it is $r_0$.

The comparison was built on $B\times D$ so that it is equivariant for the
operator: $T$ over $B\times D^*$ induces endomorphisms of $\widetilde K$,
$R\widetilde j_{B,*}\widetilde K$, $\psi K$, and every arrow respects them.
Restricting at the chosen lift identifies the action on the domain with $T_1^*$
and on the codomain with $H^*(\tau_0)$.
\end{proof}

Apply \Cref{spec:specialization} to \eqref{spec:hitchin-line}: the resulting
$\tau_0$ on $Rg_{0,*}\Q_{Y_{0,0}}$ acts on cohomology as the translation
transported to $Y_{0,0}$ by \Cref{log:restriction}, i.e.\ the operator whose
logarithm is $N_{0,0}$. A finite polynomial realizes this logarithm over the
Hitchin base: choose $M$ from \Cref{trans:logarithm} and $r\ge1$ with
$(\mathsf T^M-1)^r=0$ on total cohomology, and put
\[
 p(t)=\frac1M\sum_{a=1}^{r-1}\frac{(-1)^{a+1}}{a}(t^M-1)^a.
\]
Then $p(\tau_0)$ is an endomorphism of $Rg_{0,*}\Q_{Y_{0,0}}$ acting as
$N_{0,0}$ (no unipotence of $\tau_0$ in the derived category is needed).

\subsection{The derived shift}

\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_*\Q_X\to Rh_*\Q_X[-2]
\end{equation}
in the constructible derived category on $A$; consequently
$f'(P_kH^m(X,\Q))\subseteq P_{k-2}H^{m-2}(X,\Q)$ for all $m,k$.
\end{proposition}

\begin{proof}
Let $O_0=S\times X$ via \Cref{log:scalar-product}, $q=\id_S\times h$. Both maps of
the scalar flag correspondence \eqref{log:corr} at $\lambda=0$ are over
$\mathcal E$; the flag projection has relative dimension $b$, the scalar
inclusion codimension $b$. With
\[
 F_O=j_*Rq_*\Q_{O_0},\quad F_Y=Rg_{0,*}\Q_{Y_{0,0}},\quad
 F_Z=R(g_0\circ\iota)_*\Q_{Z_0},
\]
their pullbacks and proper Gysin maps give
$\mathsf B_0:F_O\to F_Y[2b]$ and $\mathsf A_0:F_Y\to F_O[-2b]$. Composing with the
appropriate shifts of $p(\tau_0)^2$ gives an endomorphism of $F_O$ of net shift
zero, acting on cohomology as $\mathsf A_0N_{0,0}^2\mathsf B_0$; since $j_*$ is
fully faithful this is an endomorphism of $Rq_*\Q_{O_0}$ over $S\times A$.

Let $p_S:S\times A\to A$ be the projection and $K=Rh_*\Q_X$. The product
identification gives $Rp_{S,*}Rq_*\Q_{O_0}\simeq K\otimes R\Gamma(S,\Q)$. Push
the endomorphism to $A$, precompose with the projection unit
$K\to K\otimes R\Gamma(S,\Q)$, and postcompose with the orientation trace
$K\otimes R\Gamma(S,\Q)\to K[-2]$. The result is \eqref{spec:derived-lowering};
on ordinary cohomology the unit is $\pr_0^*$, the trace is $\pr_{0,*}$, and the
middle map is the flag composite, so the operator is $f'$.

With $K_{\mathrm{norm}}=K[R]$, functoriality of perverse truncation gives
\[
 {}^p\tau_{\le k}K_{\mathrm{norm}}\to{}^p\tau_{\le k}(K_{\mathrm{norm}}[-2])
 =({}^p\tau_{\le k-2}K_{\mathrm{norm}})[-2],
\]
and taking hypercohomology in degree $m-R$ then its image in ordinary cohomology
gives the inclusion, without a splitting or multiplicativity of $P$.
\end{proof}

The derived morphisms and their shifts are collected in Table~\ref{spec:shift-table}.
The first panel holds morphisms on $\mathcal E$; composing the flag maps and
$p(\tau_0)^2$ gives an endomorphism of $F_O$, descended through $j$ and pushed by
$Rp_{S,*}$ to a degree-zero endomorphism of $K_S=K\otimes R\Gamma(S,\Q)$, and
unit/trace give $K\to K[-2]$. No perverse exactness of $Rp_{S,*}$ is used. Weight
columns refer to cohomological actions transported to $\lambda=1$.

\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
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. A weight bound $s$ means
$W_k$ maps into $W_{k+s}$ on the Betti side.}
\label{spec:shift-table}
\end{table}

% ============================================================
\section{The translation class and the double commutator}
\label{sec:commutator}
% ============================================================

The preceding sections bound how $f'$ changes both filtrations; now we compute
its double commutator with $e$. 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 to $e$.

\subsection{The characteristic class under translation}

The universal projective bundle on $C\times\mathcal Y$ defines
\[
 e_{\mathcal Y}=\frac1{2n}\int_C\ch_2(\End E)\in H^2(\mathcal Y,\Q).
\]
The ordered flag lines $Q_i$ exist on local universal-bundle charts; their
ratios $Q_i\otimes Q_j^{-1}$ descend. Define the global classes
\[
 y=c_1(Q_1\otimes Q_2^{-1}),\qquad t_C=x^*c_1(T_C),
\]
the second pulled back from $S$; retain the same notation on each fiber
$Y_{\lambda,u}$.

\begin{lemma}\label{calc:translation-formula}
For the translation $T:Y_{0,v}\to Y_{0,v}$ of $\ell=(1,-1,0,\dots,0)$ and every
integer $k\ge0$,
\begin{equation}\label{calc:translation-polynomial}
 T^{k*}e_{\mathcal Y}=e_{\mathcal Y}+k\,y+k^2\,t_C.
\end{equation}
Consequently the logarithm of \Cref{sec:translation} satisfies
\begin{equation}\label{calc:logarithm}
 N(e_{\mathcal Y})=y
\end{equation}
on $H^*(\mathcal Y,\Q)$ and on every fiber.
\end{lemma}

\begin{proof}
Work on $Y_{0,v}$. Let $i:\Delta\hookrightarrow C\times Y_{0,v}$ be the graph of
the moving point, with normal line $x^*T_C$, so
$\mathcal O(\Delta)|_\Delta\simeq x^*T_C$ and $c_1(N_\Delta)=t_C$. The residues
$v_i$ are distinct, so the Higgs operator has uniquely labeled eigenline
summands on the formal neighborhood of $\Delta$ (lift the simple residual
eigenvalues and idempotent projectors to all finite orders; a unit rescaling of
a logarithmic-canonical frame leaves the projectors fixed). Hence the blocks of
$\End E$ are global formal line bundles.

The $k$-fold translation twists the $i$th formal line lattice by
$k\ell_i\Delta$, shifting the $(i,j)$ block by $m_{ij}=k(\ell_i-\ell_j)$. For a
line block restricting to $F$ on $\Delta$,
\begin{equation}\label{calc:signed-jets}
 [\text{new}]-[\text{old}]=\sum_{a=1}^{m}\bigl[i_*(F\otimes N_\Delta^{\otimes a})\bigr],
\end{equation}
where for $m<0$ the right side is the negative of the sum over $m<a\le0$ and
vanishes for $m=0$. This follows from successive pole-order quotients
(and the dual sequence between the smaller lattice and the original for $m<0$;
$m=-1$ gives $-[i_*F]$, the kernel transformation).

This identity is global in K-theory: choose a common small lattice inside the
original and modified endomorphism bundles; their quotients are supported on a
finite formal neighborhood of $\Delta$, and the formal projectors and pole-order
filtration give associated graded terms that are graph pushforwards of descended
$Q_i\otimes Q_j^{-1}\otimes N_\Delta^{\otimes a}$. Subtracting cancels common
orders and leaves exactly \eqref{calc:signed-jets}, so the identity holds in the
Grothendieck group of coherent sheaves on $C\times Y_{0,v}$.

Grothendieck--Riemann--Roch for the graph \cite[Section~15.2]{Fulton1998} gives
$\ch(i_*F)=i_*(\ch(F)\,\mathrm{td}(N_\Delta)^{-1})$, so a summand of
\eqref{calc:signed-jets} contributes $c_1(F)+(a-\tfrac12)t_C$ to
$\int_C\ch_2$ (the inverse Todd class begins $1-t_C/2$, giving the normal sign).
The elementary identities
$\sum_{a=1}^m1=m$ and $\sum_{a=1}^m(a-\tfrac12)=m^2/2$ hold for every integer $m$.

Write $w_i=c_1(Q_i)$ on a chart; only differences $w_i-w_j$ and common-translation
invariants occur. Summing over $F=Q_i\otimes Q_j^{-1}$,
\[
 T^{k*}e_{\mathcal Y}-e_{\mathcal Y}
 =\frac1{2n}\sum_{i,j}\Bigl(k(\ell_i-\ell_j)(w_i-w_j)
 +\frac{k^2}{2}(\ell_i-\ell_j)^2t_C\Bigr).
\]
With $\sum_i\ell_i=0$,
$\sum_{i,j}(\ell_i-\ell_j)(w_i-w_j)=2n\sum_i\ell_iw_i$ and
$\sum_{i,j}(\ell_i-\ell_j)^2=2n\sum_i\ell_i^2$, which for this cocharacter are
$2ny$ and $4n$, proving \eqref{calc:translation-polynomial}; the lattice identity
was global on the coarse space before applying GRR, so this is a global
cohomological identity.

For $N$, replace $k$ by $Mq$ in \eqref{calc:translation-polynomial}. Since
$\mathsf T^{Mq}=\exp(MqN)$, both sides are polynomial in $q\ge0$ with values in a
finite-dimensional group, so equal as polynomials; comparing linear coefficients
gives $MN(e_{\mathcal Y})=My$, i.e.\ \eqref{calc:logarithm} on $Y_{0,v}$.
\Cref{log:restriction} transports it to $H^*(\mathcal Y)$ and all fibers, since
the classes in \eqref{calc:translation-polynomial} are global restrictions.
\end{proof}

The quadratic term is consistent with the flag lines: a shifted lattice satisfies
\[
 T^{k*}y=y+2k\,t_C,\qquad N(y)=2t_C,\qquad N(t_C)=0,
\]
so one must not assume $N(y)=0$.

\subsection{The flag integral}

\begin{lemma}\label{calc:flag-sum}
Let $\pi:\Fl(V)\to B$ be the full flag bundle of a rank-$n$ complex vector bundle
with formal Chern roots $w_1,\dots,w_n$. For a polynomial $F$ in the first Chern
classes of the ordered flag lines,
\[
 \pi_*\bigl(F(Q_\bullet)c_b(T_\pi)\bigr)
 =\sum_{\sigma\in\mathfrak S_n}F(w_{\sigma(1)},\dots,w_{\sigma(n)}).
\]
If $\sum_i\ell_i=0$, then
\begin{equation}\label{calc:permutation}
 \sum_{\sigma\in\mathfrak S_n}\Bigl(\sum_i\ell_iw_{\sigma(i)}\Bigr)^2
 =n(n-2)!\|\ell\|^2\Bigl(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\Bigr).
\end{equation}
Both identities, for common-translation-invariant expressions, hold for the
associated flag bundle of a principal $\PGL_n$-bundle with rational coefficients.
\end{lemma}

\begin{proof}
Iterating the projective pushforward (sum over roots with denominator the
other-root differences) along the flag tower gives a sum over ordered roots with
denominator the tangent Euler class; multiplication by $c_b(T_\pi)$ cancels it,
giving the first identity. It may be proven after passing to a splitting space
and inverting root differences, the result being a polynomial identity in the
universal Chern classes; the complex tangent Euler class fixes orientation signs.

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 positions. With $\sum_{i\ne j}\ell_i\ell_j=-\|\ell\|^2$, expanding the
square gives
\[
 (n-1)!\|\ell\|^2\textstyle\sum_iw_i^2-(n-2)!\|\ell\|^2\sum_{i\ne j}w_iw_j
 =n(n-2)!\|\ell\|^2\bigl(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\bigr).
\]
Finally, rational pullback from $B\PGL_n$ to $B\GL_n$ is injective (symmetric
polynomials in root differences into all roots on a maximal torus), so
common-translation-invariant characteristic identities can be checked after it.
\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 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
$\pi^*\pr_0^*e$ (the universal projective class is unchanged by the line twist
$J$), so the maps outside $N_{0,0}^2$ in $f'$ intertwine multiplication by $e$ and
$e_{\mathcal Y}$.

For a degree-zero derivation $N$ and an even class $a$, with $m_a$ multiplication,
$[N,m_a]=m_{N(a)}$, and direct expansion gives
\[
 [[N^2,m_a],m_a]=2m_{N(a)^2}
\]
(the first commutator is $m_{N^2(a)}+2m_{N(a)}N$; the first term commutes with
$m_a$, the second contributes $2m_{N(a)^2}$). By \Cref{calc:translation-formula},
\begin{equation}\label{calc:middle-commutator}
 [[f',e],e]=2\pr_{0,*}\mathsf A_0\,(y^2\cup-)\,\mathsf B_0\pr_0^*.
\end{equation}

Since $N_\iota=T_\pi^*$, self-intersection replaces $\iota^*\iota_*$ by
$c_b(T_\pi^*)\cup-=(-1)^b c_b(T_\pi)\cup-$. For a class pulled back from $X$, the
projection formula and \Cref{calc:flag-sum} reduce the flag integral to
\[
 \sum_{\sigma\in\mathfrak S_n}(w_{\sigma(1)}-w_{\sigma(2)})^2
 =2n(n-2)!\Bigl(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\Bigr)
 =4n(n-2)!\,\widetilde{\ch}_2(E_{x(s)}),
\]
where $E_{x(s)}$ is the universal projective data restricted to the graph of $x$
(the normalized expression is defined without a genuine universal bundle).

Under $O_0\simeq S\times X$ this class is the pullback of
$\widetilde{\ch}_2(E)$ from $C\times X$ along $x\times\id_X$; the twist $J$
contributes nothing. Integration along $S$ multiplies integration along $C$ by
$\delta$ (from $x_*[S]=\delta[C]$, or the $C$-degree-two K\"unneth term):
\[
 \pr_{0,*}\widetilde{\ch}_2(E_{x(s)})=\delta\,e,
\]
other curve degrees integrating to zero. Combining with the factor $2$ in
\eqref{calc:middle-commutator} and the self-intersection sign gives
\eqref{calc:scalar}. The integer $\delta$ is positive ($x$ a nonempty finite
\'etale cover of connected curves), so \eqref{calc:f} is a nonzero rational
rescaling giving the stated double commutator; $f'$ has degree $-2$ (flag shifts
cancel, integration over $S$ is degree $-2$), and the filtration bounds are
\Cref{spec:perverse-lowering} and \Cref{trans:weights}, unchanged by rescaling.
\end{proof}

% ============================================================
\section{Equality of the filtrations}
\label{sec:conclusion}
% ============================================================

\begin{theorem}\label{thm:main}
For every smooth projective complex curve $C$ of genus $g\ge2$, every composite
integer $n\ge4$, every integer $d$ coprime to $n$, and every $L\in\Pic^d(C)$,
\[
 P_kH^m(X^B,\Q)=W_{2k}H^m(X^B,\Q)=W_{2k+1}H^m(X^B,\Q)
\]
for all $m,k\ge0$. The equality concerns the entire cohomology, including every
nontrivial $\Gamma$-character summand after complexification.
\end{theorem}

\begin{proof}[Proof of \Cref{thm:main}]
First take $L=\mathcal O_C(dx_0)$. \Cref{norm:relative-lefschetz} gives hard
Lefschetz for cup product by $e$ on $\Gr^P H^*(X)$ (grading shift two, center
$R$); \Cref{sec:betti} gives the same on $\Gr^W H^*(X^B)$ with actual weight
$2k$ as grade $k$, and all odd weight-graded pieces vanish.

Let $\delta=\deg(S\to C)>0$ and $f=\dfrac{(-1)^{b+1}}{4n(n-2)!\,\delta}f'$.
\Cref{trans:weights} and \Cref{spec:perverse-lowering} show $f$ lowers the
half-weight filtration and the perverse filtration by two, and
\Cref{calc:double-commutator} gives $[[f,e],e]=-2e$. Apply
\Cref{alg:comparison} to $P_k$ and $W_{2k}$: $P_kH^m(X)=W_{2k}H^m(X^B)$ in every
cohomological degree, and 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, and \Cref{log:nah} shows this identification
transports the full Betti weight filtration as well.
\end{proof}

\begin{corollary}\label{cor:all-rank}
With $g\ge2$ and coprime $n,d$, and any $L\in\Pic^d(C)$, nonabelian Hodge theory
identifies
$P_kH^m(X,\Q)=W_{2k}H^m(X^B,\Q)=W_{2k+1}H^m(X^B,\Q)$ for all $m,k$, on full
rational cohomology, including its $\Pic^0(C)[n]$-variant part.
\end{corollary}
\begin{proof}
Composite ranks are \Cref{thm:main}; 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 above. Put $\Gamma=\Pic^0(C)[n]$, let $\gamma\in\Gamma$ have
order $m$, and let $\kappa\in\widehat\Gamma$,
$\kappa(\gamma')=\langle\gamma,\gamma'\rangle_\Gamma$ be the character from the
Weil pairing \cite[Section~1.3]{MSEndoscopy2021}. Let $\pi:C'\to C$ be the
associated cyclic \'etale cover, $r=n/m$, and
$d_\gamma=\operatorname{codim}_{\mathcal A^{K_C}}\mathcal A_\gamma^{K_C}$ for the
endoscopic Hitchin base. For $D=K_C$ and $D'=\pi^*K_C=K_{C'}$, let
$\mathfrak p_\kappa^B$ be the nonabelian-Hodge-transported operator of
\cite[Section~5.4]{MSEndoscopy2021}. Writing
$\widetilde{\mathcal M}_{r,d}^B(C')$ for the full degree-$d$ twisted $\GL_r$
character variety and $\mathcal M_{n,L}^B(C)=X^B$, for all $i,k\ge0$,
\[
 \mathfrak p_\kappa^B\bigl(W_{2k}H^i(\widetilde{\mathcal M}_{r,d}^B(C'),\C)\bigr)
 =W_{2k+2d_\gamma}H^{i+2d_\gamma}(\mathcal M_{n,L}^B(C),\C)_\kappa,
\]
the subscript selecting the summand with $\gamma'^*\alpha=\kappa(\gamma')\alpha$.
\end{corollary}
\begin{proof}
Since $r\mid n$, $\gcd(r,d)=1$, and \'etale Riemann--Hurwitz gives
$g(C')=m(g-1)+1\ge2$. The all-rank $\GL_r$ theorem \cite[Theorem~0.2]{MS2024}
identifies $W_{2k}$ with $P_k$ on the source (also for $r=1$). \Cref{cor:all-rank}
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 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 identifications, the definition
$\mathfrak p_\kappa^B=\eta_t^{-1}\mathfrak p_\kappa\eta_s$ 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 full
fixed-determinant cohomology.

\begin{corollary}[Multiplicativity of the perverse filtration]
\label{cor:perverse-multiplicativity}
In the setting of \Cref{cor:all-rank}, with normalization
\eqref{norm:perverse-filtration}, for all $i,j\ge0$ and $a,b\in\Z$,
\[
 P_aH^i(X,\Q)\smile P_bH^j(X,\Q)\subseteq P_{a+b}H^{i+j}(X,\Q).
\]
For isotypic parts indexed by $\chi,\psi\in\widehat\Gamma$ (complexified),
\[
 \bigl(P_aH^i(X,\C)\bigr)_\chi\smile\bigl(P_bH^j(X,\C)\bigr)_\psi
 \subseteq\bigl(P_{a+b}H^{i+j}(X,\C)\bigr)_{\chi\psi},
\]
including 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,~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
eigencharacters, giving the second.
\end{proof}

\section*{Acknowledgements}
This proof builds on the work of many authors cited above; the strategy follows
the common $\mathfrak{sl}_2$/flag-correspondence method of \cite{HMMS2025}.

\begin{thebibliography}{99}
\bibitem[MS24]{MS2024} D.~Maulik, J.~Shen, \emph{The $P=W$ conjecture for
$\GL_n$}, Ann. of Math. \textbf{200} (2024), 529--556.
\bibitem[dCMS22]{dCMS2022} M.~A.~A.~de Cataldo, D.~Maulik, J.~Shen, \emph{On the
$P=W$ conjecture for $\SL_n$}, Selecta Math. \textbf{28} (2022), 90.
\bibitem[MSE21]{MSEndoscopy2021} D.~Maulik, J.~Shen, \emph{Endoscopic
decompositions and the Hausel--Thaddeus conjecture}, Forum Math. Pi \textbf{9}
(2021), e8.
\bibitem[HMMS25]{HMMS2025} T.~Hausel, A.~Mellit, A.~Minets, O.~Schiffmann,
\emph{$P=W$ via $\mathcal H_2$}, arXiv:2209.05429.
\bibitem[Mel25]{Mellit2025} A.~Mellit, \emph{Toric stratifications of character
varieties}, Publ. Math. IH\'ES \textbf{142} (2025), 153--240.
\bibitem[Mar02]{Markman2002} E.~Markman, \emph{Generators of the cohomology ring
of moduli spaces of sheaves on symplectic surfaces}, J. reine angew. Math.
\textbf{544} (2002), 61--82.
\bibitem[She17]{Shende2017} V.~Shende, \emph{The weights of the tautological
classes of character varieties}, IMRN \textbf{2017}, 6832--6840.
\bibitem[HRV15]{HRV2015} T.~Hausel, F.~Rodriguez-Villegas, \emph{Cohomology of
large semiprojective hyperk\"ahler varieties}, Ast\'erisque \textbf{370} (2015),
113--156.
\bibitem[dCFH24]{dCFHLogHodge} M.~A.~A.~de Cataldo, A.~Fernandez Herrero,
\emph{Geometry of the logarithmic Hodge moduli space}, J. London Math. Soc.
\textbf{109} (2024), e12857.
\bibitem[FZ25]{FHZ2025} A.~Fernandez Herrero, S.~Zhang, \emph{Meromorphic Hodge
moduli spaces for reductive groups}, arXiv:2307.16755.
\bibitem[HT03]{HT2003} T.~Hausel, M.~Thaddeus, \emph{Mirror symmetry, Langlands
duality, and the Hitchin system}, Invent. Math. \textbf{153} (2003), 197--229.
\bibitem[Sim97]{Simpson1997} C.~Simpson, \emph{The Hodge filtration on
nonabelian cohomology}, Proc. Sympos. Pure Math. \textbf{62} (1997), 217--281.
\bibitem[Sai90]{SaitoExtension1990} M.~Saito, \emph{Extension of mixed Hodge
modules}, Compositio Math. \textbf{74} (1990), 209--234.
\bibitem[Tub25]{TubachStacks2025} S.~Tubach, \emph{Mixed Hodge modules on stacks},
Forum Math. Sigma \textbf{13} (2025), e175.
\bibitem[dCMSJAMS22]{dCMSJAMS2022} M.~A.~de Cataldo, D.~Maulik, J.~Shen,
\emph{Hitchin fibrations, abelian surfaces, and the $P=W$ conjecture},
J. Amer. Math. Soc. \textbf{35} (2022), 911--953.
\bibitem[BB82]{BBD1982} A.~Beilinson, J.~Bernstein, P.~Deligne, \emph{Faisceaux
pervers}, Ast\'erisque \textbf{100} (1982).
\bibitem[dCM09]{dCM2009} M.~A.~A.~de Cataldo, L.~Migliorini, \emph{The
decomposition theorem, perverse sheaves and the topology of algebraic maps},
Bull. AMS \textbf{46} (2009), 535--633.
\bibitem[Ful98]{Fulton1998} W.~Fulton, \emph{Intersection Theory}, 2nd ed.,
Springer, 1998.
\bibitem[Del70]{Deligne1970} P.~Deligne, \emph{\'Equations diff\'erentielles \`a
points singuliers r\'eguliers}, Lecture Notes in Math. \textbf{163}, Springer,
1970.
\bibitem[Del71]{Deligne1971} P.~Deligne, \emph{Th\'eorie de Hodge II}, Publ.
Math. IH\'ES \textbf{40} (1971), 5--57.
\bibitem[Del74]{Deligne1974} P.~Deligne, \emph{Th\'eorie de Hodge III}, Publ.
Math. IH\'ES \textbf{44} (1974), 5--77.
\end{thebibliography}

\end{document}