\section{Geometric coniveau and the generalized Hodge conjecture}
\label{sec:17-geometric-coniveau}

Payne asks whether the generalized Hodge conjecture holds for
$H^{15}(\Mbar_{g,n})$~\cite[Problem 6.2]{payne2026}.
We prove it by determining the geometric coniveau filtration.

Throughout this section, cohomology is rational singular cohomology
over $\mathbb C$. For a smooth proper Deligne--Mumford stack $X$ of
pure dimension with projective coarse moduli space, define
\[
 N^rH^k(X)=
 \sum_{\substack{Z\subseteq X\text{ closed algebraic}\\
                 \operatorname{codim}_X Z\ge r}}
 \ker\bigl(H^k(X)\longrightarrow H^k(X\setminus Z)\bigr),
 \qquad r\ge0.
\]
Codimension is required to be at least $r$ on every irreducible
component of $Z$; the empty support is allowed. Finite unions of
supports preserve this condition. Since $H^k(X)$ is finite-dimensional,
the subspace $N^rH^k(X)$ itself is supported on a single such $Z$.
Write $N_H^rH^k(X)$ for the largest rational sub-Hodge structure whose
Hodge types $(p,q)$ satisfy $p,q\ge r$.
The generalized Hodge conjecture in degree $k$ is the equality
$N^rH^k(X)=N_H^rH^k(X)$ for every $r\ge0$.

\begin{lemma}[Support and Hodge coniveau]
\label{lem:coniveau-support-properties}
Let $X$ and $Y$ be smooth proper Deligne--Mumford stacks of pure
dimension with projective coarse moduli spaces.
\begin{enumerate}[beginpenalty=10000]
\item One has $N^rH^k(X)\subseteq N_H^rH^k(X)$.
\item Suppose that $f:Y\to X$ is proper and
$\dim Y=\dim X-c$, with $c\ge0$. Then
\[
 f_*\bigl(N^rH^k(Y)\otimes\Lmot^c\bigr)
 \subseteq N^{r+c}H^{k+2c}(X).
\]
\item If $D$ is a rational algebraic divisor on $Y$, then its cycle
class restricts to zero on $Y\setminus|D|$. In particular, for any
smooth proper Deligne--Mumford stack $T$ and any $\alpha\in H^j(T)$, the class
$\alpha\boxtimes[D]$ is supported on $T\times|D|$.
\end{enumerate}
\end{lemma}

\begin{proof}
For the first assertion, localization identifies the kernel associated
with a closed subset $Z$ with the image of $H_Z^k(X)\to H^k(X)$.
Choose a filtration by closed subsets
$Z=Z_0\supset Z_1\supset\cdots\supset Z_m=\varnothing$ such that
each stratum $S_i=Z_i\setminus Z_{i+1}$ is smooth of pure codimension
$c_i\ge r$ in $X$ and has smooth coarse moduli space. Such a filtration
exists by noetherian induction (smooth locus of the top-dimensional
components, over the smooth locus of the coarse space). Purity gives
\[
 H^k_{S_i}(X\setminus Z_{i+1})
 \cong H^{k-2c_i}(S_i)\otimes\Lmot^{c_i}.
\]
The localization sequences for successive supports therefore express
$H_Z^k(X)$ as successive extensions of subquotients of these groups.
For each $S_i$, the coarse-space map induces an isomorphism on rational
cohomology compatible with mixed Hodge structures. Its coarse space is a
smooth locally closed algebraic subvariety of the projective coarse
space of $X$, and ordinary cohomology of a smooth algebraic variety has nonnegative
Hodge indices~\cite[Sections 3.1--3.2]{deligne1971}.
Consequently the displayed groups have Hodge indices at least $c_i$,
hence at least $r$. This property is preserved by subquotients and
extensions in the category of mixed Hodge structures, and therefore
by the image in $H^k(X)$. We use localization, purity, and their
compatibility with rational cohomology for Deligne--Mumford stacks
as in~\cite[Theorem 4.1.1 and Proposition 4.2.6]{tubach2025}.
Each kernel is a rational sub-Hodge structure, and summing proves
the first assertion.

For the second assertion, suppose that $\alpha$ restricts to zero
outside a closed subset $Z\subseteq Y$ of codimension at least $r$.
Properness makes $f(Z)$ closed, and
\[
 \dim f(Z)\le\dim Z\le\dim Y-r=\dim X-c-r.
\]
Proper pushforward commutes with restriction to the open complement
of $f(Z)$; see~\cite[Proposition 4.2.5]{tubach2025}.
The restriction of $\alpha$ to its inverse image is zero,
so $f_*\alpha$ is supported on $f(Z)$. Linearity proves the assertion.

For the third assertion, write $D$ as a rational linear combination
of irreducible divisors. Compatibility of the cycle class map with
open restriction shows that $[D]$ restricts to zero off their union.
Pullback to the product and cup product with $\alpha$ give the stated
support.
\end{proof}

\begin{proposition}[Codimension-two support for the ordinary image]
\label{prop:coniveau-ordinary-support}
For every stable pair $(g,n)$,
\[
 \mathcal I_{g,[n]}\subseteq N^2H^{15}(\Mbar_{g,n})
\]
(Definition~\ref{def:ordinary-image}).
There is a closed algebraic subset
$Z_{g,n}\subseteq\partial\Mbar_{g,n}$ of codimension at least two
supporting the entire subspace $\mathcal I_{g,[n]}$.
It may be chosen invariant under $\Sn_n$.
\end{proposition}

\begin{proof}
Put $X=\Mbar_{g,n}$ and $d=3g-3+n$. We use the complete ordinary
one-edge Gysin map of Proposition~\ref{prop:h13-input}.
For a one-edge graph $\Gamma$, its labelled normalization
$Y=X_\Gamma$ is smooth and proper of dimension $d-1$, and its
clutching map $\xi_\Gamma:Y\to X$ is proper. We show that
\[
 H^{13}(Y)=N^1H^{13}(Y).
\]

First, Theorem~\ref{thm:cited-thirteen-cohomology} expresses every
degree-thirteen class on a single moduli factor as a sum of
boundary Gysin images of degree-eleven classes. Such a class is
supported on the boundary of that factor. This proves the displayed
equality when $\Gamma$ is nonseparating, since then
$Y=\Mbar_{g-1,[n]\sqcup\{p,q\}}$.

If $\Gamma$ is separating, write $Y=Y_1\times Y_2$ for its two
moduli factors. Theorem~\ref{thm:cited-low-odd-vanishing} and the
K\"unneth formula give exactly the four summands
\[
\begin{split}
 H^{13}(Y)={}&
 H^{13}(Y_1)\otimes H^0(Y_2)
 \;\oplus\;H^0(Y_1)\otimes H^{13}(Y_2)\\
 &\oplus\;H^{11}(Y_1)\otimes H^2(Y_2)
 \;\oplus\;H^2(Y_1)\otimes H^{11}(Y_2).
\end{split}
\]
For the first two summands, the degree-thirteen presentation just
used gives a divisor support in one factor, hence a divisor support
in $Y$. The other placement is identical.

For the last two summands, Theorem~\ref{thm:cited-divisor-generation}
represents every degree-two class by a rational algebraic divisor.
If $\alpha\in H^{11}(Y_1)$ and $D$ is such a divisor on $Y_2$, then
$\alpha\boxtimes[D]$ is supported on $Y_1\times|D|$ by
Lemma~\ref{lem:coniveau-support-properties}(3), of codimension
at least one in $Y$. Interchanging the factors treats the fourth summand.

Lemma~\ref{lem:coniveau-support-properties}(2), with $c=1$, now
places every ordinary Gysin image in $N^2H^{15}(X)$. The finite
graph-automorphism quotients (Definition~\ref{def:stable-graphs}) do not
change this conclusion.
Summing over the finitely many one-edge graphs proves the inclusion.

Choose a finite basis of the ordinary image and the supports just
constructed for its basis elements. Their union is closed, has
codimension at least two, and is contained in the boundary because
every outer map is a boundary clutching map. Taking the union of
its finitely many $\Sn_n$-translates gives $Z_{g,n}$ with the stated
invariance. For a zero ordinary image, take $Z_{g,n}=\varnothing$.
\end{proof}

\begin{theorem}[The degree-fifteen coniveau filtration]
\label{thm:degree-fifteen-coniveau}
This is Theorem~\ref{thm:introduction-coniveau}.
Let $(g,n)$ be stable, put $X=\Mbar_{g,n}$, and denote by
$H^{15}(X)_{\Pmot}$ its canonical $\Pmot$-isotypic rational
sub-Hodge structure. Then, as $\Sn_n$-stable subspaces of $H^{15}(X)$,
\[
 N^1H^{15}(X)=N^2H^{15}(X)=H^{15}(X)_{\Pmot},
 \qquad N^rH^{15}(X)=0\quad(r\ge3).
\]
Moreover,
\[
 H^{15}(X)_{\Pmot}\cong\En_{g,n}\otimes\Lmot^2\Soneone.
\]
In particular, for $g\ge2$ all of $H^{15}(X)$ is supported on an
algebraic subset of codimension at least two contained in the boundary.
For $g\ge2$ with $H^{15}(X)\ne0$, its geometric coniveau is exactly two.
\end{theorem}

\begin{proof}
The ordinary image $\mathcal I_{g,[n]}$ is $H^{15}(X)_{\Pmot}$ in
genus one by Theorem~\ref{thm:genus-one}, and for $g\ge2$ it is the
whole group (Proposition~\ref{prop:genus-two-gysin-surjectivity},
Proposition~\ref{prop:genus-three-generation},
Theorem~\ref{thm:uniform-high-genus}), which is entirely
$\Pmot$-isotypic by Theorem~\ref{thm:motivic-types}; in genus zero
it vanishes, so $\mathcal I_{g,[n]}=H^{15}(X)_{\Pmot}$, which lies in
$N^2H^{15}(X)$ by Proposition~\ref{prop:coniveau-ordinary-support}.

By Theorem~\ref{thm:motivic-types} and Notation~\ref{def:motivic-pieces},
the only simple Hodge constituents are
$\Pmot$, with types $(13,2),(2,13)$, and, in genus one, $\Sonesix$,
with types $(15,0),(0,15)$. Their isotypic subspaces are canonical
(semisimplicity).
Any sub-Hodge structure of positive Hodge coniveau has zero projection
onto the $\Sonesix$-isotypic subspace: its Hodge types have both indices
positive, whereas that subspace has only the two types just listed.
The entire $\Pmot$-isotypic subspace has Hodge coniveau at least two, and none
of the Hodge types in degree fifteen has both indices at least three.
Consequently
\[
 N_H^1H^{15}(X)=N_H^2H^{15}(X)=H^{15}(X)_{\Pmot},
 \qquad N_H^rH^{15}(X)=0\quad(r\ge3).
\]
Combining the support inclusion with
Lemma~\ref{lem:coniveau-support-properties}(1) gives
\[
 H^{15}(X)_{\Pmot}\subseteq N^2H^{15}(X)
 \subseteq N^1H^{15}(X)
 \subseteq N_H^1H^{15}(X)=H^{15}(X)_{\Pmot}.
\]
The same lemma gives $N^rH^{15}(X)=0$ for $r\ge3$.
The multiplicity identification is Theorem~\ref{thm:motivic-types}, and
Proposition~\ref{prop:coniveau-ordinary-support} supplies the common
boundary support. This proves all assertions.
\end{proof}

\begin{corollary}[Generalized Hodge conjecture in degree fifteen]
\label{cor:degree-fifteen-ghc}
This is Theorem~\ref{thm:introduction-ghc}.
For every stable pair $(g,n)$, the generalized Hodge conjecture holds
for $H^{15}(\Mbar_{g,n})$. In particular, Problem 6.2
of~\cite{payne2026} has an affirmative answer.
\end{corollary}

\begin{proof}
The preceding theorem proves $N^rH^{15}(X)=N_H^rH^{15}(X)$ for
$r\ge1$, and both filtrations equal $H^{15}(X)$ at $r=0$.
\end{proof}