# Tautological projection for cycles on the moduli space of abelian varieties

Samir Canning  Department of Mathematics, ETH Zürich
<EMAIL_ADDRESS>,  Sam Molcho  Department of Mathematics, Aarhus
University<EMAIL_ADDRESS>,  Dragos Oprea  Department of Mathematics,
University of California, San Diego<EMAIL_ADDRESS>and  Rahul
Pandharipande  Department of Mathematics, ETH Zürich<EMAIL_ADDRESS>
(Date: January 2024)

###### Abstract.

We define a tautological projection operator for algebraic cycle classes on
the moduli space of principally polarized abelian varieties $\mathcal{A}_{g}$:
every cycle class decomposes canonically as a sum of a tautological and a non-
tautological part. The main new result required for the definition of the
projection operator is the vanishing of the top Chern class of the Hodge
bundle over the boundary
$\overline{\mathcal{A}}_{g}\smallsetminus\mathcal{A}_{g}$ of any toroidal
compactification $\overline{\mathcal{A}}_{g}$ of the moduli space
$\mathcal{A}_{g}$. We prove the vanishing by a careful study of residues in
the boundary geometry.

The existence of the projection operator raises many natural questions about
cycles on $\mathcal{A}_{g}$. We calculate the projections of all product
cycles $\mathcal{A}_{g_{1}}\times\ldots\times\mathcal{A}_{g_{\ell}}$ in terms
of Schur determinants, discuss Faber’s earlier calculations related to the
Torelli locus, and state several open questions.

###### Contents

  1. 1 Introduction
  2. 2 The top Chern class of the Hodge bundle
  3. 3 Tautological projection of product classes

##  1\. Introduction

###  1.1. Tautological rings of $\mathcal{A}_{g}$ and
$\overline{\mathcal{A}}_{g}$

Let $\mathcal{A}_{g}\subset\overline{\mathcal{A}}_{g}$ be a toroidal
compactification of the moduli space of principally polarized abelian
varieties. The space $\mathcal{A}_{g}$ is a nonsingular Deligne-Mumford stack
of dimension $\binom{g+1}{2}$, and the compactification
$\overline{\mathcal{A}}_{g}$ is a reduced and irreducible (but possibly
singular) proper Deligne-Mumford stack, see [FC]. The Hodge bundle

$\mathbb{E}\rightarrow{\mathcal{A}}_{g}\,$

is defined as the pullback to $\mathcal{A}_{g}$ via the zero section $s$ of
the relative cotangent bundle of the universal family $\mathcal{X}_{g}$ of
abelian varieties,

$p:\mathcal{X}_{g}\rightarrow\mathcal{A}_{g}\,,\ \ \ \
s:\mathcal{A}_{g}\rightarrow\mathcal{X}_{g}\,,\ \ \ \ \mathbb{E}\cong
s^{*}\Omega_{p}\,.$

There is a canonical extension of the Hodge bundle over
$\overline{\mathcal{A}}_{g}$ by [FC, Theorems V.2.3, VI.1.1, VI.4.2],

$\mathbb{E}\rightarrow\overline{\mathcal{A}}_{g}\,.$

By[vdG, EV], the Chern classes $\lambda_{i}$ of the Hodge bundle satisfy
Mumford’s relation111Since $\overline{\mathcal{A}}_{g}$ is possibly singular,
even as a stack, some care must be taken with the Chow theories. Here,
$\mathsf{CH}^{\mathsf{op}}$ is the $\mathbb{Q}$-algebra of operational Chow
classes. Usual Chow cycle theory, indexed by codimension, is denoted by
$\mathsf{CH}^{*}$.:

(1)
$(1+\lambda_{1}+\lambda_{2}+\ldots+\lambda_{g})(1-\lambda_{1}+\lambda_{2}-\ldots+(-1)^{g}\lambda_{g})=1\,\in\,\mathsf{CH}^{\mathsf{op}}(\overline{\mathcal{A}}_{g})\,,$

for all $g\geq 1$. Van der Geer [vdG] defined the tautological rings

$\mathsf{R}^{*}({\mathcal{A}}_{g})\subset\mathsf{CH}^{*}({\mathcal{A}}_{g})\,,\
\ \ \
\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})\subset\mathsf{CH}^{\mathsf{op}}(\overline{\mathcal{A}}_{g})$

to be the $\mathbb{Q}$-subalgebras generated by the $\lambda$-classes. Both
tautological rings are calculated by a fundamental result of [vdG].

######  Theorem 1 (van der Geer).

The following properties hold:

  1. (i)

The kernel of the quotient

$\mathbb{Q}[\lambda_{1},\lambda_{2},\lambda_{3},\ldots,\lambda_{g}]\rightarrow\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})\rightarrow
0$

is generated as an ideal by Mumford’s relation (1).

  2. (ii)

$\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$ is a Gorenstein local ring with
socle in codimension $\binom{g+1}{2}$,

$\mathsf{R}^{\binom{g+1}{2}}(\overline{\mathcal{A}}_{g})\cong\mathbb{Q}\,.$

The class $\lambda_{1}\lambda_{2}\lambda_{3}\cdots\lambda_{g}$ is a generator
of the socle.

  3. (iii)

$\mathsf{R}^{*}(\mathcal{A}_{g})\cong\mathsf{R}^{*}(\overline{\mathcal{A}_{g}})/(\lambda_{g})$
is a Gorenstein local ring with socle in codimension $\binom{g}{2}$,

$\mathsf{R}^{\binom{g}{2}}(\mathcal{A}_{g})\cong\mathbb{Q}\,.$

The class $\lambda_{1}\lambda_{2}\lambda_{3}\cdots\lambda_{g-1}$ is a
generator of the socle.

###  1.2. Tautological projection for $\overline{\mathcal{A}}_{g}$

The idea of tautological projection on $\overline{\mathcal{A}}_{g}$
(Definition 2 below) appears in work of Faber [Fa] and of Grushevsky and Hulek
[GH]. Since $\overline{\mathcal{A}}_{g}$ is proper of dimension
$\binom{g+1}{2}$, we obtain an evaluation

$\epsilon^{\mathsf{cpt}}:\mathsf{R}^{\binom{g+1}{2}}(\overline{\mathcal{A}}_{g})\to\mathbb{Q}\,,\
\quad\alpha\mapsto\int_{{\overline{\mathcal{A}}_{g}}}{\alpha}\,,$

and a pairing between classes on $\overline{\mathcal{A}}_{g}$,

(2)
$\langle\,,\,\rangle^{\mathsf{cpt}}:\,\mathsf{CH}^{k}(\overline{\mathcal{A}}_{g})\times\mathsf{R}^{\binom{g+1}{2}-k}(\overline{\mathcal{A}}_{g})\,\to\,\mathbb{Q}\,,\
\quad\langle\gamma,\delta\rangle^{\mathsf{cpt}}=\int_{{\overline{\mathcal{A}}_{g}}}{\gamma}\cdot{\delta}\,.$

Here, $\mathsf{cpt}$ stands for compact.

By Theorem 1(ii), the socle of $\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$ is
spanned by the class $\lambda_{1}\lambda_{2}\lambda_{3}\cdots\lambda_{g}$.
Equivalently,

(3)
$\gamma_{g}=\epsilon^{\mathsf{cpt}}(\lambda_{1}\lambda_{2}\lambda_{3}\cdots\lambda_{g})\neq
0\,.$

The exact evaluation222Here, $B_{2i}$ is the Bernoulli number.,

(4) $\gamma_{g}=\prod_{i=1}^{g}\frac{|B_{2i}|}{4i}\,,$

is computed in [vdG, page 9]. By the Gorenstein property of
$\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$, the pairing of tautological
classes

$\mathsf{R}^{k}(\overline{\mathcal{A}}_{g})\times\mathsf{R}^{\binom{g+1}{2}-k}(\overline{\mathcal{A}}_{g})\,\to\,\mathsf{R}^{\binom{g+1}{2}}(\overline{\mathcal{A}}_{g})\,\cong\,\mathbb{Q}$

is non-degenerate (where the last isomorphism is via
$\epsilon^{\mathsf{cpt}}$).

######  Definition 2.

Let $\gamma\in\mathsf{CH}^{*}(\overline{\mathcal{A}}_{g})$. The tautological
projection
$\mathsf{taut}^{\mathsf{cpt}}(\gamma)\in\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$
is the unique333The existence and uniqueness of
$\mathsf{taut}^{\mathsf{cpt}}(\gamma)$ follows from the Gorenstein property of
$\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$ applied to the functional
$\delta\mapsto\langle\gamma,\delta\rangle^{\mathsf{cpt}}$ on
$\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$. tautological class which
satisfies

$\langle\mathsf{taut}^{\mathsf{cpt}}(\gamma),\delta\rangle^{\mathsf{cpt}}=\langle\gamma,\delta\rangle^{\mathsf{cpt}}$

for all classes $\delta\in\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$.

$\bullet$ If $\gamma\in\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$, then
$\gamma=\mathsf{taut}^{\mathsf{cpt}}(\gamma)$, so we have a
$\mathbb{Q}$-linear projection operator:

$\mathsf{taut}^{\mathsf{cpt}}:\mathsf{CH}^{*}(\overline{\mathcal{A}}_{g})\to\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})\,,\
\ \
\mathsf{taut}^{\mathsf{cpt}}\circ\mathsf{taut}^{\mathsf{cpt}}=\mathsf{taut}^{\mathsf{cpt}}\,.$

$\bullet$ From the point of view of $\mathcal{A}_{g}$, a difficulty with the
theory on $\overline{\mathcal{A}}_{g}$ is the dependence upon
compactification. Given a subvariety

$V\subset\mathcal{A}_{g}\,,$

we can define a projection

(5)
$\mathsf{taut}^{\mathsf{cpt}}([\overline{V}])\in\mathsf{CH}^{*}(\overline{\mathcal{A}}_{g})$

with respect to the Zariski closure $V\subset\overline{V}$ in a toroidal
compactification $\mathcal{A}_{g}\subset\overline{\mathcal{A}}_{g}$, but the
projection (5) will depend upon the choice of $\overline{\mathcal{A}}_{g}$. In
order to study cycles on the moduli of abelian varieties, we would like to
construct a canonical projection operator depending just upon
$\mathcal{A}_{g}$.

###  1.3. Top Chern class of the Hodge bundle

To define a tautological projection operator on the interior
$\mathcal{A}_{g}$, we will define a pairing similar to (2). The theory depends
upon a new vanishing result for the top Chern class of Hodge bundle on
$\overline{\mathcal{A}}_{g}$.

We recall the $\lambda_{g}$-pairing on tautological classes444We refer the
reader to [FP3, P] for a review of the theory of tautological classes on the
moduli spaces of curves. Unlike the case of $\mathcal{A}_{g}$, the
tautological ring $\mathsf{R}^{*}(\mathcal{M}_{g}^{\operatorname{ct}})$ is not
a Gorenstein local ring, see [CLS, Pix] and [Pet] for the pointed case.on the
moduli space of curves of compact type
$\mathcal{M}_{g}^{\operatorname{ct}}\subset\overline{\mathcal{M}}_{g}$. On the
tautological ring of $\mathcal{M}_{g}^{\operatorname{ct}}$, the
$\lambda_{g}$-pairing is given by

$\mathsf{R}^{k}(\mathcal{M}_{g}^{\operatorname{ct}})\times\mathsf{R}^{2g-3-k}(\mathcal{M}_{g}^{\operatorname{ct}})\rightarrow\mathsf{R}^{2g-3}(\mathcal{M}_{g}^{\operatorname{ct}})\cong\mathbb{Q}\,,\
\quad(\alpha,\beta)\mapsto\int_{\overline{\mathcal{M}}_{g}}\overline{\alpha}\cdot\overline{\beta}\cdot\lambda_{g}\,,$

where $\overline{\alpha}$ and $\overline{\beta}$ are arbitrary lifts of
$\alpha$ and $\beta$ to $\overline{\mathcal{M}}_{g}$. The
$\lambda_{g}$-pairing is well-defined, independent of the lifts, because
$\lambda_{g}\in\mathsf{R}^{g}(\overline{\mathcal{M}}_{g})$ restricts trivially
to the boundary

$\lambda_{g}|_{\overline{\mathcal{M}}_{g}\smallsetminus\mathcal{M}_{g}^{\operatorname{ct}}}=0\,,$

see [FP2]. A parallel boundary vanishing for
$\lambda_{g}\in\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$ is our first
result.

######  Theorem 3.

The restriction of $\lambda_{g}$ to
$\overline{\mathcal{A}}_{g}\smallsetminus\mathcal{A}_{g}$ vanishes for every
toroidal compactification $\overline{\mathcal{A}}_{g}$.

In characteristic $p$, the Theorem 3 can be proven555We thank van der Geer for
the characteristic $p$ argument. by considering the $p$-rank zero locus in
$\overline{\mathcal{A}}_{g}.$ The $p$-rank zero locus avoids the boundary and
has class in $\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})$ equal to a multiple
of $\lambda_{g}$, [vdG, Theorem 2.4]. In all characteristics, the vanishing of
$\lambda_{g}$ over the boundary of the partial compactification
$\mathcal{A}_{g}^{\text{part}}$ of torus rank $1$ degenerations follows from
the discussion of [vdG, page 6]. The statement for the entire boundary is new.

Our proof of Theorem 3 is obtained as a consequence of the following
statements about semistable degenerations of abelian varieties:

  * (i)

The sheaf of relative log differentials has a trivial rank 1 quotient on the
singularities of the fibers of the universal family. The trivial quotient
statement is true for any semistable family, independently of abelian
structure (also applying, for example, to families of curves).

  * (ii)

For abelian schemes, the sheaf of relative log differentials is isomorphic to
the pullback of the Hodge bundle [FC].

The full proof is presented in Section 2 after a review of log structures,
semistable degenerations, and residues.

The vanishing of Theorem 3 of the top Chern class of the Hodge bundle for the
moduli of abelian varieties implies the parallel vanishing for the moduli of
curves,

$\lambda_{g}|_{\overline{\mathcal{A}}_{g}\smallsetminus\mathcal{A}_{g}}=0\ \ \
\implies\ \ \
\lambda_{g}|_{\overline{\mathcal{M}}_{g}\smallsetminus\mathcal{M}_{g}^{\operatorname{ct}}}=0\,,$

via the Torelli map from $\overline{\mathcal{M}}_{g}$ to a suitable666 The
second Voronoi compactification can be taken here [A1],[N]. toroidal
compactification $\overline{\mathcal{A}}_{g}$. We can hope for an even deeper
connection: a lifting of Pixton’s formula [HMPPS, JPPZ] for $\lambda_{g}$ on
$\overline{\mathcal{M}}_{g}$ to $\overline{\mathcal{A}}_{g}$. The natural
context for such a lifting should be the logarithmic Chow ring of the moduli
space of abelian varieties. A discussion of these ideas is presented in
Section 2.6.

###  1.4. Tautological projection for $\mathcal{A}_{g}$

Given $\alpha\in\mathsf{CH}^{*}(\mathcal{A}_{g})$, we define an evaluation,

$\epsilon^{\mathrm{ab}}:\mathsf{CH}^{\binom{g}{2}}(\mathcal{A}_{g})\to\mathbb{Q}\,,\
\quad\alpha\mapsto\int_{\overline{\mathcal{A}}_{g}}\overline{\alpha}\cdot\lambda_{g}\
=\ \deg(\lambda_{g}\cap\overline{\alpha})\,,$

where $\overline{\alpha}$ is a lift to the toroidal compactification
$\overline{\mathcal{A}}_{g}$. The answer is well-defined (independent of lift)
by the vanishing of Theorem 3. We also have an induced pairing between classes
on $\mathcal{A}_{g}$,

(6)
$\langle\,,\,\rangle:\,\mathsf{CH}^{k}(\mathcal{A}_{g})\times\mathsf{R}^{\binom{g}{2}-k}(\mathcal{A}_{g})\,\to\,\mathbb{Q}\,,\
\quad\langle\gamma,\delta\rangle=\int_{{\overline{\mathcal{A}}_{g}}}\overline{\gamma}\cdot\overline{\delta}\cdot\lambda_{g}=\deg(\overline{\delta}\cdot\lambda_{g}\cap\overline{\gamma})\,,$

which we call the $\lambda_{g}$-pairing for the moduli of abelian varieties.
Here, $\overline{\delta}$ is a lift of

$\delta\in\mathsf{R}^{*}(\mathcal{A}_{g})=\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})/\langle\lambda_{g}\rangle$

to a tautological class on $\overline{\mathcal{A}}_{g},$ while
$\overline{\gamma}$ is an arbitrary lift of $\gamma$. The lift of $\delta$ is
well-defined up to the class $\lambda_{g}$, while the lift of $\gamma$ is
well-defined up to cycles supported on the boundary. The vanishing of
$\lambda_{g}^{2}$ on $\overline{\mathcal{A}}_{g}$ and Theorem 3 ensure that
the $\lambda_{g}$-pairing is well-defined.

By Theorem 1(iii), the socle of $\mathsf{R}^{*}(\mathcal{A}_{g})$ is spanned
by the class $\lambda_{1}\lambda_{2}\lambda_{3}\cdots\lambda_{g-1}$. The
Gorenstein property of $\mathsf{R}^{*}(\mathcal{A}_{g})$ together with the
non-vanishing (3) implies that the restriction of the $\lambda_{g}$-pairing to
tautological classes

$\mathsf{R}^{k}(\mathcal{A}_{g})\times\mathsf{R}^{\binom{g}{2}-k}(\mathcal{A}_{g})\to\mathsf{R}^{\binom{g}{2}}(\mathcal{A}_{g})\cong\mathbb{Q}$

is non-degenerate (where the last isomorphism is via
$\epsilon^{\mathsf{ab}}$).

######  Definition 4.

Let $\gamma\in\mathsf{CH}^{*}(\mathcal{A}_{g})$. The tautological projection
$\mathsf{taut}(\gamma)\in\mathsf{R}^{*}(\mathcal{A}_{g})$ is the unique777The
existence and uniqueness of $\mathsf{taut}(\gamma)$ follows from the
Gorenstein property of $\mathsf{R}^{*}(\mathcal{A}_{g})$. tautological class
which satisfies

$\langle\mathsf{taut}(\gamma),\delta\rangle=\langle\gamma,\delta\rangle$

for all classes $\delta\in\mathsf{R}^{*}(\mathcal{A}_{g})$.

$\bullet$ If $\gamma\in\mathsf{R}^{*}(\mathcal{A}_{g})$, then
$\gamma=\mathsf{taut}(\gamma)$, so we have a $\mathbb{Q}$-linear projection
operator:

$\mathsf{taut}:\mathsf{CH}^{*}(\mathcal{A}_{g})\to\mathsf{R}^{*}(\mathcal{A}_{g})\,,\
\ \ \mathsf{taut}\circ\mathsf{taut}=\mathsf{taut}\,.$

$\bullet$ For $\gamma\in\mathsf{CH}^{*}({\mathcal{A}}_{g})$, tautological
projection provides a canonical decomposition

$\gamma=\mathsf{taut}(\gamma)+(\gamma-\mathsf{taut}(\gamma))$

into purely tautological and purely non-tautological parts.

$\bullet$ Tautological projection commutes with restriction: for every
toroidal compactification $\mathcal{A}_{g}\subset\overline{\mathcal{A}}_{g}$
and every class $\gamma\in\mathsf{CH}^{*}(\overline{\mathcal{A}}_{g}),$

$\mathsf{taut}^{\mathsf{cpt}}(\gamma)\big{|}_{\mathcal{A}_{g}}=\mathsf{taut}\left(\gamma\big{|}_{\mathcal{A}_{g}}\right)\,.$

To prove the restriction property, consider classes

$\gamma\in\mathsf{CH}^{*}(\overline{\mathcal{A}}_{g})\ \ \text{and}\ \
\delta\in\mathsf{R}^{*}({\mathcal{A}}_{g})\,.$

Equations (2) and (6) imply the compatibility between pairings

(7)
$\langle\gamma\big{|}_{\mathcal{A}_{g}},\delta\rangle=\langle\gamma,\overline{\delta}\,\lambda_{g}\rangle^{\textsf{cpt}},$

where $\overline{\delta}$ is any lift of $\delta$ to the compactification
$\overline{\mathcal{A}}_{g}.$ Then,

$\displaystyle\langle\mathsf{taut}^{\mathsf{cpt}}(\gamma),\overline{\delta}\,\lambda_{g}\rangle^{\mathsf{cpt}}=\langle\gamma,\overline{\delta}\lambda_{g}\rangle^{\mathsf{cpt}}\
$ $\displaystyle\implies$ $\displaystyle\
\langle\mathsf{taut}^{\mathsf{cpt}}(\gamma)\big{|}_{\mathcal{A}_{g}},\delta\rangle=\langle\gamma\big{|}_{\mathcal{A}_{g}},\delta\rangle$
$\displaystyle\implies$ $\displaystyle\ \
\mathsf{taut}^{\mathsf{cpt}}(\gamma)\big{|}_{\mathcal{A}_{g}}=\mathsf{taut}\left(\gamma\big{|}_{\mathcal{A}_{g}}\right)\,.$

Here, we have used Definition 2, equation (7), and Definition 4 (and the
argument is not possible without Theorem 3).

###  1.5. Tautological projection of product classes

As an application of the theory, we consider the tautological projections of
product loci. For $g=g_{1}+g_{2}$ with $g_{i}\geq 1$, the product map

$\mathcal{A}_{g_{1}}\times\mathcal{A}_{g_{2}}\rightarrow\mathcal{A}_{g}$

defines a class
$[\mathcal{A}_{g_{1}}\times\mathcal{A}_{g_{2}}]\in\mathsf{CH}^{*}(\mathcal{A}_{g})$.
More generally, for every partition $g=\sum_{i=1}^{\ell}g_{i}$ in positive
parts, we have a product map and an associated class:

(8) $\prod_{1=1}^{\ell}\mathcal{A}_{g_{i}}\rightarrow\mathcal{A}_{g}\,,\ \ \ \
\ \
\left[\prod_{1=1}^{\ell}\mathcal{A}_{g_{i}}\right]\in\mathsf{CH}^{*}(\mathcal{A}_{g})\,.$

Whether these product maps and classes (8) naturally extend to a
compactification $\overline{\mathcal{A}}_{g}$ depends upon the choice of
toroidal compactification. Toroidal compactifications of $\mathcal{A}_{g}$
correspond to choices of admissible fans $\Sigma_{g}$ in
$\operatorname{Sym}^{2}_{\mathrm{rc}}(\mathbb{R}^{g})$, the rational closure
of the positive-definite symmetric forms on $\mathbb{R}^{g}$. Following [GHT],
a collection of such fans $\\{\Sigma_{g}\\}_{g\in\mathbb{N}}$ is additive if
the sum $\sigma_{1}\oplus\sigma_{2}$ of any two cones
$\sigma_{1}\in\Sigma_{g_{1}}$ and $\sigma_{2}\in\Sigma_{g_{2}}$ is a cone in
$\Sigma_{g_{1}+g_{2}}$. Let $\overline{\mathcal{A}}^{\Sigma_{g}}_{g}$ be a
toroidal compactification corresponding to an additive collection of fans
$\\{\Sigma_{g}\\}$. The perfect cone compactification satisfies these
properties, see [SB]. In the additive case, the product maps extend,

$\prod_{i=1}^{\ell}\overline{\mathcal{A}}^{\Sigma_{g_{i}}}_{g_{i}}\rightarrow\overline{\mathcal{A}}^{\Sigma_{g}}_{g}\,,$

and we can therefore define cycles

$\left[\prod_{i=1}^{\ell}\overline{\mathcal{A}}^{\Sigma_{g_{i}}}_{g_{i}}\right]\in\mathsf{CH}^{*}(\overline{\mathcal{A}}_{g}^{\Sigma_{g}})\,.$

While the definition of tautological projection is independent of toroidal
compactification, natural compactifications can be used for the calculation.
We prove a closed formula for the tautological projection of the product
cycles. The result extends calculations of [GH] for $g\leq 5$.

######  Theorem 5.

For $g_{1}+\ldots+g_{\ell}=g$, the tautological projection of the product
locus
$\overline{\mathcal{A}}_{g_{1}}^{\Sigma_{g_{1}}}\times\cdots\times\overline{\mathcal{A}}_{g_{\ell}}^{\Sigma_{g_{\ell}}}$
in $\overline{\mathcal{A}}_{g}^{\Sigma_{g}}$ is given by a $g\times g$
determinant,

$\mathsf{taut}^{\mathsf{cpt}}([\overline{\mathcal{A}}_{g_{1}}^{\Sigma_{g_{1}}}\times\cdots\times\overline{\mathcal{A}}_{g_{\ell}}^{\Sigma_{g_{\ell}}}])=\frac{\gamma_{g_{1}}\cdots\gamma_{g_{\ell}}}{\gamma_{g}}\begin{vmatrix}\lambda_{\alpha_{1}}&\lambda_{\alpha_{1}+1}&\ldots&\lambda_{\alpha_{1}+g-1}\\\
\lambda_{\alpha_{2}-1}&\lambda_{\alpha_{2}}&\ldots&\lambda_{\alpha_{2}+g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{\alpha_{g}-g+1}&\lambda_{\alpha_{g}-g+2}&\ldots&\lambda_{\alpha_{g}}\end{vmatrix}\,,$

for the vector

$\alpha=(\underbrace{g-g_{1},\ldots,g-g_{1}}_{g_{1}},\underbrace{g-g_{1}-g_{2},\ldots,g-g_{1}-g_{2}}_{g_{2}},\ldots,\underbrace{g-g_{1}-\ldots-
g_{\ell},\ldots,g-g_{1}-\ldots-g_{\ell}}_{g_{\ell}}).$

We set $\lambda_{k}=0$ for $k<0$ or $k>g$ and $\lambda_{0}=1.$

In the above determinant, $\alpha_{i}$ denotes the $i^{\text{th}}$ component
of the vector $\alpha$. The last $g_{\ell}$ entries of $\alpha$ are $0$, and
contribute rows with $1$’s on the main diagonal and $0$’s below the main
diagonal. These last entries do not change the determinant, but are included
for a more symmetric formulation. The constants $\gamma_{g}$ are defined in
(4).

The proof of Theorem 5 in Section 3 relies on the connections between the
tautological ring of $\overline{\mathcal{A}}_{g}^{\Sigma_{g}}$ and the Chow
ring of the Lagrangian Grassmannian $\mathsf{LG}_{g}$ of $\mathbb{C}^{2g}$ as
explained in [vdG2]. The argument combines properties of tautological
projection, the Hirzebruch-Mumford proportionality principle, and the geometry
of $\mathsf{LG}_{g}$.

Using the restriction property of tautological projection and the relations in
Theorem 1 (iii), we prove the following result in Section 3.4.

######  Theorem 6.

For $g_{1}+\ldots+g_{\ell}=g$, the tautological projection of the product
locus ${\mathcal{A}}_{g_{1}}\times\cdots\times{\mathcal{A}}_{g_{\ell}}$ in
${\mathcal{A}}_{g}$ is given by a $(g-\ell)\times(g-\ell)$ determinant,

$\mathsf{taut}{\left([\mathcal{A}_{g_{1}}\times\cdots\times\mathcal{A}_{g_{\ell}}]\right)}=\frac{\gamma_{g_{1}}\cdots\gamma_{g_{\ell}}}{\gamma_{g}}\cdot\lambda_{g-1}\cdots\lambda_{g-\ell+1}\cdot\begin{vmatrix}\lambda_{\beta_{1}}&\lambda_{\beta_{1}+1}&\ldots&\lambda_{\beta_{1}+g^{*}-1}\\\
\lambda_{\beta_{2}-1}&\lambda_{\beta_{2}}&\ldots&\lambda_{\beta_{2}+g^{*}-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{\beta_{g^{*}}-g^{*}+1}&\lambda_{\beta_{g^{*}}-g^{*}+2}&\ldots&\lambda_{\beta_{g^{*}}}\end{vmatrix}\,,$

for the vector

$\beta=(\underbrace{g^{*}-g^{*}_{1},\ldots,g^{*}-g^{*}_{1}}_{g^{*}_{1}},\underbrace{g^{*}-g^{*}_{1}-g^{*}_{2},\ldots,g^{*}-g^{*}_{1}-g^{*}_{2}}_{g_{2}^{*}},\ldots,\underbrace{g^{*}-g_{1}^{*}-\ldots-
g_{\ell}^{*}}_{g^{*}_{\ell}})\,,$

where $g^{*}=g-\ell$ and $g_{i}^{*}=g_{i}-1$.

The tautological projections of the product loci in $\mathcal{A}_{g}$ from
Theorem 6 in the simplest cases are:

(9)
$\mathsf{taut}\left([{\mathcal{A}}_{1}\times\mathcal{A}_{g-1}]\right)=\frac{g}{6|B_{2g}|}\lambda_{g-1}\,,$
(10)
$\mathsf{taut}\left(\left[\mathcal{A}_{2}\times\mathcal{A}_{g-2}\right]\right)=\frac{1}{360}\cdot\frac{g(g-1)}{|B_{2g}||B_{2g-2}|}\cdot\lambda_{g-1}\lambda_{g-3}\,,$

(11)
$\mathsf{taut}\left(\left[\mathcal{A}_{3}\times\mathcal{A}_{g-3}\right]\right)=\frac{1}{45360}\cdot\frac{g(g-1)(g-2)}{|B_{2g}||B_{2g-2}||B_{2g-4}|}\cdot\lambda_{g-1}(\lambda_{g-4}^{2}-\lambda_{g-3}\lambda_{g-5})\,,$

(12)
$\mathsf{taut}\left(\left[\mathcal{A}_{1}\times\mathcal{A}_{2}\times\mathcal{A}_{g-3}\right]\right)=\frac{1}{90}\cdot\frac{g(g-1)(g-2)}{|B_{2g}||B_{2g-2}||B_{2g-4}|}\cdot\lambda_{g-1}\lambda_{g-2}\lambda_{g-4}\,,$

(13)
$\mathsf{taut}\left(\left[\underbrace{\mathcal{A}_{1}\times\ldots\times\mathcal{A}_{1}}_{k}\times\mathcal{A}_{g-k}\right]\right)=\left(\prod_{i=g-k+1}^{g}\frac{i}{6|B_{2i}|}\right)\lambda_{g-1}\cdots\lambda_{g-k}\,.$

In genus $g=4$, formula (13) yields

$\mathsf{taut}\left(\left[\mathcal{A}_{1}\times\mathcal{A}_{1}\times\mathcal{A}_{2}\right]\right)=420\lambda_{3}\lambda_{2}\,,\quad\mathsf{taut}\left(\left[\mathcal{A}_{1}\times\mathcal{A}_{1}\times\mathcal{A}_{1}\times\mathcal{A}_{1}\right]\right)=4200\lambda_{3}\lambda_{2}\lambda_{1}\,.$

In fact,
$\left[\mathcal{A}_{1}\times\mathcal{A}_{1}\times\mathcal{A}_{2}\right]$ and
$\left[\mathcal{A}_{1}\times\mathcal{A}_{1}\times\mathcal{A}_{1}\times\mathcal{A}_{1}\right]$
are tautological [COP].

An interesting case of (13) occurs when $k=g-1$ since the class
$\lambda_{g-1}\cdots\lambda_{1}$ generates the socle of the tautological ring
$\mathsf{R}^{\binom{g}{2}}(\mathcal{A}_{g})$. A speculation of [COP] is that
the $g$-fold product

$[\mathcal{A}_{1}\times\cdots\times\mathcal{A}_{1}]\in\mathsf{CH}^{\binom{g}{2}}(\mathcal{A}_{g})$

also lies in the socle of the tautological ring. If the speculation is
correct, then we obtain an exact evaluation

(14)
$[\mathcal{A}_{1}\times\dots\times\mathcal{A}_{1}]=\left(\prod_{i=1}^{g}\frac{i}{6|B_{2i}|}\right)\lambda_{g-1}\cdots\lambda_{1}\,.$

Question A. When is the non-tautological part of the product locus nonzero:

$\left[\prod_{1=1}^{\ell}\mathcal{A}_{g_{i}}\right]-\mathsf{taut}\left(\left[\prod_{1=1}^{\ell}\mathcal{A}_{g_{i}}\right]\right)\neq
0\,?$

The cycles
$[\mathcal{A}_{1}\times\mathcal{A}_{g-1}]\in\mathsf{CH}^{*}(\mathcal{A}_{g})$
are studied in [COP] via the Torelli map

${\mathsf{Tor}}:\mathcal{M}_{g}^{\operatorname{ct}}\to\mathcal{A}_{g}\,.$

A central result of [COP] is that $[\mathcal{A}_{1}\times\mathcal{A}_{5}]$ is
not tautological on $\mathcal{A}_{6}$, so the purely non-tautological part is
nonzero,

$\left[\mathcal{A}_{1}\times\mathcal{A}_{5}\right]-\mathsf{taut}\left(\left[\mathcal{A}_{1}\times\mathcal{A}_{5}\right]\right)\neq
0\,.$

Detection of the non-vanishing of the non-tautological part is subtle since
the class

$\Delta_{g}={\mathsf{Tor}}^{*}\left([\mathcal{A}_{1}\times\mathcal{A}_{g-1}]-\frac{g}{6|B_{2g}|}\lambda_{g-1}\right)$

lies in the kernel of the $\lambda_{g}$-pairing

$\mathsf{R}^{g-1}(\mathcal{M}_{g}^{\operatorname{ct}})\times\mathsf{R}^{g-2}(\mathcal{M}_{g}^{\operatorname{ct}})\to\mathsf{R}^{2g-3}(\mathcal{M}_{g}^{\operatorname{ct}})\cong\mathbb{Q}$

for all genera $g$ by an argument of Pixton, see [COP].

The Noether-Lefschetz loci in $\mathcal{A}_{g}$ parametrize abelian varieties
whose Picard rank jumps. The rank $2$ Noether-Lefschetz loci have been
classified in [DL] (the products
$\mathcal{A}_{g_{1}}\times\mathcal{A}_{g_{2}}$ for $g_{1}+g_{2}=g$ arise in
the classification, but there are other loci as well). After Theorem 6, we can
hope for a more general result.

Question B. Calculate the tautological projections of all Noether-Lefschetz
loci in $\mathcal{A}_{g}$.

Beyond product and Noether-Lefschetz cycles, we can consider the tautological
projection of the locus of Jacobians of genus $g$ curves of compact type,

$[\mathcal{J}_{g}]\in\mathsf{CH}^{*}(\mathcal{A}_{g})\,.$

Faber [Fa] determined these explicitly for $g\leq 7$. For all genera, in the
basis of monic square-free monomials in the $\lambda^{\prime}s$, the leading
term is given by

$\mathsf{taut}\left(\left[\mathcal{J}_{g}\right]\right)=\left(\frac{1}{g-1}\prod_{i=1}^{g-2}\frac{2}{(2i+1)|B_{2i}|}\right)\lambda_{1}\cdots\lambda_{g-3}+\ldots\,,$

as proposed in [Fa, Conjecture 1] and confirmed via [FP1, Theorem 4]. A more
complicated formula for the coefficient of the term
$\lambda_{2}\ldots\lambda_{g-4}\lambda_{g-2}$ was predicted by [Fa, Conjecture
2] and subsequently proven in [FP2, Section 5.2].

For each genus $g$, the class
$\mathsf{taut}\left(\left[\mathcal{J}_{g}\right]\right)\in\mathsf{R}^{*}(\mathcal{A}_{g})$
can be computed algorithmically by a finite number of Hodge integral
evaluations [Fa]. Finding expressions for the coefficients of the remaining
terms of $\mathsf{taut}\left(\left[\mathcal{J}_{g}\right]\right)$ is an open
question, but we could hope for more structure.

Question C. Is there a simpler way to understand the tautological projection

$\mathsf{taut}\left(\left[\mathcal{J}_{g}\right]\right)\in\mathsf{R}^{*}(\mathcal{A}_{g})\,?$

When is the non-tautological part nonzero?

### Acknowledgments

We thank Valery Alexeev, Gerard van der Geer, Sam Grushevsky, Klaus Hulek,
Aitor Iribar Lopez, Andrew Kresch, and Aaron Pixton for several related
conversations.

Discussions with Carel Faber have played a crucial role in our work. His early
calculations of tautological projections on toroidal compactifications include
an independent (unpublished) proof of formula (9). After a lecture by R.P. at
the Belgian-Dutch Algebraic Geometry Seminar in November 2023, Ben Moonen
suggested another path to the $\lambda_{g}$ vanishing of Theorem 3 by applying
rigidity results on the boundary of $\overline{\mathcal{A}}_{g}$. We have
chosen to use an approach using semistable reduction and residue maps. We
thank Dan Abramovich, Younghan Bae, and Dori Bejleri for discussions about
residues and for comments on earlier versions of the construction.

S.C. was supported by a Hermann-Weyl-Instructorship from the
Forschungsinstitut für Mathematik at ETH Zürich. S.M. was supported by
SNF-200020-182181 and ERC-2017-AdG-786580-MACI. R.P. was supported by
SNF-200020-182181, SNF-200020-219369, ERC-2017-AdG-786580-MACI, and SwissMAP.

This project has received funding from the European Research Council (ERC)
under the Euro- pean Union Horizon 2020 research and innovation program (grant
agreement No. 786580).

##  2\. The top Chern class of the Hodge bundle

###  2.1. Overview

Logarithmic geometry provides a convenient tool for considering all toroidal
compactifications of $\mathcal{A}_{g}$ simultaneously and plays an important
role in the proof of Theorem 3. A quick review of the basic language of log
geometry is given in Section 2.2. The proof of Theorem 3 relies on the residue
map constructed in Section 2.3. After a discussion of the Hodge bundle on
toroidal compactifications of $\mathcal{A}_{g}$ in Section 2.4, the proof of
Theorem 3 is presented in Section 2.5. Conjectures and future directions
related to tautological classes on toroidal compactifications of
$\mathcal{A}_{g}$ are discussed in Section 2.6.

###  2.2. The logarithmic Chow ring for toroidal embeddings

We will use the language of log geometry and assume some rudimentary
familiarity with the central definitions as given in [K]. A summary of the
relevant background information can be found in [MPS].

For a log scheme $(S,M_{S})$, we write

$\epsilon:M_{S}\to\mathcal{O}_{S}$

for the structure morphism from the monoid $M_{S}$. Let $M_{S}^{\text{gp}}$ be
the associated group, and let $\overline{M}_{S}$ be the characteristic monoid

$\overline{M}_{S}=M_{S}/\mathcal{O}_{S}^{*}\,.$

The sheaf $\overline{M}_{S}$ is constructible, and thus stratifies $S$.

For a toroidal embedding $(S,D)$, the log structure is given by the étale
sheaf of monoids

$M_{S}=\\{f\in\mathcal{O}_{S}:f\textup{ is a unit on }S\smallsetminus D\\}\,.$

For toroidal embeddings, we will denote the log structure by either $(S,D)$ or
$(S,M_{S})$ depending upon context.

An important special case is that of a normal crossings pair $(S,D)$: a smooth
Deligne–Mumford stack $S$ with a normal crossings divisor $D$. These are
precisely the log smooth log Deligne–Mumford stacks with smooth underlying
stack. The normal crossings condition is equivalent to

$\overline{M}_{S,s}=\mathbb{N}^{k}$

for each $s\in S$ and for some $k$ depending on $s$. The integer $k$ is the
number of local branches of $D$ passing though $s$.

For a morphism of log structures $f:X\to S$, let

$\overline{M}_{X/S}=\overline{M}_{X}/\overline{M}_{S}$

be the relative characteristic monoid. The morphism $f$ is _strict_ if
$\overline{M}_{X/S}=0$. For a log scheme $(X,M_{X})$ defined over a field $K$,
a (global) _chart_ for the log structure of $X$ is a finitely generated,
saturated monoid $P$ and a strict map

$X\to\textup{Spec}(K[P])\,,$

where the target carries the canonical log structure (coming from the torus
invariant divisor). We require all log schemes to admit charts étale locally.

A morphism $f:X\to S$ is _log smooth_ if, étale locally on $X$, we can find a
map of monoids $Q\to P$ such that $\text{Ker}(Q^{\text{gp}}\to P^{\text{gp}})$
and the torsion part of $\text{Coker}(Q^{\text{gp}}\to P^{\text{gp}})$ are
finite groups of order invertible in $K$, and a diagram

${X}$${S\times_{\operatorname{Spec}K[Q]}\operatorname{Spec}K[P]}$${\operatorname{Spec}K[P]}$${S}$${\operatorname{Spec}K[Q]}$$\scriptstyle{\alpha_{X}}$$\scriptstyle{\alpha_{S}}$

with $\alpha_{X},\alpha_{S}$ strict and $\alpha_{X}$ smooth. If we can find
such a diagram with $Q\to P$ of finite index and $\alpha_{X}$ étale, the
morphism is log étale. In particular, toroidal embeddings $(S,D)$ are exactly
the log smooth log schemes over $(\operatorname{Spec}K,K^{*})$.

To a toroidal embedding $(S,D)$, we can associate a cone complex
$\Sigma_{(S,D)}$. We refer the reader to [MPS, Section 4.3] for an outline of
the construction and further references. Each cone has an integral structure,
and the cone complex is built by gluing the cones together with their integral
structure. A strata blowup is a blowup of $(S,D)$ along a smooth closed
stratum. The result is a new toroidal embedding $(S^{\prime},D^{\prime})$ with
$D^{\prime}$ the total transform of $D$, so the procedure can be iterated
indefinitely. A _log modification_ of $(S,D)$ is a proper birational map
$S^{\prime}\to S$ that can be dominated by an iterated strata blowup. More
intrinsically, the log modifications of $S$ are precisely the proper,
representable, log étale monomorphisms $S^{\prime}\to S$. Combinatorially, log
modifications of $S$ correspond exactly to subdivisions of the cone complex
$\Sigma_{(S,D)}$.

Log modifications form a filtered system. Indeed, two log modifications

$S^{\prime}\to S\ \ \ \text{and}\ \ \ S^{\prime\prime}\to S$

can always be dominated by a third: the log modification corresponding to the
common refinement of the subdivisions corresponding to $S^{\prime}$ and
$S^{\prime\prime}$ together with the intersection of the integral structures.
To a toroidal embedding $(S,D)$, we can thus associate refined operational
Chow groups

$\mathsf{logCH}^{*}(S,D)=\varinjlim\mathsf{CH}^{\textup{op}}(S^{\prime})\,,$

where $S^{\prime}$ ranges over log modifications of $S$.

Toroidal compactifications of $\mathcal{A}_{g}$ correspond to admissible
decompositions of the rational closure
$\operatorname{Sym}^{2}_{\mathrm{rc}}(\mathbb{R}^{g})$ of the cone of
positive-definite symmetric quadratic forms on $\mathbb{R}^{g}$. Any two
admissible decompositions can be refined by a third. Hence,
$\mathsf{logCH}^{*}(\overline{\mathcal{A}}_{g},\partial\overline{\mathcal{A}}_{g})$
is independent of the choice of compactification. We define

$\mathsf{logCH}^{*}(\mathcal{A}_{g})=\mathsf{logCH}^{*}(\overline{\mathcal{A}}_{g},\partial\overline{\mathcal{A}}_{g})$

for any toroidal compactification $\overline{\mathcal{A}}_{g}$.

###  2.3. Semistable families and residues

For suitable families of log schemes, we prove the existence of a residue map
in Theorem 16 below. The residue map will be applied in Section 2.4 to the
universal family over the moduli space of abelian varieties in order to prove
Theorem 3.

The sheaf of relative _logarithmic differentials_ $\Omega_{X/S}^{\log}$ is
defined as the quotient of

$\Omega_{X/S}\oplus(\mathcal{O}_{X}\otimes_{\mathbb{Z}}M_{X}^{\text{gp}})$

by the subsheaf locally generated by sections of the form

$\displaystyle(d\epsilon(m),0)-(0,\epsilon(m)\otimes m)\,\ \ \ \text{and}\ \ \
(0,1\otimes n)\,,$

where $m\in M_{X}$ and $n\in\text{Im}(M_{S}^{\text{gp}})\subset
M^{\text{gp}}_{X}$, see [K]. As usual, we write

$d\log m=(0,1\otimes m)\,$

which we view as $d\epsilon(m)/\epsilon(m)$.

For a strict map $f:X\to S$, we have $\Omega_{X/S}^{\log}=\Omega_{X/S}$, and
for a log étale map $f:X\to S$, we have $\Omega_{X/S}^{\log}=0$.

######  Definition 7.

The _sheaf of residues_ is defined to be the quotient

$\mathcal{R}=\Omega_{X/S}^{\log}/\Omega_{X/S}\,.$

######  Definition 8.

([M, Definition 2.1.2]) A _logarithmic family_ $X\to S$ is a log smooth,
surjective, integral and saturated map of log schemes.

Families of stable curves and families of toroidal compactifications of semi-
abelian schemes are all examples of log families. The condition that $f$ is
integral and saturated – called weak semistability in [M] – is a technical
condition that, for log smooth $f$, implies that $f$ is flat with reduced
fibers [M, Lemma 3.1.2], [Ts, Theorem II.4.2].888When $S$ and is also log
smooth, the integrality condition is equivalent to $f$ being flat, and the
conditions that $f$ is integral and saturated together are equivalent to $f$
being flat with reduced fibers. Being integral and saturated is local on $X$
and can be understood in terms of the cone complexes $\Sigma_{X}$ and
$\Sigma_{S}$. Integrality combined with saturatedness say, locally on $X$,
that the associated map $\Sigma_{X}\to\Sigma_{S}$ maps cones of $\Sigma_{X}$
surjectively onto cones of $\Sigma_{S}$ and that the integral structure of a
cone $\sigma$ surjects onto the integral structure of its image cone.

Given a log scheme $(S,M_{S})$ and a finite index extension of sheaves
$M_{S}\to M_{S}^{\prime}$, there is a universal log DM stack
$(S^{\prime},M_{S}^{\prime})$ with a log map to $(S,M_{S})$ whose map on log
structures is given by the extension $M_{S}\to M_{S}^{\prime}$. The stack
$S^{\prime}$ is called the root of $S$ along $M_{S}\to M_{S}^{\prime}$. The
simplest instance of this operation is taking a root along a boundary stratum
of a normal crossings pair $(S,D)$. We call a composition of logarithmic
modifications and roots a _logarithmic alteration_. Log alterations of
toroidal embeddings are isomorphisms on $S\smallsetminus D$, but are not
necessarily representable. Logarithmic alterations are furthermore log étale.
See [MW] for a lengthier discussion.

######  Remark 9.

Because we work with $\mathbb{Q}$-coefficients, pullbacks via root maps induce
isomorphisms on Chow groups. Therefore, for a toroidal embedding $(S,D)$, the
logarthmic Chow groups can be equivalently defined as

$\mathsf{logCH}^{*}(S,D)=\varinjlim\mathsf{CH}^{\textup{op}}(S^{\prime})\,,$

where $S^{\prime}$ ranges over logarithmic alterations of $S$.

######  Definition 10.

Let $f:X\to S$ be a log map. A _logarithmic alteration of $f$_ is a log map
$f^{\prime}:X^{\prime}\to S^{\prime}$ and a commutative diagram

${X^{\prime}}$${X}$${S^{\prime}}$${S}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

such that $S^{\prime}\to S$ and $X^{\prime}\to X$ are logarithmic alterations.

######  Theorem 11 (Semistable Reduction Theorem, [ALT, AK, M]).

Let $f:X\to S$ be a dominant log smooth morphism of logarithmic schemes. Then
there is a log alteration $f^{\prime}:X^{\prime}\to S^{\prime}$ of $f$ which
is a log family. Furthermore, if $S$ is log smooth, then one can take
$X^{\prime}$ and $S^{\prime}$ smooth.

######  Definition 12.

A pair $(X,D)$ is called _simplest normal crossings_ if $D\subset X$ is normal
crossings in the Zariski topology and each intersection of components of $D$
is connected.

######  Remark 13.

In more geometric terms, a toroidal embedding $(X,D)$ has simplest normal
crossings if the following conditions are all satisfied:

  1. (i)

$(X,D)$ is a normal crossings pair (with $X$ smooth),

  2. (ii)

the components of $D$ have no self-intersection,

  3. (iii)

intersections of components of $D$ are connected.

We note that properties (ii) and (iii) are stable under logarithmic
alterations, but (i) is not.

######  Corollary 14.

Let $f:(X,D_{X})\to(S,D_{S})$ be a log family with $(S,D_{S})$ toroidal. Then
there is a log alteration
$f^{\prime}:(X^{\prime},D_{X^{\prime}})\to(S^{\prime},D_{S^{\prime}})$ which
is a log family where both $S^{\prime}$ and $X^{\prime}$ have simplest normal
crossings.

###### Proof.

We can make $S$ and $X$ simplest normal crossings by suitable log
modifications

$S_{1}\to S\,,\ \ \ X_{1}\to X\times_{S}S_{1}\,.$

For example, we can take the log modifications corresponding to double
barycentric subdivisions (see the discussion in [MPS, Section 5.6]). Since
logarithmic alterations are log étale and surjective, the morphism $X_{1}\to
S_{1}$ remains log smooth and surjective. Moreover, as noted in Remark 13, any
further log alteration of $X_{1}$ or $S_{1}$ which is smooth will have
simplest normal crossings. Therefore, we may apply Theorem 11 to $X_{1}\to
S_{1}$ to get the desired $X^{\prime}\to S^{\prime}$. ∎

Let $(X,M_{X})$ be a log scheme. An _orientation_ on $M_{X}$ is an ordering of
the sections of $\overline{M}_{X}(U)$ for all $U\subset X$ compatible with the
restriction maps. We say $M_{X}$ is _orientable_ if it admits an orientation.

######  Lemma 15.

Suppose $(X,D)$ is a pair with simplest normal crossings. Then the log
structure of $(X,D)$ is orientable.

###### Proof.

We choose an ordering of the components $D_{i}$ of the divisor $D$. Every
stratum is the intersection

$D_{i_{1}}\cap\cdots\cap D_{i_{k}}$

where the components appear in the ordering we have chosen. For each $x$ in
the stratum, we have a canonical isomorphism

$\overline{M}_{X,x}=\bigoplus\mathbb{N}m_{i_{k}}\,.$

Here, $m_{i_{k}}$ is the image in $\overline{M}_{X,x}$ of any element
$\widetilde{m}_{i_{k}}\in M_{X,x}$ that maps to a local defining equation for
$D_{i_{k}}$. We order the sections as $m_{i_{1}}<m_{i_{2}}\cdots<m_{i_{k}}$. ∎

######  Theorem 16.

Let $f:X\to S$ be a log family with $X$ and $S$ simplest normal crossings
pairs. Every choice of orientation on $\overline{M}_{X}$ yields a map

(15) $\mathcal{R}\to\oplus_{H}\mathcal{O}_{H}$

where $H$ ranges over the irreducible components of the stratification
determined by $\overline{M}_{X/S}$ on the locus where $\textup{rank
}\overline{M}_{X/S}\geq 1$. Furthermore, the projection

$\mathcal{R}\to\mathcal{O}_{H}$

of (15) to each summand $\mathcal{O}_{H}$ is surjective.

###### Proof.

Choose an orientation of $\overline{M}_{X}$ as in Lemma 15. We will construct
the map (15) locally on $X$, and then we will prove the gluing compatibility
required for the global definition.

Let $x$ be a point of $X$. Choose an ordered basis

$m_{1},\dots,m_{n}\in\overline{M}_{X,x}$

of $\overline{M}_{X}$ at $x$. Let $\widetilde{m}_{i}$ be arbitrary lifts in
$M_{X}$, and write $x_{i}=\epsilon(\widetilde{m}_{i})$ for their images in
$\mathcal{O}_{X}$. In other words, $x_{i}$ are local defining equations for
the divisor $D_{i}$ of $X$ at $x$. Similarly, write $t_{i}$ for the
corresponding images in $S$ near $f(x)$. Then, without loss of generality, we
may assume that the map of characteristic monoids has the form

$\mathbb{N}^{k}=\overline{M}_{S,f(x)}\to\overline{M}_{X,x}=\mathbb{N}^{n_{1}}\oplus\mathbb{N}^{n_{2}}\oplus\cdots\oplus\mathbb{N}^{n_{k}}\oplus\mathbb{N}^{\ell}\,,$

with the $j^{th}$ basis element of $\mathbb{N}^{k}$ mapping to the vector
$(\underbrace{1,\ldots,1}_{n_{j}})$ of the summand $\mathbb{N}^{n_{j}}$ on the
right. We have equations

$t_{1}=u_{1}\prod_{\alpha\in
A_{1}}x_{\alpha}\,,\,\,\ldots,\,\,t_{k}=u_{k}\prod_{\alpha\in
A_{k}}x_{\alpha}$

for disjoint sets $A_{1},\ldots,A_{k}\subset\\{1,2,\ldots,n\\}$ with
$n_{1},\ldots,n_{k}$ elements respectively and units
$u_{i}\in\mathcal{O}_{X,x}$. By the orientation assumption, the sets
$A_{1},\ldots,A_{k}$ are ordered. For convenience, we write

$A=A_{1}\cup\ldots\cup A_{k}\,.$

The additional $\ell$ parameters $y_{1},\cdots,y_{\ell}$ have vanishing loci
$V(y_{i})$ representing horizontal divisors999For example, when $X\to S$ is a
family of curves, the $V(y_{i})$ correspond to markings. In our study of the
moduli of abelian varieties, $\ell=0$. over $S$.

The logarithmic differentials $\Omega_{X,x}^{\log}$ are generated by
$\Omega_{X,x}$ and $\frac{dx_{\alpha}}{x_{\alpha}}$, $\alpha\in A$, and
$\frac{dy_{1}}{y_{1}},\ldots,\frac{dy_{\ell}}{y_{\ell}}$. We have the
relations

$\frac{du_{i}}{u_{i}}+\sum_{\alpha\in
A_{i}}\frac{dx_{\alpha}}{x_{\alpha}}=\frac{dt_{i}}{t_{i}},\,\,\,1\leq i\leq
k\,.$

The quotient $\mathcal{R}=\Omega_{X/S}^{\log}/\Omega_{X/S}$ has a presentation
as an $\mathcal{O}_{X}$-module with generators

(16)
$\frac{dx_{\alpha}}{x_{\alpha}},\frac{dy_{1}}{y_{1}},\ldots,\frac{dy_{\ell}}{y_{\ell}}\,,$

where $\alpha\in A$. The relations are

$\sum_{\alpha\in A_{i}}\frac{dx_{\alpha}}{x_{\alpha}}=0\,,\,\,\,\,1\leq i\leq
k\,,$

(since we are working with relative differentials and the $du_{i}/u_{i}$ are
in $\Omega_{X/S}$) and additionally

$\displaystyle
x_{\alpha}\frac{dx_{\alpha}}{x_{\alpha}}=0\,,\,\,y_{1}\frac{dy_{1}}{y_{1}}=0\,,\ldots,y_{\ell}\frac{dy_{\ell}}{y_{\ell}}=0\,,$

where $\alpha\in A$. The irreducible components of the stratification of
$M_{X/S}$ at $x$ (with $\text{rk }\geq 1$) are given either by

  * •

$t_{i}=0,x_{\beta}=0,x_{\gamma}=0$ for triples $(i,\beta,\gamma)$ with
$\beta<\gamma$ elements in $A_{i}$, or by

  * •

$y_{j}=0$ for some $1\leq j\leq\ell$.

Thus, we find101010Of course, $t_{i}\in(x_{\beta},x_{\gamma})$, but we have
chosen to keep $t_{i}$ in the notation to emphasize that $\beta,\gamma$ belong
to the same part $A_{i}$.

$\oplus_{H}\mathcal{O}_{H}=\bigoplus_{(i,\beta,\gamma)}\mathcal{O}_{X}/(t_{i},x_{\beta},x_{\gamma})\oplus\bigoplus_{j}\mathcal{O}_{X}/(y_{j}).$

We define a map

$\mathcal{R}\to\mathcal{O}_{X}/(t_{i},x_{\beta},x_{\gamma})$

by sending all the generators in (16) to $0$, with the exception of

$\frac{dx_{\beta}}{x_{\beta}}\mapsto
1,\quad\frac{dx_{\gamma}}{x_{\gamma}}\mapsto-1\,.$

Similarly, we define a map

$\mathcal{R}\to\mathcal{O}_{X}/(y_{j})$

by sending all generators in (16) to $0$, with the exception of

$\frac{dy_{j}}{y_{j}}\mapsto 1.$

We must verify that the map is well-defined. First, for each
$(i,\beta,\gamma)$, we see that

$\sum_{\alpha\in A_{i}}\frac{dx_{\alpha}}{x_{\alpha}}\mapsto 0$

since $\frac{dx_{\beta}}{x_{\beta}}$ and $\frac{dx_{\gamma}}{x_{\gamma}}$ map
to opposite elements in $\mathcal{O}_{X}/(t_{i},x_{\beta},x_{\gamma})$, and
the other terms map to $0$. The fact that

$x_{\alpha}\frac{dx_{\alpha}}{x_{\alpha}},\,\,y_{1}\frac{dy_{1}}{y_{1}},\ldots,y_{\ell}\frac{dy_{\ell}}{y_{\ell}},\,\alpha\in
A$

map to $0$ in $\mathcal{O}_{X}/(t_{i},x_{\beta},x_{\gamma})$ and
$\mathcal{O}_{X}/(y_{j})$ is immediate from the definitions. Surjectivity of
the map to any summand $\mathcal{O}_{H}$ is also clear, as the generator $1$
is in the image.

We now inspect how our map depended on choices; the only choices involved were
the lifts $\widetilde{m}_{i}$ of $m_{i}$, and the choice of ordering of the
$m_{i}$. A different choice of $\widetilde{m}_{i}^{\prime}$ differs from the
original one by a unit, and we have

$\frac{d(ux)}{ux}=u^{-1}du+\frac{dx}{x}.$

The term $u^{-1}du$ is an ordinary differential, and thus the residue of the
logarithmic form is independent of lift. On the other hand, the map does
depend on the ordering of the coordinates. Since we assume that the ordering
is global, however, the local maps patch uniquely to all of $X$. ∎

######  Remark 17.

In case $S$ is a point, $(X,D)$ is a simplest normal crossings pair, and only
the horizontal divisors $H=\mathcal{O}_{X}/(y_{i})$ are present in our
analysis. These are precisely the components $D_{i}$ of the divisor $D$. Our
residue map then reduces to the classical residue homomorphism

${0}$${\Omega_{X}}$${\Omega_{X}^{\log}}$${\oplus_{i}\mathcal{O}_{D_{i}}}$${0\,,}$

see [F2, Chapter 4, Proposition 1].

###  2.4. The Hodge bundle

Let $\overline{\mathcal{A}}_{g}$ be a toroidal compactification of the moduli
space of principally polarized abelian varieties. The compactification
$\overline{\mathcal{A}}_{g}$ carries a universal family of semi-abelian
schemes

$q:\mathcal{U}_{g}\to\overline{\mathcal{A}}_{g}$

together with a zero section $s:\overline{\mathcal{A}}_{g}\to\mathcal{U}_{g}$.
The Hodge bundle is the rank $g$ vector bundle on $\overline{\mathcal{A}}_{g}$
defined by

$\mathbb{E}=s^{*}\Omega_{q}\,,$

with Chern classes $\lambda_{i}=c_{i}(\mathbb{E})$.

######  Definition 18.

A compactification of $q:\mathcal{U}_{g}\to\overline{\mathcal{A}}_{g}$ is a
diagram

${\mathcal{U}_{g}}$${\mathcal{X}_{g}}$${\overline{\mathcal{A}}_{g}}$$\scriptstyle{q}$$\scriptstyle{p}$

where $p$ is a proper log smooth morphism, $\mathcal{U}_{g}$ is open and dense
in $\mathcal{X}_{g}$, and $\mathcal{U}_{g}$ acts on $\mathcal{X}_{g}$
extending the natural action of $\mathcal{U}_{g}$ on itself (and commuting
with $p$). A compactification $p$ is a compactified universal family if in
addition $p$ is a log family.

The fiber $(\mathcal{X}_{g})_{t}$ of a compactified universal family $p$ over
a point $t\in\overline{\mathcal{A}}_{g}$ contains the semiabelian scheme
$(\mathcal{U}_{g})_{t}$ as an open subscheme. More precisely, write

(17) ${0}$${T_{t}}$${(\mathcal{U}_{g})_{t}}$${A_{t}}$${0\,,}$

with $T_{t}$ a torus and $A_{t}$ an abelian scheme. Then
$(\mathcal{X}_{g})_{t}$ admits a fibration

(18) ${X(T_{t})}$${(\mathcal{X}_{g})_{t}}$${A_{t}\,,}$

where $X(T_{t})$ is a union of complete toric varieties with torus $T_{t}$.

An arbitrary toroidal compactification $\overline{\mathcal{A}}_{g}$ may not
carry a compactified universal family. However, toroidal compactifications
$\overline{\mathcal{A}}_{g}$, with compactified universal families

$p:\mathcal{X}_{g}\to\overline{\mathcal{A}}_{g}\,$

can be constructed, see [FC, Chapter VI, Section 1]. Compactifications of $q$
correspond to $\operatorname{GL}_{g}\ltimes N$-admissible decompositions
$\widetilde{\Sigma}_{g}$ of a certain subcone of
$\operatorname{Sym}^{2}_{rc}(\mathbb{R}^{g})\times\operatorname{Hom}(N,\mathbb{R})$
for a rank $g$ lattice $N$. The decomposition is required to have the property
that every cone in $\widetilde{\Sigma}_{g}$ maps into a cone of the admissible
decomposition $\Sigma_{g}$ of $\operatorname{Sym}^{2}_{rc}(\mathbb{R}^{g})$
defining $\overline{\mathcal{A}}_{g}$. A compactification $p$ is a
compactified universal family if the map

$\widetilde{\Sigma}_{g}\to\Sigma_{g}$

satisfies the additional hypotheses of Definition 8 (the cones of
$\widetilde{\Sigma}_{g}$ map onto cones of $\Sigma_{g}$, and surjectivity also
holds for their integral structure).

Both notions of compactification are stable under arbitrary base change
$\overline{\mathcal{A}}_{g}^{\prime}\to\overline{\mathcal{A}}_{g}$. For a
compactification $p:\mathcal{X}_{g}\to\overline{\mathcal{A}}_{g}$, an
arbitrary log alteration $\mathcal{X}_{g}^{\prime}\to\mathcal{X}_{g}$ of the
domain of $p$ remains a compactification. On the other hand, log alterations
of $\mathcal{X}_{g}$ are _not_ compactified families, even if the original $p$
is a family, as the composed map
$\mathcal{X}_{g}^{\prime}\to\overline{\mathcal{A}}_{g}$ is rarely a log family
(flatness and reducedness of fibers is typically destroyed). Nevertheless,
semistable reduction by Theorem 11 ensures that there is a log alteration of
the map $\mathcal{X}_{g}^{\prime}\to\overline{\mathcal{A}}_{g}$ which is a
compactified family.

The sheaf of relative logarithmic differentials of $q$ and $p$ are fiberwise
trivial of rank $g$ [FC, Chapter VI, Theorem 1.1]. Triviality follows from the
fibration descriptions (17) and (18) since the sheaf of differentials on a
semi-abelian variety and the sheaf of logarithmic differentials on a toric
variety are both trivial.

When $\overline{\mathcal{A}}_{g}$ has a compactified family $p$, we can use
$\Omega_{p}^{\log}$ to define the Hodge bundle, as

$\mathbb{E}=s^{*}\Omega_{q}=s^{*}\Omega_{q}^{\log}=s^{*}\Omega_{p}^{\log}|_{\mathcal{U}_{g}}=s^{*}\Omega_{p}^{\log}$

since the section $s$ factors through $\mathcal{U}_{g}$ and the map $q$ is
strict.

A second approach to the Hodge bundle is available. The following result can
be found in [FC, Chapter VI, Theorem 1.1].

######  Lemma 19.

Suppose $\overline{\mathcal{A}}_{g}$ carries a compactified universal family
$p:\mathcal{X}_{g}\to\overline{\mathcal{A}}_{g}$. Then,

$\Omega_{p}^{\textup{log}}=p^{*}\mathbb{E}\,\ \ \ \text{and}\ \ \
\mathbb{E}=p_{*}\Omega_{p}^{\textup{log}}\,.$

###### Proof.

Since $\Omega_{p}^{\textup{log}}$ is fiberwise trivial, cohomology and base
change implies that

$\Omega_{p}^{\textup{log}}=p^{*}p_{*}\Omega_{p}^{\textup{log}}$

Since $s^{*}p^{*}=\textup{id},$ we have

$\mathbb{E}=s^{*}\Omega_{p}^{\log}=s^{*}p^{*}p_{*}\Omega_{p}^{\log}=p_{*}\Omega_{p}^{\log}\,,$

and the result follows.

∎

######  Lemma 20.

The classes $\lambda_{i}$ extend to $\mathsf{logCH}^{*}(\mathcal{A}_{g})$.

###### Proof.

In light of Remark 9, we check compatibility of Hodge classes under
logarithmic alterations. Suppose
$\tau:\overline{\mathcal{A}}_{g}^{\prime}\to\overline{\mathcal{A}}_{g}$ is a
logarithmic alteration. Then we have a Cartesian diagram

${\mathcal{U}_{g}^{\prime}}$${\mathcal{U}_{g}}$${\overline{\mathcal{A}}_{g}^{\prime}}$${\overline{\mathcal{A}}_{g}}$$\scriptstyle{\rho}$$\scriptstyle{q^{\prime}}$$\scriptstyle{q}$$\scriptstyle{\tau}$

Since $s\circ\tau=\rho\circ s^{\prime}$ for the respective zero sections, we
have

$\tau^{*}\mathbb{E}=(s^{\prime})^{*}\rho^{*}\Omega_{q}=(s^{\prime})^{*}\Omega_{q^{\prime}}=\mathbb{E}^{\prime}\,,$

which implies the required compatibility for $\lambda_{i}$. ∎

######  Remark 21.

Toroidal compactifications of $\mathcal{A}_{g}$ with a compactified universal
family

$p:\mathcal{X}_{g}\to\overline{\mathcal{A}}_{g}$

form a cofinal system among all toroidal compactifications: given an arbitrary
toroidal compactification $\overline{\mathcal{A}}_{g}^{\prime}$, we can choose
a toroidal compactification $\overline{\mathcal{A}}_{g}$ with a compactified
universal family, and then any common refinement
$\overline{\mathcal{A}}_{g}^{\prime\prime}$ of both compactifications carries
a compactified universal family.

######  Lemma 22.

The collection of simplest normal crossings compactifications
$\mathcal{A}_{g}\subset\overline{\mathcal{A}}_{g}$ that carry a compactified
universal family with simplest normal crossings is cofinal among the toroidal
compactifications $\overline{\mathcal{A}}_{g}$.

###### Proof.

Starting with an arbitrary compactified universal family

$p:\mathcal{X}_{g}\to\overline{\mathcal{A}}_{g}$

we may apply Corollary 14 to $p$, to obtain the desired family. ∎

###  2.5. Proof of Theorem 3.

Let $\overline{\mathcal{A}}_{g}$ be a toroidal compactification of
$\mathcal{A}_{g}$. By Lemma 22, there exists a toroidal compactification
$\overline{\mathcal{A}}^{\prime}_{g}$ satisfying the following conditions:

  1. (i)

there is a proper surjection
$\overline{\mathcal{A}}^{\prime}_{g}\rightarrow\overline{\mathcal{A}}_{g}$
that is an isomorphism on the interior,

  2. (ii)

$\overline{\mathcal{A}}^{\prime}_{g}$ admits a universal family
$\mathcal{X}_{g}^{\prime}\rightarrow\overline{\mathcal{A}}^{\prime}_{g}$ of
toroidal compactifications of semi-abelian schemes,

  3. (iii)

both $\overline{\mathcal{A}}^{\prime}_{g}$ and $\mathcal{X}_{g}^{\prime}$ have
simplest normal crossings.

Because proper surjections are Chow envelopes, it suffices to show that

$\lambda_{g}\big{|}_{\partial\overline{\mathcal{A}}^{\prime}_{g}}=0\,.$

Hence, after replacing $\overline{\mathcal{A}}_{g}$ by
$\overline{\mathcal{A}}^{\prime}_{g}$, we may assume that
$\overline{\mathcal{A}}_{g}$ has properties (ii) and (iii) above.

Let $T$ be a component of the boundary divisor of $\overline{\mathcal{A}}_{g}$
and denote by $p_{T}:\mathcal{X}_{T}\rightarrow T$ the base change to $T$ of
the universal family

$p:\mathcal{X}_{g}\to\overline{\mathcal{A}}_{g}\,.$

Over $T$, either $\mathcal{X}_{T}\rightarrow T$ is smooth, or we can find a
nonempty component $H$ of the singular locus $\mathcal{X}_{T}^{\textup{sing}}$
of $p_{T}$. In the first case, $\lambda_{g}\big{|}_{T}=0$ because
$\lambda_{g}\big{|}_{\mathcal{A}_{g}}=0$.

In the second case, $H$ is an irreducible component of the locus where
$\textup{rank }\overline{M}_{\mathcal{X}_{g}/\overline{\mathcal{A}}_{g}}\geq
1$. Let

$i:H\rightarrow\mathcal{X}_{T}\,,\quad p_{T}\circ i:H\rightarrow T$

be the inclusion and the projection. The map $p_{T}\circ i$ is proper and
surjective because $p$ is a log family, so it suffices to show that
$i^{*}p_{T}^{*}\left(\lambda_{g}\big{|}_{T}\right)=0$. Using Lemma 19, we have
$\Omega_{p}^{\textup{log}}=p^{*}\mathbb{E}$. After base change and pullback by
$i$, we find

$i^{*}p_{T}^{*}\mathbb{E}\big{|}_{T}=i^{*}\Omega_{p_{T}}^{\textup{log}}\,.$

It remains to check that $c_{g}(i^{*}\Omega_{p_{T}}^{\textup{log}})=0$. By
Definition 7 and Theorem 16, we have surjections

$i^{*}\Omega_{p_{T}}^{\log}\to i^{*}\mathcal{R}\to 0\,,\ \ \
i^{*}\mathcal{R}\rightarrow\mathcal{O}_{H}\to 0\,.$

We therefore have an exact sequence of vector bundles on $H$,

${0}$${K}$${i^{*}\Omega_{p_{T}}^{\log}}$${\mathcal{O}_{H}}$${0\,.}$

We conclude
$c_{g}(i^{*}\Omega_{p_{T}}^{\log})=c_{g-1}(K)c_{1}(\mathcal{O}_{H})=0\,.$ ∎

###  2.6. Log geometry and $\lambda_{g}$.

There is a distinguished subalgebra of classes coming from the boundary in the
logarithmic Chow theory defined by the image of the algebra $\mathsf{PP}$ of
$\operatorname{GL}_{g}$-invariant piecewise polynomials111111By definition, a
piecewise polynomial on $\operatorname{Sym}^{2}_{\mathrm{rc}}(\mathbb{R}^{g})$
is an admissible decomposition together with a continuous
$\operatorname{GL}_{g}$-invariant function on the decomposition that is
polynomial on each cone. on
$\operatorname{Sym}^{2}_{\mathrm{rc}}(\mathbb{R}^{g})$,

(19) $\mathsf{PP}\to\mathsf{logCH}^{*}({\mathcal{A}}_{g})\,.$

We refer the reader to [MPS, MR] for further details regarding the
construction of the map (19). Our main conjecture concerning $\lambda_{g}$ in
the logarithmic theory is the following claim.

######  Conjecture 23.

The class $\lambda_{g}\in\mathsf{logCH}^{*}(\mathcal{A}_{g})$ lies in the
image (19) of the algebra of piecewise polynomials.

Our motivation for Conjecture 23 comes from a parallel study of $\lambda_{g}$
in the logarithmic Chow theory of the moduli space of curves
$\mathcal{M}_{g}$. Using Pixton’s formula [HMPPS, JPPZ], the class
$\lambda_{g}$ is proven in [MPS] to lie in the image of the algebra of
piecewise polynomials in $\mathsf{logCH}^{*}(\mathcal{M}_{g})$.

Question D. Find a lifting to $\mathsf{logCH}^{*}(\mathcal{A}_{g})$ of
Pixton’s formula for $\lambda_{g}\in\mathsf{logCH}^{*}(\mathcal{M}_{g})$ which
is compatible with the Torelli map.

The definition by van der Geer of the tautological ring is best suited for
studying classes on the moduli space of abelian varieties $\mathcal{A}_{g}$.
There is a larger tautological ring which takes the boundaries of the various
compactifications into account,

$\mathsf{logR}^{*}({\mathcal{A}}_{g})\subset\mathsf{logCH}^{*}(\mathcal{A}_{g})\,,$

defined to be generated by all possible piecewise polynomial and Hodge classes
on all boundary strata of all toroidal compactifications
$\mathcal{A}_{g}\subset\overline{\mathcal{A}}_{g}$.

The investigation of the structure of $\mathsf{logR}^{*}({\mathcal{A}}_{g})$
is an interesting future direction. For example, pushing forward powers of the
polarization, we can define $\kappa$ classes over
$\overline{\mathcal{A}}_{g}$, see [MOP] and [A2] for similar constructions in
the context of the moduli of K3 surfaces and over KSBA moduli respectively. In
the case of abelian varieties, we expect121212We thank V. Alexeev for a
discussion about $\kappa$ classes at the conference Higher Dimensional
Algebraic Geometry in La Jolla in January 2024. that the $\kappa$ classes lie
in $\mathsf{logR}^{*}({\mathcal{A}}_{g})$. Can a precise formula be found?

##  3\. Tautological projection of product classes

###  3.1. Product cycles

We compute here the tautological projections of all product cycles

$\mathcal{A}_{g_{1}}\times\ldots\times\mathcal{A}_{g_{\ell}}\rightarrow\mathcal{A}_{g}$

for all $g$. Calculations for product cycles for genus $g\leq 5$ can be found
in [GH].

Fix toroidal compactifications $\overline{\mathcal{A}}_{g}$ corresponding to
an additive collection of fans. The product maps

$\prod_{g_{1}+\dots+g_{\ell}=g}\mathcal{A}_{g_{1}}\times\dots\times\mathcal{A}_{g_{\ell}}\rightarrow\mathcal{A}_{g}$

then extend to maps

(20)
$\prod_{g_{1}+\dots+g_{\ell}=g}\overline{\mathcal{A}}_{g_{1}}\times\dots\times\overline{\mathcal{A}}_{g_{\ell}}\rightarrow\overline{\mathcal{A}}_{g}\,.$

For example, we could take the perfect cone compactification for every $g$ by
[SB, Lemma 2.8].

The Hodge bundle splits canonically over the product (20). Indeed, the
universal semiabelian variety restricts in the natural fashion over the
product, and the splitting of the Hodge bundle then follows by restricting the
relative cotangent bundle to the zero section.

###  3.2. Lagrangian Grassmannian

As remarked in [vdG], a consequence of Theorem 1 is the $\mathbb{Q}$-algebra
isomorphism

$\mathsf{R}^{*}(\overline{\mathcal{A}}_{g})\simeq\mathsf{CH}^{*}(\mathsf{LG}_{g})\,,$

where $\mathsf{LG}_{g}$ denotes the Lagrangian Grassmannian of
$\mathbb{C}^{2g}$ with respect to a symplectic form $\omega$. An overview of
the cohomology of the Lagrangian Grassmannian from the point of Schubert
calculus can be found, for instance, in [FPr, KT, PR].

The spaces $\overline{\mathcal{A}}_{g}$ and $\mathsf{LG}_{g}$ are further
connected by the Hirzebruch-Mumford proportionality principle. Let
$\mathsf{S}\to\mathsf{LG}_{g}$ be the universal rank $g$ subbundle, and let
$x_{i}=c_{i}(\mathsf{S}^{*}).$ Then,

(21)
$\int_{\overline{\mathcal{A}}_{g}}\lambda_{I}=\gamma_{g}\int_{\mathsf{LG}_{g}}x_{I}\,$

for every $I\subset\\{1,2,\ldots,g\\}$. Here, we use the multindex notation

$\lambda_{I}=\prod_{i\in I}\lambda_{i}\,,\quad x_{I}=\prod_{i\in I}x_{i}\,.$

The proportionality constant $\gamma_{g}$ was computed in [vdG, page 9]:

$\gamma_{g}=\int_{\overline{\mathcal{A}}_{g}}\lambda_{1}\ldots\lambda_{g}=\prod_{i=1}^{g}\frac{|B_{2i}|}{4i}\,.$

For any partition

$g_{1}+\ldots+g_{\ell}=g\,,$

we can consider the product

(22)
$\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}\to\mathsf{LG}_{g}\,.$

Finding the class of

$\overline{\mathcal{A}}_{g_{1}}\times\ldots\times\overline{\mathcal{A}}_{g_{\ell}}\rightarrow\overline{\mathcal{A}}_{g}$

is equivalent to finding the class of the product cycle (22) in
$\mathsf{CH}^{*}(\mathsf{LG}_{g})$ in terms of the Chern classes
$x_{i}=c_{i}(\mathsf{S}^{*})$ of the dual subbundle. More precisely, if

(23)
$[\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}]=\mathsf{P}(x_{1},\ldots,x_{g})\in\mathsf{CH}^{*}(\mathsf{LG}_{g})\,,$

then we have

(24)
$\mathsf{taut}^{\mathsf{cpt}}\left([\overline{\mathcal{A}}_{g_{1}}^{\Sigma_{g_{1}}}\times\ldots\times\overline{\mathcal{A}}_{g_{\ell}}^{\Sigma_{g_{\ell}}}]\right)=\frac{\gamma_{g_{1}}\cdots\gamma_{g_{\ell}}}{\gamma_{g}}\cdot\mathsf{P}(\lambda_{1},\ldots,\lambda_{g})\in\mathsf{R}^{*}(\overline{\mathcal{A}}_{g}^{\Sigma_{g}})\,.$

To derive (24) from (23), we use the Gorenstein property of
$\mathsf{R}^{*}(\overline{\mathcal{A}}_{g}^{\Sigma_{g}})$. We need only check
that polynomials of complementary degrees in the $\lambda$ classes pair
equally with both sides of (24):

  * (i)

When restricted to the product loci in
$\overline{\mathcal{A}}_{g}^{\Sigma_{g}}$ and $\mathsf{LG}_{g}$, both the
Hodge bundle $\mathbb{E}$ and the dual subbundle $\mathsf{S}^{*}$ split as
direct sums.

  * (ii)

By the Hirzebruch-Mumford proportionality principle, integrals in the
$\lambda$’s over $\overline{\mathcal{A}}_{g}^{\Sigma_{g}}$ can be evaluated in
terms of integrals in $x$’s over $\mathsf{LG}_{g}$. The answers are always
proportional (21), with proportionality constant $\gamma_{g}$.

Combining (i) and (ii), we see that the constant
$\gamma_{g_{1}}\cdots\gamma_{g_{\ell}}$ arises for all factors on the left
hand side, while the constant $\gamma_{g}$ arises for all terms on the right
hand side, showing that (23) implies (24). The purely non-tautological part of
the cycle in (24) plays no role in the argument.

###  3.3. Proof of Theorem 5

For $g_{1}+\ldots+g_{\ell}=g$, we must show that the tautological projection
of the product locus
$\overline{\mathcal{A}}_{g_{1}}\times\ldots\times\overline{\mathcal{A}}_{g_{\ell}}$
in $\overline{\mathcal{A}}_{g}$ is given by the $g\times g$ determinant

$\mathsf{taut}^{\mathsf{cpt}}([\overline{\mathcal{A}}_{g_{1}}\times\ldots\times\overline{\mathcal{A}}_{g_{\ell}}])=\frac{\gamma_{g_{1}}\ldots\gamma_{g_{\ell}}}{\gamma_{g}}\begin{vmatrix}\lambda_{\alpha_{1}}&\lambda_{\alpha_{1}+1}&\ldots&\lambda_{\alpha_{1}+g-1}\\\
\lambda_{\alpha_{2}-1}&\lambda_{\alpha_{2}}&\ldots&\lambda_{\alpha_{2}+g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{\alpha_{g}-g+1}&\lambda_{\alpha_{g}-g+2}&\ldots&\lambda_{\alpha_{g}}\end{vmatrix}$

for the vector

$\alpha=(\underbrace{g-g_{1},\ldots,g-g_{1}}_{g_{1}},\underbrace{g-g_{1}-g_{2},\ldots,g-g_{1}-g_{2}}_{g_{2}},\ldots,\underbrace{g-g_{1}-\ldots-
g_{\ell},\ldots,g-g_{1}-\ldots-g_{\ell}}_{g_{\ell}})\,.$

By the connection between product cycles on $\overline{\mathcal{A}}_{g}$ and
$\mathsf{LG}_{g}$ proven in Section 3.2, it suffices to show that the class of
the product $\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}$ in
$\mathsf{LG}_{g}$ is given by the determinant

$\begin{vmatrix}x_{\alpha_{1}}&x_{\alpha_{1}+1}&\ldots&x_{\alpha_{1}+g-1}\\\
x_{\alpha_{2}-1}&x_{\alpha_{2}}&\ldots&x_{\alpha_{2}+g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
x_{\alpha_{g}-g+1}&x_{\alpha_{g}-g+2}&\ldots&x_{\alpha_{g}}\end{vmatrix}.$

We will prove this determinantal formula using the geometry of
$\mathsf{LG}_{g}$.

Let $V=\mathbb{C}^{2g}$ with symplectic form $\omega$. We consider a splitting

(25)
$(V,\omega)\simeq(V_{1},\omega_{1})\oplus\ldots\oplus(V_{\ell},\omega_{\ell})\,,$

where $V_{1},\ldots,V_{\ell}$ are symplectic subspaces of $V$ with $\dim
V_{i}=2g_{i}$. The splitting (25) defines an embedding

$j:\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}\to\mathsf{LG}_{g}\,,\quad(P_{1},\ldots,P_{\ell})\mapsto
P=P_{1}\oplus\ldots\oplus P_{\ell}\,.$

Consider the embedding of $\mathsf{LG}_{g}$ into the usual Grassmannian
$\mathsf{G}=\mathsf{G}(g,2g)$:

$\iota:\mathsf{LG}_{g}\to\mathsf{G}\,.$

Let $\mathsf{S}\to\mathsf{G}$ be the universal subbundle (which restricts to
the universal subbundle $\mathsf{S}\rightarrow\mathsf{LG}_{g}$ via the
embedding $\iota$). Similarly, let $x_{i}$ be the Chern classes of
$\mathsf{S}^{*}$ on $\mathsf{G}$ (which restrict to the classes $x_{i}$ on
$\mathsf{LG}_{g}$). Let $\Sigma$ be the Schubert cycle in the Grassmannian
$\mathsf{G}$ associated to the partition $\alpha$ with respect to any complete
flag satisfying the property

$F_{2(g_{1}+\ldots+g_{i})}=V_{1}\oplus\ldots\oplus V_{i}\,,\quad 1\leq
i\leq\ell\,.$

By definition, $P\in\Sigma$ provided

$\dim(P\cap F_{g+j-\alpha_{j}})\geq j\,,\quad 1\leq j\leq g\,.$

For $1\leq i\leq\ell$, let $j=g_{1}+\ldots+g_{i}$, so that
$\alpha_{j}=g-(g_{1}+\ldots+g_{i}).$ We see that for $P\in\Sigma$ we have

(26) $\dim\left(P\cap F_{2(g_{1}+\ldots+g_{i})}\right)\geq
g_{1}+\ldots+g_{i}\,,\quad 1\leq i\leq\ell\,.$

The converse is also true. While there are additional requirements about
dimensions of intersections with other members of the flag, these are
automatically fulfilled by elementary linear algebra considerations.

In $\mathsf{CH}^{*}(\mathsf{G})$, we have the standard expression [F1, Chapter
14]:

$[\Sigma]=\begin{vmatrix}x_{\alpha_{1}}&x_{\alpha_{1}+1}&\ldots&x_{\alpha_{1}+g-1}\\\
x_{\alpha_{2}-1}&x_{\alpha_{2}}&\ldots&x_{\alpha_{2}+g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
x_{\alpha_{g}-g+1}&x_{\alpha_{g}-g+2}&\ldots&x_{\alpha_{g}}\end{vmatrix}.$

Moreover, we have

$\operatorname{codim}(\Sigma,\mathsf{G})=|\alpha|=\sum_{i=1}^{\ell}(g-g_{1}-\ldots-
g_{i})g_{i}=\sum_{i>j}g_{i}g_{j}\,,$

which agrees with

$\operatorname{codim}(\mathsf{LG}_{1}\times\ldots\times\mathsf{LG}_{g_{\ell}},\mathsf{LG}_{g})=\binom{g+1}{2}-\sum_{i=1}^{\ell}\binom{g_{i}+1}{2}=\sum_{i>j}g_{i}g_{j}\,.$

The scheme-theoretic claim

(27)
$\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}=i^{-1}\Sigma=\Sigma\cap\mathsf{LG}_{g}\,,$

then implies

(28)
$[\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}]=\iota^{*}[\Sigma]=\begin{vmatrix}x_{\alpha_{1}}&x_{\alpha_{1}+1}&\ldots&x_{\alpha_{1}+g-1}\\\
x_{\alpha_{2}-1}&x_{\alpha_{2}}&\ldots&x_{\alpha_{2}+g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
x_{\alpha_{g}-g+1}&x_{\alpha_{g}-g+2}&\ldots&x_{\alpha_{g}}\end{vmatrix},$

as required.

We first establish (27) set-theoretically. The left to right containment is
clear for split subspaces $P=P_{1}\oplus\ldots\oplus P_{\ell}$, so we show the
converse. Let $P\in\Sigma\cap\mathsf{LG}_{g}$. For convenience, write

$h_{i}=g_{1}+\ldots+g_{i}\,.$

We set $P_{i}=P\cap V_{i}.$ Note that $P\cap F_{2h_{i}}$ is isotropic in
$F_{2h_{i}}$, hence $\dim(P\cap F_{2h_{i}})\leq h_{i}.$ By the Schubert
condition (26), we must have

(29) $\dim(P\cap F_{2h_{i}})=h_{i}\,.$

We will prove that $\dim P_{i}=g_{i}$ for all $1\leq i\leq\ell.$

The case $i=1$ is clear by (29) since $V_{1}=F_{2h_{1}}$. For the general
case, we induct on $i$. We assume that

$\dim(P\cap V_{1})=g_{1},\,\,\ldots,\,\,\dim(P\cap V_{i})=g_{i}\,,$

and show that

$\dim(P\cap V_{i+1})=g_{i+1}\,.$

To this end, let $Q=P\cap F_{2h_{i+1}}$, so that

$\dim Q=h_{i+1}\,,\quad\dim(Q\cap F_{2h_{i}})=h_{i}$

by (29). Furthermore, $Q$ is isotropic hence Lagrangian in
$(F_{2h_{i+1}},\eta)$ where $\eta$ is the restriction of the symplectic form
$\omega$. To show

$\dim(P\cap V_{i+1})=\dim(Q\cap V_{i+1})=g_{i+1}\,,$

we compute

$\dim(Q\cap V_{i+1})=\dim Q+\dim
V_{i+1}-\dim(Q+V_{i+1})=h_{i+1}+2g_{i+1}-\dim(Q+V_{i+1})\,.$

It suffices then to show that $\dim(Q+V_{i+1})=h_{i+1}+g_{i+1}$, or
equivalently,

(30) $\dim(Q+V_{i+1})^{\eta}=2h_{i+1}-(h_{i+1}+g_{i+1})=h_{i}\,.$

Here, the complement is taken in $F_{2h_{i+1}}$. Since $Q$ is Lagrangian,
$Q^{\eta}=Q$. By construction, $V_{i+1}^{\eta}=F_{2h_{i}}$. We can therefore
rewrite (30) as

$\dim(Q\cap F_{2h_{i}})=h_{i}\,,$

which is correct by the Schubert condition (29). The inductive step is proven.

Since $P_{1}\oplus\ldots\oplus P_{\ell}\subset P$, equality must hold for
dimension reasons. Therefore,
$P\in\mathsf{LG}_{g_{1}}\times\ldots\times\mathsf{LG}_{g_{\ell}}$, and the
proof of the set-theoretic equality (27) is complete.

To show (27) holds scheme-theoretically, it suffices to prove that the scheme-
theoretic intersection $\Sigma\cap\mathsf{LG}_{g}$ is nonsingular at all
points $P\in\Sigma\cap\mathsf{LG}_{g}$. Equivalently, we will show

(31) $\dim
T_{P}\,(\Sigma\cap\mathsf{LG}_{g})\,=\dim(T_{P}\,\Sigma\,\cap\,T_{P}\,\mathsf{LG}_{g})\,\leq\dim\mathsf{LG}_{g_{1}}\times\ldots\mathsf{LG}_{g_{\ell}}=\sum_{i=1}^{\ell}\binom{g_{i}+1}{2}\,.$

We claim first that all $P\in\Sigma\cap\mathsf{LG}_{g}$ are nonsingular points
of the Schubert variety $\Sigma\subset\mathsf{G}$. We use here a result due to
[LS], [C, Corollary 2.5]: singular points of $\Sigma$ must lie in Schubert
varieties for singular partitions associated to $\alpha$, see [C, Definition
2.1] for the terminology. In our case, nonsingularity at
$P\in\Sigma\cap\mathsf{LG}_{g}$ is due to the fact that equality holds in
(29). Equality (29) prevents $P$ from satisfying the Schubert conditions for
any of the singular partitions associated to $\alpha$.

The tangent space of $\Sigma$ at nonsingular points is computed in [EH,
Theorem 4.1]: $T_{P}\Sigma$ is identified with a subspace of the space of
linear maps

$\Phi:P\to\mathbb{C}^{2g}/P$

satisfying the property

$\Phi:(P\cap F_{2h_{i}})\to(P+F_{2h_{i}})/P\,.$

For tangent space $T_{P}\mathsf{LG}_{g}$, we require

$\Phi:P\to P^{*}$

to be symmetric, where we identify $\mathbb{C}^{2g}/P\simeq P^{*}$ using the
symplectic form.

Assume $\Phi\in T_{P}\Sigma\cap T_{P}\mathsf{LG}_{g}$. A straighforward check
shows that

$(P+F_{2h_{i}})/P\simeq F_{2h_{i}}/(P\cap F_{2h_{i}})$

gets identified with $(P\cap F_{2h_{i}})^{*}$, so that

$\Phi:(P\cap F_{2h_{i}})\to(P\cap F_{2h_{i}})^{*}\,.$

We have shown above that

$(P\cap F_{2h_{i}})=(P\cap V_{1})\oplus\ldots\oplus(P\cap V_{i})\,.$

Therefore, $\Phi$ must be symmetric block diagonal with blocks of size
$g_{1},\ldots,g_{\ell}$. Equation (31) then follows. ∎

######  Example 24.

For $g_{1}+g_{2}=g$, the tautological projection of the product locus
$\overline{\mathcal{A}}_{g_{1}}\times\overline{\mathcal{A}}_{g_{2}}$ is given
by the $g_{1}\times g_{1}$ determinant

(32)
$\mathsf{taut}^{\mathsf{cpt}}([\overline{\mathcal{A}}_{g_{1}}\times\overline{\mathcal{A}}_{g_{2}}])=\frac{\gamma_{g_{1}}\gamma_{g_{2}}}{\gamma_{g}}\begin{vmatrix}\lambda_{g_{2}}&\lambda_{g_{2}+1}&\ldots&\lambda_{g-1}\\\
\lambda_{g_{2}-1}&\lambda_{g_{2}}&\ldots&\lambda_{g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{g_{2}-g_{1}+1}&\lambda_{g_{2}-g_{1}+2}&\ldots&\lambda_{g_{2}}\end{vmatrix}.$

The right hand side is the Schur determinant associated to the partition
$(\underbrace{g_{2},\ldots,g_{2}}_{g_{1}})$. The Schur determinant is in
general not preserved by exchanging $g_{1}$ and $g_{2}$ (which amounts to
transposing the partition), but it is so in the presence of Mumford’s relation
by precisely [F1, Lemma A.9.2].

$\bullet$ In case $g_{1}=1$, we obtain

$\mathsf{taut}^{\mathsf{cpt}}([\overline{\mathcal{A}}_{1}\times\overline{\mathcal{A}}_{g-1}])=\frac{g}{6|B_{2g}|}\lambda_{g-1}\,.$

$\bullet$ In case $g=2$, we obtain

$\mathsf{taut}^{\mathsf{cpt}}([\overline{\mathcal{A}}_{2}\times\overline{\mathcal{A}}_{g-2}])=\frac{1}{360}\cdot\frac{g(g-1)}{|B_{2g}||B_{2g-2}|}\cdot(\lambda_{g-2}^{2}-\lambda_{g-1}\lambda_{g-3})\,.$

######  Example 25.

For $g_{1}=\ldots=g_{k}=1$, $g_{k+1}=g-k$, Theorem 5 yields

(33)
$\mathsf{taut}^{\mathsf{cpt}}\left(\left[\underbrace{\overline{\mathcal{A}}_{1}\times\ldots\times\overline{\mathcal{A}}_{1}}_{k}\times\overline{\mathcal{A}}_{g-k}\right]\right)=\frac{\gamma_{1}^{k}\gamma_{g-k}}{\gamma_{g}}\begin{vmatrix}\lambda_{g-1}&\lambda_{g}&0&\ldots&0\\\
\lambda_{g-3}&\lambda_{g-2}&\lambda_{g-1}&\ldots&0\\\
\ldots&\ldots&\ldots&\ldots&\ldots\\\
\lambda_{g_{2}-2k+1}&\lambda_{g-2k+2}&\lambda_{g-2k+3}&\ldots&\lambda_{g-k}\end{vmatrix}.$

For example, we have

$\mathsf{taut}^{\mathsf{cpt}}\left(\left[\overline{\mathcal{A}}_{1}\times\overline{\mathcal{A}}_{1}\times\overline{\mathcal{A}}_{g-2}\right]\right)=\frac{1}{36}\cdot\frac{g(g-1)}{|B_{2g}||B_{2g-2}|}\cdot\left(\lambda_{g-1}\lambda_{g-2}-\lambda_{g}\lambda_{g-3}\right)\,.$

###  3.4. Proof of Theorem 6

Our goal is to prove that after restriction to $\mathcal{A}_{g}$, the
tautological projections of the product cycles admit further factorization:

$\mathsf{taut}{\left([\mathcal{A}_{g_{1}}\times\cdots\times\mathcal{A}_{g_{\ell}}]\right)}=\frac{\gamma_{g_{1}}\ldots\gamma_{g_{\ell}}}{\gamma_{g}}\cdot\lambda_{g-1}\cdots\lambda_{g-\ell+1}\cdot\begin{vmatrix}\lambda_{\beta_{1}}&\lambda_{\beta_{1}+1}&\ldots&\lambda_{\beta_{1}+g^{*}-1}\\\
\lambda_{\beta_{2}-1}&\lambda_{\beta_{2}}&\ldots&\lambda_{\beta_{2}+g^{*}-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{\beta_{g^{*}}-g^{*}+1}&\lambda_{\beta_{g^{*}}-g^{*}+2}&\ldots&\lambda_{\beta_{g^{*}}}\end{vmatrix}\,,$

for the vector

$\beta=(\underbrace{g^{*}-g^{*}_{1},\ldots,g^{*}-g^{*}_{1}}_{g^{*}_{1}},\underbrace{g^{*}-g^{*}_{1}-g^{*}_{2},\ldots,g^{*}-g^{*}_{1}-g^{*}_{2}}_{g_{2}^{*}},\ldots,\underbrace{g^{*}-g_{1}^{*}-\ldots-
g_{\ell}^{*}}_{g^{*}_{\ell}})\,,$

where $g^{*}=g-\ell$ and $g_{i}^{*}=g_{i}-1$.

The term $\lambda_{g-1}\cdots\lambda_{g-\ell+1}$ is expected to appear in the
formula of Theorem 6 by the following reasoning. First,

$\lambda_{g-m}\cdot[\mathcal{A}_{g_{1}}\times\ldots\times\mathcal{A}_{g_{\ell}}]=0\,,\quad
1\leq m\leq\ell-1\,.$

Indeed, the splitting of the Hodge bundle distributes a top Hodge class on at
least one of the $\ell$ factors $\mathcal{A}_{g_{i}}$, yielding the vanishing
by Theorem 1(iii). Second, we compute the annihilator ideal

$\text{Ann}\,\langle\lambda_{g-1},\ldots,\lambda_{g-\ell+1}\rangle=\langle\lambda_{g-1}\ldots\lambda_{g-\ell+1}\rangle\,.$

The right to left containment follows from the relations

(34) $\lambda_{j}^{2}\lambda_{j+1}\ldots\lambda_{g-1}=0\,,\quad 1\leq j\leq
g-1$

on $\mathcal{A}_{g}$ noted in [vdG, page 4]. The left to right inclusion can
be justified by expressing an arbitrary element $z$ of the annihilator in
terms of the square-free monomial basis in the $\lambda$’s. Using (34), in
particular $\lambda_{g-1}^{2}=0$, it follows that all monomials that appear in
$z$ must contain $\lambda_{g-1}.$ If not, $z\cdot\lambda_{g-1}$ would contain
nonzero terms in the square-free monomial basis, corresponding to the
monomials of $z$ not containing $\lambda_{g-1}$. This contradicts that $z$ is
in the annihilator ideal. Successively, we see that
$\lambda_{g-2},\ldots,\lambda_{g-\ell+1}$ must also appear in each of the
monomials of $z$, proving the claim.

###### Proof.

We only indicate the proof of Theorem 6 when $\ell=2$. The general case is an
$\ell$-fold iteration of the same argument. To start, we restrict to
$\mathcal{A}_{g}$ the expression provided by Theorem 5, see (32). Then, we
must prove

$\begin{vmatrix}\lambda_{g_{2}}&\lambda_{g_{2}+1}&\ldots&\lambda_{g-1}\\\
\lambda_{g_{2}-1}&\lambda_{g_{2}}&\ldots&\lambda_{g-2}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{g_{2}-g_{1}+1}&\lambda_{g_{2}-g_{1}+2}&\ldots&\lambda_{g_{2}}\end{vmatrix}=\lambda_{g-1}\cdot\begin{vmatrix}\lambda_{g_{2}-1}&\lambda_{g_{2}}&\ldots&\lambda_{g-3}\\\
\lambda_{g_{2}-2}&\lambda_{g_{2}-1}&\ldots&\lambda_{g-4}\\\
\ldots&\ldots&\ldots&\ldots\\\
\lambda_{g_{2}-g_{1}+1}&\lambda_{g_{2}-g_{1}+2}&\ldots&\lambda_{g_{2}-1}\end{vmatrix}$

after setting $\lambda_{g}=0.$ The parallel identity for the Lagrangian
Grassmannian is equivalent:

(35) $\begin{vmatrix}x_{g_{2}}&x_{g_{2}+1}&\ldots&x_{g-1}\\\
x_{g_{2}-1}&x_{g_{2}}&\ldots&x_{g-2}\\\ \ldots&\ldots&\ldots&\ldots\\\
x_{g_{2}-g_{1}+1}&x_{g_{2}-g_{1}+2}&\ldots&x_{g_{2}}\end{vmatrix}=x_{g-1}\cdot\begin{vmatrix}x_{g_{2}-1}&x_{g_{2}}&\ldots&x_{g-3}\\\
x_{g_{2}-2}&x_{g_{2}-1}&\ldots&x_{g-4}\\\ \ldots&\ldots&\ldots&\ldots\\\
x_{g_{2}-g_{1}+1}&x_{g_{2}-g_{1}+2}&\ldots&x_{g_{2}-1}\end{vmatrix}\mod
x_{g}\,.$

The identity does not hold in the absence of the Mumford relations.

We will derive identity (35) geometrically via an excess intersection
calculation on $\mathsf{LG}_{g}$. Fix a symplectic splitting

$V=W_{1}\oplus L_{1}\oplus W_{2}\oplus L_{2}\,,\quad\dim
W_{i}=2(g_{i}-1)\,,\quad\dim L_{i}=2\,.$

In addition, fix Lagrangian subspaces $P_{1}\subset L_{1}$ and $P_{2}\subset
L_{2}$. Let

$\iota:\mathsf{LG}_{g-1}\to\mathsf{LG}_{g}\,,\quad P\to P\oplus P_{2}\,.$

Here $\mathsf{LG}_{g-1}$ is the Lagrangian Grassmannian of $W_{1}\oplus
L_{1}\oplus W_{2}$. For this embedding, we have

(36) $\iota_{*}\,[\mathsf{LG}_{g-1}]=x_{g}\cap[\mathsf{LG}_{g}]\,,$

as can be seen by a normal bundle calculation. The reader can verify that

$\iota^{-1}(\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}})=\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}-1}\,.$

The left hand side has codimension $g_{1}g_{2}$ in $\mathsf{LG}_{g}$, while
the right hand side has codimension $g_{1}(g_{2}-1)$ in $\mathsf{LG}_{g-1}$.
Write

$j:\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}-1}\to\mathsf{LG}_{g-1}$

for the natural map determined by the pair $(W_{1}\oplus L_{1},W_{2})$. The
class $\iota^{*}(\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}})$ can be
computed via excess intersection. The excess bundle is the dual tautological
subbundle $\mathsf{S}_{g_{1}}^{*}$. Therefore,

(37)
$\iota^{*}([\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}}])=j_{*}((x_{g_{1}}\times
1)\cap[\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}-1}])=j_{*}k_{*}([\mathsf{LG}_{g_{1}-1}\times\mathsf{LG}_{g_{2}-1}])\,,$

after using (36) again. The embedding

$k:\mathsf{LG}_{g_{1}-1}\times\mathsf{LG}_{g_{2}-1}\to\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}-1}$

is defined by taking sum with $P_{1}$ on the first factor. Consider

$u:\mathsf{LG}_{g_{1}-1}\times\mathsf{LG}_{g_{2}-1}\to\mathsf{LG}_{g-2}\,,\quad
v:\mathsf{LG}_{g-2}\to\mathsf{LG}_{g-1}\,,$

where the first map is determined by the pair $(W_{1},W_{2})$ and the second
map is determined by taking sum with $P_{1}$. The equality $j\circ k=v\circ u$
follows from the definitions. By (28) in the proof of Theorem 5, we find

(38)
$u_{*}([\mathsf{LG}_{g_{1}-1}\times\mathsf{LG}_{g_{2}-1}])=v^{*}\begin{vmatrix}x_{g_{2}-1}&x_{g_{2}}&\ldots&x_{g-3}\\\
x_{g_{2}-2}&x_{g_{2}-1}&\ldots&x_{g-4}\\\ \ldots&\ldots&\ldots&\ldots\\\
x_{g_{2}-g_{1}+1}&x_{g_{2}-g_{1}+2}&\ldots&x_{g_{2}-1}\end{vmatrix}.$

Then, using (37) and (38), we have

$\displaystyle\iota^{*}([\mathsf{LG}_{g_{1}}\times\mathsf{LG}_{g_{2}}])$
$\displaystyle=$ $\displaystyle
j_{*}k_{*}([\mathsf{LG}_{g_{1}-1}\times\mathsf{LG}_{g_{2}-1}])=v_{*}u_{*}([\mathsf{LG}_{g_{1}-1}\times\mathsf{LG}_{g_{2}-1}])$
$\displaystyle=$ $\displaystyle
v_{*}v^{*}\begin{vmatrix}x_{g_{2}-1}&x_{g_{2}}&\ldots&x_{g-3}\\\
x_{g_{2}-2}&x_{g_{2}-1}&\ldots&x_{g-4}\\\ \ldots&\ldots&\ldots&\ldots\\\
x_{g_{2}-g_{1}+1}&x_{g_{2}-g_{1}+2}&\ldots&x_{g_{2}-1}\end{vmatrix}=x_{g-1}\begin{vmatrix}x_{g_{2}-1}&x_{g_{2}}&\ldots&x_{g-3}\\\
x_{g_{2}-2}&x_{g_{2}-1}&\ldots&x_{g-4}\\\ \ldots&\ldots&\ldots&\ldots\\\
x_{g_{2}-g_{1}+1}&x_{g_{2}-g_{1}+2}&\ldots&x_{g_{2}-1}\end{vmatrix}\,,$

which recovers the right hand side of (35). On the other hand, by (28), the
class on the left hand side equals

$\iota^{*}\begin{vmatrix}x_{g_{2}}&x_{g_{2}+1}&\ldots&x_{g-1}\\\
x_{g_{2}-1}&x_{g_{2}}&\ldots&x_{g-2}\\\ \ldots&\ldots&\ldots&\ldots\\\
x_{g_{2}-g_{1}+1}&x_{g_{2}-g_{1}+2}&\ldots&x_{g_{2}}\end{vmatrix},$

while the pullback
$\iota^{*}:\mathsf{CH}^{*}(\mathsf{LG}_{g})\to\mathsf{CH}^{*}(\mathsf{LG}_{g-1})$
has the effect of setting $x_{g}=0$. ∎

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