# The monodromy of families of subvarieties on abelian varieties

Ariyan Javanpeykar  Ariyan Javanpeykar
IMAPP
Radboud University Nijmegen
PO Box 9010
6500GL Nijmegen
The Netherlands<EMAIL_ADDRESS>,  Thomas Krämer  Thomas Krämer
Institut für Mathematik
Humboldt Universität zu Berlin
Rudower Chaussee 25, 12489 Berlin
Germany<EMAIL_ADDRESS>,  Christian Lehn  Christian Lehn
Fakultät für Mathamatik
Technische Universität Chemnitz
Reichenhainer Strasse 39
09126 Chemnitz
Germany<EMAIL_ADDRESS>and  Marco Maculan  Marco
Maculan
Institut de Mathématiques de Jussieu
Sorbonne Université
4, place Jussieu
75005 Paris
France<EMAIL_ADDRESS>
###### Abstract.

Motivated by recent work of Lawrence-Venkatesh and Lawrence-Sawin, we show
that non-isotrivial families of subvarieties in abelian varieties have big
monodromy when twisted by generic rank one local systems. While Lawrence-Sawin
discuss the case of subvarieties of codimension one, our results hold for
subvarieties of codimension at least half the dimension of the ambient abelian
variety. For the proof, we use a combination of geometric arguments and
representation theory to show that the Tannaka groups of intersection
complexes on such subvarieties are big.

###### Key words and phrases:

Subvarieties of abelian varieties, characteristic cycles, convolution,
monodromy, perverse sheaves, Tannaka categories.

###### 2020 Mathematics Subject Classification:

14K12, 14D05 (primary), 18M25, 20G05, 32S60 (secondary).

##  1\. Introduction

Recently, Lawrence and Venkatesh [LV20] have developed a technique to prove
nondensity of integral points on varieties that are defined over a number
field and support a geometric variation of Hodge structures with big
monodromy. They used this method to give an alternative proof of the Mordell
conjecture and to show nondensity for hypersurfaces in projective space of a
given (high) degree with good reduction outside a fixed finite set of primes.
Later, Lawrence and Sawin [LS20] applied this strategy to show that up to
translation any abelian variety over a number field contains only finitely
many smooth ample hypersurfaces with given Néron-Severi class and good
reduction outside a fixed finite set of primes. The main novelty of their work
lies in their way to control monodromy. The arguments of Lawrence and
Venkatesh have a topological flavor. For the Mordell conjecture they rely on a
judicious choice of Dehn twists; for hypersurfaces in projective space they
use the computation of the integral monodromy of the universal family by
Beauville [Bea86] (based on the work of Ebeling [Ebe84] and Janssen [Jan83]),
see also the discussion by Katz in [Kat04]. Instead, the approach by Lawrence
and Sawin involves Tannaka groups of perverse sheaves on abelian varieties
introduced by Krämer and Weissauer [KW15c]; the relation of these groups to
monodromy is reminiscent of the one between the monodromy group of a variation
of Hodge structures and its generic Mumford-Tate group [And92].

With a view towards new arithmetic applications along these lines, we prove a
big monodromy theorem for families of subvarieties of higher codimension in
abelian varieties. Our results hold for all subvarieties of codimension at
least half the dimension of the abelian variety. The geometry in this
codimension range is very different from the codimension one case in [LS20],
and the results about Tannaka groups that we obtain on the way may be of
independent interest.

###  1.1. Big monodromy

Let $S$ be a smooth irreducible variety over an algebraically closed field $k$
of characteristic zero. Let $A$ be an abelian variety of dimension $g$ over
$k$. Inside the constant abelian scheme $A_{S}:=A\times S$, let
$\mathcal{X}\subset A_{S}$ be a closed subvariety which is smooth over $S$
with connected fibers of dimension $d$. The goal of this paper is to
understand the monodromy of rank one local systems on the smooth proper family
$f\colon\mathcal{X}\to S$ in the following diagram:

${\mathcal{X}}$${A}$${A_{S}}$${S}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\scriptstyle{\operatorname{pr}_{A}}$$\scriptstyle{\operatorname{pr}_{S}}$

Our results apply both in the analytic and in the algebraic setup, using
topological local systems with coefficients in $\mathbb{F}=\mathbb{C}$ for
$k=\mathbb{C}$ resp. étale $\ell$-adic local systems with coefficients in
$\mathbb{F}=\overline{\mathbb{Q}}_{\ell}$ for a prime $\ell$ over an arbitrary
algebraically closed field $k$ of characteristic zero. Let $\pi_{1}(A,0)$ be
the topological resp. étale fundamental group with the discrete resp.
profinite topology, and denote the group of its continuous characters by

$\Pi(A,\mathbb{F})=\operatorname{Hom}(\pi_{1}(A,0),\mathbb{F}^{\times}).$

In what follows, by a _linear subvariety_ we mean a subset
$\Pi(B,\mathbb{F})\subset\Pi(A,\mathbb{F})$ for an abelian quotient variety
$A\twoheadrightarrow B$ with $\dim B<\dim A$. We say that a statement holds
for most $\chi\in\Pi(A,\mathbb{F})$ if it holds for all $\chi$ outside a
finite union of torsion translates of linear subvarieties. For
$\chi\in\Pi(A,\mathbb{F})$, let $L_{\chi}$ denote the associated rank one
local system on $A$. It follows from generic vanishing [BSS18, KW15c, Sch15]
that for most $\chi$ the higher direct images $R^{i}f_{*}\pi^{*}L_{\chi}$
vanish in all degrees $i\neq d$; we consider the local system

$V_{\chi}\;:=\;R^{d}f_{*}\pi^{*}L_{\chi}$

of rank $|e|$ where $e$ is the topological Euler characteristic of the fibers
of $\mathcal{X}\to S$. More generally, the study of finite étale covers of the
subvariety $\mathcal{X}\subset A_{S}$ induced by finite étale covers of $A$
leads to direct sums

$V_{\underline{\chi}}\;:=\;V_{\chi_{1}}\oplus\cdots\oplus V_{\chi_{n}}$

where $\underline{\chi}=(\chi_{1},\dots,\chi_{n})\in\Pi(A,\mathbb{F})^{n}$ is
an $n$-tuple of characters of the fundamental group. Using the natural
identification $\Pi(A,\mathbb{F})^{n}=\Pi(A^{n},\mathbb{F})$ we will also
apply the terminology most for such $n$-tuples of characters. Consider for
$s\in S(k)$ the monodromy representation

$\rho\colon\quad\pi_{1}(S,s)\;\longrightarrow\;\operatorname{GL}(V_{\underline{\chi},s})\quad\textnormal{on
the fiber}\quad
V_{\underline{\chi},s}\;=\;\bigoplus_{i=1}^{n}\textup{H}^{d}(\mathcal{X}_{s},L_{\chi_{i}}).$

The _algebraic monodromy group_ of the local system $V_{\underline{\chi}}$ is
the Zariski closure of the image of $\rho$. By construction it is an algebraic
subgroup of

$\operatorname{GL}(V_{\chi_{1},s})\times\cdots\times\operatorname{GL}(V_{\chi_{n},s})\;\subset\;\operatorname{GL}(V_{\underline{\chi},s}).$

This upper bound can sometimes be refined: We say that the subvariety
$\mathcal{X}\subset A_{S}$ is _symmetric up to translation_ if there exists
$a\colon S\to A$ such that $\mathcal{X}_{t}=-\mathcal{X}_{t}+a(t)$ for all
$t\in S(k)$. In this case, Poincaré duality furnishes a nondegenerate bilinear
pairing

$\theta_{\chi,s}\colon\quad V_{\chi,s}\otimes V_{\chi,s}\longrightarrow
L_{\chi,a(s)}$

for each $\chi\in\Pi(A,\mathbb{F})$, because for the dual of a rank one local
system and for its inverse image under the translation $\tau_{a(t)}\colon A\to
A,x\mapsto x+a(t)$ we have natural isomorphisms

$\displaystyle L_{\chi}^{\vee}$ $\displaystyle\simeq$
$\displaystyle[-1]^{\ast}L_{\chi},$ $\displaystyle\tau_{a(t)}^{\ast}L_{\chi}$
$\displaystyle\simeq$ $\displaystyle
L_{\chi}\otimes_{\mathbb{F}}L_{\chi,a(t)}.$

The pairing $\theta_{\chi,s}$ is symmetric if $d$ is even, and alternating
otherwise. Since the pairing is compatible with the monodromy operation on the
fiber, it follows that the algebraic monodromy group of $V_{\underline{\chi}}$
is contained in an orthogonal resp. symplectic group in the two cases. This
leads to the following definition:

######  Definition.

We say that $V_{\underline{\chi}}$ has big monodromy if its algebraic
monodromy group contains $G_{1}\times\cdots\times G_{n}$ as a normal subgroup
where $G_{i}\subset\operatorname{GL}(V_{\chi_{i},s})$ is defined by

$G_{i}\;:=\;\begin{cases}\operatorname{SL}(V_{\chi_{i},s})&\textup{if
$\mathcal{X}$ is not symmetric up to translation},\\\
\operatorname{SO}(V_{\chi_{i},s},\theta_{\chi_{i},s})&\textup{if $\mathcal{X}$
is symmetric up to translation and $d$ is even},\\\
\operatorname{Sp}(V_{\chi_{i},s},\theta_{\chi_{i},s})&\textup{if $\mathcal{X}$
is symmetric up to translation and $d$ is odd}.\end{cases}$

Note that the connected component of the algebraic monodromy group of
$V_{\underline{\chi}}$ is unaffected by base change along étale morphisms
$S^{\prime}\to S$. To take this into account we consider the fiber
$\mathcal{X}_{\bar{\eta}}$ of $\mathcal{X}\to S$ at a geometric generic point
$\bar{\eta}$ of $S$. There are four obvious cases where the local system
$V_{\underline{\chi}}$ does not have big monodromy: We say that
$\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ is

  1. (1)

_constant up to a translation_ if it is the translate of a subvariety
$Y\subset A$ along a point in $A(\bar{\eta})$. In this case the algebraic
monodromy is finite.

  2. (2)

divisible if it stable under translation by a torsion point $0\neq x\in
A(\bar{\eta})$. In this case the algebraic monodromy of each $V_{\chi_{i}}$ is
itself a group of block matrices which is normalized by the group generated by
the point $x$.

  3. (3)

a _symmetric power of a curve_ if there is a smooth curve $C\subset
A_{S,\bar{\eta}}$ such that the sum morphism $\operatorname{Sym}^{d}C\to
A_{S,\bar{\eta}}$ is a closed embedding with image $\mathcal{X}_{\bar{\eta}}$
and $d\geqslant 2$. After an étale base change over $S$, we may assume that
$C$ spreads out to a relative curve $\mathcal{C}\subset A_{S}$ which is smooth
and proper over $S$ such that the relative sum morphism
$\operatorname{Sym}_{S}^{d}\mathcal{C}\to A_{S}$ is a closed embedding with
image $\mathcal{X}$. Then we have an isomorphism compatible with monodromy:

$\textup{H}^{d}(\mathcal{X}_{s},L_{\chi})\simeq\operatorname{Alt}^{d}\textup{H}^{1}(\mathcal{C}_{s},L_{\chi}).$

  4. (4)

a _product_ if there are smooth subvarieties $X_{1},X_{2}\subset
A_{S,\bar{\eta}}$ with $\dim X_{i}>0$ such that the sum morphism $X_{1}\times
X_{2}\to A_{S,\bar{\eta}}$ is a closed embedding with image
$\mathcal{X}_{\bar{\eta}}$. Again, after an étale base change over $S$ we may
assume that $X_{i}$ spreads out to a subvariety $\mathcal{X}_{i}\subset A_{S}$
which is smooth and proper over $S$ such that the relative sum morphism
$\mathcal{X}_{1}\times\mathcal{X}_{2}\to A_{S}$ is a closed embedding with
image $\mathcal{X}$. Then we have the Künneth isomorphism which is compatible
with monodromy:

$\textup{H}^{d}(\mathcal{X}_{s},L_{\chi})\simeq\bigoplus_{i_{1}+i_{2}=d}\textup{H}^{i_{1}}(\mathcal{X}_{1,s},L_{\chi})\otimes\textup{H}^{i_{2}}(\mathcal{X}_{2,s},L_{\chi}).$

If $\mathcal{X}_{\bar{\eta}}$ is nondivisible, then condition (1) holds if and
only if the family $\mathcal{X}\to S$ is isotrivial; see corollary 4.8. In
order to avoid the appearance of the exceptional groups $E_{6}$ and $E_{7}$
and some low-dimensional half-spin groups, we assume that the topological
Euler characteristic $e$ of $\mathcal{X}_{\bar{\eta}}$ satisfies

$\displaystyle|e|\;$ $\displaystyle\neq\;27$ $\displaystyle\textup{if
$d\geqslant 2$ and $\mathcal{X}$ is not symmetric up to a translation},$ (1.1)
$\displaystyle|e|\;$ $\displaystyle\neq\;56$ $\displaystyle\textup{if
$d\geqslant 3$ is odd and $\mathcal{X}$ is symmetric up to translation},$
$\displaystyle|e|\;$ $\displaystyle\neq\;2^{2m-1}$ if $d\geqslant(g-1)/4$,
$m\in\\{3,\dots,d\\}$ has the same parity as $d$ $\displaystyle\textup{and
$\mathcal{X}$ is symmetric up to translation}.$

Note that $|e|\geqslant g$ if $\mathcal{X}_{\bar{\eta}}\subset
A_{S,\bar{\eta}}$ has ample normal bundle, see lemma 2.12. We do not know any
example of a smooth subvariety of $A_{S,\bar{\eta}}$ with ample normal bundle
and dimension $d<(g-1)/2$ whose Euler characteristic $e$ does not satisfy
(1.1).

######  Main theorem (monodromy version).

Suppose $\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ has ample normal
bundle, dimension $d<(g-1)/2$, and (1.1) holds. Then the following are
equivalent:

  1. (1)

$\mathcal{X}_{\bar{\eta}}$ is nondivisible, not constant up to translation,
not a symmetric power of a curve and not a product;

  2. (2)

$V_{\underline{\chi}}$ has big monodromy for most torsion $n$-tuples
$\underline{\chi}\in\Pi(A,\mathbb{F})^{n}$.

Smooth proper subvarieties of a simple abelian variety have ample normal
bundle. Therefore when $A$ is simple the preceding theorem is as general as it
gets for smooth subvarieties of dimension $d<(g-1)/2$, save the finite list of
exceptions in (1.1). When $A$ is arbitrary, the theorem can be applied in the
following concrete cases:

######  Corollary.

Suppose $\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ is nondivisible,
not constant up to translation, and one of the following holds:

  1. (1)

$\mathcal{X}_{\bar{\eta}}$ is a curve generating $A_{S,\bar{\eta}}$ and
$g\geqslant 4$;

  2. (2)

$\mathcal{X}_{\bar{\eta}}$ is a surface with ample normal bundle which is
neither a symmetric square of a curve nor a product, and $e\neq 27$,
$g\geqslant 6$;

  3. (3)

$\mathcal{X}_{\bar{\eta}}$ is a complete intersection of ample divisors and
$d<(g-1)/2$.

Then $V_{\underline{\chi}}$ has big monodromy for most $n$-tuples
$\underline{\chi}\in\Pi(A,\mathbb{F})^{n}$ of torsion characters.

Indeed a smooth complete intersection of ample divisors is neither a symmetric
power of a curve (corollary 2.10) nor a product (remark 6.3) and its
topological Euler characteristic satisfies $|e|\geqslant 2^{g}$ and $|e|\neq
27,56$ (corollaries 2.17, LABEL: and 2.16).

Over $k=\mathbb{C}$, the main theorem in the analytic setup is deduced from
the algebraic one by the comparison between classical and étale topology; the
hypothesis that the characters are torsion is only used here. For the proof in
the algebraic setting, we start as in [LS20] by relating the algebraic
monodromy to the Tannaka group of the rank one local systems in question, seen
as perverse sheaves on $\mathcal{X}_{\bar{\eta}}$. The idea is similar to the
study of monodromy groups via Mumford-Tate groups in the complex case [And92]:
An analog of the theorem of the fixed part due to Lawrence and Sawin says that
the monodromy will be big if we can show that the Tannaka group of the
geometric generic fiber is big (see theorem 4.10); note that the property of
the family being symmetric up to translation can be read off from its
geometric generic fiber (corollary 4.8). Thus, we are left with a question
about the Tannaka group of the geometric generic fiber of our family. In this
setting, we will reset our notation and replace $k$ by an algebraic closure of
the function field of $S$.

###  1.2. Big Tannaka groups

As before, let $A$ be an abelian variety of dimension $g$ over an
algebraically closed field $k$ of characteristic zero. Let $i\colon
X\hookrightarrow A$ be the inclusion of a smooth connected closed subvariety
of dimension $d$. We define the perverse intersection complex

$\delta_{X}\;:=\;i_{\ast}\mathbb{F}_{X}[d]$

as the pushforward of the constant sheaf, shifted in cohomological degree $-d$
so that it becomes an object of the abelian category
$\textup{Perv}(A,\mathbb{F})$ of perverse sheaves on $A$ as in [BBDG18]. As we
will recall in section 3.1, the group law on the abelian variety induces a
convolution product on perverse sheaves, and the perverse intersection complex
$\delta_{X}$ generates a neutral Tannaka category $\langle\delta_{X}\rangle$
with respect to this convolution. For the rest of this introduction, we fix a
character $\chi\in\Pi(A,\mathbb{F})$ with $\textup{H}^{i}(X,L_{\chi})=0$ for
all $i\neq d$. Such a character exists by generic vanishing. We then have a
fiber functor

$\omega\colon\quad\langle\delta_{X}\rangle\;\longrightarrow\;\operatorname{Vect}(\mathbb{F}),\quad
P\;\longmapsto\;\textup{H}^{0}(A,P\otimes L_{\chi}),$

see [KW15c, th. 13.2]. Applying this fiber functor to $P=\delta_{X}$ we
recover the vector space

$V:=\omega(\delta_{X})=\textup{H}^{0}(A,\delta_{X}\otimes
L_{\chi})=\textup{H}^{d}(X,L_{\chi}).$

The automorphisms of the fiber functor are represented by a reductive
algebraic group
$G_{X,\omega}:=G_{\omega}(\delta_{X})\subset\operatorname{GL}(V)$ which we
call the _Tannaka group of $X$_, see also definition 3.2. The definitions in
section 1.1 with $S=\operatorname{Spec}(k)$ show that if $X\subset A$ is
symmetric up to translation, then $V$ comes with a natural symmetric bilinear
form $\theta$ which is induced by Poincaré duality. This bilinear form is
symmetric or alternating depending on the parity of $d$, and it is preserved
by the action of the group $G_{X,\omega}$ as in [KW15a, lemma 2.1]. Let
$G_{X,\omega}^{\circ}\subset G_{X,\omega}$ be the connected component of the
identity and

$G_{X,\omega}^{\ast}:=[G_{X,\omega}^{\circ},G_{X,\omega}^{\circ}]$

its derived group, which is a connected semisimple algebraic group.

######  Definition.

We say that the Tannaka group $G_{X,\omega}$ of $X$ is big if the derived
group of its connected component of the identity is

$G_{X,\omega}^{*}\;=\;\begin{cases}\operatorname{SL}(V)&\textup{if $X$ is not
symmetric up to translation},\\\ \operatorname{SO}(V,\theta)&\textup{if $X$ is
symmetric up to translation and $d$ is even},\\\
\operatorname{Sp}(V,\theta)&\textup{if $X$ is symmetric up to translation and
$d$ is odd}.\end{cases}$

The main theorem from the previous section is obtained by combining the analog
of the theorem of the fixed part by Lawrence and Sawin (theorem 4.10) with the
following result, whose proof will be the main task of this paper. Again we
need to exclude a finite list of values of the topological Euler
characteristic $e$ of $X$, for which we refer to (1.1) with
$S=\mathrm{Spec}(k)$ and $\mathcal{X}=X$.

######  Main theorem (Tannaka version).

Suppose $X\subset A$ has ample normal bundle, dimension $d<(g-1)/2$, and (1.1)
holds. Then the following are equivalent:

  1. (1)

$X$ is nondivisible, not a symmetric power of a curve and not a product;

  2. (2)

The Tannaka group $G_{X,\omega}$ is big.

Similarly to the monodromy version, the preceding statement is substantially
sharp in the simple case and can be applied in the following special cases:

######  Corollary.

Suppose $X\subset A$ is nondivisible and one of the following holds:

  1. (1)

$X$ is a curve generating $A$ and $g\geqslant 4$;

  2. (2)

$X$ is a surface with ample normal bundle which is neither a product nor the
symmetric square of a curve, and $e\neq 27$, $g\geqslant 6$;

  3. (3)

$X$ is a complete intersection of ample divisors and $d<(g-1)/2$.

Then the Tannaka group $G_{X,\omega}$ is big.

The Tannaka version of the main theorem also applies when $X$ does not arise
from a family as in section 1.1, so it is stronger than the monodromy version.
We also note that over the complex numbers both versions apply in many cases
where we have no control on Mumford-Tate groups of the subvarieties. Again,
when $X$ is a complete intersection of ample divisors, then automatically
$|e|\neq 27,56$, $X$ is not a symmetric power of a curve nor a product. Before
we describe the proof of the main theorem, let us mention the following
observation to illustrate the information captured by Tannaka groups (see
corollary 3.8):

######  Fact.

Suppose that $X\subset A$ has ample normal bundle and $d<g/2$. If
$G_{X,\omega}$ is big, then the sum morphism $\operatorname{Sym}^{2}(X)\to
X+X$ is birational.

This follows from the observation that the direct image of the constant sheaf
under the sum morphism is related to the decomposition of $V\otimes
V\in\textup{Rep}_{\mathbb{F}}(G_{X,\omega})$. In fact, Larsen’s alternative
yields a necessary and sufficient criterion for the Tannaka group to be big,
using only the decomposition of the direct image of the constant sheaf under
the sum morphism. However, it seems hard to control this direct image in the
generality needed for our main theorem, so for the proof of the main theorem
we follow a different route that will be described in sections 1.3, LABEL:,
1.4, LABEL: and 1.5.

###  1.3. Simplicity of Tannaka groups

The first step in our proof of the main theorem from the previous section will
be to show that under the given assumptions, the algebraic group
$G_{X,\omega}^{*}$ is simple. We refine the methods in [Krä21, section 6] to
obtain the following simplicity criterion (see theorem 6.1):

######  Theorem A.

Suppose $X\subset A$ has ample normal bundle and is nondivisible. Then for
$g\geqslant 3$ the following are equivalent:

  1. (1)

The algebraic group $G_{X,\omega}^{\ast}$ is not simple;

  2. (2)

There are smooth subvarieties $X_{1},X_{2}\subset A$ such that the sum
morphism induces an isomorphism

$X_{1}\times X_{2}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}X.$

A smooth projective curve $C\subset A$ generating $A$ has ample normal bundle,
thus the algebraic group $G_{C,\omega}^{\ast}$ is simple for $g\geqslant 3$.
When $g=2$ the simplicity of $G_{C,\omega}^{\ast}$ remains open. More
generally A implies that $G_{X,\omega}^{\ast}$ is simple when $X\subset A$ is
nondivisible with ample normal bundle and

  1. (1)

the image of the Albanese morphism $X\to\operatorname{Alb}(X)$ is
nondegenerate in the sense of section 2.3;

  2. (2)

the natural morphism $\varphi\colon\operatorname{Alb}(X)\to A$ is an isogeny.
By Debarre’s Barth-Lefschetz theorem for abelian varieties (see [Deb95, th.
4.5] or remark 6.3) this is the case as soon as $d>g/2$ or when $X$ is a
complete intersection of ample divisors and $d\geqslant 2$.

Note that situation (2) above is a particular case of (1).

Our proof of A uses characteristic cycles on the cotangent bundle of $A$ and
their link with representation theory [Krä22, Krä21]. For convenience, we
recall some background in section 5, together with computations for the Dynkin
types $A$, $B$, $D$ to be used later. A look at characteristic cycles will
also show that the representation
$\omega(\delta_{X})\in\textup{Rep}_{\mathbb{F}}(G_{X,\omega}^{\ast})$ is
minuscule in the sense that its weights form a single Weyl group orbit, see
corollary 5.15. There are only few nontrivial minuscule representations $V$ of
a simply connected simple algebraic group $G$, all of which are listed below:

Dynkin type | G | $V$ | $\dim V$
---|---|---|---
$A_{n}$ | $\operatorname{SL}_{n+1}$ |  $r$-th wedge power | $\tbinom{n+1}{r}$
$B_{n}$ | $\operatorname{Spin}_{2n+1}$ | spin | $2^{n}$
$C_{n}$ | $\operatorname{Sp}_{2n}$ | standard | $2n$
$D_{n}$ | $\operatorname{Spin}_{2n}$ | standard of $\operatorname{SO}_{2n}$ | $2n$
$D_{n}$ | $\operatorname{Spin}_{2n}$ | half-spins | $2^{n-1}$
$E_{6}$ | $E_{6}$ |  | smallest nontrivial
---
or its dual
$27$
$E_{7}$ | $E_{7}$ | smallest nontrivial | $56$

The dimension of $\omega(\delta_{X})$ is the absolute value of the topological
Euler characteristic of $X$. Recall that the subvariety $X\subset A$ is
symmetric up to a translation if and only if the vector space
$\omega(\delta_{X})$ carries a nondegenerate bilinear form preserved by the
action of $G_{X,\omega}^{\ast}$, and this pairing is symmetric if $d$ is even
and alternating if $d$ is odd; see [KW15a, lemma 2.1]. This rules out the
occurence of $E_{6}$ for symmetric subvarieties; note that the group $E_{6}$
appears as the Tannaka group of the Fano surface in the intermediate Jacobian
of a smooth cubic threefold, but $d=(g-1)/2$ here because $d=2$ and $g=5$.
Similarly, the group $E_{7}$ preserves a nondegenerate alternating bilinear
form on its $57$-dimensional irreducible representation, so subvarieties $X$
with $G_{X,\omega}^{\ast}\simeq E_{7}$ must be odd-dimensional. However, for
$d=1$ this does not happen as we show by a direct geometric argument (see
corollary 3.11), and in higher dimension we do not any such example.
Altogether, to prove that the Tannaka group is big and conclude the proof of
the main theorem from section 1.2, we are left with wedge powers and spin
representations. The next two sections will characterize the occurence of the
former and rule out the latter.

###  1.4. Wedge powers

In contrast to the situation for hypersurfaces studied by Lawrence and Sawin
in [LS20], one cannot rule out the occurrence of nontrivial wedge powers for
subvarieties of higher codimension by numerical arguments. In fact, wedge
powers do appear, but we will use geometric arguments to obtain the following
complete classification (see theorem 7.3):

######  Theorem B.

Suppose $X\subset A$ has ample normal bundle and is nondivisible. Then for
$d<(g-1)/2$ the following are equivalent:

  1. (1)

There are integers $r$ and $n$ with $1<r\leqslant n/2$ such that
$G_{X,\omega}^{\ast}\simeq\operatorname{Alt}^{r}(\operatorname{SL}_{n})$ and
$\omega(\delta_{X})$ is the $r$-th wedge power of the standard representation.

  2. (2)

There is a nondegenerate irreducible smooth projective curve $C\subset A$ such
that

     * •

$X=C+\cdots+C\subset A$ is the sum of $r$ copies of $C$, and

     * •

the sum morphism $\operatorname{Sym}^{r}C\to X$ is an isomorphism.

###  1.5. Spin representations

Recall that for $N\geqslant 3$ the group $\operatorname{SO}_{N}(\mathbb{F})$
admits a double cover

$\operatorname{Spin}_{N}(\mathbb{F})\;\longrightarrow\;\operatorname{SO}_{N}(\mathbb{F})$

by the _spin group_ $\operatorname{Spin}_{N}(\mathbb{F})$, a simply connected
algebraic group with a faithful representation $\mathbb{S}_{N}$, the _spin
representation_. We have $\dim\mathbb{S}_{N}=2^{n}$ for $n=\lfloor
N/2\rfloor$, and if $N$ is odd, then the spin representation is irreducible.
If $N=2n$ is even, then the spin representation
$\mathbb{S}_{N}\simeq\mathbb{S}_{N}^{+}\oplus\mathbb{S}_{N}^{-}$ splits as the
direct sum of two irreducible representations called the half-spin
representations. They both have dimension
$\dim\mathbb{S}_{N}^{+}=\dim\mathbb{S}_{N}^{-}=2^{n-1}$. For odd $n=2m+1$, the
half-spin representations are both faithful and dual to each other; for even
$n=2m$, they are both self-dual and their images

$\operatorname{Spin}_{4m}^{\pm}(\mathbb{F})\;\subset\;\operatorname{GL}(\mathbb{S}_{4m}^{\pm})$

are called the half-spin groups. We show that spin or half-spin groups do not
occur for smooth nondivisible subvarieties of high enough codimension (see
theorem 8.3):

######  Theorem C.

Suppose that $X\subset A$ has ample normal bundle, is nondivisible and has
dimension $d<(g-1)/2$. Then the pair $(G_{X,\omega},\omega(\delta_{X}))$ is
not isomorphic to any of the above spin or half-spin groups with their spin or
half-spin representations unless

$(G_{X,\omega}^{\ast},\omega(\delta_{X}))\simeq(\operatorname{Spin}_{4m}^{\pm}(\mathbb{F}),\mathbb{S}_{4m}^{\pm})\quad\text{for
some $m\in\\{3,\dots,d\\}$},$

in which case $X$ has topological Euler characteristic of absolute value
$|e|=2^{2m-1}$ and is symmetric up to a translation, $d-m$ is even and
$d\geqslant(g-1)/4$.

The main theorem in section 1.2 now follows by combining theorems A, B, C, and
from this we also obtain the main theorem in section 1.1 by the analog of the
theorem of the fixed part given by theorem 4.10.

###  1.6. Conventions and notation

We always work over a field $k$ of characteristic zero. A _variety_ over $k$
is a separated finite type $k$-scheme, and a _subvariety_ is a closed
subvariety unless said otherwise. An _algebraic group_ is a finite type group
scheme over a field. For a locally free sheaf $\mathcal{E}$ (of finite rank)
on a variety $X$, we denote by
$\mathbb{P}(\mathcal{E}):=\operatorname{Proj}\operatorname{Sym}^{\bullet}\mathcal{E}^{\vee}$
the associated projective bundle. If $A$ is an abelian variety over $k$, we
denote by $\operatorname{Lie}A$ its tangent space at the identity and define
$\mathbb{P}_{A}:=\mathbb{P}((\operatorname{Lie}A)^{\vee})$. For a smooth
projective connected variety $X$, we denote by $\operatorname{Pic}^{0}(X)$ the
connected component of the identity in its Picard scheme. This is an abelian
variety, and we denote by $\operatorname{Alb}(X)$ its dual abelian variety.
Given a locally closed subvariety $Y$ of a variety $X$ over $k$, let
$\mathcal{C}_{Y/X}$ denote the _conormal sheaf_ of $Y$ in $X$, i.e., the
$\mathcal{O}_{Y}$-module $I/I^{2}$, where $I$ is the ideal sheaf of the closed
immersion $i\colon Y\to U$ for a suitable open subset $U\subset X$.

### Acknowledgments

We would like to thank Daniele Agostini, Benjamin Bakker, Yohan Brunebarbe,
Marco D’Addezio, and Olivier Debarre for helpful comments. A.J. gratefully
acknowledges support by the IHES where part of this work was completed. T.K.
was supported by the DFG research grant Kr 4663/2-1. C.L. was supported by the
DFG research grant Le 3093/3-2 and by the SMWK research grant SAXAG. M.M. was
supported by ANR grant ANR-18-CE40-0017.

##  2\. Gauss maps, positivity and nondegeneracy

In this section, we recall from the view of conormal geometry various notions
of positivity and nondegeneracy for subvarieties in abelian varieties. We
denote by $A$ an abelian variety over an algebraically closed field $k$ of
characteristic zero.

###  2.1. The stabilizer and the abelian variety generated

The _stabilizer_ of a subvariety $X\subset A$ is the algebraic subgroup
$\operatorname{Stab}_{A}(X)\subset A$ whose $k$-points are

$\operatorname{Stab}_{A}(X)(k)\;=\;\\{a\in A(k)\mid X+a=X\\}.$

Write $\operatorname{Stab}(X)=\operatorname{Stab}_{A}(X)$ if the ambient
abelian variety is clear from the context.

######  Definition 2.1.

We say $X\subset A$ is _nondivisible_ if it is integral and
$\operatorname{Stab}(X)=\\{0\\}$.

If $X\subset A$ is a connected subvariety, the _abelian subvariety generated
by $X$_ is defined to be the smallest abelian subvariety $\langle
X\rangle\subset A$ containing the image of the difference morphism $X\times
X\to A$, $(x,x^{\prime})\mapsto x-x^{\prime}$. Note that this image
$X-X\subset A$ is connected because $X\times X$ is so.

###  2.2. Conormal varieties and Gauss maps

Let us briefly recall the notion of conormal varieties and Gauss maps, which
will be crucial later. For abelian varieties, the cotangent bundle
$\Omega^{1}_{A}$ is a trivial bundle with fiber
$\textup{H}^{0}(A,\Omega^{1}_{A})={(\operatorname{Lie}{A})^{\vee}}$ of rank
$g=\dim A$. Consider the projection

$p\colon\quad\mathbb{P}(\Omega^{1}_{A})\;\longrightarrow\;\mathbb{P}_{A}=\mathbb{P}({(\operatorname{Lie}{A})^{\vee}}).$

If
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}\subset\mathbb{P}(\Omega^{1}_{A})$
is a $(g-1)$-dimensional integral subvariety, then for dimension reasons the
morphism

$\gamma_{\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}}\;:=\;p_{|\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}}\colon\quad\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}\;\longrightarrow\;\mathbb{P}_{A}$

is either dominant (and hence generically finite) or not dominant. We say that
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}$ is
_clean_ in the first case and _negligible_ in the second case. In the clean
case we denote by
$\deg\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}$
the generic degree of the generically finite dominant morphism
${\gamma}_{\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}}$,
in the negligible case we formally put
$\deg\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}=0$.

We want to apply these definitions to conormal varieties, for which we need
some more notation. For any subvariety $X\subset A$, its conormal sheaf
$\mathcal{C}_{X/A}$ fits in the exact sequence of coherent sheaves

$\mathcal{C}_{X/A}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\Omega^{1}_{A\rvert
X}\longrightarrow\Omega^{1}_{X}\longrightarrow 0.$

If $X\subset A$ is regular immersion, then $\mathcal{C}_{X/A}$ is locally free
and if $X$ is moreover integral, then $i$ is injective. If $X$ is smooth, then
all three terms are locally free and the sequence is short exact.

######  Definition 2.2.

For a reduced subvariety $X\subset A$ we define its (projective) _conormal
variety_
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\subset\mathbb{P}(\Omega^{1}_{A})$
to be the closure of $\mathbb{P}(\mathcal{C}_{X^{\textup{reg}}/A})$ in
$\mathbb{P}(\Omega^{1}_{A})$. The _Gauss map_ of $X$ is the morphism

$\gamma_{X}\;:=\;{\gamma}_{\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}}\colon\quad\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\;\longrightarrow\;\mathbb{P}_{A}$

We denote by
$\operatorname{pr}_{X}\colon\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\to
X$ the projection and
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X,x}:=\operatorname{pr}_{X}^{-1}(x)$
for $x\in X(k)$.

######  Remark 2.3.

As we almost exclusively work with the projective conormal varieties and not
with affine ones, we will usually drop the adjective _projective_. We clearly
have:

  1. (1)

The morphism
$\gamma_{X|\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X,x}}\colon\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X,x}\to\mathbb{P}_{A}$
is injective.

  2. (2)

If $X$ is smooth at a point $x$, then
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X,x}=\mathbb{P}(\mathcal{C}_{X/A,x})$.

  3. (3)

If $X$ is smooth, then
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}=\mathbb{P}(\mathcal{C}_{X/A})$.

The effect of isogenies on conormal varieties is easy to control. For an
integer $e\geqslant 1$ and an integral subvariety $X\subset A$ we denote by
$[e](X)\subset A$ its image under the isogeny $[e]\colon A\to A$. We will
always endow this image with the reduced subscheme structure, and we denote by
$e_{X}:=[e]_{\rvert X}\colon X\to[e](X)$ the finite morphism obtained by
restriction of the isogeny to the given subvariety. By abuse of notation, we
also denote by $[e]:A\times\mathbb{P}_{A}\to A\times\mathbb{P}_{A}$ the
induced morphism. Then we have:

######  Lemma 2.4.

Let $X\subset A$ be an integral subvariety, and let $Y=[e](X)\subset A$ for an
integer $e\geqslant 1$. Then we have an identity

$[e]_{\ast}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\;=\;\deg(e_{X})\cdot\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y}$

of cycles. In particular, if the subvariety $X\subset A$ is nondivisible, then
$[e]_{\ast}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}=\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y}$.

###### Proof.

The first claim follows easily from the fact that by construction the conormal
variety to any integral subvariety is integral. The second claim is then clear
because the morphism $e_{X}\colon X\to Y$ is birational if $X$ is
nondivisible. ∎

######  Corollary 2.5.

Let $X\subset A$ be a smooth integral subvariety and $Y=[e](X)$ for an integer
$e\geqslant 1$. Then the fibers of
$\operatorname{pr}_{Y}\colon\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y}\to
Y$ are pure of dimension $\operatorname{codim}_{A}Y-1$.

###### Proof.

Lemma 2.4 gives a commutative diagram

${\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}}$${\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y}}$${X}$${Y}$$\scriptstyle{[e]}$$\scriptstyle{\operatorname{pr}_{X}}$$\scriptstyle{\operatorname{pr}_{Y}}$$\scriptstyle{e_{X}}$

where the horizontal arrows are finite morphisms, and if $X$ is smooth, then
the fibers of the morphism
$\operatorname{pr}_{X}:\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\to
X$ are pure of dimension $\operatorname{codim}_{A}X-1$. ∎

###  2.3. Positivity and nondegeneracy of subvarieties

We now discuss various notions of positivity and nondegeneracy for
subvarieties of an abelian variety. We say that an integral subvariety
$X\subset A$ is degenerate if there exists a surjective morphism $\pi\colon
A\to B$ of abelian varieties with

$\dim\pi(X)<\min\\{\dim B,\dim X\\}.$

Otherwise, we say that $X$ is nondegenerate. Any closed point on the abelian
variety is a nondegenerate subvariety, and so is the abelian variety itself.
Also note that if the abelian variety $A$ is simple, then any integral
subvariety is nondegenerate. We say that a proper integral variety $X$ is _of
general type_ if there is a proper birational morphism $\nu\colon Y\to X$ from
a smooth proper connected variety $Y$ with big canonical bundle. For instance,
we have:

  1. (1)

An integral effective divisor $X\subset A$ is nondegenerate if and only if it
is ample. A curve $X\subset A$ is nondegenerate if and only if it generates
$A$. See [Deb95, §1, examples].

  2. (2)

For any elliptic curve $E$ and any simple abelian variety $B$ of dimension
$\geqslant 3$, Debarre has constructed in [Deb95, p. 189] a smooth subvariety

$X\subset A=E\times B$

of codimension $2$ which is nondegenerate but whose normal bundle is not
ample. The smooth subvariety is obtained by choosing a general ample divisor
$D\subset B$ and intersecting $E\times D$ with a general ample divisor in $A$.

  3. (3)

For $i=1,2$, let $A_{i}$ be an abelian variety and $X_{i}\subset A_{i}$ a
nondegenerate integral subvariety. By considering the projections onto the
factors, one sees that $X_{1}\times X_{2}\subset A_{1}\times A_{2}$ is of
general type but degenerate.

######  Remark 2.6.

Nondegeneracy is invariant under isogenies: Let $f\colon A\to A^{\prime}$ be
an isogeny of abelian varieties over $k$. Then an integral subvariety
$X\subset A$ is nondegenerate if and only if $f(X)\subset A^{\prime}$ is.

In what follows we often consider the sum morphism $\sigma\colon X\times Y\to
A$ for reduced subvarieties $X,Y\subset A$, and we denote by $X+Y\subset A$
its image. For nondegenerate subvarieties we have the following result by
Debarre:

######  Lemma 2.7.

Let $X,Y\subset A$ be integral subvarieties.

  1. (1)

If $X$ is nondegenerate, then $\dim(X+Y)=\min\\{\dim(X)+\dim(Y),\dim(A)\\}$.

  2. (2)

If $X$ and $Y$ are both nondegenerate, then so is $X+Y\subset A$.

###### Proof.

See [Deb05, corollary 8.11]. ∎

The relations between the various notion of nondegeneracy and positivity that
will play a role in this paper are summarized in the following diagram where
for a smooth subvariety $X\subset A$ we denote by $\mathcal{N}_{X/A}$ its
normal bundle:

$X$ smooth and
---
$\mathcal{N}_{X/A}$ ample
Gauss map $\gamma_{X}$ is
---
a finite morphism
$X$ nondegenerate
---
and $X\neq A$
$X$ of general type $\operatorname{Stab}(X)$
---
finite
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}$
clean$X$ smooth $A$ simple
---
$X$ smooth

More precisely, we have:

######  Theorem 2.8.

Let $X\subset A$ be an integral subvariety with $0<\dim X<\dim A$.

  1. (1)

The following are equivalent:

    1. (a)

the conormal cone
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}$ is
clean;

    2. (b)

the algebraic group $\operatorname{Stab}(X)$ is finite;

    3. (c)

the variety $X$ is of general type.

  2. (2)

If $X$ is nondegenerate, then $\operatorname{Stab}(X)$ is finite and $\langle
X\rangle=A$.

  3. (3)

If
$\gamma_{X}\colon\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\to\mathbb{P}_{A}$
is a finite morphism, then $X$ is nondegenerate.

  4. (4)

Suppose $X$ smooth. Then the normal bundle $\mathcal{N}_{X/A}$ is ample if and
only the Gauss map
$\gamma_{X}\colon\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\to\mathbb{P}_{A}$
is a finite morphism.

  5. (5)

If $A$ is a simple abelian variety and $X$ is of general type, then $X$ is
nondegenerate. If $X$ is moreover smooth, then $\mathcal{N}_{X/A}$ is ample.

###### Proof.

(1) The equivalence (a) $\Leftrightarrow$ (b) is shown in [Wei15a, th. 1],
while (b) $\Leftrightarrow$ (c) follows from Ueno’s fibration theorem [Uen73,
th. 3.10], [Abr94, th. 3].

(2) For the finiteness of the stabilizer, denote by $p\colon A\to
B:=A/\operatorname{Stab}(X)$ the quotient morphism. This quotient morphism is
not surjective, since by construction we have $p^{-1}(p(X))=X\neq A$. The
nondegeneracy of $X$ then forces $p\colon X\to\pi(X)$ to be generically
finite, and it follows that $\operatorname{Stab}(X)$ is finite as desired. To
show that $\langle X\rangle=A$, consider the quotient morphism $q\colon A\to
A/\langle X\rangle$. The image $q(X)$ is a point, hence the nondegeneracy of
$X$ and the assumption $\dim X>0$ imply that $\dim A/\langle X\rangle=0$,
which shows that we have $\langle X\rangle=A$.

(3) We prove the contrapositive. If $X\subset A$ is degenerate, then there is
a surjective morphism $\pi\colon A\to B$ of abelian varieties such that $\dim
Y<\min\\{\dim B,\dim X\\}$, where $Y:=\pi(X)$. We have the following
commutative of $\mathcal{O}_{X}$-modules with exact rows

${(\pi^{\ast}\mathcal{C}_{Y/B})_{\rvert
X}}$${(\pi^{\ast}\Omega^{1}_{B})_{\rvert
X}}$${(\pi^{\ast}\Omega^{1}_{Y})_{\rvert
X}}$${0}$${\mathcal{C}_{X/A}}$${\Omega^{1}_{A\rvert
X}}$${\Omega^{1}_{X}}$${0}$$\scriptstyle{j}$$\scriptstyle{\varepsilon}$$\scriptstyle{\textup{d}\pi}$$\scriptstyle{i}$

where $\textup{d}\pi$ is the pull-back of differential forms along $\pi$. Here
$i$ is injective over the smooth locus $X^{\textup{reg}}\subset X$, and
likewise $j$ is injective over $\pi^{-1}(Y^{\textup{reg}})$: Indeed, the short
exact sequence

$0\longrightarrow(\mathcal{C}_{Y/B})_{\rvert
Y^{\textup{reg}}}\longrightarrow\Omega^{1}_{B\rvert
Y^{\textup{reg}}}\longrightarrow\Omega^{1}_{Y^{\textup{reg}}}\longrightarrow
0$

of $\mathcal{O}_{Y^{\textup{reg}}}$-modules is locally split because the
$\mathcal{O}_{Y^{\textup{reg}}}$-module $\Omega^{1}_{Y^{\textup{reg}}}$ is
locally free; the pull-back along $\pi$ of the above short exact sequence
hence stays exact. It follows that $\varepsilon$ is also injective over the
nonempty open subset

$U:=X^{\textup{reg}}\cap\pi^{-1}(Y^{\textup{reg}}).$

The hypothesis $\dim Y<\dim X$ implies that the induced morphism $\pi_{\rvert
U}\colon U\to Y^{\textup{reg}}$ is not generically finite. Thus, for $y$
ranging over a dense open subset of $Y^{\textup{reg}}$, the fiber
$Z:=\pi^{-1}(y)\cap U$ is positive-dimensional. Pick a nonzero vector
$v\in\mathcal{C}_{Y,y}$, which exists because $\dim Y<\dim B$. Then

$0\;\neq\;j(v)\;\in\;\bigcap_{x\in Z}\mathcal{C}_{X,x}.$

Thus, if we denote by $F:=\operatorname{pr}_{X}(\gamma_{X}^{-1}([j(v)])\subset
X$ the image of $\gamma_{X}^{-1}([j(v)])$ under the projection
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\to
X$, then the subset $Z$ is contained in $F$. This shows that the dimension of
$\gamma_{X}^{-1}([j(v)])$ is positive.

(4) Since $X$ is smooth we have
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}=\mathbb{P}(\mathcal{C}_{X/A})$.
The normal bundle $\mathcal{N}_{X/A}$ is globally generated, thus the
equivalence is [Laz04b, Example 6.1.5].

(5) When $A$ is a simple abelian variety, any integral subvariety is
nondegenerate, and the ampleness of the normal bundle of a smooth subvariety
$X$ in $A$ follows from [Har71, prop. 4.1]. ∎

###  2.4. Symmetric powers of curves in abelian varieties

We show here that symmetric powers of a (smooth projective) curve $C$ cannot
be embedded as a complete intersection of ample divisors as claimed in section
1.4. Recall that the curve $C$ has gonality $\geqslant n+1$ if and only if the
sum map $\operatorname{Sym}^{n}C\to
X:=C+\cdots+C\subset\operatorname{Pic}^{0}(C)$ is an isomorphism. If so the
normal bundle of $X$ is ample [Deb95, §1, Examples (2)]. Imposing further
positivity properties to the normal bundle is far more restrictive:

######  Proposition 2.9.

Let $C\subset A$ a smooth irreducible projective curve such that the sum
morphism $\operatorname{Sym}^{n}C\to X:=C+\cdots+C\subset A$ is an isomorphism
for some $n\geqslant 2$. Then $C$ is nonhyperelliptic of genus $g\geqslant 3$
and the following hold:

  1. (1)

If the normal bundle
$\mathcal{N}_{X/A}=\mathcal{V}_{1}\oplus\dots\oplus\mathcal{V}_{r}$ is a
direct sum of ample vector bundles, then

$n\leqslant\max_{i=1,\dots,r}\operatorname{rk}\mathcal{V}_{i}+1.$

  2. (2)

The normal bundle $\mathcal{N}_{X/A}$ is a direct sum of ample line bundles if
and only if $g=3$, $n=2$, and $A$ is isomorphic to
$\operatorname{Pic}^{0}(C)$.

###### Proof.

By Lefschetz’s principle, we may assume $k=\mathbb{C}$. First of all, the
curve $C$ is nonhyperelliptic of genus $g\geqslant 3$. Otherwise, $C$ would be
symmetric when suitably embedded in its Jacobian. In particular, the sum
morphism would contract the antidiagonal and thus would not induce an
isomorphism $\operatorname{Sym}^{n}C\simeq C+\cdots+C$.

(1) Arguing by contradiction, suppose the inequality in the statement does not
hold. Then we can apply the Barth-Lefschetz theorem [Deb95, th. 4.5] to obtain
isomorphisms

$\textup{H}^{i}(A)\simeq\textup{H}^{i}(X),\qquad i=1,2,$

of rational cohomology groups. On the other hand, the computation of
cohomology of symmetric powers of curves [Mac62, 1.2] yields the following
expressions:

$\displaystyle\textup{H}^{1}(X)$
$\displaystyle=\textup{H}^{1}(\operatorname{Sym}^{n}C)\simeq\textup{H}^{1}(C^{n})^{\mathfrak{S}_{n}}=\textup{H}^{0}(C)\otimes\textup{H}^{1}(C),$
$\displaystyle\textup{H}^{2}(X)$
$\displaystyle=\textup{H}^{2}(\operatorname{Sym}^{n}C)\simeq\textup{H}^{2}(C^{n})^{\mathfrak{S}_{n}}=\operatorname{Alt}^{2}\textup{H}^{1}(C)\oplus\textup{H}^{0}(C)\otimes\textup{H}^{2}(C)^{n-1}.$

Recalling the equality
$\textup{H}^{2}(A)=\operatorname{Alt}^{2}\textup{H}^{1}(A)$ we obtain a
contradiction.

(2) If $C$ has genus $g=3$, the subvariety
$C+C\subset\operatorname{Pic}^{0}(C)$ is a theta divisor and hence ample.
Conversely, suppose that the normal bundle $\mathcal{N}_{X/A}$ is a direct sum
of ample line bundles. We first claim that then $A$ is isogenous to
$\operatorname{Pic}^{0}(C)$. Indeed, as above we have isomorphisms

$\textup{H}^{1}(A)\simeq\textup{H}^{1}(X)\simeq\textup{H}^{1}(C).$

Now we cannot conclude as in (1) because the Barth-Lefschetz theorem here only
says that $\textup{H}^{2}(A)\to\textup{H}^{2}(X)$ is injective. Instead, write
$\mathcal{N}_{X/A}=\mathcal{L}_{1}\oplus\cdots\oplus\mathcal{L}_{g-2}$ for
ample line bundles $\mathcal{L}_{i}$ on $X$. By looking at the short exact
sequence

$0\longrightarrow\textup{T}_{X}\longrightarrow\operatorname{Lie}A\otimes\mathcal{O}_{X}\longrightarrow\mathcal{N}_{X/A}=\mathcal{L}_{1}\oplus\cdots\oplus\mathcal{L}_{g-2}\longrightarrow
0,$

we see that the line bundles $\mathcal{L}_{i}$ are globally generated and

(2.1)
$\mathcal{L}_{1}\otimes\cdots\otimes\mathcal{L}_{g-2}\simeq\mathcal{K}_{X}$

where $\mathcal{K}_{X}=\operatorname{Alt}^{2}\Omega^{1}_{X}$ is the canonical
bundle on $X$. We identify $X$ with $\operatorname{Sym}^{2}C$ and write
$\pi\colon C\times C\to X$ for the quotient morphism. Since $\pi$ ramifies
exactly on the diagonal $\Delta$ of $C\times C$, we have
$\pi^{\ast}\mathcal{K}_{X}=\mathcal{K}_{C\times C}(-\Delta)$. Let us fix a
point $p\in C(k)$ and consider the embedding $f\colon C\to C\times C$,
$x\mapsto(x,p)$. Then

(2.2) $f^{\ast}\pi^{\ast}\mathcal{K}_{X}=\mathcal{K}_{C}(-p).$

On the other hand, for $i=1,\dots,g-2$, the line bundle
$\mathcal{M}_{i}:=f^{\ast}\pi^{\ast}\mathcal{L}_{i}$ on $C$ is ample and
globally generated. Moreover, the curve $C$ being nonhyperelliptic, we
necessarily have $\deg\mathcal{M}_{i}\geqslant 3$. By combining (2.1) and
(2.2) and then by taking degrees, we obtain the inequality

$2g-3=\deg\mathcal{K}_{C}(-p)=\sum_{i=1}^{g-2}\deg\mathcal{M}_{i}\geqslant
3(g-2).$

This forces $g=3$. For a suitable Abel-Jacobi embedding
$C\hookrightarrow\operatorname{Pic}^{0}(C)$, there exists an isogeny
$\varphi\colon\operatorname{Pic}^{0}(C)\to A$ such that the following diagram
commutes:

${\operatorname{Sym}^{2}C}$${\Theta}$${\operatorname{Pic}^{0}(C)}$${\operatorname{Sym}^{2}C}$${X}$${A}$$\scriptstyle{\sim}$

$\scriptstyle\sim$

$\scriptstyle{\varphi}$$\scriptstyle{\sim}$

Here the leftmost horizontal arrows are induced by the sum and
$\Theta\subset\operatorname{Pic}^{0}(C)$ is a theta divisor. The preimage
$\varphi^{-1}(X)$ is smooth, thus its connected components are irreducible. As
$\Theta$ is one of them, the others are $\Theta+a$ for
$a\in\operatorname{Ker}\varphi$. Since any two translates of an ample divisor
meet, we have $\Theta=\varphi^{-1}(X)$. But the isogeny $\varphi$ induces an
isomorphism $\Theta\simeq X$, thus $\varphi$ must be injective. ∎

######  Corollary 2.10.

Let $C\subset A$ be a smooth irreducible projective curve such that the sum
morphism $\operatorname{Sym}^{n}C\to X:=C+\cdots+C\subset A$ is an isomorphism
for some $n\geqslant 2$. Then $C$ is nonhyperelliptic of genus $g\geqslant 3$
and the following are equivalent:

  1. (1)

The subvariety $X\subset A$ is a complete intersection of ample divisors.

  2. (2)

We have $g=3$, $n=2$, and $A$ is isomorphic to $\operatorname{Pic}^{0}(C)$.

###### Proof.

For complete intersections of ample divisors, the normal bundle is a direct
sum of ample line bundles. Hence, proposition 2.9 (2) applies. ∎

As an amusing aside, of no use in what follows, note that proposition 2.9
implies the classical bound for the gonality of a smooth projective curve:

######  Corollary 2.11.

A smooth projective curve of genus $g$ has gonality $\leqslant(g+3)/2$.

###### Proof.

As already mentioned, the curve $C$ has gonality $\geqslant n+1$ if and only
if the sum morphism $\operatorname{Sym}^{n}C\to
X:=C+\cdots+C\subset\operatorname{Pic}^{0}(C)$ is an isomorphism, and if this
is the case, then $X$ has ample normal bundle in $\operatorname{Pic}^{0}(C)$.
Since the normal bundle has rank $g-n$ proposition 2.9 (1) implies $n\leqslant
g-n+1$, that is $n+1\leqslant(g+3)/2$. ∎

In particular proposition 2.9 (1) is sharp in the two extremal cases—that of
an indecomposable ample normal bundle and that of a sum of ample line bundles.

###  2.5. Bounds for the topological Euler characteristic

We now pass to some numerics concerning the topological Euler characteristic
of complete intersections. To ease notation below, we define $g:=\dim A$. For
a smooth subvariety $X\subset A$, let $e_{X}$ denote its topological Euler
characteristic. By definition it is the top Chern class of the tangent bundle
$T_{X}$ of $X$. Consider the short exact sequence of vector bundles on $X$,

$0\longrightarrow T_{X}\longrightarrow T_{A\rvert
X}\longrightarrow\mathcal{N}_{X/A}\longrightarrow 0.$

Since the total Chern class is multiplicative in short exact sequences and the
tangent bundle of $A$ is trivial, we have

$c(T_{X})=c(T_{A\rvert
X})c(\mathcal{N}_{X/A})^{-1}=c(\mathcal{N}_{X/A})^{-1}.$

First of all, note that we have the following lower bound whenever the normal
bundle is ample.

######  Lemma 2.12.

Let $X\subset A$ be a $d$-dimensional smooth subvariety with ample normal
bundle. Then

$|e_{X}|\geqslant\max\\{g,2^{\min\\{d,\lfloor\sqrt{g-1}\rfloor\\}}\\}.$

###### Proof.

We may suppose $k=\mathbb{C}$. By definition the inverse of the total Chern
class is the total Segre class. We have
$|e_{X}|=(-1)^{d}s_{d}(\mathcal{N}_{X/A})=s_{d}(\mathcal{N}_{X/A}^{\vee})$
where $d=\dim X$ and $s_{d}$ is the $d$-th Segre class. Now the normal bundle
$\mathcal{N}_{X/A}$ is ample and globally generated. Since
$\textup{H}^{1}(X,\mathbb{C})\neq 0$ we have $|e_{X}|\geqslant g$ by [BSS93,
Theorem 4]. According to [EIL00, Prop. 2.4] we also have $|e_{X}|\geqslant
2^{\min\\{d,\lfloor\sqrt{g-1}\rfloor\\}}$ because the cotangent bundle of $X$
is nef.111Beware that in both references the authors adopt the convention dual
to the one in [Ful98] for the definition of Segre classes. ∎

The previous lower bound is doubtlessly not sharp. Indeed for a smooth
projective curve $X$ generating $A$ we have $|e_{X}|\geqslant 2g-2$. For
surfaces we have:

######  Lemma 2.13.

Let $X\subset A$ be a smooth projective surface generating $A$ and with finite
stabilizer. Then

$e_{X}\geqslant 3g-3.$

###### Proof.

Write $c_{1}=c_{1}(T_{X})$ and $c_{2}=c_{2}(T_{X})=e_{X}$ and
$\chi=\chi(X,\mathcal{O}_{X})$ as usual. By Theorem 2.8 the surface $X$ is of
general type. Thus the Bogomolov-Miyaoka-Yau inequality gives
$c_{1}^{2}\leqslant 3c_{2}$ which is equivalent to $3\chi\leqslant c_{2}$ by
Noether’s formula. On the other hand, let us write
$q=h^{1}(X,\mathcal{O}_{X})$ and $p=h^{2}(X,\mathcal{O}_{X})$ so that
$\chi=1-q+p$. The surface $X$ is minimal, thus we can apply the inequality
$p\geqslant 2q-4$ (see Beauville’s appendix to [Deb82] for a proof), which is
equivalent to $\chi\geqslant q-3$. Combining these inequalities yields
$c_{2}\geqslant 3(q-1)$. Since $X$ generates $A$ by hypothesis, we have
$q\geqslant g$ which concludes the proof. ∎

When the subvariety is a complete intersection of ample divisors the previous
lower bounds can be drastically improved. In order to show this, for integers
$n\geqslant 2$ and $r\in\\{1,\dots,n-1\\}$, consider the following subset of
partitions of $n$,

$P(n,r)\;:=\;\\{a=(a_{1},\dots,a_{r})\in\mathbb{Z}^{r}\mid
a_{1},\dots,a_{r}\geqslant 1,a_{1}+\cdots+a_{r}=n\\}.$

Note that $P(n,r)$ has cardinality $\binom{n-1}{n-r}$.

######  Lemma 2.14.

Let $X$ be a smooth complete intersection of ample divisors
$D_{1},\dots,D_{r}$ in $A$. Then

$e_{X}\;=\;(-1)^{\dim X}\sum_{a\in P(g,r)}D_{1}^{a_{1}}\cdots D_{r}^{a_{r}}.$

###### Proof.

The hypothesis of $X$ being a complete intersection of the divisors
$D_{1},\dots,D_{r}$ implies that the normal bundle $\mathcal{N}_{X/A}$ is the
direct sum of (the restriction to $X$ of) the line bundles
$\mathcal{O}(D_{1}),\dots,\mathcal{O}(D_{r})$. In particular,

$c(\mathcal{N}_{X/A})=c(\mathcal{O}(D_{1}))\cdots
c(\mathcal{O}(D_{r}))=(1+D_{1})\cdots(1+D_{r})\in\operatorname{CH}(X).$

By inverting formally $1+D_{i}$ we find the following expression

$c(T_{X})=\sum_{n=0}^{g-r}(-1)^{n}\sum_{\begin{subarray}{c}a_{1},\dots,a_{r}\geqslant
0\\\ a_{1}+\cdots+a_{r}=n\end{subarray}}D_{1}^{a_{1}}\cdots
D_{r}^{a_{r}}\in\operatorname{CH}(X).$

Looking at it in the Chow ring of $A$ amounts to multiplying it by
$D_{1}\cdots D_{r}$. We conclude by then taking the piece of degree $g$. ∎

Recall that, for an ample divisor $D\subset A$, the self-intersection $D^{g}$
is positive and divisible by $g!$, as the ratio $D^{g}/g!$ is given by
$h^{0}(A,\mathcal{O}(D))$.

######  Lemma 2.15.

For ample divisors $D_{1},\dots,D_{g}\subset A$, we have $D_{1}\cdots
D_{g}\geqslant g!$.

###### Proof.

The Khovanskii-Teissier inequality [Laz04a, Theorem 1.6.1] states that the
lower bound $(D_{1}\cdots D_{g})^{g}\geqslant D_{1}^{g}\cdots D_{g}^{g}$
holds. Since each factor on the right-hand side is a positive multiple of
$g!$, this concludes the proof. ∎

######  Proposition 2.16.

Let $X\subset A$ be a smooth complete intersection of ample divisors of
dimension $d\geqslant 1$. Then $e_{X}$ is even and

$|e_{X}|\geqslant g!\tbinom{g-1}{d}.$

###### Proof.

By assumption $X$ is a complete intersection of ample divisors, say
$D_{1},\dots,D_{r}$ where $r=g-d$ is the codimension of $X$. Lemma 2.14 shows

$e_{X}\;=\;(-1)^{d}\sum_{a\in P(g,r)}D_{1}^{a_{1}}\cdots D_{r}^{a_{r}}.$

Since the divisors $D_{1},\dots,D_{r}$ are ample, by lemma 2.15 we have
$D_{1}^{a_{1}}\cdots D_{r}^{a_{r}}\geqslant g!$ for each $a\in P(g,r)$. Since
the cardinality of $P(g,g-d)$ is $\binom{g-1}{d}$, the inequality in the
statement follows. For the parity of $e_{X}$, by the Lefschetz principle, we
may assume $k=\mathbb{C}$. Then each
$[D_{i}]^{a_{i}}\in\textup{H}^{2a_{i}}(A,\mathbb{Z})$ is divisible by
$a_{i}!$. Since $d\geqslant 1$, for each $a\in P(g,r)$ we have $a_{i}\geqslant
2$ for some $i$, thus we conclude that $e_{X}$ is even. ∎

By Proposition 2.16, the absolute value of the Euler characteristic of a
smooth connected complete intersection of ample divisors in $A$ is never equal
to $27$. We now prove that $|e_{X}|\neq 56$, except in the case of curves in
abelian surfaces and abelian threefolds (in which case there are examples).

######  Corollary 2.17.

If $X\subset A$ is a smooth complete intersection of ample divisors of
dimension $d\geq 1$ and $(d,g)\neq(1,2),(1,3)$, then $|e_{X}|\neq 56$.

###### Proof.

Proposition 2.16 implies $|e_{X}|\geqslant g!\binom{g-1}{d}$ which settles the
matter for $g\geqslant 5$. On the other hand, if $X$ is itself a divisor, that
is $d=g-1$, then $|e_{X}|=X^{g}$ is divisible by $g!$. The only two cases left
are $(d,g)=(1,4),(2,4)$ for which $g!\binom{g-1}{d}=72$. ∎

Note that proposition 2.16 furnishes another proof of corollary 2.10. Indeed
the $n$-th symmetric power of a smooth projective curve of genus $g\geqslant
2$ has topological Euler characteristic of absolute value $\binom{2g-2}{n}$;
see [Mac62, 4.4]. Using that the gonality is $\leqslant(g+3)/2$ we conclude
because, for $g\geqslant 4$ and $n\leqslant(g+1)/2$, we have

$\tbinom{2g-2}{n}<g!\tbinom{g-1}{n}.$

##  3\. Perverse sheaves on abelian varieties

In this section, we collect some general results about perverse sheaves on
abelian varieties. We work over a field $k$ with $\operatorname{char}(k)=0$,
but as in [LS20, section 3] we do not require this field to be algebraically
closed; for the relation with monodromy groups we will later need to work over
function fields. For any variety $X$ over $k$ we denote by

$\textup{Perv}(X,\mathbb{F})\;\subset\;\textup{D}^{b}_{c}(X,\mathbb{F})$

the abelian category of perverse sheaves with coefficients in
$\mathbb{F}=\overline{\mathbb{Q}}_{\ell}$ for a fixed prime number $\ell$. For
$k=\mathbb{C}$, we will later also consider perverse sheaves in the analytic
sense with coefficients in $\mathbb{F}=\mathbb{C}$, and we will use the above
notation also in this case. The results below work both in the $\ell$-adic
setting over any field $k$ and in the analytic setting with
$k=\mathbb{F}=\mathbb{C}$. We let $\pi_{1}(A,0)$ be the étale resp.
topological fundamental group in the two settings, with the profinite resp.
discrete topology, and write
$\Pi(A,\mathbb{F})=\operatorname{Hom}(\pi_{1}(A,0),\mathbb{F}^{\times})$ for
the group of its continuous characters.

###  3.1. Convolution on abelian varieties

For convenience, let us briefly recall the Tannakian description of perverse
sheaves on abelian varieties $X=A$ given in [KW15c]. The sum morphism
$\sigma\colon A\times A\to A$ induces a convolution product

$*:{\mathrm{D}^{b}_{c}}(A,\mathbb{F})\times{\mathrm{D}^{b}_{c}}(A,\mathbb{F})\longrightarrow{\mathrm{D}^{b}_{c}}(A,\mathbb{F}),\quad
K_{1}*K_{2}:=R\sigma_{*}\left(K_{1}\boxtimes K_{2}\right)$

which endows the derived category with the structure of a rigid symmetric
monoidal category [Wei11] (in loc. cit. this is stated only over algebraically
closed fields $k$, but the proof works in the general case without changes).
The subcategory of perverse sheaves is not stable under the convolution
product, but it becomes so after passing to a certain quotient category. To
explain this, recall that for any $P\in\textup{Perv}(A,\mathbb{F})$ we have

$\chi(A,P)\;:=\;\sum_{i\in\mathbb{Z}}\,(-1)^{i}\dim_{\mathbb{F}}H^{i}(A,P)\;\geqslant\;0.$

Indeed, over $k=\mathbb{C}$ this was observed by Franecki and Kapranov [FK00,
cor. 1.4]; the general case can be reduced to the complex case by choosing a
model over some algebraically closed subfield of $k$ which embeds into the
complex numbers, see lemma A.1. The additivity of the Euler characteristic in
short exact sequences then implies that perverse sheaves of Euler
characteristic zero form a Serre subcategory

$\textup{S}(A,\mathbb{F})\;:=\;\\{P\in\textup{Perv}(A,\mathbb{F})\mid\chi(A,P)=0\\}\;\subset\;\textup{Perv}(A,\mathbb{F})$

inside the abelian category of perverse sheaves. Let
$\textup{T}(A,\mathbb{F})\subset{\mathrm{D}^{b}_{c}}(A,\mathbb{F})$ be the
full subcategory of sheaf complexes whose perverse cohomology sheaves are in
$\textup{S}(A,\mathbb{F})$; its objects will be called negligible sheaf
complexes.

######  Proposition 3.1.

The triangulated quotient category
$\overline{{\mathrm{D}^{b}_{c}}}(A,\mathbb{F}):={\mathrm{D}^{b}_{c}}(A,\mathbb{F})/\textup{T}(A,\mathbb{F})$
inherits from the perverse $t$-structure on the derived category a
$t$-structure whose heart

$\overline{\textup{Perv}}(A,\mathbb{F})\;\subset\;\overline{{\mathrm{D}^{b}_{c}}}(A,\mathbb{F})$

is equivalent to the abelian quotient category
$\textup{Perv}(A,\mathbb{F})/\textup{S}(A,\mathbb{F})$. It also inherits the
structure of a rigid symmetric monoidal category with respect to a convolution
product

$*:\overline{{\mathrm{D}^{b}_{c}}}(A,\mathbb{F})\times\overline{{\mathrm{D}^{b}_{c}}}(A,\mathbb{F})\longrightarrow\overline{{\mathrm{D}^{b}_{c}}}(A,\mathbb{F})$

induced by the convolution product on the derived category. On the
triangulated quotient category, this product is $t$-exact in both of its
arguments. Thus, it restricts to a product

$*:\overline{\textup{Perv}}(A,\mathbb{F})\times\overline{\textup{Perv}}(A,\mathbb{F})\longrightarrow\overline{\textup{Perv}}(A,\mathbb{F}),$

and $\overline{\textup{Perv}}(A,\mathbb{F})$ is a neutral Tannaka category
with respect to this product.

###### Proof.

Fix an algebraic closure $K\supset k$. Then the functor
${\mathrm{D}^{b}_{c}}(A,\mathbb{F})\to{\mathrm{D}^{b}_{c}}(A_{K},\mathbb{F})$
is exact for the perverse $t$-structure, compatible with the convolution
product, and preserves the subcategories of negligible objects. Hence, the
result follows from the statement over algebraically closed fields in [KW15c,
Krä14]; note that by Deligne’s internal characterization of neutral Tannaka
categories [Cou20, §6.4], it suffices to construct a fiber functor on every
finitely generated tensor subcategory. ∎

In what follows, by an abelian tensor category we mean a rigid symmetric
monoidal abelian $\mathbb{F}$-linear category.

###  3.2. Tannaka groups of perverse sheaves

Let $\mathcal{C}\subset\overline{\textup{Perv}}(A,\mathbb{F})$ be a full
abelian tensor subcategory and

$\omega\colon\quad\mathcal{C}\;\longrightarrow\;\operatorname{Vect}(\mathbb{F})$

a given fiber functor on this subcategory. The existence of such fiber
functors is guaranteed by proposition 3.1; there is no canonical choice of
such a fiber functor, but any two fiber functors on a neutral Tannaka category
over an algebraically closed field $\mathbb{F}$ are noncanonically isomorphic
[DMOS82, th. 3.2.(b)]. Once we have chosen a fiber functor, we get an
equivalence of abelian tensor categories between $\mathcal{C}$ and the
category $\textup{Rep}_{\mathbb{F}}(G_{\omega}(\mathcal{C}))$ of finite-
dimensional algebraic representations of the affine group scheme

$G_{\omega}(\mathcal{C})\;:=\;\operatorname{Aut}^{\otimes}(\omega)$

over $\mathbb{F}$ called the _Tannaka group_ of $\mathcal{C}$. We are
interested in algebraic quotients of this proalgebraic group scheme:

######  Definition 3.2.

For any $P\in\mathcal{C}$, we obtain from the above construction an affine
algebraic group

$G_{\omega}(P)\;:=\;\operatorname{Im}\bigl{(}G_{\omega}(\mathcal{C})\to\operatorname{GL}(\omega(P))\bigr{)}$

over $\mathbb{F}$ with a faithful representation on the vector space
$\omega(P)\in\operatorname{Vect}(\mathbb{F})$ whose dimension is the Euler
characteristic

$\dim_{\mathbb{F}}(\omega(P))\;=\;\chi(A,P),$

see [KW15c, proof of cor. 4.2]. Let us denote by $\iota\colon\langle
P\rangle\hookrightarrow\mathcal{C}$ the smallest abelian tensor subcategory
which contains the object $P$ and is stable under subobjects and quotients.
Then $G_{\omega}(P)=G_{\omega\circ\iota}(\langle P\rangle)$ for the fiber
functor $\omega\circ\iota\colon\langle
P\rangle\to\operatorname{Vect}(\mathbb{F})$ and we have a commutative diagram
of abelian tensor categories:

${\langle
P\rangle}$${\textup{Rep}_{\mathbb{F}}(G_{\omega}(P))}$${\mathcal{C}}$${\textup{Rep}_{\mathbb{F}}(G_{\omega}(\mathcal{C}))}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

If $P\in\mathcal{C}$ is a simple object, then the faithful representation
$\omega(P)\in\textup{Rep}_{\mathbb{F}}(G_{\omega}(P))$ is irreducible and then
$G_{\omega}(P)$ is reductive by [Hum78, 19.1 prop. (b)]. This is in particular
the case when $P=\delta_{X}$ is the intersection complex of an integral
subvariety $X\subset A$, in which case we write

$G_{X,\omega}:=G_{\omega}(\delta_{X}).$

For the rest of this section, we fix a full abelian tensor subcategory
$\mathcal{C}\subset\overline{\textup{Perv}}(A,\mathbb{F})$ and a fiber functor
$\omega\colon\mathcal{C}\to\operatorname{Vect}(\mathbb{F})$. When there is no
risk of confusion, we also write $\omega$ for the restriction of the given
fiber functor to any subcategory of $\mathcal{C}$.

###  3.3. The derived group of the connected component

It is often convenient to pass from arbitrary reductive groups to connected
semisimple groups: For a reductive group $G$, let $G^{\circ}\subset G$ be its
connected component of the identity, and note that the derived group

$G^{\ast}\;:=\;[G^{\circ},G^{\circ}]$

is a connected semisimple group. For the reductive Tannaka groups from section
3.2 we will understand the connected components and the center in terms of
direct images of perverse sheaves under the morphisms $[d]\colon A\to
A,x\mapsto dx$ for $d\in\mathbb{N}$ and $t_{a}\colon A\to A,x\mapsto x+a$ for
$a\in A(k)$. For a perverse sheaf $Q\in\textup{Perv}(A,\mathbb{F})$ and a
point $a\in A(k)$, we define

$Q_{a}\;:=\;t_{a*}P$

and we say that $Q$ is nondivisible if it is simple and satisfies
$Q_{a}\not\simeq Q$ for all $a\in A(k)$ with $a\neq 0$. We denote by

$\Gamma_{P}:=\\{a\in A(k)_{\textup{tors}}\mid\delta_{a}\in\langle P\rangle\\}$

the abelian group of torsion points whose associated skyscraper sheaf appears
in the Tannaka category $\langle P\rangle$ generated by a perverse sheaf
$P\in\mathcal{C}$. Note that $\Gamma_{P}$ is finite: Indeed, every skyscraper
sheaf $\delta_{a}\in\langle P\rangle$ defines a character of the Tannaka group
$G_{\omega}(P)$ and algebraic groups have only finitely many torsion
characters. In fact the first part of the following result shows that all
torsion characters of the Tannaka group are given by skyscraper sheaves in
torsion points:

######  Proposition 3.3.

Let $k$ be algebraically closed and $P\in\mathcal{C}$ a simple perverse sheaf.

  1. (1)

The group of connected components of the Tannaka group $G:=G_{\omega}(P)$ is
given by

$G/G^{\circ}\;\simeq\;\operatorname{Hom}(\Gamma_{P},\mathbb{G}_{m}).$

  2. (2)

Fix an integer $d\geqslant 1$ with $d\cdot\Gamma_{P}=\\{0\\}$. Then for all
$Q,Q^{\prime}\in\langle P\rangle$ we have:

$\displaystyle\omega(Q)_{\rvert G^{\circ}}\;\simeq\;\omega(Q^{\prime})_{\rvert
G^{\circ}}$ $\displaystyle\quad\Longleftrightarrow\quad$
$\displaystyle[d]_{*}Q\;\simeq\;[d]_{*}Q^{\prime},$ $\omega(Q)_{\rvert
G^{\circ}}$ is irreducible $\displaystyle\quad\Longleftrightarrow\quad$
$\displaystyle\textnormal{\em$Q$ is nondivisible}.$

  3. (3)

Let $\det(P)\in\langle P\rangle$ be the unique simple perverse sheaf which
corresponds to the top wedge power of $V:=\omega(P)$. Then $P$ is a skyscraper
sheaf. If $V_{\rvert G^{\circ}}$ is irreducible, we have:

$\textnormal{\em$G^{\circ}$
semisimple}\quad\Longleftrightarrow\quad\textnormal{\em$\operatorname{Supp}(\det(P))$
is a torsion point}.$

###### Proof.

For $k=\mathbb{C}$, parts (1) and (2) are due to Weissauer [Wei15b] who also
shows that every invertible object in the Tannaka category of perverse sheaves
is a skyscraper sheaf (this in particular applies to $\det(P)$); alternatively
one could use the Riemann-Hilbert correspondence and the results for holonomic
$\mathcal{D}$-modules in [Krä22, section 3.c]. From $k=\mathbb{C}$ one can
pass to an arbitrary algebraically closed field of characteristic zero because
the Tannaka group is invariant under extensions of algebraically closed fields
and any perverse sheaf is defined over the algebraic closure of a finitely
generated field, see corollary 4.4 resp. lemma A.1. The claim about
semisimplicity in (3) follows since by Schur’s lemma the center
$Z=Z(G^{\circ})$ acts on $V$ by scalars and hence $\det(V)$ has finite order
if and only if $Z$ is finite. ∎

######  Definition 3.4.

For perverse sheaves $P\in\mathcal{C}$ we denote the derived group of the
connected component of the Tannaka group $G=G_{\omega}(P)$ by

$G_{\omega}^{\ast}(P)\;:=\;[G^{\circ},G^{\circ}].$

If $P=\delta_{X}$ is the intersection complex of a subvariety $X\subset A$, we
put $G_{X,\omega}^{\ast}:=G_{\omega}^{\ast}(P)$.

Proposition 3.3 allows us to realize this group as the Tannaka group of
another perverse sheaf:

######  Corollary 3.5.

Suppose $k$ is algebraically closed. Let $P\in\mathcal{C}$ be a simple
perverse sheaf. Then for any integer $d\geqslant 1$ with
$[d]_{*}P\in\mathcal{C}$ and any $a\in A(k)$ with $P_{a}\in\mathcal{C}$ the
following properties hold:

  1. (1)

$G_{\omega}^{\ast}(P_{a})\simeq G_{\omega}^{\ast}(P)$.

  2. (2)

$G_{\omega}^{\circ}([d]_{*}P)\simeq G_{\omega}^{\circ}(P)$.

  3. (3)

$G_{\omega}([d]_{*}P)$ is connected if and only if $d\cdot\Gamma_{P}=0$.

  4. (4)

Suppose $P$ is nondivisible with $[d]_{*}\det(P_{a})=\delta_{0}$ and
$d\cdot\Gamma_{P_{a}}=0$. If $[d]_{*}P_{a}$ belongs to $\mathcal{C}$, then

$G_{\omega}([d]_{*}P_{a})\;\simeq\;G_{\omega}^{\ast}(P).$

###### Proof.

(1) By [Krä21, lemma 4.3.2] the inclusions $\langle P\rangle\subset\langle
P\oplus\delta_{a}\rangle\supset\langle P_{a}\rangle$ induce isomorphisms
$G_{\omega}^{\ast}(P)\simeq G_{\omega}^{\ast}(P\oplus\delta_{a})\simeq
G_{\omega}^{\ast}(P_{a})$.

(2) By [Wei15b] or [Krä22, cor. 1.6], the pushforward $[d]_{*}\colon\langle
P\rangle\to\langle[d]_{*}P\rangle$ is a tensor functor which induces an
isomorphism between the connected components of the identity of the respective
Tannaka groups.

(3) This follows from proposition 3.3 (1) applied to $[d]_{*}P$ since
$d\cdot\Gamma_{P}=\Gamma_{[d]_{*}P}$.

(4) By the previous two steps, the group $G_{\omega}([d]_{*}P_{a})$ is
connected. One easily sees that the perverse sheaf $[d]_{*}P_{a}$ is
nondivisible with $\det([d]_{*}P_{a})=[d]_{*}\det(P_{a})=\delta_{0}$ so that
$G_{\omega}([d]_{*}P_{a})$ is a semisimple group by the last part of
proposition 3.3. It is therefore equal to the derived group of its connected
component of the identity, which by (1) and (2) coincides with
$G_{\omega}^{\ast}(P)$. ∎

######  Remark 3.6.

The isomorphism $G_{\omega}^{\circ}([d]_{*}P)\simeq G_{\omega}^{\circ}(P)$ in
corollary 3.5 (2) is not canonical, it involves the choice of an isomorphism
between the two fiber functors $\omega$ and $\omega\circ[d]_{*}$ on the tensor
category $\langle P\rangle$. But we can choose the isomorphism in a
contravariant functorial way with respect to monomorphisms in the full tensor
subcategory

$\mathcal{C}\cap[d]_{*}^{-1}(\mathcal{C})\;:=\;\\{Q\in\mathcal{C}\mid[d]_{*}Q\in\mathcal{C}\\}\;\subset\;\mathcal{C}$

by fixing an isomorphism between the fiber functors $\omega$ and
$\omega\circ[d]_{*}$ on this category.

###  3.4. Larsen’s alternative

Let $X\subset A$ be a subvariety such that $\delta_{X}\in\mathcal{C}$. We are
interested in criteria under which the Tannaka group $G_{X,\omega}$ is big.
Suppose that $X\subset A$ is nondegenerate and $2\dim X<\dim A$, so that by
lemma 2.7 the sum morphism

$\sigma\colon\quad X\times X\;\longrightarrow\;W\;:=\;X+X\;\subset\;A$

is generically finite onto its image, and this image is nondegenerate. Let
$U\subset W$ be a smooth open dense subset over which $\sigma$ is a finite
étale cover. By adjunction, we have an inclusion
$\delta_{U}\subset\sigma_{*}(\delta_{X\times X})_{|U}$ as a direct summand.
The decomposition theorem [BBDG18] extends this to an inclusion
$\delta_{W}\subset\delta_{X}*\delta_{X}=\sigma_{*}(\delta_{X\times X})$ as a
direct summand in the derived category of constructible sheaf complexes. More
precisely, there exists a unique semisimple perverse sheaf
$\varepsilon_{W}\in\textup{Perv}(A,\mathbb{F})$ without negligible direct
summands, and a unique negligible complex
$\nu_{X}\in{\mathrm{D}^{b}_{c}}(A,\mathbb{F})$, such that

$\delta_{X}*\delta_{X}\;=\;\delta_{W}\oplus\varepsilon_{X}\oplus\nu_{X}.$

With this notation, we obtain the following criterion for big Tannaka groups:

######  Lemma 3.7.

For any nondegenerate subvariety $X\subset A$ with $2\dim X<\dim A$, the
following are equivalent:

  1. (1)

$G_{X,\omega}$ is big in the sense of section 1.2.

  2. (2)

$\varepsilon_{X}$ is either a simple perverse sheaf, or a direct sum of a
simple perverse sheaf and a skyscraper sheaf of rank one.

###### Proof.

Recall that $W\subset A$ is a proper nondegenerate subvariety, so it cannot be
the support of a negligible sheaf complex. On the other hand,
$\operatorname{Supp}(\varepsilon_{X}\oplus\nu_{X})=W$ since the morphism
$\sigma\colon X\times X\to W$ has generic degree two. It follows that
$\operatorname{Supp}(\varepsilon_{X})=W$. In particular, the representation
$V=\omega(\delta_{X})\in\textup{Rep}_{\mathbb{F}}(G_{X,\omega})$ must have
dimension $\dim V>2$, since otherwise $\varepsilon_{X}$ would be the
skyscraper sheaf corresponding to $\det(V)$ by proposition 3.3 (3).

By applying the fiber functor $\omega$, one sees that the condition (2) is
equivalent to saying that in the decomposition of the tensor square $V\otimes
V$ there are only two irreducible direct summands of dimension $>1$. Since
$\dim(V)>2$, this is equivalent to (1) by Larsen’s alternative [Kat01, p. 113]
for the subgroup $G_{X,\omega}\subset\operatorname{GL}(V)$. ∎

######  Corollary 3.8.

Let $X\subset A$ be nondegenerate with $2\dim X<\dim A$. If $G_{X,\omega}$ is
big, then the sum morphism $Y=\operatorname{Sym}^{2}(X)\to W=X+X$ is
birational.

###### Proof.

Consider the following commutative diagram, where $q$ denotes the quotient
morphism:

$Z=X\times
X$$W=X+X$$Y=\operatorname{Sym}^{2}X$$\scriptstyle\sigma$$\scriptstyle q$
$\scriptstyle\tau$

Since $q\colon Z\to Y$ is a finite branched cover of degree $\deg(q)=2$, the
decomposition theorem shows that

$q_{*}(\delta_{Z})\;\simeq\;\delta_{Y}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-}}\oplus\delta_{Y}^{-}$

where $\delta_{Y}^{-}\in\textup{Perv}(Y,\mathbb{F})$ is a semisimple perverse
sheaf of generic rank one. It then follows that

$\delta_{X}*\delta_{X}\;\simeq\;R\sigma_{*}(\delta_{Z})\;\simeq\;R\tau_{*}(\delta_{Y}^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-}})\oplus
R\tau_{*}(\delta_{Y}^{-}).$

Since $\tau$ is generically finite, the decomposition theorem shows
$R\tau_{*}(\delta_{Y})\simeq\delta_{W}\oplus K$ where the complex
$K\in{\mathrm{D}^{b}_{c}}(W,\mathbb{F})$ is a direct sum of shifts of
semisimple perverse sheaves. Over any smooth open dense subset $U\subset W$
over which $\sigma$ is a finite étale cover, we have

$R\tau_{*}(\delta_{Y}^{-})^{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}-}}_{|U}\;\simeq\;M[d]\quad\textnormal{and}\quad
K_{|U}\;\simeq\;N[d]\quad\textnormal{for}\quad d\;=\;\dim X,$

where $M$, $N$ are local systems of rank $\operatorname{rk}(M)=\deg(\tau)$ and
$\operatorname{rk}(N)=\deg(\tau)-1$. So in the decomposition

$\delta_{X}*\delta_{X}\;\simeq\;\delta_{W}\oplus
R\tau_{*}(\delta_{Y}^{-})\oplus K$

all three summands have support $W$ unless $\deg(\tau)=1$ (in which case
$N=0$). Our assumptions imply that $W$ is of general type, hence a sheaf
complex with support $W$ is not negligible. But if the Tannaka group
$G_{X,\omega}$ is big, then by lemma 3.7 we can have only two summands in
$\delta_{X}*\delta_{X}$ that are not negligible and not skyscraper sheaves.
Hence, in this case it follows that $\deg(\tau)=1$ as required. ∎

In the above we have used the full decomposition of the tensor square of the
defining representation. However, it suffices in fact to consider only the
alternating or the symmetric square:

######  Corollary 3.9.

Let $X\subset A$ be nondegenerate with $2\dim X<\dim A$, and consider the
representation

$V\;:=\;\begin{cases}\operatorname{Alt}^{2}(\omega(\delta_{X}))&\text{if
$2\nmid\dim X$},\\\ \operatorname{Sym}^{2}(\omega(\delta_{X}))&\text{if
$2\mid\dim X$}.\end{cases}$

If $V\in\textup{Rep}_{\mathbb{F}}(G_{X,\omega}^{*})$ has at most one
irreducible direct summand of dimension $>1$, then the sum morphism
$\tau\colon\operatorname{Sym}^{2}X\to X+X$ is birational.

###### Proof.

Let $Y=\operatorname{Sym}^{2}X$ and $W=X+X$. We know
$V=\omega(R\tau_{*}(\delta_{Y}))$. Recall that by [Wei15b] or [Krä22, section
3.c] all one-dimensional representations of the Tannaka group arise from
skyscraper sheaves. Hence our assumption on the representation $V$ means that
$R\tau_{*}(\delta_{Y})\in{\mathrm{D}^{b}_{c}}(W,\mathbb{F})$ has at most one
non-negligible direct summand $P$ which is not a skyscraper sheaf. But from
the proof of the previous corollary we know that
$R\tau_{*}(\delta_{Y})=\delta_{W}\oplus K$ where $K$ is a direct sum of a
semisimple perverse sheaf and a negligible complex. As in that proof it
follows that $K_{|U}\simeq 0$ over some open dense subset $U\subset W$ and
hence that $\deg(\tau)=1$. ∎

######  Corollary 3.10.

Let $X\subset A$ be a smooth irreducible curve generating $A$, and assume
$\dim A\geqslant 3$. If the representation
$V=\operatorname{Alt}^{2}(\omega(\delta_{X}))\in\textup{Rep}_{\mathbb{F}}(G_{X,\omega}^{*})$
is a sum of an irreducible representation and a one-dimensional trivial
representation, then

  1. (1)

$X=p-X$ for some point $p\in X$,

  2. (2)

$\tau\colon Y=\operatorname{Sym}^{2}X\to W=X+X$ is finite birational over
$U=W\setminus\\{p\\}$,

  3. (3)

$G_{X,\omega}^{*}=\operatorname{Sp}(\omega(\delta_{X}),\theta)$ for the
natural symplectic form $\theta$ on $\omega(\delta_{X})$.

###### Proof.

By assumption $\operatorname{Alt}^{2}(\omega(\delta_{X}))$ contains a one-
dimensional trivial representation, so the representation $\omega(\delta_{X})$
is isomorphic to its dual. Therefore $X=p-X$ for some point $p\in X$. Now for
dimension reasons $\tau\colon Y\to W$ restricts to a finite morphism over the
complement $U=W\setminus\Sigma$ of a finite set $\Sigma\subset X$ of points.
Note that $Y=\operatorname{Sym}^{2}X$ is smooth for a smooth curve $X$, so we
have $\delta_{Y}=\mathbb{F}_{Y}[2]$. Base change then shows that for any point
$q$ we have

$\mathcal{H}^{0}(R\tau_{*}(\delta_{Y}))_{q}\;\simeq\;H^{2}(\tau^{-1}(q),\mathbb{F})\;\begin{cases}\;=\;0&\text{if
$q\notin\Sigma$},\\\ \;\neq\;0&\text{if $q\in\Sigma$}.\end{cases}$

Since $R\tau_{*}(\delta_{Y})$ is a direct sum of a semisimple perverse sheaf
$P$ and a negligible sheaf complex and since negligible sheaf complexes cannot
have cohomology sheaves which are skyscraper sheaves, it follows that $P$
contains the skyscraper sheaves $\delta_{q}$ in all points $q\in\Sigma$. But
by assumption $R\tau_{*}(\delta_{Y})$ contains a unique skyscraper summand,
hence it follows that $\Sigma=\\{p\\}$ and thus $\tau$ is finite over
$U=X\setminus\\{p\\}$.

In particular $R\tau_{*}(\delta_{Y}^{-})_{|U}$ is a perverse sheaf, and we
have $\mathcal{H}^{i}(R\tau_{*}(\delta_{Y}^{-}))|_{|U}=0$ in all degrees
$i\neq-2$ because $\delta_{Y}^{-}$ is a constructible sheaf placed in degree
$-2$. But any semisimple perverse sheaf on a surface with cohomology sheaves
only in degrees $-2$ is the minimal extension of a local system on any open
dense subset of the surface. In our case that local system has rank one
because $\delta_{Y}^{-}$ has generic rank one and $\deg(\tau)=1$. Local
systems of rank one are simple, hence it follows that the minimal extension
$R\tau_{*}(\delta_{Y}^{-})$ is a simple perverse sheaf.

In conclusion, this shows that
$\delta_{X}*\delta_{X}=R\tau_{*}(\delta_{Y})\oplus R\tau_{*}(\delta_{Y}^{-})$
is a sum of two simple perverse sheaves and a skyscraper sheaf. It then
follows by the same argument as in [KW15b, th. 6.1] that
$G_{X,\omega}^{*}=\operatorname{Sp}(\omega(\delta_{X}),\theta)$; note that
$\dim(\omega(\delta_{X}))=\chi(\delta_{X})\geqslant g>2$ since the curve $X$
generates $A$. ∎

######  Corollary 3.11.

Let $X\subset A$ be a smooth irreducible curve generating $A$, and assume
$\dim A\geqslant 3$. Then the group $G_{X,\omega}^{\ast}$ is not isomorphic to
$E_{7}$ acting on $\omega(\delta_{X})$ via its irreducible representation of
dimension $56$.

###### Proof.

For the $56$-dimensional irreducible representation $W$ of the group $E_{7}$
the alternating square $\operatorname{Alt}^{2}(W)$ is a sum of an irreducible
and a one-dimensional trivial representation. However, corollary 3.10 says
that $\operatorname{Alt}^{2}(\omega(\delta_{X}))$ can be a sum of an
irreducible and a one-dimensional trivial representation only if
$G_{X,\omega}^{\ast}\simeq\operatorname{Sp}_{56}(\mathbb{F})$. ∎

###  3.5. Character twists

Recall that
$\Pi(A,\mathbb{F})=\operatorname{Hom}(\pi_{1}(A,0),\mathbb{F}^{\times})$
denotes the group of continuous characters of the étale resp. topological
fundamental group of the abelian variety. For $\chi\in\Pi(A,\mathbb{F})$, let
$L_{\chi}$ be the local system of rank one with monodromy representation given
by the character $\chi$. For $P\in\textup{Perv}(A,\mathbb{F})$ we call
$P_{\chi}:=P\otimes_{\mathbb{F}}L_{\chi}\in\textup{Perv}(A,\mathbb{F})$ the
twist of the given perverse sheaf by the character. Such twists of perverse
sheaves appear in the generic vanishing theorem of [KW15c, Sch15, BSS18]: Let
us say that a subset of $\Pi(A,\mathbb{F})$ is a proper subtorus if it has the
form

$\Pi(A/B,\mathbb{F})\;\subset\;\Pi(A,\mathbb{F})$

where $B\subset A$ is a nonzero abelian subvariety. Then the generic vanishing
theorem says that there is a finite union
$\mathcal{S}(P)\subset\Pi(A,\mathbb{F})$ of translates of proper subtori such
that

$H^{i}(A,P_{\chi})\;=\;0\quad\textnormal{for all $i\neq 0$ and all
$\chi\in\Pi(A,\mathbb{F})\smallsetminus\mathcal{S}(P)$}.$

We will use this in section 4.3 to write down explicit fiber functors with a
natural Galois action. Up to noncanonical isomorphism, the Tannaka group of a
perverse sheaf does not change under twists:

######  Lemma 3.12.

Let $P\in\mathcal{C}$. Then for every character $\chi\in\Pi(A,\mathbb{F})$
with $P_{\chi}\in\mathcal{C}$ we have

$G_{\omega}(P_{\chi})\;\simeq\;G_{\omega}(P).$

###### Proof.

By [KW15c, prop. 4.1], twisting by $\chi$ gives rise to an equivalence of
tensor categories

$\langle P\rangle\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\langle
P_{\chi}\rangle,\quad Q\;\longmapsto\;Q_{\chi}$

in $\overline{\textup{Perv}}(A,\mathbb{F})$. This equivalence need not be
compatible with the fiber functor $\omega$ on the source and target, but since
$\mathbb{F}$ is algebraically closed, any two fiber functors are
noncanonically isomorphic; hence the same holds for the Tannaka groups. ∎

##  4\. Galois theory for perverse sheaves

In this section we discuss the behavior of Tannaka groups of perverse sheaves
under extension of the base field and recall the connection between such
Tannaka groups and classical monodromy groups in [LS20, section 5]. We mostly
follow the arguments in loc. cit. but remove the assumption of geometric
semisimplicity in the Galois exact sequence by using a result of D’Addezio and
Esnault [DE21].

###  4.1. Extension of the base field

Let $K/k$ be a field extension, and consider the base change functor

$(-)_{K}\colon\quad\textup{Perv}(A,\mathbb{F})\;\longrightarrow\;\textup{Perv}(A_{K},\mathbb{F}),\quad
P\;\longmapsto\;P_{K}.$

Passing to the abelian quotient categories by the subcategories of perverse
sheaves of Euler characteristic zero, we have:

######  Lemma 4.1.

The base change functor descends to a faithful exact $\mathbb{F}$-linear
tensor functor

$(-)_{K}\colon\quad\overline{\textup{Perv}}(A,\mathbb{F})\;\longrightarrow\;\overline{\textup{Perv}}(A_{K},\mathbb{F}).$

###### Proof.

The functor
$(-)_{K}\colon\textup{Perv}(A,\mathbb{F})\to\textup{Perv}(A_{K},\mathbb{F})$
is a faithful $\mathbb{F}$-linear exact functor. Let $q_{K}=q\circ(-)_{K}$
denote its composite with the quotient functor $q$ as shown below:

$\textup{Perv}(A,\mathbb{F})$$\textup{Perv}(A_{K},\mathbb{F})$$\overline{\textup{Perv}}(A_{K},\mathbb{F})$$\scriptstyle(-)_{K}$$\scriptstyle
q_{K}$ $\scriptstyle q$

Since $q_{K}$ is an exact functor between abelian categories which sends all
objects of the Serre subcategory
$\textup{S}(A,\mathbb{F})\subset\textup{Perv}(A,\mathbb{F})$ to zero, it
factors by the universal property of abelian quotient categories [Gab62, cor.
2, p. 368] through a unique exact functor

$(-)_{K}\colon\quad\overline{\textup{Perv}}(A,\mathbb{F})\longrightarrow\overline{\textup{Perv}}(A_{K},\mathbb{F}).$

This functor is clearly $\mathbb{F}$-linear, and it admits the structure of a
tensor functor with respect to the natural isomorphisms $(P*Q)_{K}\simeq
P_{K}*Q_{K}$ inherited from the derived category. Any exact
$\mathbb{F}$-linear tensor functor of rigid abelian tensor categories with
$\operatorname{End}(\mathbf{1})=\mathbb{F}$ is automatically faithful [DMOS82,
prop. 1.19], so the claim follows. ∎

Starting from a given full abelian tensor subcategory
$\mathcal{C}\subset\overline{\textup{Perv}}(A,\mathbb{F})$, let us now denote
by

$\mathcal{C}_{K}\;=\;\\{Q\mid\textnormal{$\exists P\in\mathcal{C}$ such that
$Q$ is a subquotient of
$P_{K}$}\\}\;\subset\;\overline{\textup{Perv}}(A_{K},\mathbb{F})$

the full abelian tensor subcategory generated by the essential image of
$\mathcal{C}$ under the functor $(-)_{K}$ from lemma 4.1. The category
$\mathcal{C}_{K}$ is again neutral Tannaka as it is a full abelian tensor
subcategory of the neutral Tannaka category
$\overline{\textup{Perv}}(A_{K},\mathbb{F})$. In what follows, we fix a fiber
functor

$\omega\colon\quad\mathcal{C}_{K}\;\longrightarrow\;\operatorname{Vect}(\mathbb{F}).$

Precomposing with the base extension functor $(-)_{K}$ we get a fiber functor
on $\mathcal{C}$ and we denote by

$\displaystyle G_{\omega}(\mathcal{C}_{K})$ $\displaystyle\ =$
$\displaystyle\operatorname{Aut}^{\otimes}(\omega\mid\mathcal{C}_{K}),$
$\displaystyle G_{\omega}(\mathcal{C})$ $\displaystyle:=$
$\displaystyle\operatorname{Aut}^{\otimes}(\omega\mid\mathcal{C}),$

the corresponding Tannaka groups.

######  Corollary 4.2.

We have a closed immersion $G_{\omega}(\mathcal{C}_{K})\hookrightarrow
G_{\omega}(\mathcal{C})$.

###### Proof.

By construction the faithful exact $\mathbb{F}$-linear tensor functor
$(-)_{K}\colon\mathcal{C}\to\mathcal{C}_{K}$ is compatible with our chosen
fiber functors, hence it defines a homomorphism of Tannaka groups. The latter
is a closed immersion by [DMOS82, prop. 2.21(b)], since every object of
$\mathcal{C}_{K}$ is isomorphic to a subquotient of $P_{K}$ for some
$P\in\mathcal{C}$. ∎

###  4.2. The Galois sequence

Let $k^{\prime}\subset K$ be the algebraic closure of $k$ in $K$. The category

$\textup{Rep}_{\mathbb{F}}(\operatorname{Aut}(k^{\prime}/k))$

of continuous finite-dimensional representations of the profinite group
$\operatorname{Aut}(k^{\prime}/k)$ over $\mathbb{F}$ is a neutral Tannaka
category. If $k^{\prime}/k$ is Galois, then
$\operatorname{Aut}(k^{\prime}/k)=\operatorname{Gal}(k^{\prime}/k)$ is a
quotient of the absolute Galois group of $k$. In this case we can identify
objects of the above category with sheaves on $\operatorname{Spec}(k)$ and
hence the pushforward under the neutral element
$e\colon\operatorname{Spec}(k)\to A$ gives a fully faithful embedding

${e_{*}\colon}$${\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(k^{\prime}/k))}$${\overline{\textup{Perv}}(A,\mathbb{F}).}$

We will view Galois representations as a full subcategory of skyscraper
sheaves and drop the $e_{*}$ from the notation. Our chosen fiber functor on
$\mathcal{C}$ restricts to a fiber functor

$\omega\colon\quad\mathcal{C}\cap\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(k^{\prime}/k))\;\longrightarrow\;\operatorname{Vect}(\mathbb{F}).$

Let

$G_{\omega,\mathcal{C}}(k^{\prime}/k)\;:=\;\operatorname{Aut}^{\otimes}(\omega\,|\,\mathcal{C}\cap\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(k^{\prime}/k)))$

denote its Tannaka group. Representations of this group correspond to
skyscraper sheaves $P\in\mathcal{C}$ in the origin, and we have a homomorphism
$\operatorname{Aut}(k^{\prime}/k)\to G_{\omega,\mathcal{C}}(k^{\prime}/k)$.

######  Theorem 4.3.

Assume as above that $k^{\prime}/k$ is Galois. Then we have a short exact
sequence of proalgebraic groups

(4.1) $1\longrightarrow G_{\omega}(\mathcal{C}_{K})\longrightarrow
G_{\omega}(\mathcal{C})\longrightarrow
G_{\omega,\mathcal{C}}(k^{\prime}/k)\longrightarrow 1.$

###### Proof.

Corollary 4.2 gives a closed immersion $i\colon G_{\omega}(\mathcal{C}_{K})\to
G_{\omega}(\mathcal{C})$. Moreover, since $k^{\prime}/k$ is a Galois
extension, we have by the above an embedding as a full tensor subcategory

${\mathcal{C}\cap\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(k^{\prime}/k))}$${\mathcal{C}.}$

which is stable under subobjects, and this embedding is compatible with the
chosen fiber functors on the source and target. By [DMOS82, prop. 2.21(a)] we
then have an epimorphism

${p\colon}$${G_{\omega}(\mathcal{C})}$${G_{\omega,\mathcal{C}}(k^{\prime}/k).}$

By construction, $p\circ i$ is trivial. Thus, to complete the proof, by [DE21,
prop. A.13], it suffices to check that

  1. (1)

the functor $(-)_{K}\colon\mathcal{C}\to\mathcal{C}_{K}$ is observable [DE21,
Appendix A], and

  2. (2)

for every $P\in\mathcal{C}$ the maximal trivial subobject of $P_{K}$ lies in
the essential image of the functor
$e_{*}\colon\mathcal{C}\cap\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(k^{\prime}/k))\to\mathcal{C}$.

For part (1) it suffices by lemma A.4(1) in loc. cit. to show that, for
$P\in\mathcal{C}$, any rank one subobject

$S\;\subset\;P_{K}$

is a direct summand in a semisimple object $Q_{K}$ with $Q\in\mathcal{C}$. To
check this, note that the rank one objects in the Tannaka category of perverse
sheaves are rank one skyscraper sheaves, and that the sum of all perverse rank
one skyscraper subsheaves of $P_{K}$ is semisimple, being a sum of simple
objects. To conclude the proof of (1), it suffices to show that this direct
sum descends to a perverse subsheaf $Q\subset P$, as it then follows that $S$
is a direct summand of $Q_{K}$ as desired. To prove that the sum of all rank
one skyscraper subsheaves descends to $k$, we first show that the maximal
skyscraper subsheaf of $P_{K}$ descends to a subsheaf of $P$. Indeed, the
Verdier dual of the sum of all perverse skyscraper subsheaves is the maximal
perverse skyscraper quotient of the Verdier dual $D(P_{K})$, which is
$\mathcal{H}^{0}(D(P_{K}))=\mathcal{H}^{0}(D(P))_{K}$. Hence, the maximal
skyscraper subsheaf descends. Replacing the given perverse sheaf $P$ by the
maximal skyscraper subsheaf supported at the origin, we are reduced to the
case $A=\operatorname{Spec}k$. Then $P$ is given by a Galois representation
$V\in\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(\bar{k}/k))$ and the claim
reduces to the following to facts:

  * •

A subspace of $V$ is stable under $\operatorname{Gal}(\bar{K}/K)$ if and only
if it is so under $\operatorname{Gal}(\bar{k}/k^{\prime})$ (since
$\operatorname{Gal}(\bar{K}/K)\to\operatorname{Gal}(\bar{k}/k^{\prime})$ is
surjective for $k^{\prime}$ algebraically closed in $K$).

  * •

The sum of all one-dimensional subrepresentations of
$V_{|\operatorname{Gal}(\bar{k}/k^{\prime})}$ is stable under
$\operatorname{Gal}(\bar{k}/k)$ (since
$\operatorname{Gal}(\bar{k}/k^{\prime})$ is a normal subgroup of
$\operatorname{Gal}(\bar{k}/k)$).

For (2) we argue similarly: The unit object of the tensor category
$\mathcal{C}_{K}$ is the skyscraper sheaf $\delta_{0}$ of rank one supported
in the origin. So the maximal trivial subobject of $P_{K}$ is the maximal
subobject of the form $\delta_{0}^{\oplus n}$ for some integer $n\geqslant 0$,
and this subobject descends to a subobject $Q\subset P$ as before. ∎

######  Corollary 4.4.

If $k$ is algebraically closed, then for every extension $K/k$ we have a
natural isomorphism

$G_{\omega}(\mathcal{C}_{K})\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;G_{\omega}(\mathcal{C}).$

In particular, for every perverse sheaf $P\in\mathcal{C}$, we have
$G_{\omega}(P_{K})\simeq G_{\omega}(P)$.

###### Proof.

If $k$ is algebraically closed, then $k^{\prime}=k$ and hence
$G_{\omega}(k^{\prime}/k)\simeq\\{1\\}$. ∎

###  4.3. A splitting of the sequence

We now apply the above when $K=\bar{k}$ is an algebraic closure of $k$. In the
Galois sequence in theorem 4.3 we have used the fully faithful functor

${e_{*}\colon}$${\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(\bar{k}/k))}$${\overline{\textup{Perv}}(A,\mathbb{F}).}$

that identifies a Galois representation with the corresponding skyscraper
sheaf at the origin. We now describe a splitting of the sequence in theorem
4.3 for a special category $\mathcal{C}$ such that the functor
$e_{*}\colon\mathcal{C}\cap\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(\bar{k}/k))\hookrightarrow\mathcal{C}$
has a left inverse. More precisely, let

$\mathcal{C}\;=\;\textup{Perv}_{0}(A,\mathbb{F})$

be the full subcategory of all $P\in\textup{Perv}(A,\mathbb{F})$ for which all
simple subquotients $Q$ of $P_{\bar{k}}$ satisfy

$H^{i}(A_{\bar{k}},Q)\;=\;0\quad\textnormal{for all}\quad i\;\neq\;0.$

Its image

$\overline{\textup{Perv}}_{0}(A,\mathbb{F})\;\subset\;\overline{\textup{Perv}}(A,\mathbb{F})$

is a full abelian tensor subcategory which is equivalent to
$\textup{Perv}_{0}(A,\mathbb{F})/S_{0}(A,\mathbb{F})$, where
$S_{0}(A,\mathbb{F}):=S(A,\mathbb{F})\cap\textup{Perv}_{0}(A,\mathbb{F})$ is
the full subcategory of perverse sheaves $P$ with the property that all the
subquotients $Q$ of $P_{\bar{k}}$ satisfy $H^{\bullet}(A_{\bar{k}},Q)=0$. We
then get a functor

$\omega\colon\quad\overline{\textup{Perv}}_{0}(A,\mathbb{F})\;=\;\textup{Perv}_{0}(A,\mathbb{F})/S_{0}(A,\mathbb{F})\;\longrightarrow\;\operatorname{Vect}(\mathbb{F}),\quad
Q\;\longmapsto\;H^{0}(A_{\bar{k}},Q)$

which is exact by definition of the source category. Moreover, $\omega$ is a
tensor functor by the Künneth isomorphism

$H^{\bullet}(A_{\bar{k}},P*Q)\;\simeq\;H^{\bullet}(A_{\bar{k}},P)\otimes
H^{\bullet}(A_{\bar{k}},Q),$

since for $P,Q\in\textup{Perv}_{0}(A,\mathbb{F})$ only the cohomology in
degree zero contributes. For the fiber functor obtained in this way, we
obtain:

######  Lemma 4.5.

For $\mathcal{C}=\overline{\textup{Perv}}_{0}(A,\mathbb{F})$ with the fiber
functor $\omega:=H^{0}(A_{\bar{k}},-)$ we have a splitting

$G_{\omega}(\mathcal{C})\;\simeq\;G_{\omega}(\mathcal{C}_{\bar{k}})\rtimes
G_{\omega,\mathcal{C}}(\bar{k}/k).$

So for any $P\in\overline{\textup{Perv}}_{0}(A,\mathbb{F})$, the action of
$\operatorname{Gal}(\bar{k}/k)$ on $V=\omega(P)$ factors through the
normalizer

$N(G_{\omega}(P_{\bar{k}}))\;\subset\;\operatorname{GL}(V).$

###### Proof.

While the fiber functor $\omega=H^{0}(A_{\bar{k}},-)$ is only defined on
$\mathcal{C}:=\overline{\textup{Perv}}_{0}(A,\mathbb{F})$, it comes with a
natural Galois action in the sense that we have a commutative diagram

$\mathcal{C}$$\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(\bar{k}/k))$$\operatorname{Vect}(\mathbb{F})$$\scriptstyle\exists$$\scriptstyle\omega$

where the top row is a left inverse of the functor
$e_{*}\colon\textup{Rep}_{\mathbb{F}}(\operatorname{Gal}(\bar{k}/k))\to\mathcal{C}$.
∎

###  4.4. Big monodromy from big Tannaka groups

Now again assume that $k$ is an algebraically closed field of characteristic
zero. Consider the constant abelian scheme $A_{S}:=A\times_{k}S$, where $S$ is
an integral scheme over $k$. We denote by $\bar{\eta}$ a geometric point over
the generic point $\eta$ of $S$. Let $\mathcal{X}\subset A_{S}$ be an
irreducible closed subscheme which is smooth over $S$. We want to control the
monodromy of the family $\mathcal{X}\to S$ twisted by a generic rank one local
system as in [LS20]. In this context, the following terminology will be
useful.

######  Definition 4.6.

We say that $\mathcal{X}\subset A_{S}$ is _constant up to translation in
$A(S)$_ if there is a subvariety $Y\subset A$ and a point $a\in A(S)$ such
that $\mathcal{X}=Y_{S}+a$.

In favorable situations, this condition can be read off from the geometric
generic fiber of $\mathcal{X}\to S$ via the following descent result:

######  Lemma 4.7.

Suppose $S$ is a smooth and irreducible variety. Let
$\mathcal{Y},\mathcal{Z}\subset A_{S}$ be subvarieties which are flat over
$S$. If the subvariety $\mathcal{Y}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ has
trivial stabilizer, then the following are equivalent:

  1. (1)

$\mathcal{Z}=\mathcal{Y}+a$ for some $a\in A(S)$.

  2. (2)

$\mathcal{Z}_{\bar{\eta}}=\mathcal{Y}_{\bar{\eta}}+a$ for some $a\in
A(\bar{\eta})$.

###### Proof.

Clearly, the first property implies the second. Conversely, suppose that we
have $\mathcal{Z}_{\bar{\eta}}=\mathcal{Y}_{\bar{\eta}}+a$ for some point
$a\in A(\bar{\eta})$. First, we claim that the point $a$ comes from a point
$a\in A(\eta)$. Indeed, let $F$ be the function field of $S$ and let
$y=[\mathcal{Y}_{\eta}]$ and $z=[\mathcal{Z}_{\eta}]$ be the $F$-points of the
Hilbert scheme $\operatorname{Hilb}(A)$ defined by the generic fibers of
$\mathcal{Y}\to S$ and $\mathcal{Z}\to S$, seen as subvarieties of
$A_{S,\eta}$. Now, the abelian variety $A$ acts on the Hilbert scheme by
translation. The transporter

$T=\\{t\in A_{S,\eta}\mid z=y+t\\}$

is a subvariety of $A_{S,\eta}$. Note that $T(\bar{\eta})$ is nonempty, as it
contains the point $a$. Actually, the point $a$ is the only one of
$T(\bar{\eta})$. For, note that the stabilizer of the subvariety
$\mathcal{Y}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ acts freely and
transitively on the base-change of $T$ to $\bar{\eta}$. On the other hand, the
stabilizer of $\mathcal{Y}_{\bar{\eta}}$ is trivial by assumption, so the
transporter $T(\bar{\eta})$ must be a singleton. The variety $T$ is defined
over $F$ and has only one point over an algebraically closed field, thus
$T=\operatorname{Spec}F$ which proves the claim.

The point $a\in A(\eta)$ can be seen as a rational map $a\colon
S\dashrightarrow A$, which is moreover everywhere defined by smoothness of $S$
[Mil86, th. 3.1]. To conclude the proof, note that the generic fibers of
$\mathcal{Y}+a$ and $\mathcal{Z}$ coincide, hence $\mathcal{Z}=\mathcal{Y}+a$
by flatness. ∎

######  Corollary 4.8.

If $S$ is a smooth irreducible variety and if the subvariety
$\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ is nondivisible, then the
following are equivalent:

  1. (1)

$\mathcal{X}\subset A_{S}$ is constant (resp. symmetric) up to translation in
$A(S)$.

  2. (2)

$\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ is constant (resp.
symmetric) up to translation in $A(\bar{\eta})$.

Moreover, the subvariety $\mathcal{X}\subset A_{S}$ is constant up to
translation in $A(S)$ if and only if the family $\mathcal{X}\to S$ is
isotrivial.

###### Proof.

The equivalence of (1) and (2) follows directly from lemma 4.7. Now suppose
that the family $\mathcal{X}\to S$ is isotrivial. In order to prove that the
subvariety $\mathcal{X}\subset A_{S}$ is constant up to translation, we may by
the equivalence of (1) and (2) replace $S$ by an étale cover and hence assume
$\mathcal{X}\simeq Y_{S}$ for some $Y\subset A$. Fixing $y\in Y(k)$, we get a
section $x\colon S\to\mathcal{X}$ that gives rise to a commutative diagram:

${\mathcal{X}}$${\operatorname{Alb}(\mathcal{X}/S)}$${A_{S}}$${A_{S}}$${Y_{S}}$${\operatorname{Alb}(Y_{S}/S)}$${A_{S}}$${A_{S}}$$\scriptstyle{\operatorname{alb}_{x}}$

$\scriptstyle\sim$

$\scriptstyle\sim$

$\scriptstyle{z\mapsto
z+x}$$\scriptstyle{\operatorname{alb}_{y}}$$\scriptstyle{z\mapsto z+y}$

Here $\operatorname{alb}_{a}$ and $\operatorname{alb}_{y}$ are the relative
Albanese morphisms and the composite of the horizontal arrows are the
inclusions $\mathcal{X}\subset A_{S}$ resp. $Y_{S}\subset A_{S}$. Hence,
$\mathcal{X}\subset A_{S}$ is constant up to translation. ∎

######  Example 4.9.

The primitivity is needed in the above: Let $Y\subset A$ be a subvariety with
finite stabilizer $\operatorname{Stab}(Y)\neq\\{0\\}$. Viewing
$S:=A/\operatorname{Stab}(Y)$ as the orbit of the point $[Y]$ in
$\operatorname{Hilb}(A)$ under the translation action of $A$, we get by
restriction of the universal subvariety of $A\times_{k}\operatorname{Hilb}(A)$
a subvariety $\mathcal{X}\subset A_{S}$ with fiber $Y+a$ over a point $[a]\in
S(k)$. Then the family $\mathcal{X}\to S$ is not constant up to translation in
$A(S)$, but it is so up to translation by a section in $A(\bar{\eta})$.

We now assume that $\mathcal{X}\subset A_{S}$ is not constant up to
translation in $A(S)$. Then the monodromy of the smooth family $\mathcal{X}\to
S$ twisted by a generic rank one local system is related to the Tannaka group
of the perverse sheaf
$\delta_{X}\in\textup{Perv}(A_{S,\bar{\eta}},\mathbb{F})$ on the geometric
generic fiber

$X\;:=\;\mathcal{X}_{\bar{\eta}}$

as follows. For $\chi\in\Pi(A,\mathbb{F})$, let $L_{\chi}$ denote the
corresponding rank one local system on $A$. The generic vanishing theorem for
perverse sheaves [BSS18, KW15c, Sch15] shows that

$\delta_{X,\chi}\;:=\;\delta_{X}\otimes
L_{\chi}\;\in\;\overline{\textup{Perv}}_{0}(A_{S,\bar{\eta}},\mathbb{F})$

for most $\chi\in\Pi(A,\mathbb{F})$, where most means all characters $\chi$
outside a finite union of torsion translates of linear subvarieties of
$\Pi(A,\mathbb{F})$. From section 4.3 we get a fiber functor

$\omega\;:=\;H^{0}(A_{S,\bar{\eta}},-)\colon\quad\langle\delta_{X,\chi}\rangle\;\longrightarrow\;\operatorname{Vect}(\mathbb{F}),$

and we denote by

$G^{*}_{X,\chi}\;:=\;[G_{\omega}^{\circ}(\delta_{X,\chi}),G_{\omega}^{\circ}(\delta_{X,\chi})]$

the derived group of the connected component of the Tannaka group. Note that
by lemma 3.12 the isomorphism type of this group does not depend on the chosen
character; we say that $X$ has a simple derived connected Tannaka group if
$G_{X,\chi}^{\ast}$ is simple for some (hence every) character $\chi$ with the
above vanishing properties.

To define the monodromy of the family $f\colon\mathcal{X}\to S$ twisted by a
rank one local system, let $\pi\colon\mathcal{X}\to A$ be the projection to
the abelian variety. Using generic vanishing on the geometric generic fiber of
$X\subset A_{S,\bar{\eta}}$ and proceeding by Noetherian induction, one sees
that for most $\chi$ the higher direct images $R^{i}f_{*}\pi^{*}L_{\chi}$
vanish in all degrees $i\neq d$, where $d$ denotes the relative dimension of
the family $f\colon\mathcal{X}\to S$. For such $\chi$ the remaining direct
image

$V_{\chi}\;:=\;R^{d}f_{*}\pi^{*}L_{\chi}$

is a local system. More generally we consider for
$\underline{\chi}=(\chi_{1},\dots,\chi_{n})\in\Pi(A,\mathbb{F})^{n}$ the
direct sum

$V_{\underline{\chi}}\;:=\;V_{\chi_{1}}\oplus\cdots\oplus V_{\chi_{n}}.$

Let
$\rho\colon\pi_{1}(S,\bar{\eta})\rightarrow\operatorname{GL}(V_{\underline{\chi},\bar{\eta}})$
be the corresponding monodromy representation on the geometric generic fiber.
We define the _algebraic monodromy group_ of $V_{\underline{\chi}}$ as the
Zariski closure

$M(V_{\underline{\chi}})\;:=\;\overline{\operatorname{Im}(\rho)}\;\subset\;\operatorname{GL}(V_{\underline{\chi},\bar{\eta}}).$

The link between our main theorem from the introduction and the Tannaka groups
introduced above is the following result by Lawrence and Sawin, an analog of
the theorem of the fixed part:

######  Theorem 4.10.

Let $S$ be a smooth integral variety over $k$, and let $\mathcal{X}\subset
A_{S}$ an integral subvariety such that

  1. (1)

the family $f\colon\mathcal{X}\to S$ is smooth of relative dimension $d$, it
is not constant up to translation in $A(S)$, and

  2. (2)

the geometric generic fiber $X=\mathcal{X}_{\bar{\eta}}\subset
A_{S,\bar{\eta}}$ is nondivisible and has a simple derived connected Tannaka
group.

Then for most $\underline{\chi}\in\Pi(A,\mathbb{F})^{n}$ we have

$G^{\ast}_{X,\chi_{1}}\times\cdots\times
G^{\ast}_{X,\chi_{n}}\;\unlhd\;M(V_{\underline{\chi}}).$

###### Proof.

In [LS20, th. 5.6] this is stated for hypersurfaces, but the proof works for
smooth subvarieties of any codimension. For convenience, we recall the main
ideas in our setup: The fiber

$V_{\underline{\chi},\bar{\eta}}\;=\;\bigoplus_{i=1}^{n}\textup{H}^{d}(X,L_{\chi_{i}})$

comes with a monodromy action of $\pi_{1}(S,{\bar{\eta}})$ preserving the
summands on the right-hand side; the algebraic monodromy is the Zariski
closure of the image of $\pi_{1}(S,{\bar{\eta}})$ inside

$\operatorname{GL}(V_{\chi_{1},\bar{\eta}})\times\cdots\times\operatorname{GL}(V_{\chi_{n},\bar{\eta}})\quad\textnormal{where}\quad
V_{\chi_{i},\bar{\eta}}\;=\;\textup{H}^{d}(X,L_{\chi_{i}}).$

Since $S$ is smooth, this algebraic monodromy is the Zariski closure of the
image of the absolute Galois group of the function field of $S$. By lemma 4.5
the Galois action normalizes the subgroups
$G_{X,\chi_{i}}^{\ast}\subset\operatorname{GL}(V_{\chi_{i}})$, in fact the
algebraic monodromy is a subgroup

$M(V_{\underline{\chi}})\;\subset\;G_{X_{0},\chi_{1}}\times\cdots\times
G_{X_{0},\chi_{n}}$

where $X_{0}:=\mathcal{X}_{\eta}$ denotes the generic fiber and
$G_{X_{0},\chi_{i}}:=G_{\omega}(\delta_{X_{0},\chi_{i}})$. We must show that
this upper bound on the algebraic monodromy is almost sharp in the sense that
for most $\underline{\chi}=(\chi_{1},\dots,\chi_{n})$, the algebraic monodromy
contains the normal subgroup

$G^{\ast}_{X,\chi_{1}}\times\cdots\times
G^{\ast}_{X,\chi_{n}}\;\unlhd\;G_{X_{0},\chi_{1}}\times\cdots\times
G_{X_{0},\chi_{n}}.$

In what follows, it will be convenient to identify all factors on the left-
hand side with a fixed simple algebraic group. For this, we fix a fiber
functor $\xi\colon\langle\delta_{X}\rangle\to\operatorname{Vect}(\mathbb{F})$
and pick an isomorphism between $\textup{H}^{0}(A_{\bar{\eta}},-)$ and the
fiber functor obtained as the composite

${\langle\delta_{X_{0},\chi_{i}}\rangle}$${\langle\delta_{X_{0}}\rangle}$${\langle\delta_{X}\rangle}$${\operatorname{Vect}(\mathbb{F}),}$$\scriptstyle{\sim}$$\scriptstyle{(-)_{\bar{\eta}}}$$\scriptstyle{\xi}$

where the isomorphism on the left is the inverse of $P\mapsto P_{\chi_{i}}$.
We get a commutative diagram

${G_{X,\chi_{1}}^{*}\times\cdots\times
G_{X,\chi_{n}}^{*}}$${(G^{*}_{X})^{n}}$${M(V_{\underline{\chi}})}$${G_{X_{0},\chi_{1}}\times\cdots\times
G_{X_{0},\chi_{n}}}$${(G_{X_{0}})^{n}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where
$G_{X}^{*}:=[G^{\circ}_{\xi}(\delta_{X}),G^{\circ}_{\xi}(\delta_{X})]\subset
G_{X_{0}}:=G_{\xi}(\delta_{X_{0}})$. Note that $G_{X_{0}}$ is contained in the
normalizer

$N(G_{X}^{*})\subset\operatorname{GL}(\xi(\delta_{X_{0}}))$

by lemma 4.5. Now we use the following general observation [LS20, lemma 5.4]:

######  Fact 4.11.

Let $G\subset\operatorname{GL}(V)$ be a simple algebraic group, and
$N(G)\subset\operatorname{GL}(V)$ its normalizer. Then for every integer
$n\geqslant 1$ there exists a finite list of irreducible representations

$W_{\alpha}\;=\;W_{\alpha,1}\boxtimes\cdots\boxtimes
W_{\alpha,n}\;\in\;\textup{Rep}_{\mathbb{F}}(N(G)^{n})\quad(\alpha\in\\{1,\dots,N\\})$

such that for any reductive subgroup $H\subset N(G)^{n}$ the following two
properties are equivalent:

  1. (1)

$G^{n}\subset H$.

  2. (2)

$H$ has no invariants on any of the representations $W_{\alpha}$.

In particular, the group $G^{n}$ has no invariants on any of the
representations $W_{\alpha}$.

We apply this to $V=\xi(\delta_{X_{0}})$, $G=G_{X}^{*}$ and
$H=M(V_{\underline{\chi}})$. Since $G_{X_{0}}\subset N(G_{X}^{*})$, each
$W_{\alpha,i}\in\textup{Rep}_{\mathbb{F}}(N(G))$ defines a representation of
the Tannaka group $G_{X_{0}}$ and hence a perverse sheaf

$P_{\alpha,i}\;\in\;\langle\delta_{X_{0}}\rangle.$

The representation obtained by pullback under the isomorphism
$G_{X_{0},\chi_{i}}\to G_{X_{0}}$ then corresponds to the perverse sheaf
$(P_{\alpha,i})_{\chi_{i}}\in\langle\delta_{X_{0},\chi_{i}}\rangle$. By
construction, we have an isomorphism

(4.2) $\displaystyle W_{\alpha}$ $\displaystyle\;=\;$ $\displaystyle
W_{\alpha,1}\boxtimes\cdots\boxtimes W_{\alpha,n}$ $\displaystyle\;\simeq\;$
$\displaystyle\textup{H}^{0}(A_{S,\bar{\eta}},(P_{\alpha,1})_{\chi_{1}})\boxtimes\cdots\boxtimes\textup{H}^{0}(A_{S,\bar{\eta}},(P_{\alpha,n})_{\chi_{n}})$

of representations of $N(G_{X,\chi_{1}}^{*})\times\cdots\times
N(G_{X,\chi_{n}}^{*})$. To keep track of how the Galois action on the right-
hand side depends on the chosen characters, it will be convenient to pass to
$A_{S,\bar{\eta}}^{n}=A_{S,\bar{\eta}}\times\cdots\times A_{S,\bar{\eta}}$ via
the Künneth isomorphism. Consider the perverse sheaf

$K_{0}\;:=\;e_{1*}\delta_{X_{0}}\oplus\cdots\oplus
e_{n*}\delta_{X_{0}}\;\in\;\overline{\textup{Perv}}_{0}(A_{S,\eta}^{n},\mathbb{F})$

where $e_{i}\colon A_{S,\eta}\hookrightarrow A_{S,\eta}^{n}$ denotes the
$i$-th coordinate inclusion. Let
$K\in\overline{\textup{Perv}}_{0}(A_{S,\bar{\eta}}^{n},\mathbb{F})$ be the
base change of the perverse sheaf $K_{0}$ to the geometric generic fiber. Note
that $G^{*}_{\zeta}(K)=(G_{X}^{*})^{n}$ for the fiber functor
$\zeta:=\xi\boxtimes\cdots\boxtimes\xi\colon\langle
K\rangle\to\operatorname{Vect}(\mathbb{F})$ and that we have

$Q_{\alpha}\;:=\;e_{1*}P_{\alpha,1}*\cdots*e_{n*}P_{\alpha,n}\;=\;P_{\alpha,1}\boxtimes\cdots\boxtimes
P_{\alpha,n}\;\in\;\langle K_{0}\rangle.$

Returning to character twists again, consider the local system
$L_{\underline{\chi}}:=L_{\chi_{1}}\boxtimes\cdots\boxtimes L_{\chi_{n}}$ and
put

$K_{0,\underline{\chi}}\;:=\;K_{0}\otimes L_{\underline{\chi}},\quad
K_{\underline{\chi}}\;:=\;K\otimes
L_{\underline{\chi}}\quad\textnormal{and}\quad
Q_{\alpha,\underline{\chi}}\;:=\;Q_{\alpha}\otimes L_{\underline{\chi}}.$

Then we have

$G_{\omega}^{*}(K_{\underline{\chi}})\;=\;G_{X,\chi_{1}}^{*}\times\cdots\times
G_{X,\chi_{n}}^{*}\quad\textnormal{and}\quad
Q_{\alpha,\underline{\chi}}\;\in\;\langle K_{0,\underline{\chi}}\rangle$

Combining (4.2) with the Künneth isomorphism we obtain a Galois equivariant
isomorphism

$W_{\alpha}\;\simeq\;\textup{H}^{0}(A^{n}_{S,\bar{\eta}},Q_{\alpha,\underline{\chi}}),$

where the Galois group acts on the left-hand side via
$\operatorname{Gal}(\bar{\eta}/\eta)\to M(V_{\underline{\chi}})$ and on the
right-hand side by the natural Galois action.

Now recall that by the last claim in 4.11, the group $(G_{X}^{*})^{n}$ has no
invariants on $W_{\alpha}$. But $(G_{X}^{*})^{n}=G_{\zeta}^{*}(K)$ is the
derived group of the connected component of the Tannaka group of the perverse
sheaf $K$, and $W_{\alpha}=\zeta(Q_{\alpha})$ is the representation defined by
$Q_{\alpha}\in\langle K_{0}\rangle$. Hence, we can apply [LS20, lemma 5.2]:
The vanishing of invariants of the derived connected Tannaka group on the
geometric generic fiber implies that $Q_{\alpha}$ has no perverse subquotient
coming by pullback from $A$ via $A_{S,\eta}\to A$. By a spreading out argument
[LS20, lemma 5.3] the last property implies that for most $\underline{\chi}$
the Galois invariants of
$\textup{H}^{0}(A_{S,\bar{\eta}},Q_{\alpha,\underline{\chi}})$ vanish. Thus,
the algebraic monodromy has no invariants on any of the representations
$W_{\alpha}$ and hence by the equivalence of (1) and (2) above it contains all
of $(G_{X}^{*})^{n}$ as required. ∎

######  Remark 4.12.

For $n\geqslant 2$, the above proof gives more precise information on the
dependence of $n$ of the locus of characters on which the conclusion of
theorem 4.10 holds: There exists a finite union
$\Sigma\subset\Pi(A,\mathbb{F})^{2}$ of torsion translates of proper linear
subvarieties such that the conclusion of the theorem holds for all $n\geqslant
2$ and all
$\underline{\chi}=(\chi_{1},\dots,\chi_{n})\in\Pi(A,\mathbb{F})^{n}$ with

$(\chi_{i},\chi_{j})\notin\Sigma\quad\textnormal{for all}\quad i\neq j.$

This follows from the fact that the list of representations constructed in the
proof of 4.11 arises from a finite list of representations of $N(G)^{2}$ by
pullback under the various projections $N(G)^{n}\to N(G)^{2}$.

##  5\. From representations to geometry

In this section, we explain the link between representations and
characteristic cycles, which will be our main tool to show that under certain
assumptions the Tannaka group of a smooth subvariety will be big. We work over
an algebraically closed field $k$ with $\operatorname{char}(k)=0$, and
starting from section 5.3 we assume $k=\mathbb{C}$.

###  5.1. The ring of clean cycles

Over the complex numbers an important invariant of a perverse sheaf is its
characteristic cycle, which is a formal sum of conormal varieties adapted to a
suitable Whitney stratification. As we recall in section 5.3, the convolution
product of perverse sheaves is mirrored by a ‘convolution product’ on their
characteristic cycles. To define the latter, we need to introduce a
convolution product of conormal varieties, which can be done over any
algebraically closed field $k$ of characteristic zero as follows.

Recall that, for an integral subvariety $Z\subset A$, its conormal variety
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}$ is
said to be _clean_ if its Gauss map
$\gamma_{Z}\colon\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}\to\mathbb{P}_{A}$
is dominant—by theorem 2.8 this is the case if and only if the variety $Z$ is
of general type—and _negligible_ otherwise.

######  Definition 5.1.

The group of clean cycles $\mathcal{L}(A)$ is the free abelian group generated
by the projective conormal cones
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}$ of
integral subvarieties $Z\subset A$, modulo the subgroup generated by the
negligible ones. The projection onto the quotient induces an isomorphism

$\bigoplus_{Z\subset
A}\mathbb{Z}\cdot\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\mathcal{L}(A),$

where the direct sum ranges over the integral subvarieties of general type
$Z\subset A$.

A _clean cycle_ is an element of $\mathcal{L}(A)$ and, by means of the
preceding isomorphism, will always be seen as a finite formal sum
$\sum_{Z}m_{Z}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}$,
$m_{Z}\in\mathbb{Z}$, indexed by the integral subvarieties $Z\subset A$ of
general type.

Recall that in definition 2.2 we defined the conormal variety
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}$
for a reduced but not necessarily irreducible subvariety $Z\subset A$. For
simplicity, we still write
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}$
for the conormal variety seen as a cycle on $A\times\mathbb{P}_{A}$, or merely
as a clean cycle. In particular, in the latter case, we have

$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}=\sum_{Z^{\prime}\subset
Z}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z^{\prime}},$

the sum ranging over the irreducible components $Z^{\prime}\subset Z$ of
general type. We will consistently perpetrate this abuse of notation by
writing

$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}=m_{1}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z_{1}}+\cdots+m_{n}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z_{n}}$

for a cycle $Z=m_{1}Z_{1}+\cdots+m_{n}Z_{n}$ on $A$, with $m_{i}\in\mathbb{Z}$
and $Z_{i}\subset A$ integral.

######  Definition 5.2.

Let
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{1}},\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{2}}$
be clean conormal varieties. Let $U\subset\mathbb{P}_{A}$ be an open dense
subset of the projective cotangent space to the abelian variety such that over
this open subset the Gauss maps
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{i}\mid
U}:=\gamma_{X_{i}}^{-1}(U)\to U$ are finite étale covers. The fiber product of
these two finite étale covers embeds into $A\times A\times U\subset A\times
A\times\mathbb{P}_{A}$, and we denote by

$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}\;:=\;\overline{\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{1}\rvert
U}\times_{U}\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{2}\rvert
U}}\;\subset\;A\times A\times\mathbb{P}_{A}$

its Zariski closure. We define the convolution of the conormal varieties to be
the clean cycle

$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{1}}\circ\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X_{2}}\;:=\;\sigma_{*}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}})\;\in\;\mathcal{L}(A)$

arising by pushforward under the sum morphism $\sigma\colon A\times
A\times\mathbb{P}_{A}\to A\times\mathbb{P}_{A}$. We extend this product
$\circ$ on conormal varieties bilinearly to a product on the group of clean
cycles

$\circ\colon\quad\mathcal{L}(A)\times\mathcal{L}(A)\;\longrightarrow\;\mathcal{L}(A).$

This endows the group $\mathcal{L}(A)$ with a natural ring structure. The
product $\circ$ should not be confused with an intersection of cycles, indeed
the intersection product of any two cycles in $\mathcal{L}(A)$ is zero for
dimension reasons. For any integer $n\neq 0$ the pushforward

$[n]_{*}\colon\quad\mathcal{L}(A)\;\longrightarrow\;\mathcal{L}(A)$

is a ring homomorphism. For
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}\in\mathcal{L}$
we denote by
$\langle\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}\rangle\subset\mathcal{L}(A)$
the smallest subring of $\mathcal{L}(A)$ which contains
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}$ and is
stable under passing from a clean cycle to its irreducible components.

###  5.2. A reminder on Segre classes

In the discussion of wedge powers and spin representations to be carried out
in sections 7 to 8, we will need to control the effect that certain tensor
constructions on clean cycles have on the dimension of their base. For this we
recall in this section some basic facts about Segre classes, or Chern-Mather
classes222In the case of abelian varieties Segre classes and Chern-Mather
classes are the same, since the cotangent bundle to abelian varieties is
trivial. in the terminology of [Krä21, section 3]:

######  Definition 5.3.

The Segre classes of a cycle
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}$ on
$A\times\mathbb{P}_{A}$ of pure dimension $g-1$ are defined as the cycle
classes

$s_{d}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}})\;:=\;(\operatorname{pr}_{A})_{*}([\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}]\cdot[A\times
H_{d}])\;\in\;\operatorname{CH}_{d}(A)$

where $H_{d}\subset\mathbb{P}_{A}$ is a general linear subspace of dimension
$d<g=\dim A$ and $\operatorname{CH}_{d}(A)$ denotes the Chow group of
dimension $d$ algebraic cycles with $\mathbb{Z}$-coefficients, modulo rational
equivalence.

The following observation allows to control the dimension of the base of a
clean cycle in terms of its Segre classes:

######  Remark 5.4.

For any subvariety $Z\subset A$ we have
$s_{d}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z})=0$
for all $d>\dim Z$. On the other hand, if $Z$ has a top-dimensional
irreducible component of general type, then the Segre classes $s_{d}(Z)$ are
represented by nonzero effective cycles for all $d\in\\{0,1,\dots,\dim Z\\}$;
this follows from the dominance of the Gauss map $\gamma_{Z}$ and Kleiman’s
generic transversality theorem [Krä21, lemma 3.1.2 (3)]. The top degree Segre
class is the fundamental class

$s_{\dim Z}(Z)\;=\;[Z].$

Since clean cycles live on the projective cotangent bundle, there is no Segre
class in degree $g=\dim A$. We view the total Segre class
$s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}):=s_{0}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}})+\cdots+s_{g-1}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}})$
as an element of quotient

$\operatorname{CH}_{<g}(A)\;:=\;\operatorname{CH}_{\bullet}(A)/\operatorname{CH}_{g}(A).$

To define a ring structure on this quotient, recall that the additive group
$\operatorname{CH}_{\bullet}(A)$ comes with a natural ring structure where the
product is given by the Pontryagin product

$[X]*[Y]\;:=\;\sigma_{*}[X\times Y]\quad\textnormal{for the sum
morphism}\quad\sigma\colon X\times Y\to X+Y\subset A.$

and that $\operatorname{CH}_{g}(A)\subset\operatorname{CH}_{\bullet}(A)$ is an
ideal for the Pontryagin product. Working with the truncated Chow ring has the
advantage that the total Segre class is compatible with the convolution
product of clean cycles in the following sense:

######  Lemma 5.5.

Let
$\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{1},\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{2}\in\mathcal{L}(A)$.
If both Gauss maps
$\gamma_{\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{i}}\colon\operatorname{Supp}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{i})\to\mathbb{P}$
are finite morphisms, then

$s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{1}\circ\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{2})=s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{1})*s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{2})\quad\textnormal{\em
in}\quad\operatorname{CH}_{<g}(A).$

###### Proof.

See [Krä21, lemma 3.3.1]. ∎

Thus, the convolution product of clean cycles can be controlled via Pontryagin
products of Segre classes. For the latter, one can use the following
observation:

######  Lemma 5.6.

Let $X,Y\subset A$ be proper reduced subvarieties. Suppose that every
irreducible component of maximal dimension in $Y$ is of general type and that
at least one irreducible component of maximal dimension in $X$ is
nondegenerate. Then the cycle

$s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X})\ast
s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})$

is nonzero and effective in all degrees $\leqslant\min\\{\dim X+\dim Y,\dim
A-1\\}$.

###### Proof.

Since the Pontryagin product is bilinear and the Pontryagin product of two
effective cycles is effective or zero, it suffices to show the statement when
$X$ and $Y$ are both integral. Let $d=\dim X$, $e=\dim Y$ and $g=\dim A$, and
consider the Segre class

$s_{m-d}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})\;\in\;\operatorname{CH}_{m-d}(A)\quad\textnormal{for}\quad
d\;\leqslant\;m\;\leqslant\;\min\\{d+e,g-1\\}.$

This class is represented by an effective cycle since
$m-d\in\\{0,1,\dots,e\\}$. For any irreducible component $Z_{m-d}\subset A$ of
an effective cycle representing this class, we have

$s_{d}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X})*s_{m-d}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})\;=\;[X]*s_{m-d}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})\;=\;[X]*[Z_{m-d}]+\cdots\;\in\;\operatorname{CH}_{m}(A)$

where $\cdots$ stands for a cycle which is effective or zero. Since by
assumption $X$ is nondegenerate, we furthermore know from lemma 2.7 that the
sum morphism

$\sigma\colon\quad X\times Z_{m-d}\;\longrightarrow\;X+Z_{m-d}\;\subset\;A$

is generically finite onto its image. So $[X]*[Z_{m-d}]=\sigma_{*}([X\times
Z_{m-d}])$ is an effective class in $\operatorname{CH}_{m}(A)$. In conclusion,
we see that the Pontryagin product
$s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X})*s(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})$
is nonzero and effective in all degrees $m$ with $d\leqslant
m\leqslant\min\\{d+e,g-1\\}$. In the remaining range $0\leqslant m<d$ the
effectivity of the Pontryagin product is trivial because in that range we can
look at
$s_{m}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X})*s_{0}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})=\deg(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})\cdot
s_{m}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X})$;
note that
$\deg(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y})>0$
because $Y$ is of general type. ∎

######  Corollary 5.7.

Let $X,Y\subset A$ be reduced subvarieties, possibly reducible. If the Gauss
maps $\gamma_{X}$, $\gamma_{Y}$ are both finite morphisms, then

$\dim\pi(\operatorname{Supp}(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{X}\circ\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Y}))\;=\;\min\\{\dim
X+\dim Y,\dim A-1\\}.$

###### Proof.

Combine lemma 5.5 and lemma 5.6. ∎

###  5.3. Clean characteristic cycles

From now on and until the end of this section, we work over $k=\mathbb{C}$.
Recall that to any perverse sheaf $P\in\textup{Perv}(A,\mathbb{C})$ one may
attach a characteristic cycle [Dim04, definition 4.3.19], a finite formal sum
of conormal varieties

$\operatorname{CC}(P)\;=\;\sum_{Z\subset
A}m_{Z}(P)\cdot\Lambda_{Z}\quad\textnormal{with}\quad
m_{Z}(P)\;\in\;\mathbb{N}.$

Here the sum runs over all integral subvarieties $Z\subset A$, and only
finitely many $m_{Z}(P)$ are nonzero. These cycles contain a lot of
information, e.g., the Dubson-Kashiwara index formula shows that we can read
off the topological Euler characteristic as

$\chi(A,P)\;=\;\sum_{Z\subset
A}m_{Z}(P)\cdot\deg(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z})$

where
$\deg(\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z})$
is the degree of the Gauss map from section 2.2, see [FK00]. Passing from
$CC(P)$ to its projectivization and discarding all components which are not
clean, we define the clean characteristic cycle by

$\operatorname{cc}(P)\;:=\;\sum_{\begin{subarray}{c}\textup{$Z\subset A$
of}\\\ \textup{general
type}\end{subarray}}m_{Z}(P)\cdot\scalerel*{{\rotatebox[origin={c}]{180.0}{$\mathbb{V}$}}}{\mathbb{V}}_{Z}.$

It contains all information needed for the Dubson-Kashiwara index formula.
This index formula implies that for $P\in\textup{Perv}(A,\mathbb{C})$ we have
$\operatorname{cc}(P)=0$ if and only if $P\in\textup{S}(A,\mathbb{C})$. So the
clean characteristic cycle of perverse sheaves is defined on the abelian
quotient category
$\overline{\textup{Perv}}(A,\mathbb{C})=\textup{Perv}(A,\mathbb{C})/\textup{S}(A,\mathbb{C})$.
By additivity in short exact sequences we then obtain a group homomorphism
from the Grothendieck group of this abelian quotient category to the group of
clean cycles:

$\operatorname{cc}\colon\quad
K(\overline{\textup{Perv}}(A,\mathbb{C}))\;\longrightarrow\;\mathcal{L}(A)$

The Grothendieck group of an abelian tensor category is not just an abelian
group, but also a ring with the product given by the tensor product, which in
our case is the convolution product $*$ of perverse sheaves. By [Krä21, th.
2.1.1 and ex. 1.3.2] we have