# The Tannaka group $E_{6}$ only arises
from cubic threefolds

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 Mathematik
Ruhr-Universität Bochum
Universitätsstrasse 150
Postfach IB 45
44801 Bochum, Germany<EMAIL_ADDRESS>and  Marco Maculan  Marco Maculan
Institut de Mathématiques de Jussieu
Sorbonne Université
4 place Jussieu
75005 Paris
France<EMAIL_ADDRESS>
###### Abstract.

We show that under mild assumptions, the Fano surfaces of lines on smooth
cubic threefolds are the only smooth subvarieties of abelian varieties whose
Tannaka group for the convolution of perverse sheaves is an exceptional simple
group. This in particular leads to a considerable strengthening of our
previous work on the Shafarevich conjecture. A key idea is to control the
Hodge decomposition on cohomology by a cocharacter of the Tannaka group of
Hodge modules, and to play this off against an improvement of the Hodge number
estimates for irregular varieties by Lazarsfeld-Popa and Lombardi.

###### Contents

  1. 1 Introduction
  2. 2 Tannaka groups of Hodge modules
  3. 3 The Hodge cocharacter
  4. 4 Subvarieties with exceptional Tannaka groups
  5. 5 Nonnormal cubic hypersurfaces
  6. 6 On Fano varieties of lines
  7. 7 Subvarieties with Tannaka group $E_{6}$
  8. A Hodge number estimates

##  1\. Introduction

Any subvariety of an abelian variety gives rise to a reductive group via the
convolution of perverse sheaves. For smooth subvarieties these Tannaka groups
have recently been used to obtain arithmetic finiteness results for varieties
over number fields [LS20, KM23] and the big monodromy criterion in [JKLM23].
For both applications one has to exclude exceptional Tannaka groups. Such
exceptional groups are known to occur for Fano surfaces of smooth cubic
threefolds, which have Tannaka group $E_{6}$ by [Krä16]. We show that in the
range relevant for the above applications, these Fano surfaces are the only
examples with exceptional Tannaka groups; this leads to a considerable
strengthening of [KM23, JKLM23]. The key idea is to upgrade the Tannakian
formalism from perverse sheaves to complex Hodge modules in the sense of
Sabbah-Schnell [SS18]. In particular, we prove a comparison theorem between
the Tannaka groups for perverse sheaves and for Hodge modules which implies
that the Hodge decomposition is defined by a cocharacter of the perverse
Tannaka group. This puts a strong restriction on the arising Tannaka groups
that will be useful also for other applications.

###  1.1. Tannaka groups of subvarieties

Let $A$ be a $g$-dimensional abelian variety over an algebraically closed
field $k$ of characteristic zero, and consider an integral smooth subvariety
$X\subset A$ of dimension $d<g$. Our results apply both in an analytic and in
an algebraic setup: For complex varieties we use perverse sheaves with
coefficients in $\mathbb{F}=\mathbb{C}$, while for varieties over general
fields $k$ we use $\ell$-adic perverse sheaves with coefficients in
$\mathbb{F}=\overline{\mathbb{Q}}_{\ell}$ for a prime $\ell$. We define the
perverse intersection complex

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

as the pushforward of the constant sheaf under the closed immersion $i\colon
X\hookrightarrow A$, 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]. The group law on the abelian variety induces a
convolution product on perverse sheaves so that the convolution powers of
$\delta_{X}$ generate a neutral Tannaka category $\langle\delta_{X}\rangle$,
see for instance [JKLM23, sect. 3]. In what follows, we fix a fiber functor

$\omega\colon\quad\langle\delta_{X}\rangle\;\longrightarrow\;\operatorname{Vect}(\mathbb{F}).$

The automorphisms of this fiber functor are represented by a reductive
algebraic group

$G_{X,\omega}\;:=\;G_{\omega}(\delta_{X})\;\subset\;\operatorname{GL}(\omega(\delta_{X}))$

which we call the _Tannaka group of the subvariety $X\subset A$_. If $X$ is
invariant under translation by a nonzero point of $A$, then it descends to the
quotient of $A$ modulo the subgroup generated by that point. Hence, in what
follows we assume that $X$ is  _nondivisible_ in the sense that it is not
stable under the translation by any nonzero point of $A$. In this case, the
connected component of the identity $G_{X,\omega}^{\circ}\subset G_{X,\omega}$
acts irreducibly on the representation $\omega(\delta_{X})$ by [JKLM23, prop.
3.3]. We are interested in its derived group

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

which is a connected semisimple algebraic group. The smoothness of $X$ implies
that $\omega(\delta_{X})$ is a minuscule representation [Krä22, cor. 1.10]
[JKLM23, cor 5.15] of dimension equal to the absolute value of the topological
Euler caracteristic of $X$:

$\dim\omega(\delta_{X})\;=\;|\chi_{\textup{top}}(X)|.$

This leaves only very few possibilities because the Tannaka group is known to
be simple in most cases: Suppose that $g\geqslant 3$ and that the normal
bundle of $X\subset A$ is ample. Then by [JKLM23, th. A], the group
$G_{X,\omega}^{\ast}$ fails to be simple if and only if there exist smooth
positive-dimensional subvarieties $X_{1},X_{2}\subset A$ such that the sum
morphism induces an isomorphism

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

Beyond the standard representations of classical groups, there are only few
other minuscule representations of simple algebraic groups: Wedge powers of
the standard representation of $\operatorname{SL}_{n}$, (half)spin
representations of spin groups, and the smallest irreducible representations
of the exceptional groups $E_{6}$ and $E_{7}$. In [JKLM23] we completely
characterized wedge powers and ruled out spin representations; the main goal
of this paper is to understand when exceptional groups occur, with a focus on
$E_{6}$.

###  1.2. Exceptional Tannaka groups

The group $E_{6}$ does occur as the Tannaka group of the Fano surface of lines
on a smooth cubic threefold:

######  Example.

Let $Y\subset\mathbb{P}^{4}$ be a smooth cubic threefold. The Fano variety of
lines on $Y$ is a smooth projective surface $X$ whose Albanese variety
$A=\operatorname{Alb}(X)$ has dimension $5$. Moreover, for any chosen base
point the corresponding Albanese morphism is a closed embedding
$X\hookrightarrow\operatorname{Alb}(X)$. In what follows, we fix a base point
and view $X$ as a subvariety of its Albanese variety. By [Krä16, th. 2] then

$(G^{\ast}_{X,\omega},\omega(\delta_{X}))\;\simeq\;(E_{6},V)$

where $V$ is one of the two irreducible representations of $E_{6}$ with $\dim
V=27$. It is clear from the description of $X$ as a Fano variety of lines that
the classical Gauss map

$\quad
X\;\lhook\joinrel\longrightarrow\;\operatorname{Gr}_{2}(\operatorname{Lie}A)$

is an embedding, hence $X\subset A$ is nondivisible. Recall that $A$ is
principally polarized and its theta divisor is the image of the difference
morphism

$X\times X\;\longrightarrow\;X-X\;\subset\;A$

by [CG72, th. 13.4] [Bea82], which permits to recover the cubic threefold: The
theta divisor has an isolated singularity at the origin and its tangent cone
$C$ there can be identified with the cubic threefold $Y$ in suitable linear
coordinates on $\operatorname{Lie}A$. In the same coordinates $Y$ is also
identified with the image of the projection

$\mathbb{P}(T_{X})\;\longrightarrow\;\mathbb{P}(\operatorname{Lie}A).$

The goal of this paper is to show that under mild assumptions, the above is
the only example of exceptional Tannaka groups. In particular, in the
dimension range relevant for [KM23, JKLM23] we obtain (see corollary 7.2):

######  Theorem A.

Let $X\subset A$ be a smooth irreducible subvariety with ample normal bundle
and dimension $<g/2$. Then the following are equivalent:

  1. (1)

$X\subset A$ is nondivisible with Tannaka group $G_{X,\omega}^{\ast}\simeq
E_{6}$.

  2. (2)

$X$ is isomorphic to the Fano surface of lines on a smooth cubic threefold,
and the canonical morphism $\operatorname{Alb}(X)\to A$ is an isogeny.

The Fano surface $X$ of lines on a smooth cubic threefold
$Y\subset\mathbb{P}^{4}$ does not always have ample normal bundle in the
Albanese variety: By proposition 6.14 the normal bundle of an Albanese
embedding $X\hookrightarrow\operatorname{Alb}(X)$ fails to be ample if and
only if there is a hyperplane $H\subset\mathbb{P}^{4}$ such that the cubic
surface $H\cap Y\subset H$ is a cone over an elliptic curve; this happens for
instance for the cubic threefold with affine equation

$y^{2}-f(x)+z^{2}t+t^{3}\;=\;0,$

where $f(x)$ is a degree $3$ polynomial without multiple roots. But $X$ still
has a weaker positivity property: As in [JKLM23], we say that an integral
subvariety $Z\subset A$ of an abelian variety $A$ is _nondegenerate_ if for
every surjective morphism of abelian varieties $\varphi\colon A\to B$ we have

$\dim\varphi(Z)=\min\\{\dim Z,\dim B\\}.$

The Fano surface of lines on any smooth cubic threefold is nondegenerate
inside its Albanese variety by [Sch17, th. 1], since it is a summand of a
theta divisor; in fact loc. cit. refers to a stronger notion of nondegeneracy,
which implies the above one by [Ran81, lemma II.12]. Our proof of A proceeds
by reduction to the case of surfaces. In this case we only need to assume
nondegeneracy, hence we obtain the following result which now includes the
Fano surfaces of _all_ smooth cubic threefolds (see theorem 7.1):

######  Theorem B.

Suppose $g\geqslant 5$. For any smooth irreducible surface $X\subset A$, the
following three properties are equivalent:

  1. (1)

$X\subset A$ is nondivisible and nondegenerate with Tannaka group
$G_{X,\omega}^{\ast}\simeq E_{6}$.

  2. (2)

$X\subset A$ is nondivisible and nondegenerate with

$\chi(X,\mathcal{O}_{X})=6,\qquad c_{2}(X)=27,$

and the difference morphism $X\times X\to X-X$ has generic degree $6$.

  3. (3)

$X$ is isomorphic to the Fano surface of lines on a smooth cubic threefold,
and the canonical morphism $\operatorname{Alb}(X)\to A$ is an isogeny.

Note that in both theorems A and B the canonical morphism
$\operatorname{Alb}(X)\to X$ need not be an isomorphism:

######  Example.

Let $X$ be the Fano surface of lines on a smooth cubic threefold, and let

$\Theta\;=\;X-X\;\subset\;\operatorname{Alb}(X).$

be its theta divisor. The upper bound on the number of 2-torsion points on
theta divisors in [MP12] shows that for any theta divisor on a principally
polarized abelian variety, there exists a 2-torsion point outside that theta
divisor. Applying this to our case, we can find a point
$a\in\operatorname{Alb}(X)$ of order two with $a\notin\Theta=X-X$. We then
have

$X\cap(X+a)\;=\;\varnothing$

and it follows that the isogeny

$p\colon\quad\operatorname{Alb}(X)\;\longrightarrow\;A\;:=\;\operatorname{Alb}(X)/\langle
a\rangle$

maps $X\subset\operatorname{Alb}(X)$ isomorphically onto its image
$\bar{X}:=p(X)\subset A$. So this image is still an integral smooth
nondivisible subvariety. The pushforward of perverse sheaves under an isogeny
does not change the derived connected component of their Tannaka groups
[JKLM23, cor. 3.5], so we still have

$(G_{\omega}^{\ast}(\bar{X}),\omega(\delta_{\bar{X}}))\simeq(E_{6},V)$

although the canonical morphism $p\colon\operatorname{Alb}(\bar{X})\to A$ is
not an isomorphism.

###  1.3. The Hodge filtration

A key ingredient in our proof of theorems A and B is the theory of complex
Hodge modules by Sabbah-Schnell [SS18]. To explain the idea, let $A$ be a
complex abelian variety and $\operatorname{HM}(A)$ the category of direct sums
of complex Hodge modules of arbitrary weight on it, see section 2 for details.
The category $\operatorname{HM}(A)$ is semisimple $\mathbb{C}$-linear abelian
and comes equipped with a faithful exact $\mathbb{C}$-linear functor

$\operatorname{DR}\colon\quad\operatorname{HM}(A)\;\longrightarrow\;\textup{Perv}(A,\mathbb{C}).$

To any $M\in\operatorname{HM}(A)$ one may attach a Tannaka group
$G_{\omega}(M)$ in the same way as for perverse sheaves; in corollary 2.5 we
show a comparison theorem between this group and the Tannaka group
$G_{\omega}(P)$ of the perverse sheaf $P=\operatorname{DR}(M)$. In particular,
the derived groups of their connected components of the identity are the same:

$G_{\omega}^{\ast}(M)\;=\;G_{\omega}^{\ast}(P).$

This allows us to enrich with Hodge theoretic data all the previous Tannakian
constructions for perverse sheaves, so for the rest of this section we will
work only with Hodge modules. It is now time to be more specific about our
choice of fiber functors: Fix a Hodge module $M\in\operatorname{HM}(A)$, and
let $L$ be a unitary local system of rank one on $A$ with the property that
all perverse subquotients of $\operatorname{DR}(M)\otimes L$ have their
cohomology concentrated in degree zero; there are plenty of such local systems
by generic vanishing for perverse sheaves on abelian varieties [KW15, Sch15].
Then the functor

$\omega\colon\quad\langle
M\rangle\;\longrightarrow\;\operatorname{Vect}(\mathbb{C}),\qquad
N\;\longmapsto\;\textup{H}^{0}(A,\operatorname{DR}(N)\otimes L)$

is a fiber functor. Now $L$ underlies a complex variation of Hodge structures
$\mathbb{L}$, so from the natural identification $\operatorname{DR}(M)\otimes
L=\operatorname{DR}(M\otimes\mathbb{L})$ we obtain a Hodge decomposition

$\textup{H}^{0}(A,\operatorname{DR}(M)\otimes
L)\;=\;\bigoplus_{(p,q)\in\mathbb{Z}^{2}}\textup{H}^{p,q}(M\otimes\mathbb{L})$

We show that this Hodge decomposition is compatible with the convolution
product, which allows to enrich $\omega$ to a tensor functor
$\omega^{H}\colon\langle
M\rangle\rightarrow\operatorname{Vect}_{\mathbb{Z}\times\mathbb{Z}}(\mathbb{C})$
with values in the category of finite-dimensional bigraded complex vector
spaces. This leads to the following result (see corollary 3.2):

######  Theorem C.

There is a natural morphism $\lambda\colon\mathbb{G}_{m}^{2}\to G_{\omega}(M)$
such that the Hodge decomposition

$\omega^{H}(M)\;\;=\bigoplus_{(p,q)\in\mathbb{Z}^{2}}\textup{H}^{p,q}(M\otimes\mathbb{L})$

is the decomposition in weight spaces for the characters
$(p,q)\in\mathbb{Z}^{2}=\operatorname{Hom}(\mathbb{G}_{m}^{2},\mathbb{G}_{m})$.

We call $\lambda$ the _Hodge cocharacter_ of $M$. The fact that this
cocharacter takes values in the subgroup

$G_{\omega}^{\ast}(M)\;\subset\;\operatorname{GL}(\omega(M))$

imposes strong restrictions on the possible Hodge filtrations if the subgroup
is known to be small. This in particular applies to the Tannaka group of
smooth subvarieties $X\subset A$: The perverse intersection complex
$\delta_{X}\in\textup{Perv}(A,\mathbb{C})$ naturally lifts to a complex Hodge
module $\delta_{X}^{H}\in\operatorname{HM}(A)$ of weight $d=\dim X$ and its
Hodge numbers are

$\dim\textup{H}^{p,d-p}(\delta_{X}^{H}\otimes\mathbb{L})\;=\;(-1)^{d-p}\chi(X,\Omega^{p}_{X}),$

see section 3. Comparing this with an improvement to the Hodge estimates by
Lazarsfeld-Popa [LP10] and Lombardi [Lom13] explained in appendix A, we obtain
the following result (see proposition 4.6):

######  Theorem D.

Let $X\subset A$ be a smooth nondivisible irreducible subvariety with ample
normal bundle and dimension $d$ such that $G_{X,\omega}^{\ast}$ is a simple
exceptional group. Then, either

  1. (1)

$G_{X,\omega}^{\ast}\simeq E_{6}$, $|\chi_{\textup{top}}(X)|=27$ and
$d\in\\{2,4,6\\}$, or

  2. (2)

$G_{X,\omega}^{\ast}\simeq E_{7}$, $|\chi_{\textup{top}}(X)|=56$ and
$d\in\\{3,\dots,15\\}$ is odd.

Moreover $g\leqslant g_{\max}$ for the following upper bound $g_{\max}$
depending on $d$:

$\displaystyle(1)\quad\begin{array}[]{c|ccccc}d&2&4&6\\\ \hline\cr
g_{\max}&7&6&8\end{array}$
$\displaystyle\quad(2)\quad\begin{array}[]{c|ccccccc}d&3&5&7&9&11&13&15\\\
\hline\cr g_{\max}&9&10&12&13&15&16&18\end{array}$

In each case we also know the Hodge numbers
$h^{p}(X):=(-1)^{d-p}\chi(X,\Omega^{p}_{X})$, as they must be among those
listed in propositions 4.3, LABEL: and 4.5. This essentially reduces the proof
of theorem A to the case of surfaces treated in theorem B. For the proof of
the latter we apply the representation theory of $E_{6}$ to show that the
difference morphism

$X\times X\;\longrightarrow\;D\;:=\;X-X\;\subset\;A,$

is generically finite of degree $\geqslant 6$ over its image (see section
4.6), and then use this numerical information to show by direct geometric
arguments that the projective tangent cone to $D\subset A$ at the origin is a
smooth cubic threefold whose associated Fano surface of lines is isomorphic to
$X$ (see section 7).

###  1.4. Big Monodromy

As an application of the above we obtain a significant strengthening of the
big monodromy criterion in [JKLM23]. More precisely, let $S$ be a smooth
complex variety. Let $A$ be a complex abelian variety of dimension $g$ and
$\mathcal{X}\subset A_{S}:=A\times S$ a subvariety such that the projection
$f\colon\mathcal{X}\to S$ is a smooth morphism with connected fibers of
dimension $d$. The situation is resumed in the following diagram:

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

For a character $\chi\colon\pi_{1}(A,0)\to\mathbb{C}^{\times}$ let $L_{\chi}$
denote the associated rank one local system on $A$. Given an $n$-tuple of such
characters $\underline{\chi}=(\chi_{1},\dots,\chi_{n})$, we consider the local
system

$V_{\underline{\chi}}\;:=\;R^{d}f_{\ast}\pi^{*}L_{\underline{\chi}}\quad\textup{where}\quad
L_{\underline{\chi}}\;:=\;L_{\chi_{1}}\oplus\cdots\oplus L_{\chi_{n}}.$

The fiber at $s$ of this local system comes with a linear action of the group
$\pi_{1}(S,s)$ via the monodromy representation. With terminology as in loc.
cit. we obtain the following strengthening of the big monodromy result in
[JKLM23, sect. 1.1]:

######  Big Monodromy Criterion.

Suppose $X:=\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ has ample normal
bundle and dimension $d<(g-1)/2$. If $\mathcal{X}$ is symmetric up to
translation, assume that (1.1) below holds. Then the following are equivalent:

  1. (1)

$X$ 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 of
characters $\underline{\chi}$.

Here $\mathcal{X}$ is said to be _symmetric up to translation_ if there is a
morphism $a\colon S\to A$ with the property that
$\mathcal{X}_{t}=a(t)-\mathcal{X}_{t}$ for all $t\in S(\mathbb{C})$. Only in
this case we need to exclude some very specific numerics by assuming that the
topological Euler characteristic $e=\chi_{\textup{top}}(X)$ of $X$ has
absolute value

(1.1) $|e|\;\neq\;\begin{cases}56&\text{if\leavevmode\nobreak\ $d=3$,
$g\in\\{8,9\\}$ and $\chi(X,\mathcal{O}_{X})=-7$,}\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ 2^{2m-1}&\text{if\leavevmode\nobreak\ $d\geqslant(g-1)/4$
and\leavevmode\nobreak\ $m\in\\{3,\dots,d\\},m\equiv d$ modulo
$2$.}\end{cases}$

This assumption is empty as soon as $d<(g-1)/4$. Apart from curves, in
[JKLM23] we had no control whatsoever on the dimension and irregularity of
varieties with exceptional Tannaka group, so we coarsely assumed $|e|\neq
27,56$ for all $(d,g)$ in the relevant range. Instead, theorems A and D now
only leave the cases $d=3$ and $g=8,9$ within the range $d<(g-1)/2$. For
future applications outside this range, it will be sufficient to exclude the
finite list of $(d,g)$ in D.

### Acknowledgements

We thank Victor Gonzalez-Alonso, Manfred Lehn, Luigi Lombardi, Claude Sabbah,
Christian Schnell and Mads Villadsen for discussions related to this project.
The computer-aided searches in propositions 4.3, LABEL: and 4.5 were carried
out by Mads Villadsen. C. L. was supported by the DFG through the research
grants Le 3093/3-2, Le 3093/5-1. T. K. was supported by the DFG through the
research grant Kr 4663/2-1.

### Conventions

A _variety_ over a field $k$ is a separated $k$-scheme of finite type, and a
_subvariety_ is a closed subvariety. For a vector bundle $\mathcal{V}$ on a
variety we denote by $\mathbb{P}(\mathcal{V})$ the projective bundle of lines
in $\mathcal{V}$.

##  2\. Tannaka groups of Hodge modules

In this section we introduce Tannaka categories of Hodge modules on complex
abelian varieties and relate their Tannaka groups to those for perverse
sheaves.

###  2.1. Complex Hodge modules

For a complex manifold $X$ and $w\in\mathbb{Z}$, we denote by
$\operatorname{HM}(X,w)$ the category of polarizable complex Hodge modules of
weight $w$ on $X$ in the sense of [SS18, def. 14.2.2]. This is a semisimple
abelian $\mathbb{C}$-linear category, and it comes with a faithful exact
$\mathbb{C}$-linear functor
$\operatorname{DR}\colon\operatorname{HM}(X,w)\rightarrow\textup{Perv}(X)$ to
the category $\textup{Perv}(X)$ of perverse sheaves on $X$ with complex
coefficients. We denote by

$\operatorname{HM}(X)\;:=\;\bigoplus_{w\in\mathbb{Z}}\operatorname{HM}(X,w)$

the category whose objects are the formal direct sums
$M=\bigoplus_{w\in\mathbb{Z}}M_{w}$ of Hodge modules
$M_{w}\in\operatorname{HM}(X,w)$ such that $M_{w}=0$ for all but finitely many
weights $w$, with morphisms

$\operatorname{Hom}_{\operatorname{HM}(X)}(M,N)\;:=\;\bigoplus_{w\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{HM}(X,w)}(M_{w},N_{w}).$

This is again a semisimple abelian $\mathbb{C}$-linear category with a
faithful exact $\mathbb{C}$-linear functor

$\operatorname{DR}\colon\quad\operatorname{HM}(X)\;\longrightarrow\;\textup{Perv}(X),\quad
M\;\longmapsto\;\bigoplus_{w\in\mathbb{Z}}\operatorname{DR}(M_{w}).$

######  Example 2.1.

If $X=\\{\mathrm{pt}\\}$ is a point, then
$\operatorname{HM}(\\{\mathrm{pt}\\})$ is the category
$\operatorname{Vect}_{\mathbb{Z}\times\mathbb{Z}}(\mathbb{C})$ of finite-
dimensional bigraded complex vector spaces

$V\;=\;\bigoplus_{p,q\in\mathbb{Z}}V^{p,q}$

and $\operatorname{HM}(\\{\mathrm{pt}\\},w)$ is its subcategory of _pure
$\mathbb{C}$-Hodge structures of weight $w$_, by which we mean those bigraded
vector spaces that satisfy $V^{p,q}=0$ for all $p,q\in\mathbb{Z}$ with
$p+q\neq w$. Moreover,

$\operatorname{DR}\colon\quad\operatorname{HM}(\\{\mathrm{pt}\\})=\operatorname{Vect}_{\mathbb{Z}\times\mathbb{Z}}(\mathbb{C})\;\longrightarrow\;\textup{Perv}(\\{\mathrm{pt}\\})=\operatorname{Vect}(\mathbb{C})$

is the functor that forgets the grading.

By a _variation of $\mathbb{C}$-Hodge structures_ of weight $w$ on $X$ we mean
a $\mathcal{C}^{\infty}$-bundle $\mathcal{V}$ together with a flat connection
and a decomposition into $\mathcal{C}^{\infty}$-subbundles

$\mathcal{V}\;=\;\bigoplus_{p+q=w}\mathcal{V}^{p,q}$

that satisfies Griffiths transversality. In what follows we only consider
_polarizable_ variations of $\mathbb{C}$-Hodge structures in the sense of
[SS18, def. 4.1.9]; by theorem 14.6.1 in loc. cit. any such polarizable
variation of $\mathbb{C}$-Hodge structures can be viewed as an object of
$\operatorname{HM}(X,w+\dim X)$. More generally, for any integral subvariety
$Z\subset X$ and any pure polarizable variation of $\mathbb{C}$-Hodge
structures $\mathbb{V}$ of weight $w$ on an open dense $U\subset
Z^{\textup{reg}}$ there is a unique simple Hodge module

$M\;=\;\operatorname{IC}_{Z}(\mathbb{V})\;\in\;\operatorname{HM}(X,w+\dim Z)$

with $M_{|U}=\mathbb{V}$. Conversely every simple Hodge module
$M\in\operatorname{HM}(X)$ arises like this for a unique germ of a simple pure
polarizable variation of $\mathbb{C}$-Hodge structures on a dense open subset
of an integral subvariety of $X$ [SS18, th. 16.2.1].

######  Example 2.2.

For any integral subvariety $Z\subset X$, the trivial variation
$\mathbb{C}_{U}$ of Hodge structures of rank one and weight zero on the smooth
locus $U=Z^{\textup{reg}}$ defines a pure Hodge module

$\delta_{Z}^{H}\;=\;\operatorname{IC}_{Z}(\mathbb{C}_{U})\;\in\;\operatorname{HM}(X,d).$

of weight $d=\dim Z$. Its image under the functor $\operatorname{DR}$ is the
perverse intersection complex

$\delta_{Z}\;:=\;\operatorname{DR}(\delta_{Z}^{H})\in\textup{Perv}(X).$

If $Z$ is smooth, then this intersection complex is given by
$\delta_{Z}=\mathbb{C}_{Z}[d]$.

###  2.2. Hodge structure on cohomology

Passing from perverse sheaves to Hodge modules will enrich the cohomology
groups with $\mathbb{C}$-Hodge structures. Indeed, with the notation of [SS18,
def. 12.7.28], we have for any proper morphism $f\colon X\to Y$ direct image
functors

${{}_{\scriptscriptstyle{T}}f}_{*}^{(k)}\colon\quad\operatorname{HM}(X)\;\longrightarrow\;\operatorname{HM}(Y),$

for $k\in\mathbb{Z}$. These lift the perverse direct image functors
${}^{p}\mathcal{H}^{k}\circ Rf_{*}$ in the sense that the following diagram
commutes:

${\operatorname{HM}(X)}$${\operatorname{HM}(Y)}$${\textup{Perv}(X)}$${\textup{Perv}(Y)}$$\scriptstyle{{{}_{\scriptscriptstyle{T}}f}_{*}^{(k)}}$$\scriptstyle{\operatorname{DR}}$$\scriptstyle{\operatorname{DR}}$$\scriptstyle{{}^{p}\mathcal{H}^{k}\circ\textup{R}f_{*}}$

Taking $f\colon X\to Y=\\{\mathrm{pt}\\}$, we see that for any
$M\in\operatorname{HM}(X)$ the groups $\textup{H}^{k}(X,\operatorname{DR}(M))$
come with a natural bigrading. We call this the _Hodge decomposition_ and
denote it by

$\textup{H}^{k}(X,\operatorname{DR}(M))\;=\;\bigoplus_{p,q\in\mathbb{Z}}\textup{H}^{p,q}(M).$

Note that this is a pure $\mathbb{C}$-Hodge structure of weight $w+k$ if
$M\in\operatorname{HM}(X,w)$.

###  2.3. Tannaka categories

Now let $X=A$ be an abelian variety. In this case every perverse sheaf has
nonnegative Euler characteristic by [FK00, cor. 1.4], so inside the abelian
category of perverse sheaves, those of Euler characteristic zero form a Serre
subcategory

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

The objects of this subcategory are called _negligible_ perverse sheaves. On
the abelian quotient category
$\overline{\textup{Perv}}(A)=\textup{Perv}(A)/\textup{S}_{\textup{Perv}}(A)$
the group law $\sigma\colon A\times A\to A$ gives rise to a convolution
product

$*\colon\quad\overline{\textup{Perv}}(A)\times\overline{\textup{Perv}}(A)\;\longrightarrow\;\overline{\textup{Perv}}(A),\quad
P_{1}*P_{2}\;:=\;{}^{p}\mathcal{H}^{0}(R\sigma_{*}(P_{1}\boxtimes P_{2}))$

making $\overline{\textup{Perv}}(A)$ a neutral Tannaka category as recalled in
[JKLM23, sect. 3.1]. The same constructions work also for the category of
Hodge modules: Consider the Serre subcategory

$\textup{S}_{\operatorname{HM}}(A)\;:=\;\\{M\in\operatorname{HM}(A)\mid\chi(A,\operatorname{DR}(M))=0\\}\;\subset\;\operatorname{HM}(A).$

Then the functor $\operatorname{DR}$ descends to an exact faithful
$\mathbb{C}$-linear functor on the abelian quotient category

$\operatorname{DR}\colon\quad\operatorname{\overline{\operatorname{HM}}}(A):=\operatorname{HM}(A)/\textup{S}_{\operatorname{HM}}(A)\;\longrightarrow\;\overline{\textup{Perv}}(A),$

and this functor naturally underlies a tensor functor with respect to the
convolution product

$*\colon\quad\operatorname{\overline{\operatorname{HM}}}(A)\times\operatorname{\overline{\operatorname{HM}}}(A)\;\longrightarrow\;\operatorname{\overline{\operatorname{HM}}}(A),\quad
M_{1}*M_{2}\;:=\;{{}_{\scriptscriptstyle{T}}{\sigma}}_{*}^{(0)}(M_{1}\boxtimes
M_{2}).$

So for $M_{1},M_{2}\in\operatorname{\overline{\operatorname{HM}}}(A)$ we have
isomorphisms

$\operatorname{DR}(M_{1}\ast
M_{2})\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\operatorname{DR}(M_{1})\ast\operatorname{DR}(M_{2})$

compatible with the associativity, commutativity and unit of the respective
tensor categories; note that to avoid sign issues in the commutativity, we
will always use right $\mathcal{D}$-modules for the definition of
$\operatorname{HM}(A)$.

######  Corollary 2.3.

$\operatorname{\overline{\operatorname{HM}}}(A)$ is again a neutral Tannaka
category.

###### Proof.

Pick any fiber functor on the neutral Tannaka category
$\overline{\textup{Perv}}(A)$. Composing it with the faithful exact
$\mathbb{C}$-linear functor $\operatorname{DR}$, which is a tensor functor by
the above, we get a fiber functor on
$\operatorname{\overline{\operatorname{HM}}}(A)$. ∎

###  2.4. An exact sequence of Tannaka groups

We want to compare the Tannaka groups for Hodge modules with those for the
underlying perverse sheaves. For the rest of this section, we fix a full
abelian tensor subcategory
$\mathcal{C}\subset\operatorname{\overline{\operatorname{HM}}}(A)$ and denote
by

$\operatorname{DR}(\mathcal{C})\subset\overline{\textup{Perv}}(A)$

the smallest full abelian tensor subcategory containing all subquotients of
$\operatorname{DR}(M)$ for $M\in\mathcal{C}$. Suppose that on this subcategory
we are given a fiber functor, i.e. a faithful exact $\mathbb{C}$-linear tensor
functor

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

Precomposing with the exact faithful $\mathbb{C}$-linear tensor functor
$\operatorname{DR}\colon\mathcal{C}\to\operatorname{DR}(\mathcal{C})$ we get a
fiber functor $\omega\circ\operatorname{DR}$ on the original category:

${\mathcal{C}}$${\operatorname{Vect}(\mathbb{C})}$${\operatorname{DR}(\mathcal{C})}$$\scriptstyle{\omega\circ\operatorname{DR}}$$\scriptstyle{\operatorname{DR}}$$\scriptstyle{\omega}$

Let
$G_{\omega}(\mathcal{C})=\operatorname{Aut}^{\otimes}(\omega\circ\operatorname{DR})$
and
$G_{\omega}(\operatorname{DR}(\mathcal{C}))=\operatorname{Aut}^{\otimes}(\omega)$
be the corresponding Tannaka groups. It follows from the definition of the
category $\operatorname{DR}(\mathcal{C})$ that $\operatorname{DR}$ induces an
embedding

$G_{\omega}(\operatorname{DR}(\mathcal{C}))\;\lhook\joinrel\longrightarrow\;G_{\omega}(\mathcal{C})$

as a closed subgroup [DMOS82, prop. 2.21b]. In fact this subgroup is normal:
To understand this, let

$\mathcal{C}_{\\{0\\}}\;:=\;\\{M\in\mathcal{C}\mid\operatorname{Supp}(M)=\\{0\\}\\}\;\subset\;\mathcal{C}$

be the full subcategory of objects in $\mathcal{C}$ supported at the origin;
it can be seen as a tensor subcategory of the category of bigraded vector
spaces via example 2.1, hence its Tannaka group

$G_{\omega}(\mathcal{C}_{\\{0\\}})\;:=\;\operatorname{Aut}^{\otimes}(\omega\circ\operatorname{DR}|\mathcal{C}_{\\{0\\}})$

is a quotient of a torus of rank two; in particular, it is a torus. We then
have the following Hodge theoretic analog of the Galois sequence in [JKLM23,
th. 4.3]:

######  Proposition 2.4.

We have an exact sequence

$1\longrightarrow G_{\omega}(\operatorname{DR}(\mathcal{C}))\longrightarrow
G_{\omega}(\mathcal{C})\longrightarrow
G_{\omega}(\mathcal{C}_{\\{0\\}})\longrightarrow 1.$

###### Proof.

We have an embedding $\mathcal{C}_{\\{0\\}}\hookrightarrow\mathcal{C}$ as a
full abelian tensor subcategory which is stable under subobjects. By Tannaka
duality, any such embedding corresponds to an epimorphism
$G_{\omega}(\mathcal{C})\twoheadrightarrow G_{\omega}(\mathcal{C}_{\\{0\\}})$
[DMOS82, prop. 2.21a]. The exactness of the sequence of Tannaka groups follows
by the same argument as in the proof of [JKLM23, th. 4.3]. Indeed, the
argument there only needs that for any $M\in\mathcal{C}$ the following two
properties hold:

  1. (1)

Any rank one subobject of $\operatorname{DR}(M)$ lies in a semisimple
subobject of $\operatorname{DR}(M)$.

  2. (2)

The maximal trivial subobject of $\operatorname{DR}(M)$ lies in the essential
image of the functor
$\operatorname{DR}\colon\langle\mathcal{C}\rangle\to\operatorname{DR}(\mathcal{C})$.

Both properties are trivial in the case at hand: The rank one objects of the
tensor category $\mathcal{C}$ are the perverse skyscraper sheaves, and the
maximal trivial subobject is the direct sum of all skyscraper subsheaves
supported at the origin. Hence both claims follow from the fact that for any
$M\in\operatorname{HM}(A)$, every simple subobject of the perverse sheaf
$\operatorname{DR}(M)$ (or equivalently, of the underlying holonomic
$\mathcal{D}$-module) has the property that a direct sum of twists of that
subobject underlies a Hodge submodule of $M$ by [SS18, prop. 16.3.1]. ∎

For $M\in\mathcal{C}$ with underlying perverse sheaf $P=\operatorname{DR}(M)$,
let $V=\omega(P)$. Then the groups

$\displaystyle G_{\omega}(M)$ $\displaystyle:=$
$\displaystyle\operatorname{Im}(G_{\omega}(\mathcal{C})\to\operatorname{GL}(V))$
$\displaystyle G_{\omega}(P)$ $\displaystyle:=$
$\displaystyle\operatorname{Im}(G_{\omega}(\operatorname{DR}(\mathcal{C}))\to\operatorname{GL}(V))$

can be identified with the Tannaka groups of the full tensor subcategories
$\langle M\rangle\subset\mathcal{C}$ and $\langle
P\rangle\subset\operatorname{DR}(\mathcal{C})$ generated by $M$ and $P$
respectively. These Tannaka groups are algebraic groups, by construction they
are subgroups of $\operatorname{GL}(V)$. They are reductive since all our
Hodge modules and perverse sheaves are semisimple. So the derived group of
their connected component of the identity is a connected semisimple group
which we denote by

$G^{*}_{\omega}(-)\;:=\;[G^{\circ}_{\omega}(-),G^{\circ}_{\omega}(-)].$

The groups for Hodge modules and for perverse sheaves are then related as
follows:

######  Corollary 2.5.

For $M\in\mathcal{C}$ consider the perverse sheaf $P=\operatorname{DR}(M)$.
Let $V=\omega(P)$ as above. Then

$G_{\omega}(P)\;\subset\;G_{\omega}(M)\;\subset\;N_{\operatorname{GL}(V)}(G_{\omega}(P))\quad\text{and}\quad
G_{\omega}^{*}(P)\;=\;G_{\omega}^{*}(M).$

###### Proof.

Applying proposition 2.4 to the tensor category $\mathcal{C}=\langle M\rangle$
generated by $M$, we get an exact sequence

$1\longrightarrow G_{\omega}(P)\longrightarrow G_{\omega}(M)\longrightarrow
G_{\omega}(\mathcal{C}_{0})\longrightarrow 1.$

So $G_{\omega}(P)\unlhd G_{\omega}(M)$ is a normal subgroup. Since both are
subgroups of $\operatorname{GL}(V)$, this implies the claim about the
containment in the normalizer. For the claim about the derived groups of the
connected component, recall that $G_{\omega}(\mathcal{C}_{0})$ is a torus; so
the reductive Lie algebras $\operatorname{Lie}G_{\omega}(M)$ and
$\operatorname{Lie}G_{\omega}(P)$ have the same maximal semisimple Lie
subalgebra. ∎

###  2.5. Character twists

Any unitary local system $L$ of rank one on $A$ naturally underlies a unique
pure polarizable variation $\mathbb{L}$ of Hodge structures of weight zero,
concentrated in bidegree $(0,0)$. We then have an endofunctor

$\operatorname{HM}(A)\;\longrightarrow\;\operatorname{HM}(A),\quad
M\;\longmapsto\;M\otimes\mathbb{L}$

which descends to the quotient categories modulo negligibles, where
$M\in\operatorname{HM}(A)$ is called _negligible_ if $\operatorname{DR}(M)$ is
a negligible perverse sheaf in the sense of section 2.3. The induced functor

$\operatorname{\overline{\operatorname{HM}}}(A)\;\longrightarrow\;\operatorname{\overline{\operatorname{HM}}}(A)$

is a tensor functor with respect to the convolution product: For
$M_{1},M_{2}\in\operatorname{\overline{\operatorname{HM}}}(A)$ we have natural
isomorphisms

$\displaystyle(M_{1}*M_{2})\otimes\mathbb{L}$
$\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{\sigma}}_{*}^{(0)}(M_{1}\boxtimes
M_{2})\otimes\mathbb{L}$
$\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{\sigma}}_{*}^{(0)}((M_{1}\boxtimes
M_{2})\otimes\sigma^{-1}(\mathbb{L}))$ by the projection formula
$\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{\sigma}}_{*}^{(0)}((M_{1}\otimes\mathbb{L})\boxtimes(M_{2}\otimes\mathbb{L}))$
since $\sigma^{-1}(\mathbb{L})\simeq\mathbb{L}\boxtimes\mathbb{L}$
$\displaystyle\;\simeq\;(M_{1}\otimes\mathbb{L})*(M_{2}\otimes\mathbb{L}).$

Let $\mathcal{C}$ and $\omega$ be as in the previous section. Then up to
noncanonical isomorphism, the Tannaka group of a Hodge module in $\mathcal{C}$
does not change under twists:

######  Lemma 2.6.

For $M\in\mathcal{C}$ and $\mathbb{L}$ a unitary rank one local system with
$M\otimes\mathbb{L}\in\mathcal{C}$, we have

$G_{\omega}(M\otimes\mathbb{L})\;\simeq\;G_{\omega}(M).$

###### Proof.

As in [KW15, prop. 4.1], twisting by $\mathbb{L}$ gives rise to an equivalence
of tensor categories

$\langle M\rangle\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\langle
M\otimes\mathbb{L}\rangle,\quad N\;\longmapsto\;N\otimes\mathbb{L}$

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

##  3\. The Hodge cocharacter

We now show that the Hodge decomposition on cohomology is induced by a
cocharacter of the Tannaka group, and we gather some general estimates for the
Hodge level of intersection complexes that will be useful later.

###  3.1. The Hodge decomposition

Let $\textup{Perv}_{0}(A)\subset\textup{Perv}(A)$ be the full subcategory of
all perverse sheaves $P$ with the property that all the perverse subquotients
$Q$ of $P$ satisfy

$\textup{H}^{i}(A,Q)\;=\;0\quad\text{for all}\quad i\;\neq\;0.$

As in [JKLM23, sect. 4.3], its image

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

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

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

This is a tensor functor. In particular, for all perverse sheaves
$P_{1},P_{2}\in\overline{\textup{Perv}}(A)$ we have natural isomorphisms

$\vartheta_{P_{1},P_{2}}\colon\quad\omega(P_{1}*P_{2})\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\omega(P_{1})\otimes\omega(P_{2})$

which are defined as follows: Let $f\colon A\to\\{\operatorname{pt}\\}$ be the
map to a point, then

$\omega(P)\;=\;\mathcal{H}^{0}(\textup{R}f_{*}(P))\quad\text{for all}\quad
P\in\textup{Perv}_{0}(A).$

Now, the complex $\textup{R}f_{\ast}(P)$ has cohomology at most in degree $0$
so we identify it with $\omega(P)$. With this in mind, using the Künneth
isomorphism and the fact that $f\circ\sigma=f\times f$, we obtain

$\displaystyle\omega(P_{1}*P_{2})$
$\displaystyle\;\simeq\;Rf_{*}(P_{1}*P_{2})$
$\displaystyle\;\simeq\;Rf_{*}R\sigma_{*}(P_{1}\boxtimes P_{2})$
$\displaystyle\;\simeq\;R(f\times f)_{*}(P_{1}\boxtimes P_{2})$
$\displaystyle\;\simeq\;Rf_{*}(P_{1})\otimes Rf_{*}(P_{2})$
$\displaystyle\;\simeq\;\omega(P_{1})\otimes\omega(P_{2}).$

These constructions can be upgraded to Hodge modules: Let
$\operatorname{HM}_{0}(A)\subset\operatorname{HM}(A)$ be the full subcategory
of all Hodge modules $M$ with $\operatorname{DR}(M)\in\textup{Perv}_{0}(A)$.
Then $\omega$ lifts to the functor

$\omega^{H}\colon\quad\operatorname{HM}_{0}(A)\;\longrightarrow\;\operatorname{Vect}_{\mathbb{Z}\times\mathbb{Z}}(\mathbb{C}),\quad
M\;\longmapsto\;\textup{H}^{0}(A,\operatorname{DR}(M))$

where the bigrading on the values of the functor is the Hodge decomposition
from section 2.2:

$\textup{H}^{0}(A,\operatorname{DR}(M))\;=\;\bigoplus_{p,q\in\mathbb{Z}}\textup{H}^{p,q}(M).$

######  Proposition 3.1.

For $M_{1},M_{2}\in\mathcal{C}$ the isomorphism
$\vartheta_{\operatorname{DR}(M_{1}),\operatorname{DR}(M_{2})}$ is compatible
with the Hodge decomposition. Hence the functor $\omega^{H}$ is a tensor
functor.

###### Proof.

The construction is the same as for perverse sheaves, but working on the level
of Hodge modules. For a morphism $g$ we denote by
${{}_{\scriptscriptstyle{T}}{g}}_{*}$ the pushforward functor of [SS18, def.
12.7.35], which replaces the derived pushforward $Rg_{*}$ of constructible
sheaves. As above, let $f\colon A\to\\{\operatorname{pt}\\}$ be the map to a
point, then

$\omega^{H}(M)\;=\;{{}_{\scriptscriptstyle{T}}{f}}^{(0)}_{*}(M)\quad\text{ for
all\leavevmode\nobreak\ $M\in\operatorname{HM}_{0}(A)$.}$

As above, the complex of bigraded vector space
${{}_{\scriptscriptstyle{T}}{f}}_{*}(M)$ has cohomology at most in degree $0$
so we identify it with $\omega^{H}(M)$. Let $\sigma\colon A\times A\to A$ be
the group law. Then we have $f\circ\sigma=f\times f$, hence the Leray spectral
sequence [SS18, cor. 12.7.38] gives

${{}_{\scriptscriptstyle{T}}{f}}_{*}\circ{{}_{\scriptscriptstyle{T}}{\sigma}}_{*}={{}_{\scriptscriptstyle{T}}{(f\times
f)}}_{*}.$

Thus in
$\operatorname{HM}(\\{\operatorname{pt}\\})=\operatorname{Vect}_{\mathbb{Z}\times\mathbb{Z}}(\mathbb{C})$
we obtain

$\displaystyle\omega^{H}(M_{1}*M_{2})$
$\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{f}}_{*}(M_{1}*M_{2})$
$\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{f}}_{*}{{}_{\scriptscriptstyle{T}}{\sigma}}_{*}(M_{1}\boxtimes
M_{2})$ $\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{(f\times
f)}}_{*}(M_{1}\boxtimes M_{2})$
$\displaystyle\;\simeq\;{{}_{\scriptscriptstyle{T}}{f}}_{*}(M_{1})\otimes{{}_{\scriptscriptstyle{T}}{f}}_{*}(M_{2})$
$\displaystyle\;\simeq\;\omega^{H}(M_{1})\otimes\omega^{H}(M_{2}),$

where in the third step we have used the Künneth isomorphism. ∎

The above can be applied to any Hodge module $M\in\operatorname{HM}(A)$ after
twisting by a suitable rank one local system. More precisely, by generic
vanishing on abelian varieties [KW15, Sch15], we can find a unitary local
system $L$ of rank one on $A$ such that

$M\otimes\mathbb{L}\;\in\;\operatorname{HM}_{0}(A).$

Recall that by lemma 2.6, the isomorphism class of the Tannaka group is
unaffected by the twist. After the twist, we obtain:

######  Corollary 3.2.

For $M$, $\mathbb{L}$ as above, there is a morphism
$\lambda\colon\mathbb{G}_{m}^{2}\to G_{\omega}(M\otimes\mathbb{L})$ of
algebraic groups such that the Hodge decomposition

$\omega^{H}(M\otimes\mathbb{L})\;=\;\bigoplus_{(p,q)\in\mathbb{Z}^{2}}\textup{H}^{p,q}(M\otimes\mathbb{L})$

is the decomposition in weight spaces for the characters
$(p,q)\in\mathbb{Z}^{2}=\operatorname{Hom}(\mathbb{G}_{m}^{2},\mathbb{G}_{m})$.

In classical Hodge theory the bigrading is given by a representation of
$\mathbb{S}_{\mathbb{C}}=\mathbb{G}_{m}^{2}$ for
$\mathbb{S}=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(\mathbb{G}_{m})$. By analogy,
we give the following definition:

######  Definition 3.3.

The group morphism $\lambda\colon\mathbb{G}_{m}^{2}\to
G_{\omega}(\mathcal{C})$ in the above corollary is called the _Hodge
cocharacter_.

The fact that $\lambda$ factors over the Tannaka group will allow us to obtain
information about Tannaka groups from information about Hodge numbers.

### âÄ¢

Hodge level of intersection complexes Any object of $\operatorname{HM}(A)$ is
a direct sum of simple objects, and any simple Hodge module is the
intersection complex $\operatorname{IC}_{Z}(\mathbb{V})$ of a variation of
Hodge structures $\mathbb{V}$ on an open dense subset of an integral
subvariety $Z\subset A$. We are interested in the twisted Hodge numbers

$h^{p}(M)\;:=\;h^{p,w-p}(M)\;:=\;\dim\textup{H}^{p,w-p}(M\otimes\mathbb{L})$

where the twist is by a unitary local system L of rank one with
$\textup{H}^{i}(A,\operatorname{DR}(M)\otimes L)=0$ for all $i\neq 0$; we drop
the local system from the notation since it does not affect the Hodge numbers.
If $\mathbb{V}=\mathbb{C}_{U}$ is the trivial variation of Hodge structures on
the smooth locus $U\subset X$ of an integral subvariety $X\subset A$, we also
write

$\delta_{X}^{H}\;:=\;\operatorname{IC}_{X}(\mathbb{C}_{U})\;\in\;\operatorname{HM}(A,d).$

The underlying perverse sheaf is the perverse intersection complex
$\delta_{X}:=\operatorname{DR}(\delta_{X}^{H})$ of the subvariety, and we
write

$h^{p}(X)\;:=\;h^{p}(\delta_{X}^{H}).$

For smooth $X$ these twisted Hodge numbers have the following properties:

######  Lemma 3.4.

If $X\subset A$ is a smooth subvariety of dimension $d$, then for all
$p\in\mathbb{Z}$ we have

$h^{p}(X)\;=\;(-1)^{d-p}\chi(X,\Omega^{p}_{X})\;=\;h^{d-p}(X).$

If moreover $X\subset A$ has ample normal bundle and $d<g=\dim A$, then these
Hodge numbers satisfy

$h^{p}(X)\;\geqslant\;\begin{cases}g-d+1&\text{for\leavevmode\nobreak\
$p\in\\{0,d\\}$},\\\\[1.00006pt] 2&\text{for\leavevmode\nobreak\
$p\in\\{1,d-1\\}$},\\\\[1.00006pt] 1&\text{for\leavevmode\nobreak\ $2\leqslant
p\leqslant d-2$.}\end{cases}$

###### Proof.

If $X$ is smooth, then $h^{p}(X)=(-1)^{d-p}\chi(X,\Omega^{p}_{X})$ by [KM23,
appendix B], thus $h^{p}(X)=h^{d-p}(X)$ by Serre duality. For the inequalities
see corollary A.2. ∎

######  Corollary 3.5.

For any smooth subvariety $X\subset A$ with nonzero ample normal bundle, its
dimension is equal to the level of the Hodge decomposition:

$\dim X\;=\;\max\\{|p-2d|\colon h^{p}(X)\neq 0\\}.$

For perverse intersection complexes of singular subvarieties $X\subset A$ we
still have an upper bound. More generally, for intersection complexes of
variations $\mathbb{V}$ of Hodge structures, denote by

$\ell(\mathbb{V})\;:=\;\max\\{|p-q|\colon\mathbb{V}^{p,q}\neq 0\\}$

the _level_ of the Hodge structure. We then obtain:

######  Lemma 3.6.

Let $X\subset A$ be an integral subvariety of dimension $d$. Then for any pure
polarizable variation of $\mathbb{C}$-Hodge structures on an open dense subset
$U\subset X^{\textup{reg}}$ we have

$h^{p,q}(\operatorname{IC}_{X}(\mathbb{V}))\;=\;0\quad\text{for}\quad|p-q|\;>\;\ell(\mathbb{V})+d,$

###### Proof.

By definition
$h^{p,q}(\operatorname{IC}_{X}(\mathbb{V}))=\dim\textup{H}^{p,q}(\operatorname{IC}_{X}(\mathbb{V})\otimes\mathbb{L})$,
where $\mathbb{L}$ is a unitary local system of rank one. Since

$\operatorname{IC}_{X}(\mathbb{V})\otimes\mathbb{L}\;=\;\operatorname{IC}_{X}(\mathbb{V}\otimes\mathbb{L}),$

we may replace $\mathbb{V}$ by the variation of $\mathbb{C}$-Hodge structures
$\mathbb{V}\otimes\mathbb{L}$ which has the same level. Hence in what follows
we will assume that the unitary local system $\mathbb{L}$ is trivial.

Now $\mathbb{V}$ embeds into a variation of pure polarizable
$\mathbb{R}$-Hodge structures of the same level. To see this, write
$\mathcal{V}$ for the flat $\mathcal{C}^{\infty}$-bundle underlying
$\mathbb{V}$. Define $\bar{\mathbb{V}}$ to be the variation of
$\mathbb{C}$-Hodge structures whose underlying flat
$\mathcal{C}^{\infty}$-bundle is the complex conjugate of $\mathcal{V}$ and
whose component of bidegree $(p,q)$ is the complex conjugate of
$\mathcal{V}^{q,p}$. This defines a variation of $\mathbb{C}$-Hodge structures
by [SS18, sect. 4.1.a]. Note that a polarization on $\mathbb{V}$ induces a
natural polarization on $\bar{\mathbb{V}}$. Therefore
$\mathbb{W}=\mathbb{V}\oplus\bar{\mathbb{V}}$ is the complexification of the
sought-for polarizable variation of $\mathbb{R}$-Hodge structures, see also
[SS18, rem. 4.1.12]. Replacing $\mathbb{V}$ by $\mathbb{W}$ we assume in what
follows that $\mathbb{V}$ comes from a polarizable variation of
$\mathbb{R}$-Hodge structures.

Now let $k=\max\\{p\in\mathbb{Z}\mid F^{p}\mathbb{V}\neq 0\\}$, so that
$F^{k}\mathbb{V}\subset\mathbb{V}$ is the smallest piece in the Hodge
filtration. If the variation of Hodge structures has weight $w$, then by Hodge
symmetry the Hodge numbers of its fibers can be nonzero at most in bidegrees
$(i,w-i)$ with $k\leqslant i\leqslant w-k$. So the level of the Hodge
structure on the fibers is

$\ell(\mathbb{V})\;=\;2k-w.$

Now by the conventions of [SS18, appendix A.7], the right
$\mathcal{D}_{U}$-module $\mathcal{M}_{U}:=\omega_{U}\otimes\mathbb{V}$
associated to the variation of Hodge structures comes with the increasing
Hodge filtration

$F_{p}\mathcal{M}_{U}\;:=\;\omega_{U}\otimes F^{p-d}\mathbb{V}$

which starts in degree $p=k+d$. By the theory of Hodge modules this filtration
naturally extends to an increasing filtration on the right
$\mathcal{D}_{A}$-module $\mathcal{M}$ underlying the Hodge module
$M=\operatorname{IC}_{X}(\mathbb{V})$. Moreover, the lowest piece
$F_{p}\mathcal{M}$ of this filtration is still in degree $p=k+d$, see for
instance [SY23, sect. 2.4] in the context of Saito’s Hodge modules. Now
consider the de Rham complex

$\operatorname{DR}(\mathcal{M})\;=\;\bigl{[}\cdots\to\mathcal{M}\otimes\wedge^{2}\mathcal{T}_{A}\to\mathcal{M}\otimes\mathcal{T}_{A}\to\mathcal{M}\bigr{]}$

where the last term $\mathcal{M}$ is placed in degree zero. It comes with the
increasing filtration

$F_{p}\operatorname{DR}(\mathcal{M})\;=\;\bigl{[}\cdots\to
F_{p-2}(\mathcal{M})\otimes\wedge^{2}\mathcal{T}_{A}\to
F_{p-1}(\mathcal{M})\otimes\mathcal{T}_{A}\to F_{p}(\mathcal{M})\bigr{]}$

which by the above starts in degree $p=k+d$. Therefore, the Hodge filtration
on de Rham cohomology

$F_{p}\textup{H}^{0}(A,\operatorname{DR}(M))\;=\;\operatorname{Im}\bigl{(}\textup{H}^{0}(A,F_{p}\operatorname{DR}(M))\to\textup{H}^{0}(A,\operatorname{DR}(M)\bigr{)}$

can only start in degree $p=k+d$ or higher. Since $M$ is a pure Hodge module
of weight $w+d$ by [SS18, th. 14.6.1], the de Rham cohomology
$\textup{H}^{0}(A,\operatorname{DR}(M))$ in degree zero is a pure Hodge
structure of weight $w+d$ by th. 14.3.2(2) in loc. cit. So it follows again by
Hodge symmetry that its level is

$\ell(\textup{H}^{0}(A,\operatorname{DR}(M))\;\leqslant\;2(k+d)-(w+d)\;=\;(2k-w)+d\;=\;\ell(\mathbb{V})+d,$

which concludes the proof. ∎

The above then also gives an estimate on the level of the Hodge structures
arising from the decomposition theorem applied to intersection complexes:

######  Lemma 3.7.

Let $Y$ be a smooth variety with a proper morphism $f\colon Y\to A$. Then for
every direct summand
$M\subset{{}_{\scriptscriptstyle{T}}f}_{*}^{(0)}(\delta_{Y}^{H})$, its twisted
Hodge numbers satisfy

$h^{p,q}(M)\;=\;0\quad\text{for}\quad|p-q|\;>\;\dim
f^{-1}(\operatorname{Supp}M).$

###### Proof.

Recall that by $S$-decomposability of Hodge modules [SS18, th. 14.2.17(1)], we
have

${{}_{\scriptscriptstyle{T}}f}_{*}^{(0)}(\delta_{Y}^{H})\;=\;\bigoplus_{X\subset
A}M_{X}$

where $X\subset A$ runs through all integral subvarieties and
$M_{X}\subset{{}_{\scriptscriptstyle{T}}f}_{*}^{(0)}(\delta_{Y}^{H})$ denotes
the maximal Hodge submodule with strict support $X$. It suffices to prove the
lemma when $M=M_{X}$ for some $X\subset A$. Since the strict support
decomposition is compatible with the natural functor from Saito’s
$\mathbb{Q}$-Hodge modules to complex Hodge modules, the summand
$M\subset{{}_{\scriptscriptstyle{T}}f}_{*}^{(0)}(\delta_{Y}^{H})$ then arises
from a direct summand in the category of $\mathbb{Q}$-Hodge modules; this will
allow us to embed its generic stalk cohomology in the mixed Hodge structure of
a singular variety even though the mixed case is not yet written for complex
Hodge modules [SS18].

Since $M$ has strict support $X$, we have
$M=\operatorname{IC}_{X}(\mathbb{V})$ for a polarizable variation $\mathbb{V}$
of $\mathbb{C}$-Hodge structures on an open dense subset of $X$. Let $d=\dim
X$, then at a general point $x\in X$ the fiber of the local system $V$
underlying the variation $\mathbb{V}$ is a direct summand

$V_{x}\;=\;\mathcal{H}^{-d}(\operatorname{DR}(M))_{x}\;\hookrightarrow\;\mathcal{H}^{-d}(Rf_{*}(\delta_{Y}))_{x}\;=\;\textup{H}^{\dim
Y-d}(f^{-1}(x)).$

By the discussion in the first paragraph, this inclusion is induced by an
inclusion of mixed $\mathbb{Q}$-Hodge structures. The fibers $f^{-1}(x)$ may
be singular, but since for any singular variety the mixed Hodge structure on
its cohomology has level at most the dimension of the variety, our variation
of Hodge structures has level

$\ell(\mathbb{V})\;\leqslant\;\ell(\textup{H}^{\bullet}(f^{-1}(Y))\;\leqslant\;\dim
f^{-1}(x)\;=\;\dim f^{-1}(X)-d$

where the last equality uses that $x\in X$ is general. Since
$M=\operatorname{IC}_{X}(\mathbb{V})$, the previous lemma 3.6 gives
$h^{p,q}(M)=0$ for $|p-q|>(\dim f^{-1}(X)-d)+d=\dim f^{-1}(X)$. ∎

##  4\. Subvarieties with exceptional Tannaka groups

We now discuss the restrictions which the above results impose on subvarieties
with Tannaka group $E_{6}$ and $E_{7}$. As above we work over the complex
numbers.

###  4.1. Reduction to semisimple groups

For Hodge modules of nonzero weight the Tannaka group is not semisimple: The
weight is given by a central cocharacter of this group. But for fixed weight
we can still describe the Hodge decomposition by a cocharacter with values in
a connected semisimple group as follows. To fix notation, suppose we are given

  * •

a finite-dimensional vector space $V$ over $\mathbb{C}$,

  * •

a connected semisimple subgroup $G\subset\operatorname{GL}(V)$,

  * •

a cocharacter $\lambda\colon\mathbb{G}_{m}^{2}\to Z\cdot G$ where
$Z=\mathbb{G}_{m}\cdot\operatorname{id}_{V}\subset\operatorname{GL}(V)$,

such that the weight spaces

$V^{p,q}:=\\{v\in V\mid\lambda(x,y)(v)=x^{p}y^{q}\cdot v\;\;\text{for
all}\;\;(x,y)\in\mathbb{G}_{m}^{2}\\}$

satisfy Hodge symmetry in the sense that $\dim V^{p,q}=\dim V^{q,p}$ for all
$p,q$. The Hodge symmetry implies $\lambda(z,z^{-1})\in\operatorname{SL}(V)$
for all $z\in\mathbb{G}_{m}$. On the other hand we have

$G\;=\;(Z\cdot G)\cap\operatorname{SL}(V)$

since $G$ is a connected semisimple group. So $\lambda$ restricts on the
antidiagonal to a cocharacter

$\rho\colon\quad\mathbb{G}_{m}\;\longrightarrow\;G,\quad
z\;\longmapsto\;\lambda(z,z^{-1})$

with values in the connected semisimple subgroup
$G\subset\operatorname{GL}(V)$. For $v\in V^{p,q}$ we have

$\rho(z)(v)\;=\;z^{p-q}\cdot v\;=\;z^{2p-w}\cdot v\quad\text{for the
weight}\quad w\;:=\;p+q.$

We will apply this in the following situation:

######  Example 4.1.

Let $M\in\operatorname{HM}(A,w)$ be a simple Hodge module on a complex abelian
variety $A$. Assume that the Hodge decomposition

$V\;:=\;\textup{H}^{0}(A,\operatorname{DR}(M\otimes\mathbb{L}))\;=\;\bigoplus_{p+q=w}V^{p,q}$

from section 3.1 satisfies Hodge symmetry in the sense that $\dim V^{p,q}=\dim
V^{q,p}$ for all $p,q\in\mathbb{Z}$. Recall that this decomposition is given
by the weights for the Hodge cocharacter

$\lambda\colon\quad\mathbb{G}_{m}^{2}\;\longrightarrow\;G_{\omega}(M)\;\subset\;\operatorname{GL}(V)\quad\text{with}\quad\lambda(z,w)_{|V^{p,q}}\;=\;z^{p}w^{q}\cdot\operatorname{id}_{V^{p,q}}.$

Since $M$ is a simple Hodge module, the group $G_{\omega}(M)$ acts on $V$ by
an irreducible representation. By Schur’s lemma its center then acts on $V$ by
scalars. Since any connected reductive group is the almost direct product of
its center and its derived group, it follows that

$G_{\omega}^{\circ}(M)\;\subset\;Z\cdot G\quad\text{for}\quad
Z\;:=\;\mathbb{G}_{m}\cdot\operatorname{id}_{V}\;\subset\;\operatorname{GL}(V).$

So we are in the situation described above.

###  4.2. Cocharacters with Hodge properties

Let $G$ be a complex semisimple algebraic group, and let $V$ be a finite-
dimensional representation of $G$ with finite kernel. Any cocharacter
$\lambda\colon\mathbb{G}_{m}\rightarrow G$ induces a weight space
decomposition

$V\;=\;\bigoplus_{n\in\mathbb{Z}}V^{n}(\lambda)$

where $\lambda$ acts on $V^{n}(\lambda)$ via the character
$n\in\mathbb{Z}=\operatorname{Hom}(\mathbb{G}_{m},\mathbb{G}_{m})$. We want to
determine all choices of the cocharacter $\lambda$ such that the Hodge numbers
$\dim V^{n}(\lambda)$ have the following properties:

  1. (H1)

Hodge symmetry: $\dim V^{n}(\lambda)=\dim V^{-n}(\lambda)$ for all
$n\in\mathbb{Z}$.

  2. (H2)

Gap freeness: There exists an integer $\ell\geqslant 0$ such that

$V^{n}(\lambda)\;\neq\;0\quad\Longleftrightarrow\quad
n\;=\;2i-\ell\;\text{with}\;i\;\in\;\\{0,1,\dots,\ell\\}.$

  3. (H3)

Outer Hodge number estimate: For $\ell$ as above we have

$\dim V^{\ell-1}(\lambda)\geqslant 2\quad\textup{and}\quad\dim
V^{\ell}(\lambda)\geqslant 3.$

The set of such cocharacters is a finite union of conjugacy classes:

######  Lemma 4.2.

For any given $(G,V)$, there exist up to $G$-conjugacy only finitely many
cocharacters $\lambda$ that satisfy the property (H2).

###### Proof.

Fix a maximal torus $T\subset G$. Any cocharacter is conjugate to one with
values in this torus, so it suffices to consider cocharacters $\lambda\in
X_{\ast}(T)=\operatorname{Hom}(\mathbb{G}_{m},T)$. The weight space
decomposition for $\lambda$ is then obtained as follows: Consider the weight
space decomposition

$V\;=\;\bigoplus_{\chi\in X^{\ast}(T)}V(\chi)$

for our maximal torus, where the action of the torus on the subspace
$V(\chi)\subset V$ is given by $\chi\in
X^{\ast}(T)=\operatorname{Hom}(T,\mathbb{G})$. Then the weight $n$ space for
the cocharacter $\lambda$ is the direct sum

$V^{n}(\lambda)\;=\;\bigoplus_{\langle\lambda,\chi\rangle=n}V(\chi)$

where the sum is over all characters $\chi\in X^{\ast}(T)$ with
$\langle\chi,\lambda\rangle=n$. To make use of this, let

$S\;:=\;\\{\chi\in X^{\ast}(T)\mid V(\chi)\neq 0\\}$

be the set of characters that enter in the given representation. Condition
(H2) then implies

$|\langle\lambda,\chi\rangle|\;\leqslant\;\ell\;<\;\dim V\quad\text{for
all}\quad\chi\;\in\;S.$

This condition leaves only finitely many choices of $\lambda\in
X_{\ast}(T)=\operatorname{Hom}(\mathbb{G}_{m},T)$ in the cocharacter lattice,
since in the dual character lattice the occurring weights $\chi\in S$ span a
subgroup

$\langle S\rangle_{\mathbb{Z}}\;\subset\;X^{\ast}(T)$

of finite index: Indeed, if the subgroup spanned by those weights had smaller
rank than $X^{\ast}(T)$, then we could find $\mu\in
X_{\ast}(T)\smallsetminus\\{0\\}$ with $\langle\mu,\chi\rangle=0$ for all
$\chi\in S$. But then the image of

$\mu\colon\quad\mathbb{G}_{m}\;\longrightarrow\;G$

would be a subgroup of positive dimension acting trivially on the
representation $V$ which is impossible since $V$ has finite kernel. ∎

The above proof allows to enumerate all cocharacters $\lambda$ with (H1),
(H2), (H3) for a given $(G,V)$ by an exhaustive computer search. Below, we
will list the results for the exceptional groups $E_{6}$ and $E_{7}$ in their
minimal representations.

###  4.3. Cocharacters of $G=E_{6}$

Consider the simply connected group $G=E_{6}$. Fix a system of positive roots
for a maximal torus $T\subset G$. We label the fundamental dominant weights

$\varpi_{1},\dots,\varpi_{6}\;\in\;X^{\ast}(T)$

as in [Bou02, ch. VI, sect. 4.12], and the dual basis by
$\varpi_{1}^{\vee},\dots,\varpi_{6}^{\vee}\in X_{\ast}(T)$. An exhaustive
computer search gives:

######  Proposition 4.3.

Let $V$ be a $27$-dimensional irreducible representation of $E_{6}$, then for
any cocharacter

$\lambda\colon\quad\mathbb{G}_{m}\;\longrightarrow\;E_{6}$

with (H1), (H2), (H3), the Hodge numbers $h^{i}(\lambda):=\dim
V^{2i-\ell}(\lambda)$ for $0\leqslant i\leqslant\ell$ are given by one of the
rows of the following table:

$\begin{array}[]{c|c|c|c|c|c|c}h^{0}&h^{1}&h^{2}&h^{3}&h^{4}&h^{5}&h^{6}\\\
\hline\cr 6&15&6&&&&\\\ 3&6&9&6&3&&\\\ 3&3&3&9&3&3&3\\\ \end{array}\vskip
6.0pt plus 2.0pt minus 2.0pt$

Moveover, up to conjugacy the only cocharacter giving the Hodge numbers
$6,15,6$ in the first row of the table is the cocharacter
$\lambda=\varpi_{1}^{\vee}$.

######  Remark 4.4.

On the adjoint representation $W=\operatorname{Lie}E_{6}$, the cocharacter
$\lambda=\varpi_{1}^{\vee}$ induces a grading of length $\ell=4$ with

$\dim W^{2i-\ell}(\lambda)\;=\;\begin{cases}1&\text{for\leavevmode\nobreak\
$i=0,4$},\\\ 20&\text{for\leavevmode\nobreak\ $i=1,3$},\\\
36&\text{for\leavevmode\nobreak\ $i=2$}.\end{cases}$

This information will be useful in the proof of proposition 4.8 below.

###  4.4. Cocharacters of $G=E_{7}$

Now take the simply connected group $G=E_{7}$, then as above an exhaustive
computer search gives:

######  Proposition 4.5.

Let $V$ be a $56$-dimensional irreducible representation of $E_{7}$, then for
any cocharacter

$\lambda\colon\quad\mathbb{G}_{m}\;\longrightarrow\;E_{7}$

with (H1), (H2), (H3) whose associated Hodge decomposition has odd length
$\ell$, the corresponding Hodge numbers $h^{i}(\lambda)=\dim
V^{2i-\ell}(\lambda)$ for $0\leqslant i\leqslant\ell$ are given by one of the
rows of the following table:

$\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}h^{0}&h^{1}&h^{2}&h^{3}&h^{4}&h^{5}&h^{6}&h^{7}&h^{8}&h^{9}&h^{10}&h^{11}&h^{12}&h^{13}&h^{14}&h^{15}\\\
\hline\cr 7&21&21&7&&&&&&&&&&&&\\\ 6&7&15&15&7&6&&&&&&&&&&\\\
6&6&1&15&15&1&6&6&&&&&&&&\\\ 5&3&10&10&10&10&3&5&&&&&&&&\\\
5&2&6&5&10&10&5&6&2&5&&&&&&\\\ 4&3&3&12&6&6&12&3&3&4&&&&&&\\\
5&2&5&1&5&10&10&5&1&5&2&5&&&&\\\ 4&2&3&5&8&6&6&8&5&3&2&4&&&\\\
4&2&2&5&1&8&6&6&8&1&5&2&2&4&\\\ 4&2&2&4&1&1&8&6&6&8&1&1&4&2&2&4\\\
\end{array}\vskip 6.0pt plus 2.0pt minus 2.0pt$

###  4.5. Hodge numbers of exceptional subvarieties

Putting together the above, we obtain the following restriction on smooth
subvarieties $X\subset A$ of complex abelian varieties $A$ such that the
Tannaka group

$G_{X,\omega}^{\ast}\;:=\;G_{\omega}^{\ast}(\delta_{X})$

of their perverse intersection complex is a simple exceptional group:

######  Proposition 4.6.

Let $A$ be an abelian variety of dimension $g$, and $X\subset A$ a smooth
nondivisible irreducible subvariety of dimension $d$ with ample normal bundle
such that $G_{X,\omega}^{\ast}$ is a simple exceptional group. Then, either

  1. (1)

$G_{X,\omega}^{\ast}\simeq E_{6}$, $\dim V_{X}=27$ and $d\in\\{2,4,6\\}$, or

  2. (2)

$G_{X,\omega}^{\ast}\simeq E_{7}$, $\dim V_{X}=56$ and $d\in\\{3,\dots,15\\}$
is odd.

Moreover $g\leqslant g_{\max}$ for the following upper bound $g_{\max}$
depending on $d$:

$\displaystyle(1)\quad\begin{array}[]{c|ccccc}d&2&4&6\\\ \hline\cr
g_{\max}&7&6&8\end{array}$
$\displaystyle\quad(2)\quad\begin{array}[]{c|ccccccc}d&3&5&7&9&11&13&15\\\
\hline\cr g_{\max}&9&10&12&13&15&16&18\end{array}$

###### Proof.

Since $X\subset A$ is a smooth subvariety, the Tannaka group
$G_{X,\omega}^{\ast}$ acts on $V_{X}$ via a minuscule representation [Krä22,
cor. 1.10]. For the simple exceptional groups the only minuscule
representations are the irreducible representations of dimension $27$ for
$E_{6}$ and of dimension $56$ for $E_{7}$. The latter representation admits a
nondegenerate bilinear form which is alternating, so in this case $X$ must
have odd dimension by [JKLM23, sect. 1.2]; hence for $E_{7}$ we only need to
consider cocharacters whose associated Hodge decomposition has odd length $d$.
Also, when $X$ is a divisor, the topological Euler characteristic of $X$ is
divisible by $g!$, hence cannot be $27$ or $56$ up to sign. Therefore, we can
assume $d\leqslant g-2$. Since $X\subset A$ has ample normal bundle, lemma 3.4
and corollary 3.5 imply that the Hodge cocharacter induces a cocharacter of
$G_{X,\omega}^{\ast}$ satisfying properties (H1), (H2) and (H3). Lemma 3.4
also implies the following bound on the codimension:

$g-d+1\;\leqslant\;h^{d}(X)\;:=\;\chi(X,\Omega^{d}_{X}).$

So it follows that the Hodge numbers
$h^{p}(X)=(-1)^{d-p}\chi(X,\Omega^{p}_{X})$ must be among those in
propositions 4.3, LABEL: and 4.5. ∎

######  Corollary 4.7.

If in the above proposition we moreover assume that $d<g/2$, then the
topological Euler characteristic $\chi_{\textup{top}}(X)=(-1)^{d}\dim V_{X}$
and the Hodge numbers $h^{p}(X)=(-1)^{d-p}\chi(X,\Omega^{p}_{X})$ are among
one of the following cases:

$\begin{array}[]{c|c|c|c|c|c|c|c}G_{X}^{\ast}&\dim
V_{X}&d&g&h^{0}&h^{1}&h^{2}&h^{3}\\\ \hline\cr E_{6}&27&2&5,6,7&6&15&6&\\\
E_{7}&56&3&7,8,9&7&21&21&7\end{array}\vskip 6.0pt plus 2.0pt minus 2.0pt$

In the rest of this paper we will show that the $E_{6}$ case in the above
table occurs only when $X$ is the Fano surface of lines on a smooth cubic
threefold, and $A$ is isogenous to its Albanese variety (which has $g=5$).

###  4.6. Surfaces with group $E_{6}$ and the difference morphism

Let us take a closer look at the first row of the table in corollary 4.7. The
corollary in particular says that if $X\subset A$ is a smooth nondivisible
irreducible subvariety with ample normal bundle and dimension $d<g/2$ such
that $G_{X,\omega}^{\ast}\simeq E_{6}$, then $X$ must be a surface with

$\chi(X,\mathcal{O}_{X})=6,\qquad\chi(X,\Omega^{1}_{X})=-15,\qquad\chi(X,\Omega^{2}_{X})=6.$

We also know the Chern numbers: The top Chern class is
$c_{2}(X)=\dim\omega(\delta_{S})=27$, hence $c_{1}(X)^{2}=45$ by Noether’s
formula. To show that $X$ is indeed the Fano surface of lines on a smooth
cubic threefold, we will need one more piece of numerical information: The
degree of the difference morphism

$d\colon\quad X\times X\;\longrightarrow\;X-X\;\subset\;A.$

The representation theory of the exceptional group $E_{6}$ implies:

######  Proposition 4.8.

Let $A$ be an abelian variety of dimension $g\geqslant 5$ and $X\subset A$ a
smooth nondegenerate nondivisible surface with $G_{X,\omega}^{\ast}\simeq
E_{6}$. Then the difference morphism $d\colon X\times X\to X-X$ is generically
finite of degree $\deg(d)\geqslant 6$.

###### Proof.

Let $D:=X-X$. Since $X$ is nondegenerate, the morphism $d\colon X\times X\to
D$ is generically finite, hence over some open dense subset $U\subset
D^{\textup{reg}}$ it restricts to a finite étale cover of degree $\deg(d)$.
The direct image $\delta_{X}*\delta_{-X}=Rd_{*}(\delta_{X\times X})$ therefore
restricts on $U$ to a local system of rank $\deg(d)$, placed in cohomological
degree $-\dim D=-4$. By adjunction that local system contains the trivial
local system of rank one as a direct summand. The decomposition theorem
therefore shows that

(1a) $\delta_{X}*\delta_{-X}\;\simeq\;\delta_{D}\oplus P$

where $P$ is a semisimple perverse sheaf on $D$ which restricts on $U\subset
D$ to a local system of rank $\deg(d)-1$, placed in cohomological degree $-4$.
Passing to the clean characteristic cycles of these perverse sheaves in the
sense of [JKLM23, sect. 5.3], we get

(1b) $\displaystyle\operatorname{cc}(\delta_{D})$
$\displaystyle\;=\;\Lambda_{D}+\sum_{Z}m_{Z}(\delta_{D})\cdot\Lambda_{Z},$
(1c) $\displaystyle\operatorname{cc}(P)$
$\displaystyle\;=\;(\deg(d)-1)\cdot\Lambda_{D}+\sum_{Z}m_{Z}(P)\cdot\Lambda_{Z}$

with certain integers $m_{Z}(\delta_{D}),m_{Z}(P)\geqslant 0$. On the right-
hand side of these equations the sums are taken over all integral proper
subvarieties $Z\subsetneq D$, and $\Lambda_{Z}\subset T^{*}A$ denotes the
conormal variety to $Z$ inside $A$. Hence we only need to show that the
conormal variety $\Lambda_{D}$ enters in
$\operatorname{cc}(\delta_{X}*\delta_{-X})$ with multiplicity at least six.

To interpret this in terms of representation theory, we need some notation.
Fix a maximal torus $T\subset E_{6}$. We use the fundamental weights
$\varpi_{1},\dots,\varpi_{6}\in X^{\ast}(T)$ from section 4.3. The dominant
integral weights are precisely those of the form

$\lambda\;=\;a_{1}\varpi_{1}+\cdots+a_{6}\varpi_{6}\quad\text{with
integers}\quad a_{1},\dots,a_{6}\;\geqslant\;0,$

and for any such weight we will denote by $V_{\lambda}$ the irreducible
representation of $E_{6}$ of highest weight $\lambda$. The classes of these
irreducible representations form a basis for the representation ring
$\textup{R}(E_{6})$, the Grothendieck ring of the tensor category of all
finite-dimensional representations of $E_{6}$. We identify the representation
ring $\textup{R}(E_{6})$ with the Weyl group invariants in the group algebra
$\mathbb{Z}[X^{\ast}(T)]$ by the map sending a representation $V$ to its
character

$\operatorname{char}(V)\;\in\;\mathbb{Z}[X^{\ast}(T)].$

In these terms the representation ring $\textup{R}(E_{6})$ has a basis
consisting of the Weyl group orbits of the dominant integral weights. For a
weight $\lambda\in X^{\ast}(T)$ we denote by $[\lambda]\in\textup{R}(E_{6})$
its Weyl group orbit. The character of a representation is a single Weyl group
orbit only if the representation is minuscule, so the basis given by Weyl
group orbits differs from the one given by the irreducible representations. In
what follows we are interested in the irreducible representations that appear
in the tensor product of the 27-dimensional representation $V=V_{\varpi_{1}}$
with its dual $V^{\vee}=V_{\varpi_{6}}$:

(2a) $V\otimes V^{\vee}\;\simeq\;V_{\alpha}\oplus
V_{\beta}\oplus\bf{1}\quad\text{with}\quad\begin{cases}\alpha\;=\;\varpi_{2},\\\
\beta\;=\;\varpi_{1}+\varpi_{6}.\end{cases}$

Here $V_{\alpha}$ is the adjoint representation of $E_{6}$ and $V_{\beta}$ is
an irreducible representation of dimension 650. Their characters are

(2b) $\displaystyle\operatorname{char}(V_{\alpha})$
$\displaystyle\;=\;\phantom{0}6\cdot[0]+1\cdot[\alpha],$ (2c)
$\displaystyle\operatorname{char}(V_{\beta})$
$\displaystyle\;=\;20\cdot[0]+5\cdot[\alpha]+1\cdot[\beta].$

Now we come back to geometry. Fix an isomorphism $G_{X,\omega}^{\ast}\simeq
E_{6}$. By corollary 2.5 we have
$G_{\omega}^{\ast}(\delta_{X}^{H})=G_{X,\omega}^{\ast}$, so we may work as
well with Hodge modules. Since $\omega(\delta_{X}^{H})$ is a minuscule
representation, it must be one of the two irreducible representations of
dimension 27, which are dual to each other. So we may assume
$\omega(\delta_{X}^{H})\simeq V$. Then eq. 2a translates to

(3a) $\delta_{X}^{H}*\delta_{-X}^{H}\;\simeq\;M_{\alpha}\oplus
M_{\beta}\oplus\delta_{0}^{H}$

where for $\lambda\in\\{\alpha,\beta\\}$, we denote by
$M_{\lambda}\in\langle\delta_{X}^{H}\rangle$ the simple Hodge module with
associated representation $\omega(M_{\lambda})\simeq V_{\lambda}$. The
characteristic cycles of these Hodge modules can be computed as follows: By
[JKLM23, th. 5.11, lemma 5.14] there is a natural way to attach to any Weyl
group orbit $[\lambda]$ an effective Lagrangian cycle $\Lambda_{\lambda}$ on
the cotangent bundle $T^{*}A$ such that if we extend this map additively to
the representation ring $\textup{R}(E_{6})$, then it sends the character of
any representation to the characteristic cycle of the corresponding perverse
sheaf. Thus eqs. 2b, LABEL: and 2c translate to

(3b) $\displaystyle\operatorname{cc}(M_{\alpha})$
$\displaystyle\;=\;\phantom{0}6\cdot\Lambda_{0}+1\cdot\Lambda_{\alpha},$ (3c)
$\displaystyle\operatorname{cc}(M_{\beta})$
$\displaystyle\;=\;20\cdot\Lambda_{0}+5\cdot\Lambda_{\alpha}+1\cdot\Lambda_{\beta},$

where all summands on the right-hand side are effective Lagrangian cycles.

Comparing eq. 1a and eq. 3a, we see that $\delta_{D}^{H}\simeq M_{\lambda}$
for some $\lambda\in\\{\alpha,\beta\\}$. We claim that

$\delta_{D}^{H}\;\simeq\;M_{\alpha}.$

Indeed, if $\delta_{D}^{H}\simeq M_{\beta}$, then
$\operatorname{cc}(M_{\beta})$ would contain the irreducible component
$\Lambda_{D}$ with multiplicity one. By eq. 3c this component would then have
to enter in the summand $\Lambda_{\beta}$, since the other summands enter with
higher multiplicity. Then by eq. 1b the other summand $\Lambda_{\alpha}$ could
involve only conormal varieties $\Lambda_{Z}$ with

$\dim(Z)<\dim(D)=4.$

But then by eq. 3b the cycle $\operatorname{cc}(M_{\alpha})$ would also only
involve conormal varieties $\Lambda_{Z}$ with $\dim(Z)<4$. This would imply

$\dim\operatorname{Supp}M_{\alpha}\;<\;\dim\operatorname{Supp}D\;=\;4,$

hence $\dim d^{-1}(\operatorname{Supp}M_{\alpha})<4$ because $d\colon X\times
X\to D$ is dominant and $X\times X$ is irreducible. By lemma 3.7 then

$h^{p,q}(M_{\alpha})\;=\;0\quad\text{for}\quad|p-q|\;>\;3.$

But $\omega(M_{\alpha})\simeq V_{\alpha}$ is the adjoint representation of
$E_{6}$, and by remark 4.4 the Hodge cocharacter induces on the adjoint
representation a Hodge decomposition of level $4$, a contradiction. This
proves that $\delta_{D}^{H}\simeq M_{\alpha}$ as claimed. But then by eq. 3b
the conormal variety $\Lambda_{D}$ enters as a component in the cycle
$\Lambda_{\alpha}$. It then follows that the cycle

$\operatorname{cc}(\delta_{X}^{H}*\delta_{-X}^{H})\;=\;26\cdot\Lambda_{0}+6\cdot\Lambda_{\alpha}+1\cdot\Lambda_{\beta}$

contains $\Lambda_{D}$ with multiplicity at least six and we are done. ∎

The above numerical results in fact characterize Fano surfaces of smooth cubic
threefolds, as we will see in the classification of subvarieties with Tannaka
group $E_{6}$ in theorem 7.1. In particular, the degree of the morphism
$d\colon X\times X\to X-X$ is precisely $6$. However, for this will not use
any representation theory; instead, we will show by direct geometric arguments
that any smooth projective surface with the above numerical properties is
isomorphic to the Fano surface of a smooth cubic threefold. As a preparation
for this, we gather in the next two sections some general facts about cubic
hypersurfaces and Fano varieties of lines.

##  5\. Nonnormal cubic hypersurfaces

In this section, we give a classification of nonnormal cubic hypersurfaces
over an arbitrary algebraically closed field $k$.

###  5.1. Statement

In order to understand the Fano variety of lines on a singular cubic threefold
it will be useful to have a complete classification of the nonnormal ones.
Examples of these are cones over nonnormal cubic plane curves and cones over
nonnormal cubic surfaces, where by a cone we mean the following:

######  Definition 5.1.

Let $\pi\colon V\to W$ be a linear map of finite-dimensional $k$-vector spaces
and $Y\subset\mathbb{P}(W)$ a subvariety. If $\pi$ is surjective but not an
isomorphism, then the scheme-theoretic closure in $\mathbb{P}(V)$ of the
preimage of $Y$ under the induced morphism
$\mathbb{P}(V)\smallsetminus\mathbb{P}(\operatorname{Ker}\pi)\to\mathbb{P}(W)$
is called the _cone_ over $Y$ with respect to $\pi$.

The result of the classification of nonnormal cubic threefolds will be that
apart from cones, the only one is the following:

######  Example 5.2.

A subvariety $X\subset\mathbb{P}^{4}$ is said to be a _twisted plane_ if it is
defined by the equation

$x_{0}^{2}x_{1}+x_{0}x_{2}x_{4}+x_{3}x_{4}^{2}=0$

for some homogeneous coordinates $x_{0},\dots,x_{4}$ on $\mathbb{P}^{4}$
[FP01, sect. 2.2.9]. The singular locus of $X$ is the plane $x_{0}=x_{4}=0$,
thus $X$ is nonnormal.

For cubic surfaces we correct some arguments proposed by Dolgachev [Dol12] in
the discussion after theorem 9.2.1. To this end, let $\pi\colon
S\to\mathbb{P}^{2}$ be the blow-up in a point $o$, with exceptional divisor
$E\subset S$. Consider the closed embedding

$S\;\lhook\joinrel\longrightarrow\;\mathbb{P}(V^{\vee})\quad\textup{where}\quad
V=\textup{H}^{0}(S,\mathcal{O}_{S}(2)(-E)),$

and let $\mathcal{O}_{S}(i):=\pi^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(i)$ for
$i\geqslant 0$. The purpose of this section is to prove the following result,
which is probably well-known but for which we could not find a complete proof
in the literature:

######  Proposition 5.3.

Let $X\subset\mathbb{P}^{n+1}$ be an integral nonnormal cubic hypersurface.

  * •

If $n=1$, then $X$ is a nodal or a cuspidal curve.

  * •

If $n=2$, then $X$ is either a cone over a nonnormal cubic integral plane
curve or a projection of $S\subset\mathbb{P}(V^{\vee})$ from a point not in
$S$.

  * •

If $n=3$, then $X$ is either a cone over a nonnormal cubic surface or a
twisted plane as in example 5.2.

  * •

If $n\geqslant 4$, then $X$ is a cone over a nonnormal integral cubic
threefold.

The rest of the section will be devoted to the proof of this statement.

###  5.2. Reduction to surfaces

We begin with the following:

######  Lemma 5.4.

Let $X\subset\mathbb{P}^{n+1}$ be a nonnormal integral cubic hypersurface with
$n\geqslant 1$, then $X^{\mathrm{sing}}$ is a linear subspace of dimension
$n-1$.

###### Proof.

We argue by induction on $n\geqslant 1$. For $n=1$ the only nonnormal integral
cubic plane curves are the nodal and the cuspidal plane cubics, which both
have only one singular point. Suppose $n\geqslant 2$. By Serre’s criterion of
normality the singular locus of $X$ has dimension $n-1$. Let $Y\subset
X^{\mathrm{sing}}$ be an irreducible component of dimension $n-1$ and suppose
that $Y$ is not a linear subspace. Pick a point $y\in Y$ such that the line
through $y$ and a general point $y^{\prime}\in Y\smallsetminus\\{y\\}$ is not
contained in $Y$. As $\deg X=3$ and $Y\subset X^{\mathrm{sing}}$, every such
line is contained in $X$. An open subset of $X$ is thus covered by lines going
through $y$. It follows that $X$ is a cone with vertex $y$ over an integral
cubic hypersurface $Z\subset\mathbb{P}^{n}$, thus $Y^{\mathrm{sing}}$ is a
cone with vertex $y$ over $Z^{\mathrm{sing}}$. In particular we have $\dim
Z^{\mathrm{sing}}=n-2$, hence the cubic hypersurface $Z$ is not normal. By the
induction hypothesis, $Z^{\mathrm{sing}}$ is a linear subspace of dimension
$n-2$, thus $Y^{\mathrm{sing}}$ is a linear subspace of dimension $n-1$. ∎

######  Lemma 5.5.

Let $X\subset\mathbb{P}^{n+1}$ be a nonnormal integral cubic hypersurface with
$n\geqslant 3$.

  1. (1)

If $n=3$, then $X$ is either a cone over a nonnormal cubic surface or the
twisted plane from example 5.2.

  2. (2)

If $n\geqslant 4$, then $X$ is a cone over a nonnormal integral cubic
threefold.

###### Proof.

By lemma 5.4 the singular locus is linear subspace of dimension $n-1$, which
we may assume to have equations $x_{0}=x_{1}=0$. In such coordinates, the
hypersurface $X$ is defined by an equation of the form

$x_{0}^{2}f_{1}+x_{0}x_{1}f_{2}+x_{1}^{2}f_{3}=0$

for some linear forms $f_{1},f_{2},f_{3}$. Let $m+2$ be the dimension of the
linear span of the linear forms $x_{0}$, $x_{1}$, $f_{1}$, $f_{2}$ and
$f_{3}$. If $n>m$, then $X$ is the cone over a cubic hypersurface in
$\mathbb{P}^{m+1}$ which is necessarily integral and nonnormal. Since
$m\leqslant 3\leqslant n$ the equality $n=m$ holds only for $m=3$ in which
case we may choose $f_{1},f_{2},f_{3}$ as coordinate functions and obtain that
$X$ is the twisted plane in example 5.2. ∎

With the above lemma, the proof of proposition 5.3 will be reduced at the end
of the next section to the study of nonnormal cubic surfaces given below.

###  5.3. Nonnormal cubic surfaces

In order to conclude the proof of proposition 5.3 we need to describe
nonnormal cubic surfaces which are not cones over nonnormal plane cubic
curves. To do this, for a point $v\in\mathbb{P}(V^{\vee})\smallsetminus S$ let
$W_{v}\subset V$ be the subspace of linear forms vanishing at $v$. Let

$p_{v}\colon\quad\mathbb{P}(V^{\vee})\smallsetminus\\{v\\}\;\longrightarrow\;\mathbb{P}(W_{v}^{\vee})$

be the corresponding projection. The geometry of the projection $p_{v}\colon
S\to p_{v}(S)$ depends on the image of the point $v$ via the projection with
center $E$ that we denote by

$q\colon\quad\mathbb{P}(V^{\vee})\smallsetminus
E\;\longrightarrow\;\mathbb{P}(V^{\prime\vee})\quad\textup{where}\quad
V^{\prime}:=\textup{H}^{0}(S,\mathcal{O}_{S}(2)(-2E)).$

More precisely, the line bundle $\mathcal{L}:=\mathcal{O}_{S}(1)(-E)$ is
globally generated on $S$, hence the restriction of $q$ to $S\smallsetminus E$
extends to a morphism $q\colon S\to\mathbb{P}(V^{\prime\vee})$. The latter is
the composite morphism

$S\;\longrightarrow\;\mathbb{P}(\textup{H}^{0}(S,\mathcal{L})^{\vee})\;\lhook\joinrel\longrightarrow\;\mathbb{P}(\textup{H}^{0}(S,\mathcal{L}^{\otimes
2})^{\vee})=\mathbb{P}(V^{\prime\vee})$

where the second map is the second Veronese embedding. Note that the first map
can be identified with the projection $S\to E$ onto the exceptional divisor.
Let

$C\;:=q(S)\quad\subset\;\mathbb{P}(V^{\prime\vee})$

be the smooth conic which is the image of the previous composite morphism. For
any $v^{\prime}\in\mathbb{P}(V^{\prime\vee})$, the closure of
$q^{-1}(v^{\prime})$ in $\mathbb{P}(V^{\prime})$ is a plane $P_{v^{\prime}}$
which intersects $S$ in the following way:

  * •

If $v^{\prime}\in C$, then $S\cap P_{v^{\prime}}=L\cup E$ where $L\subset S$
is the strict transform of a line in $\mathbb{P}^{2}$ passing through the
point $o$.

  * •

If $v^{\prime}\not\in C$, then $S\cap P_{v^{\prime}}=E$, because $C=q(S)$ and
$E$ is the center of the projection $q$.

Recall that the strict transform of a line $L\subset\mathbb{P}^{2}$ is mapped
to a line if $L$ contains the point $o$, and to a smooth conic otherwise. With
this in mind, we have:

######  Proposition 5.6.

The image $X_{v}:=p_{v}(S)\subset\mathbb{P}(V^{\prime\vee})$ is a nonnormal
cubic surface and its singular locus
$X_{v}^{\mathrm{sing}}\subset\mathbb{P}(V^{\prime\vee})$ is a line. The
morphism $p\colon S\to X_{v}$ is finite birational and on the ramification
locus $B\subset S$ it is of degree $2$. More precisely:

  * •

If $q(v)\in C$, then $B=L\cup E$ where $L\subset S$ is the strict transform of
a line in $\mathbb{P}^{2}$ passing through $o$ and $p_{v}\colon L\cup E\to
X_{v}^{\mathrm{sing}}$ identifies the two lines.

  * •

If $q(v)\not\in C$, then $B$ is a smooth conic and $q\colon B\to
X_{v}^{\mathrm{sing}}$ is a degree $2$ map.

The situation is resumed in fig. 1.

$E$$L$$X_{v}^{\mathrm{sing}}$$S$$X_{v}$$p_{v}$

$B$$X_{v}^{\mathrm{sing}}$$p_{v}$

Figure 1.  The two possibilities for $p_{v}\colon S\to X$ in proposition 5.6:
The first line shows the case $q(v)\in C$, the second the case $q(v)\not\in
C$.

###### Proof.

The nonnormality of $X_{v}$ follows from the description of $p_{v}\colon S\to
X_{v}$. Indeed, if $X_{v}$ were to be normal, then $p_{v}$ would be an
isomorphism because $p_{v}$ is finite birational and $S$ is smooth. But we
will show that $p_{v}$ is not an isomorphism, thus the surface $X_{v}$ cannot
be normal. The line bundle $\mathcal{O}_{S}(2)(-E)$ has self-interesction $3$
thus $S\subset\mathbb{P}(V^{\vee})$ has degree $3$ and the same is true for
$X_{v}\subset\mathbb{P}(W_{v}^{\vee})$. By lemma 5.4 the singular locus
$X_{v}^{\mathrm{sing}}$ is a line.

Suppose $q(v)\in C$. In this case, $B=P\cap S$ where $P$ is the closure of
$q^{-1}(q(v))$. Indeed, as recalled above, $P$ meets $S$ in $L\cup E$ where
$L$ is the strict transform of a line in $\mathbb{P}^{2}$ passing through $o$.
When $x\in L\cup E$, the line $D$ joining $x$ and $v$ is contained in the
plane $P$. Therefore the scheme-theoretic intersection $D\cap(L\cup E)$ is the
cycle $[x]+[y]$ where $y\in E$ is equal to $x$ if and only if $x$ is the
intersection point of $L$ and $E$. This shows that $p_{v}\colon L\cup E\to
X_{v}$ identifies the lines $L$ and $E$. To conclude it suffices to prove that
$p_{v}\colon S\smallsetminus B\to X_{v}$ is injective and takes values in
$X_{v}\smallsetminus p_{v}(B)$. Indeed, we show that given $x\in
S\smallsetminus(L\cup E)$ the line $D$ joining $x$ and $v$ meets $S$ only in
$x$. To see this, first notice that $D\cap E=\varnothing$: Otherwise, the line
$D$ would lie in the closure $P^{\prime}$ of $q^{-1}(q(x))$ and we would have
$q(x)=q(v)$. Now $D\cap S$ is contained in the preimage of $q(D)\cap C$. The
line $q(D)$ meets the smooth conic $C$ in the two distinct points $q(x)$ and
$q(v)$, thus

$D\cap S\;\subset\;q^{-1}(q(D)\cap C)=L\cup L^{\prime}$

where $P^{\prime}\cap S=L^{\prime}\cup E$ and $L^{\prime}\neq L$ is the strict
transform of a line in $\mathbb{P}^{2}$ passing through $o$. The lines $D$ and
$L$ do not meet (otherwise $v$ would belong to $S$) and $D$ meets $L^{\prime}$
only in $x$.

Suppose $q(v)\not\in C$. We claim that there is a unique plane
$P\subset\mathbb{P}(V^{\vee})$ containing $v$ and the strict transform $Q$ of
a line in $\mathbb{P}^{2}$ not passing through $o$. For the uniqueness, let
$P$ and $P^{\prime}$ distinct be planes in $\mathbb{P}(V^{\vee})$ meeting $S$
in the strict transform $Q$ and $Q^{\prime}$ respectively of distinct lines in
$\mathbb{P}^{2}$ not passing through $o$. Then $P\cap P^{\prime}$ is the
meeting point of $Q$ and $Q^{\prime}$, and in particular it is contained in
$S$. To see this, notice that $P\cup P^{\prime}$ is not contained in any
hyperplane. If it were contained in some hyperplane $H$ say, then we would
have

$Q\cup Q^{\prime}\;=\;(P\cup P^{\prime})\cap S\;\subset\;H\cap P.$

But by construction $H\cap S$ is the strict transform of a (possibly singular)
conic in $\mathbb{P}^{2}$ that _passes_ through $o$. It follows that $P\cap
P^{\prime}=\\{v\\}$ is contained $S$, which contradicts the hypothesis
$v\not\in S$. For the existence, note that every line $L$ in $\mathbb{P}^{2}$
determines a plane $P_{L}\subset\mathbb{P}(V^{\vee})$ and these planes cover
$\mathbb{P}(V^{\vee})$. Moreover, by construction the points in the cone in
$\mathbb{P}(V^{\vee})$ over $C$ are exactly the points lying in $P_{L}$ for
some $L$ passing through $o$. Since $v$ is not one of those, we obtain the
wanted plane.

We are now ready to conclude the proof. Let $P$ be the plane containing $v$
and the strict transform $B$ of a line in $\mathbb{P}^{2}$ not passing through
$o$. Then $B\subset P$ is a smooth conic and any line through $v$ meets $B$ in
two points (taking in account multiplicities). This shows that $B\to X_{v}$ is
of degree $2$ onto its image. To conclude it suffices to show that
$p_{v}\colon S\smallsetminus B\to X_{v}$ takes values in $X_{v}\smallsetminus
p_{v}(B)$ and is injective. To do this, let $x,y\in S$ be distinct points such
that $p_{v}(x)=p_{v}(y)$ and let $D$ be the line joining $x$, $y$ and $v$.
Note that $x$ and $y$ cannot lie both in the exceptional divisor, otherwise we
would have $D=E$ hence $v\in S$. It follows that $x$ and $y$ have distinct
images under the blow-up map $\pi\colon S\to\mathbb{P}^{2}$. Let
$L\subset\mathbb{P}^{2}$ be the line joining $\pi(x)$ and $\pi(y)$, and let
$Q\subset S$ be its strict transform. If the blow-up point $o$ were to belong
to $L$, then $Q$ would be mapped to a line in $\mathbb{P}(V^{\vee})$
containing $x$ and $y$, implying $D=Q$ and $v\in S$ again. Therefore $L$ does
not contain $o$, so $Q$ is embedded as a smooth conic in
$\mathbb{P}(V^{\vee})$. To conclude, notice that the plane $P^{\prime}$
determined by $Q$ must be $P$: Otherwise, as argued above, $P$ and
$P^{\prime}$ would meet in a single point which belongs to $S$; but $v\in
P\cap P^{\prime}$ so that would imply $v\in S$ once more. Finally, we have
$P=P^{\prime}$ hence $Q=B$. ∎

###### Proof of proposition 5.3.

We choose homogeneous coordinates $x_{0},x_{1},x_{2}$ on $\mathbb{P}^{2}$ and
identify $S$ with the blow-up of $\mathbb{P}^{2}$ in $o=[1:0:0]$. Then a basis
of $V$ is given by the monomials

$u_{ij}=x_{i}x_{j}\qquad\textup{with}\quad 0\leqslant i\leqslant j\leqslant
2\quad\textup{and}\quad(i,j)\neq(0,0),$

and $S$ is cut out by the vanishing of the polynomials

$\displaystyle h_{1}$ $\displaystyle=u_{02}u_{11}-u_{01}u_{12},$
$\displaystyle h_{2}$ $\displaystyle=u_{01}u_{22}-u_{02}u_{12},$
$\displaystyle h_{3}$ $\displaystyle=u_{11}u_{22}-u_{12}^{2}.$

With this notation the vector space $V^{\prime}$ is spanned by $u_{11}$,
$u_{12}$ and $u_{22}$ and the conic $C$ has equation $h_{3}=0$. If
$v\in\mathbb{P}(V)$ is defined by the vanishing of $u_{01}$, $u_{12}$,
$u_{22}$ and $u_{02}+u_{11}$, then $q(v)\in C$ and

(5.1) $X_{v}:\quad u_{12}^{3}+u_{22}^{2}u_{01}-u_{12}u_{22}(u_{02}+u_{11})=0.$

If instead $v$ is defined by the vanishing of $u_{ij}$ for $(i,j)\neq(1,2)$,
then $q(v)\not\in C$ and

(5.2) $X_{v}:\quad u_{02}^{2}u_{11}-u_{01}^{2}u_{22}=0.$

The discussion preceding theorem 9.2.1 in [Dol12] shows that up to a linear
change of variables, any nonnormal integral cubic surface
$X\subset\mathbb{P}^{3}$ which is not a cone over a nonnormal plane cubic
curve has equation (5.1) or (5.2). Together with lemma 5.5 and proposition 5.6
this concludes the proof. ∎

##  6\. On Fano varieties of lines

We now prove some facts about Fano varieties of lines that will be used for B.
Let $k$ be an algebraically closed field. For a finite-dimensional $k$-vector
space $V$, the Fano variety of lines on a subvariety $X\subset\mathbb{P}(V)$
is the $k$-scheme $F_{X}$ whose points with values in a $k$-scheme $S$ are
rank $2$ vector bundles $\mathcal{V}\subset V\otimes_{k}\mathcal{O}_{S}$ on
$S$ with locally free quotient such that
$\mathbb{P}(\mathcal{V})\subset\mathbb{P}(V)\times S$ is contained in $X\times
S$. Equivalently, $F_{X}$ is the Hilbert scheme parametrizing subvarieties of
$X$ with Hilbert polynomial $P(z)=z+1$ with respect to line bundle
$\mathcal{O}(1)$.

###  6.1. Rational normal scrolls

For an integer vector $a=(a_{1},\dots,a_{r})\in\mathbb{Z}^{r}$ we put
$\mathcal{O}_{\mathbb{P}^{1}}(a)=\mathcal{O}_{\mathbb{P}^{1}}(a_{1})\oplus\cdots\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{r})$.
Let

$p_{a}\colon\quad\mathbb{P}(a)=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(-a))\;\longrightarrow\;\mathbb{P}^{1}$

be the projective bundle of lines in $\mathcal{O}_{\mathbb{P}^{1}}(-a)$. If
$0\leqslant a_{1}\leqslant\cdots\leqslant a_{r}$ with $a_{r}>0$, then the line
bundle $\mathcal{O}_{\mathbb{P}(a)}(1)$ on $\mathbb{P}(a)$ is generated by
global sections. This defines a morphism

$f_{a}\colon\quad\mathbb{P}(a)\;\longrightarrow\;\mathbb{P}(V(a)^{\vee})\quad\text{with}\quad
V(a)\;=\;V(a_{1})\oplus\cdots\oplus V(a_{r})$

where
$V(a_{i})=\textup{H}^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(a_{i}))$.
The morphism $f_{a}$ is birational and embeds each fiber of $\pi$ linearly.
Moreover, $f_{a}$ is a closed immersion if and only if $a_{i}>0$ for all
$i=1,\dots,r$. The image of $f_{a}$ is called the _rational normal scroll_ of
type $a$ and will be denoted $S(a)$. The rational normal scroll $S(a)$ is
smooth in the following cases [EH87, p. 6]:

  * •

$a_{i}>0$ for all $i$, in which case $f_{a}\colon\mathbb{P}(a)\to S_{a}$ is an
isomorphism;

  * •

$a=(0,\dots,0,1)$, in which case $S(a)$ is a projective $r$-plane in
$\mathbb{P}(V(a)^{\vee})$.

Let $F(a)=F_{S(a)}$ be the Fano variety of lines in $S(a)$, and denote by
$\operatorname{Gr}_{2}(\mathcal{O}_{\mathbb{P}^{1}}(-a))$ the Grassmannian of
planes in $\mathcal{O}_{\mathbb{P}^{1}}(-a)$.

######  Proposition 6.1.

Suppose $a_{i}>0$ for all $i=1,\dots,r$. Then $F(a)=F(a)_{v}\sqcup F(a)_{h}$
with

  * •

$F(a)_{v}=\operatorname{Gr}_{2}(\mathcal{O}_{\mathbb{P}^{1}}(-a))$,

  * •

$F(a)_{h}=\mathbb{P}(W^{\vee})$ where $W=V(1)^{n}$ and $n=|\\{i:a_{i}=1\\}|$.

Moreover $F(a)_{h}\subset\operatorname{Gr}_{2}(V(a)^{\vee})$ is linearly
embedded via the Plücker embedding.

###### Proof.

Since $f_{a}$ is a closed embedding, a line in $S(a)$ is the image of a closed
embedding $\varepsilon\colon\mathbb{P}^{1}\hookrightarrow\mathbb{P}(a)$ such
that the line bundle $\varepsilon^{\ast}\mathcal{O}_{\mathbb{P}(a)}(1)$ has
degree $1$. The morphism
$g:=p_{a}\circ\varepsilon\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ is either
dominant or constant. We say that the corresponding line is _horizontal_ in
the first case, and _vertical_ in the second. The Fano variety of lines
decomposes correspondingly as

$F(a)=F(a)_{h}\sqcup F(a)_{v}$

where the subscript $h$ resp. $v$ stands for horizontal resp. vertical.
Vertical lines are parametrized by the relative Grassmannian
$\operatorname{Gr}_{2}(\mathcal{O}_{\mathbb{P}^{1}}(-a))$. For horizontal
lines, the condition $\deg\varepsilon^{\ast}\mathcal{O}_{\mathbb{P}(a)}(1)=1$
implies that the morphism $g$ is an isomorphism, hence up to precomposing
$\varepsilon$ by $g^{-1}$, we may assume $g=\operatorname{id}$. It follows
that horizontal lines in $S(a)$ are given by degree $1$ line bundles on
$\mathbb{P}^{1}$ which are quotients of $\mathcal{O}_{\mathbb{P}^{1}}(-a)$.
Such quotients are necessarily quotients of
$\mathcal{O}_{\mathbb{P}^{1}}(a_{1},\dots,a_{n})=\mathcal{O}_{\mathbb{P}^{1}}(1)^{n}$
where $n$ is as in the statement. Finally, twisting by
$\mathcal{O}_{\mathbb{P}^{1}}(-1)$, this discussion shows that vertical lines
in $S(a)$ are given by surjective morphisms
$\mathcal{O}_{\mathbb{P}^{1}}^{n}\to\mathcal{O}_{\mathbb{P}^{1}}$. This
identifies $F(a)_{h}$ with $\mathbb{P}(W^{\vee})$ for $W=V(1)^{n}$. For the
second claim write $V:=V(a)=W\oplus V^{\prime}$ with

$V^{\prime}=V(a_{n+1})\oplus\cdots\oplus V(a_{r}).$

The Plücker embedding of
$\mathbb{P}(W^{\vee})\hookrightarrow\mathbb{P}(\operatorname{Alt}^{2}V^{\vee})$
is given by $L\mapsto L\otimes\operatorname{Alt}^{2}(H^{\vee})$ which is
clearly linear. ∎

######  Example 6.2.

The case when $S(a)$ is smooth with $a_{1}+\cdots+a_{r}=3$ will be relevant to
us. Because of the previous characterization, there are three possibilities:

  * •

$a=(3)$. In this case $S(3)\subset\mathbb{P}^{3}$ is the twisted cubic, i.e.
the triple Veronese embedding of the projective line. Hence

$F(3)=\varnothing.$

  * •

$a=(1,2)$. In this case $S(1,2)\subset\mathbb{P}^{4}$ is the blow-up
$\pi\colon S\to\mathbb{P}^{2}$ of $\mathbb{P}^{2}$ in a point and it is
embedded via the line bundle $\pi^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(2)(-E)$,
where $E\subset S$ is the exceptional divisor. Thus

$\displaystyle F(1,2)_{v}$ $\displaystyle=\\{\textup{fibers of the
projection\leavevmode\nobreak\ $S\to E$}\\},$ $\displaystyle F(1,2)_{h}$
$\displaystyle=\\{E\\}.$

  * •

$a=(1,1,1)$. In this case $S(1,1,1)$ is $\mathbb{P}^{1}\times\mathbb{P}^{2}$
embedded in $\mathbb{P}^{5}$ via the Segre embedding. Then, the Plücker
embedding

$F(1,1,1)_{v}\hookrightarrow\mathbb{P}(\operatorname{Alt}^{2}\mathcal{O}_{\mathbb{P}^{1}}(-1,-1,-1))=\mathbb{P}(2,2,2)\cong\mathbb{P}^{1}\times\mathbb{P}^{2}$

is an isomorphism and $F(1,1,1)_{h}=\mathbb{P}^{2}$.

###  6.2. The Fano variety of a cone

Let $\pi\colon V\to W$ be a surjective linear map between finite-dimensional
$k$-vector spaces. For $i=0,1,2$ consider the following closed subvariety of
the Grassmannian of planes in $V$:

$\operatorname{Gr}^{i}_{2}(V,\pi)\;:=\;\\{E\subset
V\mid\dim(E\cap\operatorname{Ker}\pi)\geqslant
i\\}\;\subset\;\operatorname{Gr}_{2}(V).$

We have $\operatorname{Gr}^{0}_{2}(V,\pi)=\operatorname{Gr}_{2}(V)$ and
$\operatorname{Gr}^{2}_{2}(V,\pi)=\operatorname{Gr}_{2}(\operatorname{Ker}\pi)$.
For $i=0,1$ the induced morphism

$\pi_{i}\colon\quad\operatorname{Gr}^{i}_{2}(V,\pi)\smallsetminus\operatorname{Gr}^{i+1}_{2}(V,\pi)\;\longrightarrow\;\operatorname{Gr}_{2-i}(W),\quad
E\;\longmapsto\;\pi(E)$

is surjective and smooth. Moreover, given a plane $E\subset W$, the closure of
the fiber of $\pi_{0}$ in the corresponding point
$[E]\in\operatorname{Gr}_{2}(W)$ is $\operatorname{Gr}_{2}(\pi^{-1}(E))$.

Let $Y\subset\mathbb{P}(W)$ be a nonempty subvariety and
$X\subset\mathbb{P}(V)$ the cone over $Y$. Consider the respective Fano
varieties of lines $F_{X}\subset\operatorname{Gr}_{2}(V)$ and
$F_{Y}\subset\operatorname{Gr}_{2}(W)$. At the set-theoretical level we have

$F_{X}\quad=\quad\pi_{0}^{-1}(F_{Y})\;\sqcup\;\pi_{1}^{-1}(Y)\;\sqcup\;\operatorname{Gr}_{2}(\operatorname{Ker}\pi).$

At the topological level, note that
$\operatorname{Gr}_{2}(\operatorname{Ker}\pi)$ is contained in the closure of
any fiber of $\pi_{1}$. For the other components, we have:

######  Proposition 6.3.

With the notation above,

  1. (1)

If $Y$ is covered by lines, then $\pi_{0}^{-1}(F_{Y})$ is dense in $F_{X}$ and
$\pi_{0}$ sets up a bijection between the irreducible components of $F_{X}$
and those of $F_{Y}$.

  2. (2)

If $Y^{\prime}\subset Y$ is an irreducible component that is not covered by
lines, then the closure of $\pi_{1}^{-1}(Y^{\prime})$ is an irreducible
component of $F_{X}$.

  3. (3)

If $Y$ contains no lines, then $\pi_{1}$ sets up a bijection between the
irreducible components of $F_{X}$ and those of $Y$.

###### Proof.

(1) The subset $\pi_{1}^{-1}(Y)\subset F_{X}$ parametrizes lines contained in
the closure of fibers of $X\dashrightarrow Y$. By hypothesis any $y\in Y$
belongs to some line $L\subset Y$. Therefore lines in the closure of fiber of
$X\dashrightarrow Y$ in $y$ are contained in
$\operatorname{Gr}_{2}(\pi^{-1}(E))=\overline{\pi_{0}^{-1}([E])}$ where
$L=\mathbb{P}(E)$ and $[E]\in\operatorname{Gr}_{2}(W)$ is the point defined by
$E$.

(2) Lines in the closure of fiber of $X\dashrightarrow Y$ in a point $y\in
Y^{\prime}$ belonging to no line contained in $Y$ form a subset of
$\pi_{1}^{-1}(Y^{\prime})$ not lying in the closure of $\pi_{0}^{-1}(F_{Y})$.

(3) is clear. ∎

It will be useful to combine proposition 6.3 with the following remark: For
each irreducible component $F^{\prime}\subset F_{Y}$ and $Y^{\prime}\subset
Y$, we have

$\displaystyle\dim\pi_{0}^{-1}(F^{\prime})\;$ $\displaystyle=\;\dim
F^{\prime}+2(\dim V-\dim W),$ $\displaystyle\dim\pi_{1}^{-1}(Y^{\prime})\;$
$\displaystyle=\;\dim Y^{\prime}+2(\dim V-\dim W-1).$

######  Example 6.4.

Suppose $\dim W=3$ and that $Y$ is an integral cubic plane curve. Then the
preceding statement says that the Fano variety of lines in the cone $X$ is
irreducible of dimension $2(\dim V-4)+1$.

######  Example 6.5.

Suppose $\dim W=4$ and that $Y\subset\mathbb{P}(W)$ is an integral cubic
surface with only finitely many lines. This is the case if $Y$ is smooth, or
if it has only isolated singularities and is not a cone over an elliptic curve
in $\mathbb{P}^{2}$ [BW79, Sak10]. Then $F_{X}$ is pure of dimension $2(\dim
V-4)$ and its irreducible components (with the reduced structure) are the
following: the closure of $\pi_{1}^{-1}(F_{Y})$ in $\operatorname{Gr}_{2}(V)$
and, for each line in $Y$, one component isomorphic to $\mathbb{P}^{2(\dim
V-4)}$ linearly embedded with respect to the Plücker embedding.

######  Example 6.6.

Suppose $\dim W=4$ and that $Y\subset\mathbb{P}(W)$ is a nonnormal integral
cubic surface that is not a cone over a cubic plane curve. Borrow notation
from section 5.3. By proposition 5.3 the cubic surface $Y$ is obtained from
the blow-up $S$ of $\mathbb{P}^{2}$ in a point,

$S=S(1,2)\;\subset\;\mathbb{P}(V^{\vee})\quad\textup{where}\quad V=V(1,2),$

by projecting it from a point $v\in\mathbb{P}(V^{\vee})$ not in $S$. Recall
from example 6.2 that

$F_{S}\;=\;F_{S,h}\sqcup F_{S,v}\qquad\textup{with}\quad
F_{S,h}=\\{\textup{pt}\\},\quad F_{S,v}\cong\mathbb{P}^{1}.$

It follows from proposition 5.6 that such a projection induces a morphism
between Fano varieties of lines $f\colon F_{S}\to F_{Y}$. The morphism $f$ is
a closed embedding once restricted to irreducible components. Moreover,
following the case distinction in proposition 5.6 there are two possibilities:

  * •

if $q(v)\in C$, then $F_{Y,{\mathrm{red}}}=f(F_{S,v})$ has one irreducible
component;

  * •

otherwise $F_{Y,{\mathrm{red}}}=f(F_{S,v})\sqcup
f(F_{S,h})\sqcup\\{Y^{\mathrm{sing}}\\}$ has three irreducible components
(recall by lemma 5.4 that the singular locus $Y^{\mathrm{sing}}$ of $Y$ is a
line).

The surface $Y$ is covered by lines, thus by proposition 6.3, the morphism
$\pi_{0}$ sets up a bijection between the irreducible components of the Fano
variety of $X$ and that of $Y$. Continuing with the above distinction, we have
the following cases:

  * •

if $q(v)\in C$, then $F_{X}$ is irreducible of dimension $1+2(\dim V-4)$;

  * •

otherwise $F_{X}$ has one irreducible component of dimension $1+2(\dim V-4)$
and two irreducible components isomorphic to $\mathbb{P}^{2(\dim V-4)}$
linearly embedded with respect to the Plücker embedding.

###  6.3. The Fano variety of singular cubic hypersurfaces

Let $X\subset\mathbb{P}^{n+1}$ be an integral cubic hypersurface and $F=F_{X}$
its Fano variety of lines. Let $\mathcal{O}_{F}(1)$ be the restriction to
$F\subset\operatorname{Gr}_{2}(V)$ of the line bundle defining the Plücker
embedding.

######  Proposition 6.7.

If $n=3$ and $\dim F\leqslant 2$, then, for all integer $i\geqslant 0$,

$\chi(F,\mathcal{O}_{F}(i))=\textstyle 45\binom{i+1}{2}-45i+6.$

###### Proof.

When $X$ has isolated singularities, this is equation (1.21.1) in [AK77],
whose proof relies on corollary 1.4, lemma 1.14 and prop. 1.15 (i) in loc.
cit.. However, with the notation therein, the proof of these facts only uses
that $F$ is the zero subscheme of a regular section of the locally free sheaf
$\operatorname{Sym}^{3}Q$. By theorem 3.3 (iv) in loc. cit., this latter fact
is true more generally when $\dim F\leqslant 2$, hence the result follows.
Note that the statement of theorem 3.3 (iv) gives a somewhat weaker statement:
for each $x\in F$ at which $F$ has dimension $\leqslant 2$, the subvariety $F$
is the zero subscheme of a section $s_{x}$ of $\operatorname{Sym}^{3}Q$ which
is regular at $x$. Anyway, it is clear from its proof that the section $s_{x}$
can be taken to be the same for all points $x\in F$. ∎

The main goal of this section is to complement [AK77] by describing precisely
the cases where $F$ is not of dimension $2$. When $X$ is a cone over a cubic
surface with finitely many lines $F$ has dimension $2$; see example 6.5. On
the other hand, in examples 6.6, LABEL: and 6.4 we saw that the Fano variety
of a cone over a nonnormal cubic surface or over a normal cubic plane curve
has dimension $3$. The twisted plane in example 5.2 also furnishes such an
example:

######  Example 6.8.

Let $X\subset\mathbb{P}^{4}$ be the twisted plane from example 5.2, given by

$x_{0}^{2}x_{1}+x_{0}x_{2}x_{4}+x_{3}x_{4}^{2}=0.$

The discussion in [FP01, sect. 2.2.9] shows that the Fano variety of lines
$F=F_{X}$ on $X$ has three irreducible components:

  * •

one $3$-dimensional component is given by the lines in the $1$-dimensional
family of planes in $X$ with equations, for $[t_{0}:t_{1}]\in\mathbb{P}^{1}$,

$t_{1}x_{0}-t_{0}x_{4}=0,\qquad
t_{0}^{2}x_{1}+t_{0}t_{1}x_{2}+t_{1}^{2}x_{3}=0;$

  * •

one $2$-dimensional component is the dual of the plane $X^{\mathrm{sing}}$ and
is linearly embedded via the Plücker embedding;

  * •

one $2$-dimensional component obtained as the closure of the image of the
morphism
$\mathbb{P}^{2}\smallsetminus\\{[0:1:0],[0:0:1]\\}\to\operatorname{Gr}_{2}(5)$
associating to $[t_{0}:t_{1}:t_{2}]$ the line joining the points

$[t_{0}^{2}:t_{1}t_{2}:-t_{0}t_{2}:0:t_{0}t_{1}],\quad[0:t_{1}^{2}:-2t_{0}t_{1}:t_{0}^{2}:0]\quad\in\;\mathbb{P}^{4}.$

This component has degree $4$ with respect to the Plücker embedding.

The main result of this section states that these are the only cases of
$3$-dimensional Fano varieties. The argument goes through in any dimension and
yields:

######  Proposition 6.9.

For $n=3$ we have $\dim F\leqslant 3$ and equality holds if and only if $X$ is
either

  * •

the twisted plane in example 6.8, or

  * •

a cone over a nonnormal integral cubic surface, or

  * •

a cone over a normal cubic plane curve.

For $n\geqslant 4$ we have $\dim F\leqslant 2n-3$ and if equality holds then
$X$ is a cone.

This is the combination of the following two lemmas with the classification of
nonnormal cubic hypersurfaces in proposition 5.3.

######  Lemma 6.10.

Let $x\in X$ and $F_{X,x}\subset F_{X}$ the subvariety of lines passing
through $x$. If $X$ is not a cone, then

$\dim F_{X,x}\leqslant n-2.$

###### Proof.

Suppose there is an irreducible component $Z\subset F_{X,x}$ of dimension
$\geqslant n-1$. Consider the universal line $\pi\colon P\to Z$ and the
evaluation morphism $f\colon P\to X$. The morphism $f\colon P\smallsetminus
f^{-1}(x)\to X$ is injective because there is a unique line passing through
two distinct points. As $P\neq f^{-1}(x)$, this implies

$n=\dim X\geqslant\dim P\smallsetminus f^{-1}(x)=\dim Z+1\geqslant n,$

hence $\dim Z=n-1$ and $f$ is surjective. It follows that the lines in $Z$
cover $X$ contradicting the assumption that $X$ is not a cone. ∎

######  Lemma 6.11.

If $\dim X^{\mathrm{sing}}\leqslant n-2$, then either $\dim F\leqslant 2n-3$
or $X$ is a cone.

###### Proof.

Suppose that $X$ is not a cone. By [AK77, th. 4.2] the variety $F$ is smooth
of dimension $2n-4$ at the point corresponding to any line not meeting the
singular locus. Therefore, it remains to treat the case of lines going through
singular points. Since $\dim X^{\mathrm{sing}}\leqslant n-2$ by hypothesis,
lemma 6.10 implies that $F$ at a line going through a singular point has
dimension $\leqslant 2(n-2)$. This concludes the proof. ∎

###### Proof of proposition 6.9.

By lemma 6.11 it remains to treat the case when $X$ is not normal. We then
conclude by proposition 5.3 and examples 6.6, LABEL:, 6.4, LABEL: and 6.8. ∎

###  6.4. Positivity of Fano surfaces of cubic threefolds

Let $Y\subset\mathbb{P}^{4}$ be a smooth cubic threefold, and $X$ the Fano
surface of lines on $Y$. We fix an Albanese embedding $X\hookrightarrow
A:=\operatorname{Alb}(X)$. The subvariety $X\subset A$ is nondegenerate, but
in general its normal bundle $\mathcal{N}$ fails to be ample. Since
$\mathcal{N}$ is globally generated, the ampleness of $\mathcal{N}$ is
equivalent to the finiteness of the Gauss map

$\gamma\colon\quad\mathbb{P}(\mathcal{C})\;\longrightarrow\;\mathbb{P}((\operatorname{Lie}A)^{\vee})$

where $\mathcal{C}=\mathcal{N}^{\vee}$ is the conormal bundle [Laz04, ex.
6.1.5]. On the other hand, the hypersurface $Y$ can be recovered from $X$ as
the image of the morphism

$\pi\colon\quad\mathbb{P}(T_{X})\;\longrightarrow\;\mathbb{P}(\operatorname{Lie}A)$

for suitable coordinates on $\operatorname{Lie}A$. Via this identification,
any line in $Y$ is the image of $\mathbb{P}(T_{X,x})$ for a unique $x\in X$.
For any point $p\in\mathbb{P}((\operatorname{Lie}A)^{\vee})$, let
$H_{p}\subset\mathbb{P}(\operatorname{Lie}A)$ denote the corresponding
hyperplane. Then by the above discussion, the following are equivalent:

  * •

the fiber $\gamma^{-1}(p)$ is finite, and

  * •

the cubic surface $Y\cap H_{p}$ contains only finitely many lines.

Note that in any case, we have:

######  Lemma 6.12.

For any hyperplane $H\subset\mathbb{P}^{4}$ the surface $H\cap Y$ is integral
and normal.

###### Proof.

If the cubic surface $S:=H\cap Y$ is not integral, it contains a plane $P$.
Since the dual plane $P^{\vee}$ parametrizes lines on $P$, we have a closed
embedding $P^{\vee}\hookrightarrow X$ which is necessarily an isomorphism
because $X$ is a surface. This is absurd since for instance $X$ is of general
type. Similarly, if $S$ were nonnormal, then the Fano variety of lines $F_{S}$
on $S$ would be of dimension $3$ by examples 6.4, LABEL: and 6.6. But $F_{S}$
embeds into $F_{Y}=X$ which is of dimension $2$, a contradiction. ∎

######  Lemma 6.13.

For a point $p\in\mathbb{P}((\operatorname{Lie}A)^{\vee})$ the following are
equivalent:

  1. (1)

the fiber $\gamma^{-1}(p)$ is not finite;

  2. (2)

the cubic surface $H_{p}\cap Y\subset H$ is a cone over an elliptic curve in
$\mathbb{P}^{2}$.

###### Proof.

For (1) $\Rightarrow$ (2) note that the cubic surface $H_{p}\cap Y$ is normal
by lemma 6.12, and the only normal cubic surfaces containing infinitely many
lines are cones over elliptic curves [BW79, Sak10]. The converse (2)
$\Rightarrow$ (1) is clear by example 6.4. ∎

As an immediate consequence, we obtain the following characterization:

######  Proposition 6.14.

The following are equivalent:

  1. (1)

The normal bundle of $X\subset\operatorname{Alb}(X)$ is not ample.

  2. (2)

There is a hyperplane $H\subset\mathbb{P}^{4}$ such that the cubic surface
$H\cap Y\subset H$ is a cone over an elliptic curve in $\mathbb{P}^{2}$.

The condition (2) in the above statement occurs if and only if for a suitable
choice of homogeneous coordinates on $\mathbb{P}^{4}$ the equation of
$Y\subset\mathbb{P}^{4}$ takes the form

$Y\colon\quad f(x_{0},x_{1},x_{2})+x_{4}q(x_{0},\dots,x_{4})\;=\;0,$

where $f$ is a homogeneous cubic that defines an elliptic curve
$E\subset\mathbb{P}^{2}$ and $q$ is a quadratic form. We also remark that the
cubic threefolds in question are parametrized by a $32$-dimensional subvariety
of $\mathbb{P}(\operatorname{Sym}^{3}(k^{5})^{\vee})\simeq\mathbb{P}^{34}$. In
particular, if $Y$ is a generic cubic threefold, then $\mathcal{N}$ is ample.
In general, we have the following:

######  Corollary 6.15.

The morphism $\gamma$ is finite outside a finite set in
$\mathbb{P}((\operatorname{Lie}A)^{\vee})$.

###### Proof.

If the normal bundle $\mathcal{N}$ is ample, the morphism $\gamma$ is finite.
Henceforth suppose that $\mathcal{N}$ is not ample and let
$p\in\mathbb{P}((\operatorname{Lie}A)^{\vee})$ be a point such that $H_{p}\cap
Y$ is a cone over an elliptic curve $E_{p}$. Note that the cubic surface
$H_{p}\cap Y$ determines the hyperplane $H_{p}$ uniquely as its linear span.
Now each point of $E_{p}$ gives a line on the threefold, hence we have a
closed embedding $E_{p}\hookrightarrow X$. Since a nondegenerate surface
$X\subset A$ cannot be covered by translates of elliptic curves, there are at
most finitely many possible $E_{p}\hookrightarrow X$. But $E_{p}$ determines
the cubic surface $H_{p}\cap Y$ by taking the union of all the lines
parametrized by $E_{p}$. Since the cubic surface determines $H_{p}$, this
leaves only finitely many possibilities for $p$. ∎

######  Remark 6.16.

The previous statement corrects an error in the remark after (iii) on p. 1462
in [Krä20] where it had been assumed that the Gauss map for the Fano surface
is a finite morphism. However, the result in loc. cit. is still true with the
same proof, since it only uses [Krä21, lemma 3.3.1] which can still be applied
when $\gamma$ is finite over the complement of finitely many points.

###  6.5. A construction

Let $X$ be a smooth projective variety over $k$, $V$ a finite-dimensional
$k$-vector space and

(6.1) $0\longrightarrow\mathcal{T}\longrightarrow
V\otimes_{k}\mathcal{O}_{X}\longrightarrow\mathcal{N}\longrightarrow 0$

be a short exact sequence of vector bundles over $X$. Consider the composite
morphism

$\pi\colon\quad\mathbb{P}(\mathcal{T})\lhook\joinrel\longrightarrow\mathbb{P}(V)\times
X\longrightarrow\mathbb{P}(V).$

Let $Y\subset\mathbb{P}(V)$ be the image of $\mathbb{P}(\mathcal{T})$ and
$F=F_{Y}$ the Fano variety of lines of $Y$. The short exact sequence (6.1)
defines a morphism

$\gamma\colon\quad X\;\longrightarrow\;F.$

Assuming the generic finiteness of $\pi$, the main result of this section
states that the generic line of $Y$ parametrized by $X$ cannot be deformed
into a line that is not of the same nature:

######  Proposition 6.17.

Suppose $\dim V>\dim X+2$ and $\pi$ generically finite. Then there is a
nonempty open subset $X^{\prime}\subset X$ such that for every $x\in
X^{\prime}$ the tangent map of $\gamma$ is surjective:

$\textup{T}_{x}\gamma\colon\quad\textup{T}_{x}X\;\longrightarrow\longrightarrow\;\textup{T}_{\gamma(x)}F.$

If $\gamma$ is generically finite, then the image of $\gamma$ is an
irreducible component of $F$.

###### Proof.

Let $Y^{\prime}\subset Y$ be a nonempty open subset over which $\pi$ is finite
flat. We claim that $X^{\prime}=p(\pi^{-1}(Y^{\prime}))$ does the job, where
$p\colon\mathbb{P}(\mathcal{T})\to X$ is the projection. To show this, given
$x\in X^{\prime}$, let us consider a tangent vector to $\gamma(x)$ in $F$. By
interpreting tangent vectors as points of $F$ with values in
$R=k[\varepsilon]$, this consists in a free $R$-submodule $\mathcal{V}\subset
V\otimes_{k}R$ of rank $2$ with free cokernel and such that
$\mathbb{P}(\mathcal{V})$ is contained in $Y_{R}$. Moreover, saying that such
a tangent vector lies over $\gamma(x)$ means that

$\mathbb{P}(\mathcal{V})_{{\mathrm{red}}}=\mathbb{P}(\mathcal{T}_{x})\subset\mathbb{P}(L).$

Finally, the fact that $x$ belongs to $X^{\prime}$ boils down to saying that
$\mathbb{P}(\mathcal{V})_{{\mathrm{red}}}$ meets $Y^{\prime}$. It follows that
the fiber product

$P:=\mathbb{P}(\mathcal{T})_{R}\times_{Y_{R}}(\mathbb{P}(\mathcal{V})\cap
V_{R})$

is nonempty and flat over $R$. Better, an irreducible component of
$P_{{\mathrm{red}}}$ is the open subset

$\mathbb{P}(\mathcal{V})_{{\mathrm{red}}}\cap
Y^{\prime}\;=\;\mathbb{P}(\mathcal{T}_{x})\cap
Y^{\prime}\quad\subset\;\mathbb{P}(\mathcal{T}_{x}).$

In particular, the open subset
$P\smallsetminus\overline{P\smallsetminus\mathbb{P}(\mathcal{T}_{x})}$ is
nonempty and flat over $R$. Consider its scheme-theoretic closure $P^{\prime}$
in $\mathbb{P}(\mathcal{T})_{R}$. Then $P^{\prime}$ is flat over $R$ and
$P^{\prime}_{{\mathrm{red}}}=\mathbb{P}(\mathcal{T}_{x})$. It follows that the
morphism $\pi_{\rvert P^{\prime}}\colon P^{\prime}\to\mathbb{P}(\mathcal{V})$
is an isomorphism. Indeed the morphism $\pi_{\rvert P^{\prime}}$ is finite
hence the line bundle
$\mathcal{L}=\pi^{\ast}\mathcal{O}_{\mathbb{P}(\mathcal{V})}(1)$ is ample on
$P^{\prime}$. Since $\pi$ induces an open immersion
$P\smallsetminus\overline{P\smallsetminus\mathbb{P}(\mathcal{T}_{x})}\hookrightarrow\mathbb{P}(\mathcal{V})$,
for each $d\geqslant 0$ the morphism

$\pi^{\ast}\colon\textup{H}^{0}(\mathbb{P}(\mathcal{V}),\mathcal{O}(d))\longrightarrow\textup{H}^{0}(P^{\prime},\mathcal{L}^{\otimes
d})$

is injective. By Nakayama’s lemma, it is actually an isomorphism because it is
an isomorphism modulo $\varepsilon$. Therefore, it induces an isomorphism

$\pi\textstyle\colon\quad
P^{\prime}=\operatorname{Proj}\Bigl{(}\bigoplus_{d\geqslant
0}\textup{H}^{0}(P^{\prime},\mathcal{L}^{\otimes
d})\Bigr{)}\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\mathbb{P}(\mathcal{V})=\operatorname{Proj}\Bigl{(}\bigoplus_{d\geqslant
0}\textup{H}^{0}(\mathbb{P}(\mathcal{V}),\mathcal{O}(d))\Bigr{)}.$

Now $p\colon P^{\prime}\to X$ factors through a morphism
$\operatorname{Spec}k[\varepsilon]\to X$ with the unique point of
$\operatorname{Spec}k$ being sent to $x$: Indeed, the coherent sheaf of
$\mathcal{O}_{X}$-algebras $\mathcal{A}=p_{\ast}\mathcal{O}_{P^{\prime}}$ is
set-theoretically supported only on $x$, so $\mathcal{A}$ is the skyscraper
sheaf on $x$ with value
$\textup{H}^{0}(X,p_{\ast}\mathcal{O}_{P^{\prime}})=\textup{H}^{0}(P^{\prime},\mathcal{O}_{P^{\prime}})=k[\varepsilon]$.
Seeing this $R$-valued point of $X$ as a tangent vector to $X$ in $x$ gives
the desired lifting and concludes the proof. ∎

As a sanity check, note that for $X=\mathbb{P}^{1}$ the hypotheses of
proposition 6.17 are never fulfilled, hence this is not in contradiction to
proposition 6.1.

##  7\. Subvarieties with Tannaka group $E_{6}$

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed
field $k$ of characteristic zero. Putting together the results from the
previous sections, we now complete our classification of smooth subvarieties
with Tannaka group $E_{6}$.

###  7.1. Main result

As recalled in the introduction, the Fano surfaces of lines on cubic
threefolds are examples of subvarieties $X\subset A$ with Tannaka group
$E_{6}$. The main result of this section is that these are the only possible
examples among nondegenerate surfaces:

######  Theorem 7.1.

Suppose $g\geqslant 5$. For any smooth irreducible surface $X\subset A$, the
following three properties are equivalent:

  1. (1)

$X\subset A$ is nondivisible and nondegenerate with Tannaka group
$G_{X,\omega}^{\ast}\simeq E_{6}$.

  2. (2)

$X\subset A$ is nondivisible and nondegenerate with

$\chi(X,\mathcal{O}_{X})=6,\qquad c_{2}(X)=27,$

and the difference morphism $d\colon X\times X\to X-X$ has generic degree
$\geqslant 6$.

  3. (3)

$X$ is isomorphic to the Fano surface of lines on a smooth cubic threefold,
and the canonical morphism $\operatorname{Alb}(X)\to A$ is an isogeny.

Combining this result with corollary 4.7, we obtain the following
classification of subvarieties with Tannaka group $E_{6}$:

######  Corollary 7.2.

Let $X\subset A$ be a smooth irreducible subvariety with ample normal bundle
and dimension $<g/2$. Then the following are equivalent:

  1. (1)

$X\subset A$ is nondivisible with Tannaka group $G_{X,\omega}^{\ast}\simeq
E_{6}$.

  2. (2)

$X$ is isomorphic to the Fano surface of lines on a smooth cubic threefold,
and the canonical morphism $\operatorname{Alb}(X)\to A$ is an isogeny.

The rest of the section is devoted to the proof of theorem 7.1. By the
Lefschetz principle and [JKLM23, cor. 4.4] we may assume $k=\mathbb{C}$. In
this case we already know from section 4.6 that (1) $\Rightarrow$ (2), and (3)
$\Rightarrow$ (1) is clear from the example in Example of the introduction. So
the only remaining point is (2) $\Rightarrow$ (3), which gives a numerical
characterization of Fano surfaces of smooth cubic threefolds.

###  7.2. Strategy of the proof

Before coming to the details, let us outline the main steps in the proof of
(2) $\Rightarrow$ (3). The idea is to recover the cubic threefold as in the
example in Example. By assumption, the subvariety $X\subset A$ is a surface
with

$\chi(X,\mathcal{O}_{X})=\chi(X,\Omega^{2}_{X})=6,\qquad c_{2}(X)=27.$

By Noether’s formula we then also know $c_{1}(X)^{2}=45$ and by the
Hirzebruch-Riemann-Roch theorem we have $\chi(X,\Omega^{1}_{X})=-15$. Now we
can use the short exact sequence

(7.1)
$0\longrightarrow\mathcal{T}=T_{X}\longrightarrow\operatorname{Lie}A\otimes_{k}\mathcal{O}_{X}\longrightarrow\mathcal{N}=\mathcal{N}_{X/A}\longrightarrow
0,$

to compute the Chern classes of the normal bundle $\mathcal{N}$ and find:

$c_{1}(\mathcal{N})^{2}=45,\qquad
c_{2}(\mathcal{N})=c_{1}(X)^{2}-c_{2}(X)=18.$

Borrow notation from section 6.5 with the short exact sequence (6.1) being
(7.1) and $V=\operatorname{Lie}A$. Since $X$ is nondivisible, the Gauss map
$\gamma\colon X\to\operatorname{Gr}_{2}(\operatorname{Lie}A)$ is a finite
morphism [Deb95, prop. 3.1]. By interpreting $c_{2}(\mathcal{N})$ as the top
Segre class of $T_{X}$, the positivity of $c_{2}(\mathcal{N})$ implies that
the morphism $\pi\colon\mathbb{P}(T_{X})\to\mathbb{P}(\operatorname{Lie}A)$ is
generically finite. The bootstrap result is the computation of the degree of
$\pi$. To do this, we use that the difference morphism

$d\colon\quad X\times X\;\longrightarrow\;X-X\;\subset\;A$

is generically finite of degree $\geqslant 6$ by assumption. It is then a
completely general fact that $\deg\pi\geqslant\deg d$ within this framework.
More precisely:

######  Lemma 7.3.

Let $Z\subset A$ be an integral smooth subvariety, and suppose that the
difference morphism $d_{Z}\colon Z\times Z\to A$ and the morphism
$\pi_{Z}\colon\mathbb{P}(T_{Z})\to\mathbb{P}(\operatorname{Lie}A)$ are
generically finite onto their images. Then

$\deg\pi_{Z}\geqslant\deg d_{Z}.$

In particular we have $\deg\pi\geqslant 6$. On other hand, by construction the
degree of $\pi$ has to divide $c_{2}(\mathcal{N})=18$. Therefore, there are
only three possibilities:

$\deg\pi\in\\{6,9,18\\}.$

This constrains also the degree of $\gamma$, as it must divide the degree of
$\pi$. Moreover, the pull-back on $X$ of the line bundle $\mathcal{O}(1)$ on
$\mathbb{P}(\operatorname{Alt}^{2}\operatorname{Lie}A)$ via the composite of
$\gamma$ and of the Plücker embedding is
$\operatorname{Alt}^{2}(T_{X}^{\vee})=\Omega^{2}_{X}$. It follows that the
degree of $\gamma$ needs to divide also
$c_{1}(\Omega^{2}_{X})=c_{1}(X)^{2}=45$. As a consequence, we have

(7.2) $\deg\gamma\in\\{1,3,9\\}.$

As in section 6.5 let $Y\subset\mathbb{P}(\operatorname{Lie}A)$ be the image
of $\pi$. Then $Y$ has dimension $3$ and degree

$\deg Y=c_{2}(\mathcal{N})/\deg\pi\in\\{1,2,3\\}.$

On the other hand, the nondegeneracy of $X$ implies that $X$ generates the
abelian variety $A$. In particular $Y$ is not contained in any hyperplane,
which rules out the possibility $\deg Y=1$. Furthermore, the classical lower
bound (see [EH87, prop. 0] for instance)

$\deg Y\geqslant 1+\operatorname{codim}Y$

implies that $Y$ has codimension $1$ or $2$. Moreover, if
$\operatorname{codim}Y=2$, then equality holds in the above inequality. In
loc.cit. subvarieties for which the above inequality is an identity are
completely classified. We rule out the codimension $2$ by comparing the Fano
variety in each of these cases with the Fano variety $F$ of lines on $Y$:

######  Proposition 7.4.

We have $g=5$ and $Y$ is a cubic hypersurface.

We are left with showing that $Y$ is smooth. We do this by comparing the
Hilbert polynomials (with respect to the Plücker embedding) of $X$ and of the
Fano variety $F$ of lines on $Y$, which will show:

######  Proposition 7.5.

The hypersurface $Y$ is smooth and $\gamma\colon X\to F$ is an isomorphism.

The rest of the section is devoted to the proof of lemmas 7.3, LABEL:, 7.4,
LABEL: and 7.5.

###  7.3. The projection to the tangent space: Proof of lemma 7.3

Since there is no risk of confusion here, let us write $d=d_{Z}$ and
$\pi=\pi_{Z}$. Let $D:=Z-Z$ be the image of the difference morphism. Consider
the blow-up $\tilde{A}$ of $A$ in $0$, the blow-up $\tilde{D}$ of $D$ in $0$
and the blow-up $\tilde{P}$ of $P=Z\times Z$ in the diagonal. The normal
bundle of the diagonal in $Z\times Z$ is by definition the tangent bundle of
$Z$. It follows that the exceptional divisor of $\tilde{P}\to P$ is
$\mathbb{P}(T_{Z})$. On the other hand the exceptional divisor of
$\tilde{A}\to A$ is $\mathbb{P}(\operatorname{Lie}A)$ and that of
$\tilde{D}\to D$ is the tangent cone $C$ of $D$ at $0$. Let
$\tilde{d}\colon\tilde{P}\to\tilde{D}$ be the morphism induced by the
difference map. The situation is resumed in the following commutative diagram:

(7.3)
${\mathbb{P}(T_{Z})}$${C}$${\mathbb{P}(\operatorname{Lie}A)}$${\tilde{P}}$${\tilde{D}}$${\tilde{A}}$${P=Z\times
Z}$${D=Z-Z}$${A}$$\scriptstyle{\tilde{d}}$$\scriptstyle{d}$

######  Lemma 7.6.

The composite morphism $\mathbb{P}(T_{Z})\to\mathbb{P}(\operatorname{Lie}A)$
is $\pi$.

###### Proof.

We reduce to the case $Z=A$ by noticing that $d$ is the restriction to
$Z\times Z$ of the difference morphism $A\times A\to A$ and $\pi$ is the
restriction to $\mathbb{P}(T_{Z})$ the projection

$\mathbb{P}(T_{A})=\mathbb{P}(\operatorname{Lie}A)\times
A\longrightarrow\mathbb{P}(\operatorname{Lie}A).$

Write $d$ as the composite of the automorphism $(x,y)\mapsto(x-y,y)$ of
$A\times A$ and the second projection $\operatorname{pr}_{2}\colon A\times
A\to A$. In this way we reduce to the situation where we replace $d$ by the
second projection $\operatorname{pr}_{2}$ and $\tilde{P}$ by the blow-up of
$A\times A$ in $A\times\\{0\\}$. In this case the result to prove is that the
morphism
$A\times\mathbb{P}(\operatorname{Lie}A)\to\mathbb{P}(\operatorname{Lie}A)$
induced by $\operatorname{pr}_{2}$ is the second projection. This is of course
clear and concludes the proof. ∎

To conclude the proof of lemma 7.3, notice first that the top-left square in
diagram (7.3) is cartesian. Indeed, this boils down to the fact that the
scheme-theoretic fiber of $d$ in $0$ is the diagonal $\Delta_{Z}\colon Z\to
Z\times Z$. To see this, notice that the difference morphism $A\times A\to A$
is smooth and the scheme-theoretic preimage of $0$ is the diagonal
$\Delta_{A}\colon A\hookrightarrow A\times A$ with its reduced structure; the
claim then follows because the following square is cartesian:

${Z}$${Z\times Z}$${A}$${A\times
A}$$\scriptstyle{\Delta_{Z}}$$\scriptstyle{\Delta_{A}}$

######  Lemma 7.7.

Let $f\colon S\to T$ be a proper surjective morphism between varieties and
$T^{\prime}\subset T$ a subvariety whose scheme-theoretic preimage
$S^{\prime}=S\times_{T}T^{\prime}$ is integral. If the morphisms $f$ and
$f_{\rvert S^{\prime}}\colon S^{\prime}\to T^{\prime}_{{\mathrm{red}}}$ are
generically finite, then

$\deg(f_{\rvert S^{\prime}})\geqslant\deg(f).$

###### Proof.

The locus of $T$ where $f$ is finite is open and meets $T^{\prime}$. By
replacing $T^{\prime}$ with such an open subset, we may assume that $f$ is
finite. Now, degree of $f$ is computed as the generic rank of the coherent
$\mathcal{O}_{T}$-module $\mathcal{F}=f_{\ast}\mathcal{O}_{S}$. Let
$f^{\prime}\colon S^{\prime}\to T^{\prime}$ be the morphism induced by $f$.
Since the morphism $f$ is affine, we have

$\mathcal{F}_{\rvert T^{\prime}}=f^{\prime}_{\ast}\mathcal{O}_{S^{\prime}}.$

The subvariety $S^{\prime}\subset S$ is assumed to be integral, thus its
scheme-theoretic image is $T^{\prime}_{{\mathrm{red}}}\subset T^{\prime}$. It
follows that the coherent $\mathcal{O}_{T^{\prime}}$-module
$f^{\prime}_{\ast}\mathcal{O}_{S^{\prime}}$ is supported in
$T^{\prime}_{{\mathrm{red}}}$. Therefore, there is no loss of generality in
assuming that $T^{\prime}$ is reduced, hence integral. By the above
discussion, the generic degree of $f^{\prime}$ is the generic rank of
$\mathcal{F}_{\rvert T^{\prime}}$. The lower semicontinuity of the function
$T\ni t\mapsto\operatorname{rk}\mathcal{F}_{t}$ gives the result. ∎

We apply the preceding lemma with $S=\tilde{P}$, $T=\tilde{D}$, $f=\tilde{d}$
and $T^{\prime}=C$. The hypotheses are fulfilled because $\tilde{d}$ is
generically finite, the scheme-theoretic preimage of $C$ is
$\mathbb{P}(T_{Z})$, and the restriction of $\tilde{d}$ to $\mathbb{P}(T_{Z})$
is $\pi$ by lemma 7.6 hence generically finite by hypothesis. Moreover, the
morphism $\tilde{d}$ is generically finite of same degree of $d$ because blow-
ups are birational maps. ∎

###  7.4. The cubic hypersurface: Proof of proposition 7.4

By hypothesis $X$ is nondegenerate, thus it generates the abelian variety $A$.
It follows that $Y$ cannot be contained in any hyperplane. Therefore the lower
bound $\deg Y\geqslant 1+\operatorname{codim}Y$ holds by [EH87, prop. 0]. As
already argued, we have $\deg\pi\in\\{3,9\\}$ hence

$\deg Y=c_{2}(\mathcal{N})/\deg\pi\in\\{2,3\\}.$

We start by excluding the case $\deg Y=2$. In this case we have
$\operatorname{codim}Y=1$ hence $g=5$ and $Y$ is a $4$-dimensional quadric. To
rule out the case, let $F$ be the Fano variety of lines in $Y$ and recall by
proposition 6.17 that the image of $\gamma\colon X\to F$ is a $2$-dimensional
irreducible component of $F$. The rank $r$ of the quadric $Y$ is necessarily
$3$, $4$ or $5$ because $Y$ is integral. We proceed by distinguishing case by
case:

  * •

If $r=3$, then $Y$ is a cone over a smooth plane conic $Q$ and $F$ is
irreducible of dimension $\dim Q+2(5-3-1)=3$ by proposition 6.3.
Contradiction.

  * •

If $r=4$, then $Y$ is a cone over the normal rational scroll
$S(1,1)=\mathbb{P}^{1}\times\mathbb{P}^{1}$ embedded in $\mathbb{P}^{3}$ via
the Segre embedding [Har92, p. 285]. By proposition 6.1 the Fano variety of
lines in $S(1,1)$ is $\mathbb{P}^{1}\sqcup\mathbb{P}^{1}$, thus $F$ has two
irreducible components of dimension $\dim\mathbb{P}^{1}+2(5-4)=3$ by
proposition 6.3. This is a contradiction.

  * •

If $r=5$, then $Y$ is smooth quadric threefold in $\mathbb{P}^{4}$ and
$F\cong\mathbb{P}^{3}$; see for instance [Har92, ex. 22.6]. Contradiction.

The above discussion shows

$\deg Y=3.$

The inequality $\deg Y\geqslant 1+\operatorname{codim}Y$ implies that
$Y\subset\mathbb{P}(\operatorname{Lie}A)$ has codimension $1$ or $2$. It
remains to rule out the codimension $2$ case. If this is the case, the variety
$Y$ is of minimal degree, hence by [EH87, th. 1] it is either a cone over a
smooth rational scroll or a cone over the Veronese surface, that is,
$\mathbb{P}^{2}$ embedded in $\mathbb{P}^{5}$ by the second Veronese
embedding. Now taking a cone with respect to some linear projection preserves
the codimension. This rules out at once the case of the Veronese surface since
it has codimension $3$ while $Y$ has codimension $2$. Suppose $Y$ is a smooth
rational normal scroll $S=S(a)$ or a cone over such a scroll. By proposition
6.17 the scheme-theoretic image of the finite morphism $\gamma\colon X\to F$
is one of the irreducible components of $F$. A case by case distinction shows
that this is a contradiction. More precisely, the tuple
$a=(a_{1},\dots,a_{r})$ satisfies $1\leqslant a_{1}\leqslant\cdots\leqslant
a_{r}$ and $a_{1}+\cdots+a_{r}=3$. This leaves us with the $3$ possibilities
described in detail in example 6.2:

$a=(3),(1,2),(1,1,1).$

Suppose $\dim S=1$. In this case the scroll $S$ is the image of the triple
Veronese embedding $\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{3}$. As $S$
contains no lines, proposition 6.3 implies that $F$ is irreducible of
dimension $3$. But $X$ dominates a component of $F$. Contradiction.

Suppose $\dim S=2$. In this case the Fano variety is
$\mathbb{P}^{1}\sqcup\\{\textup{pt}\\}$. By proposition 6.3 the component of
$F$ over $\mathbb{P}^{1}$ has dimension $\dim\mathbb{P}^{1}+2(6-5)=3$ hence
$X$ cannot dominate it. Instead again by proposition 6.3 the component of $F$
over the singleton is a projective plane embedded linearly via the Plücker
embedding. If $X$ were to dominate it, we would have $\deg\gamma=45$,
contradicting (7.2).

Suppose $\dim S=3$. In this case $Y=S$ and