# Relative and absolute Lefschetz standard conjectures for some Lagrangian
fibrations

Giuseppe Ancona  ,  Mattia Cavicchi  ,  Robert Laterveer  and  Giulia Saccà

###### Abstract.

We show that the hyper-Kähler varieties of OG10-type constructed by
Laza–Saccà–Voisin (LSV) verify the Lefschetz standard conjecture. This is an
application of a more general result, stating that certain Lagrangian
fibrations verify this conjecture. The main technical assumption of this
general result is that the Lagrangian fibration satisfies the hypotheses of
Ngô’s support theorem. Verifying that the LSV tenfolds do satisfy those
hypotheses is of independent interest. Another point of independent interest
of the paper is the definition and the study of the Lefschetz standard
conjecture in the relative setting, and its relation to the classical absolute
case.

###### Contents

  1. 1 Introduction
  2. 2 Conventions
  3. 3 Statement of the main result
  4. 4 The relative Lefschetz standard conjecture
  5. 5 Good decomposition: from relative to absolute
  6. 6 Ngô fibrations and relative Lefschetz
  7. 7 The Shen–Yin decomposition is good
  8. 8 Voisin’s trick and the end of the proof
  9. 9 An application to LSV tenfolds
    1. 9.1 LSV tenfolds and their Ngô fibration
    2. 9.2 Polarizability of the descended relative Prym
    3. 9.3 Application of Theorem 3.1 to LSV tenfolds

##  1\. Introduction

The standard conjectures, formulated by Grothendieck and Bombieri in the
sixties, have been hugely influential in the development of the theory of
motives and algebraic cycles [Gro69, Kle68, Kle94]. Yet, in spite of their
pervasive presence111Grothendieck presented them as the most urgent task in
algebraic geometry., not a great deal more is known about them nowadays than
in the sixties.

In this paper we will work over the complex numbers. In this setting it is
known that the Lefschetz standard conjecture implies all the other standard
conjectures. Let us recall its formulation. Let $L$ be an ample divisor on a
smooth projective variety $X$ of dimension $d_{X}$. The hard Lefschetz theorem
says that the map

(1.1) $\displaystyle\cup L^{n}:H^{d_{X}-n}(X,\mathbb{Q})\xrightarrow{\hskip
1.85pt\sim\hskip 1.85pt}H^{d_{X}+n}(X,\mathbb{Q})$

is an isomorphism. The Lefschetz standard conjecture predicts that the inverse
of this map is induced by an algebraic correspondence. This project is guided
by the basic question: how well does this conjecture behave under fibration?

The starting point of this note is Voisin’s paper [Voi22], where the Lefschetz
standard conjecture in degree 2 is proved for HK manifolds which are covered
by Lagrangian subvarieties. Particularly relevant for our paper is the case of
Lagrangian fibrations: in this case, Voisin proves the conjecture in degree
two, under the assumption that the SYZ conjecture holds. Here, we use Voisin’s
idea, extending it thanks to the use of Ngô’s Support Theorem.

To explain how this theorem enters the picture, let us assume that $X$ is
endowed with a surjective morphism $f:X\rightarrow B$ to a smooth projective
variety $B$ (with $f$ non-constant and non-finite). We ask ourselves how much
the Lefschetz standard conjecture for $B$ and for the smooth fibers may help
to show the Lefschetz standard conjecture for $X$.

Let us be more precise and let us read this through the decomposition theorem.
To ease notation, we will work with the constant complex
$\mathcal{C}=\mathbb{Q}[d_{X}]$ on $X$ and write
$H^{n}(X)=H(X,\mathcal{C})=H^{d_{X}+n}(X,\mathbb{Q})$. Moreover, let $\eta$ be
a relatively ample divisor on $X$, $L_{B}$ an ample divisor on $B$ and $\beta$
its pullback on $X$. The decomposition theorem gives in particular a
decomposition

(1.2) $\displaystyle
H^{n}(X)=\bigoplus_{a+b=n}H^{a}(B,^{p}\\!\\!R^{b}f_{*}\mathcal{C}),$

where the ${}^{p}\\!\\!R^{b}f_{*}\mathcal{C}$ denote the perverse cohomology
objects of $Rf_{*}\mathcal{C}$, as well as the following isomorphisms:

(1.3)
$\displaystyle\cup\eta^{b}:H^{a}(B,^{p}\\!\\!R^{-b}f_{*}\mathcal{C})\xrightarrow{\hskip
1.85pt\sim\hskip 1.85pt}H^{a}(B,^{p}\\!\\!R^{+b}f_{*}\mathcal{C}),$ (1.4)
$\displaystyle\cup\beta^{a}:H^{-a}(B,^{p}\\!\\!R^{b}f_{*}\mathcal{C})\xrightarrow{\hskip
1.85pt\sim\hskip 1.85pt}H^{+a}(B,^{p}\\!\\!R^{b}f_{*}\mathcal{C}).$

The combination of (1.2), (1.3) and (1.4) may suggest that the Lefschetz
standard conjecture can be reduced to showing that the inverses of (1.3) and
(1.4) are algebraic. Additionally, at first sight one might think that an
algebraic construction of the inverse of (1.3) can be reduced to the Lefschetz
standard conjecture for the smooth fibers of $f$ and similarly that an
algebraic construction of the inverse of (1.4) can be reduced to the Lefschetz
standard conjecture for $B$. However, a number of subtleties appear.

  1. (1)

The perverse sheaf ${}^{p}\\!R^{b}f_{*}\mathcal{C}$ might have direct factors
supported on the singular locus of $f$. In this case the information on the
smooth fibers will give no control on these direct factors.

  2. (2)

Inverting the action of $L_{B}$ on the cohomology of $B$ is not enough for
inverting $\beta$. Indeed $f^{*}H(B)\subset H(X)$ coincides only with the
factor $H^{*}(B,^{p}\\!\\!R^{-d_{X}}f_{*}\mathcal{C})$ in the decomposition
$\eqref{BBD}$. This is the most tragic point.

  3. (3)

Even if one were able to construct $\Lambda_{\eta^{b}}$ and
$\Lambda_{\beta^{a}}$ acting as the inverses of (1.3) resp. (1.4), it is
unclear how to put all of them together and deduce an algebraic correspondence
$\Lambda_{n}:H^{n}(X)\xrightarrow{\hskip 1.85pt\sim\hskip 1.85pt}H^{-n}(X).$
For example one could think of the formula
$\Lambda_{n}=\sum_{a+b=n}\Lambda_{\eta^{b}}\circ\Lambda_{\beta^{a}}$ but such
a sum could lead to cancellations. It is indeed unclear how, for example,
$\Lambda_{\eta^{b}}$ acts on perverse degrees different from $b$. Already the
actions of $\eta$ and $\beta$ are not in general bigraded with respect to the
decomposition (1.2).

In the case where $X$ is a hyper-Kähler variety (hence $f$ ia a Lagrangian
fibration) one can partially bypass these problems.

  1. (1)

Ngô’s support theorem [Ngô10] can be used to control the support of
${}^{p}\\!R^{b}f_{*}\mathcal{C}$ and avoid problem (1).

  2. (2)

An idea of Voisin [Voi22] can be used to (roughly speaking) construct a second
Lagrangian fibration inverting the role of $\eta$ and $\beta$. (Hence problem
(2) reduces to problem (1).)

  3. (3)

A general result of Shen and Yin shows that the operators $\eta$ and $\beta$
are actually bigraded in the setting of Lagrangian fibrations [SY].

These ingredients allow us to prove the following general criterion.

######  Theorem 1.1.

Let $X$ be a hyper-Kähler variety admitting a Lagrangian fibration
$f:X\rightarrow B$. Assume that:

  1. (1)

The Picard rank of $X$ is $2$.

  2. (2)

There is only one isotropic line in the Picard group which is in the boundary
of the birational Kähler cone (the line generated by $\beta$).

  3. (3)

The fibration $f$ is an Ngô fibration (see Definition 6.3).

  4. (4)

The fibers of $f$ are irreducible.

Then the Lefschetz standard conjecture holds for $X$.

The main application is the following. Some other applications will be given
in the text.

######  Theorem 1.2.

Let $M$ be a smooth cubic fourfold, and let $X$ be the associated Lagrangian
fibered hyper-Kähler manifold constructed in [LSV17, Voi18, Sac22]. Then $X$
satisfies the standard conjectures.

In the same spirit as [Voi22], we give a generalization of Theorem 1.1.

######  Theorem 1.3.

Let $X$ be a hyper-Kähler variety admitting a Lagrangian fibration
$f:X\rightarrow B$. Assume that:

  1. (1)

The Picard rank of $X$ is $2$.

  2. (2)

The SYZ conjecture holds for $X$.

  3. (3)

The Lagrangian fibrations on any birational model of $X$ are Ngô fibrations.

  4. (4)

The fibers of these fibrations are irreducible.

Then the Lefschetz standard conjecture holds for $X$.

Let us comment on the above theorems, their assumptions, their applications
and their possible generalizations.

Assumption (1) is very reasonable. Indeed, by [Mat16] and local Torelli any
hyper-Kähler variety admitting a Lagrangian fibration is the specialization of
a Picard rank $2$ hyper-Kähler admitting a Lagrangian fibration and,
furthermore, the Lefschetz standard conjecture for a variety implies the
conjecture for all of its specializations.

If assumption (1) holds, then there are exactly two isotropic lines with
respect to the Beauville–Bogomolov form. One, namely the line generated by the
lagrangian class $\beta$, is certainly in the boundary of the birational
Kähler cone. As far as the second isotropic line is concerned, there are two
cases, depending on whether it is also on this boundary. Theorems 1.1 and 1.3
correspond precisely to these two cases. If the other line is also on this
boundary, then by the SYZ conjecture there is a second Lagrangian fibration
associated with it. Notice that this conjecture has been proved for all known
deformation types of hyper-Kähler varieties. When there is only one line there
is no second fibration, since the second ray of the birational Kähler cone
corresponds to a birational contraction. There is, however, a second fibration
up to an automorphism of the motive, as was observed by Voisin [Voi22].

The restrictive assumptions are (3) and (4). Note that in [Voi22], this result
is proved in degree $2$ without these two assumptions on the Lagrangian
fibration, since singular fibers end up not mattering for the degree $2$ case.
Assumption (3) is needed to apply Ngô’s support theorem and assumption (4) is
needed to deduce through Ngô’s support theorem that the perverse sheaves
${}^{p}\\!R^{b}f_{*}\mathcal{C}$ have full support. For the LSV tenfolds it
takes some work to verify assumption (3); this part of the paper (Section 9)
is of independent interest.

Despite the recent growth of interest in the use of Ngô’s support theorem for
Lagrangian fibrations, not many are known to be Ngô fibrations (i.e. to
satisfy the assumptions required to implement the support theorem). It is a
folklore conjecture that all Lagrangian fibrations are Ngô fibrations. Hence,
we expect that in the future our methods will be applied to more general
Lagrangian fibrations.

Our strategy is to first deal with a relative version of the Lefschetz
standard conjecture and then to pass from the relative to the classical
absolute case. This relative version is new and contains some subtleties with
respect to the absolute one. The correct setting for this relative version is
a category of relative homological motives which we define taking inspiration
from the one defined in [CH00]. Our category seems to be the correct one for
dealing with problems such as the construction of algebraic classes, but of
course it is very far from the conjectural abelian category of relative pure
motives.

These foundational results should be interesting in their own right. In a
sequel to this paper we will give a stratified version of these constructions
to deal with the case when the perverse sheaves
${}^{p}\\!R^{b}f_{*}\mathcal{C}$ have direct factors strictly supported on the
base. This will allow us to get rid of assumption (4), leading to even more
applications.

To end this introduction let us mention that, with very different methods,
Charles and Markman proved the Lefschetz standard conjecture for hyper-Kähler
varieties of K3[n]-type [CM13] and more recently Foster proved it for those of
Kummer type in some degrees [Fos23]. Together with [Voi22] mentioned above,
these are among the rare previous results on this conjecture and not only in
the realm of hyper-Kähler varieties.

### Organization of the paper

Section 2 gives the general conventions and contains in particular the
construction of the category of relative homological motives. In Section 3 we
state the main results and recall the basic definitions needed. Section 4
gives several equivalent formulations of the relative Lefschetz standard
conjecture; we also highlight the subtleties which appear in this relative
setting and not in the absolute one. In Section 5 we recall the notion of good
decomposition of cohomology due to de Cataldo [dC13]. In the presence of such
a good decomposition the subtleties explained in the previous section
disappear. In general, the decompositions induced by the decomposition theorem
are not good, but it turns out that they are so for Lagrangian fibrations.
This is a particular instance of a result of Shen and Yin [SY] and it is
explained in Section 7. The second tool is Ngô support theorem. We recall it
for the convenience of the reader in Section 6 and explain some applications.
In Section 8 we recall the argument of Voisin that allows us to reason as if
there were a second Lagrangian fibration, and use this to conclude the proof
of the main result, Theorem 8.2 (Theorem 1.3 in the introduction, of which
Theorem 1.1 is a special case). The long Section 9 is dedicated to LSV
tenfolds; the aim is to deduce Theorem 9.12 (Theorem 1.2 in the introduction)
from our main result. The first subsection recalls their geometry, the second
proves the polarizability of the underlying group scheme and the third
concludes the proof of Theorem 9.12.

### Acknowledgments

We thank Salvatore Floccari, Lie Fu, Emanuele Macrì, Claire Voisin and Qizheng
Yin for their interest in our work and for numerous fruitful discussions.

This research was partly supported by the grants ANR–18–CE40–0017 and
ANR–20–CE40–0023 of the Agence National de la Recherche. GS acknowledges
support from NSF CAREER grant DMS-2144483 and NSF FRG grant DMS-2052750.

##  2\. Conventions

Throughout the paper we will work with the following notation and conventions.

  1. (1)

Varieties. We work with algebraic varieties over fields of characteristic
zero, and most of the time with algebraic varieties over the complex numbers.
Whenever a field of definition different from $\mathbb{C}$ appears (for
example the residue field of the generic point of a variety) standard
techniques allow us to reduce to $\mathbb{C}$ (take a model and embed the
field of definition).

  2. (2)

Rational coefficients. All the invariants we consider (Chow rings,
cohomology,$\ldots$) will be with rational coefficients.

  3. (3)

Cohomology. For a variety $X$ of dimension $d_{X}$ we denote by
$C_{X}=\mathbb{Q}[d_{X}]$ the constant sheaf concentrated in degree $-d_{X}$.
We write $H^{n}(X)$ for $H^{n}(X,C_{X})$. This is the usual singular
cohomology with rational coefficients with degrees shifted so that the
Poincaré duality is centered in zero.

  4. (4)

Derived categories. Fix a base variety $B$, and consider either the analytic
topology on $B(\mathbb{C})$, or the étale topology on $B$. We will denote by
$D(B)$ the derived category of sheaves over $B$ for either one of these two
topologies, and by $D^{b}(B)$ its full subcategory of bounded objects. We will
use $\mathbb{Q}$-coefficients in the analytic case, and
$\overline{\mathbb{Q}}_{\ell}$-coefficients, for $\ell$ a prime, in the étale
case. The context will make clear which category we are using. We will
consider either the _classical_ or the _perverse_ $t$-structure on $D(B)$ (see
[BBD82] for generalities). For an integer $n\in\mathbb{Z}$, the classical
cohomology objects of an object $K$ in $D(B)$ will be denoted by $H^{n}K$. If
$f:Z\rightarrow B$ is a morphism and $K$ a complex over $Z$, write $F$ for
either one of the functors $f_{*},f_{!}$. We will denote by $R^{n}F(K)$ the
classical cohomology objects of the complex $RF(K)$, and by
${}^{p}\\!R^{n}F(K)$ its perverse cohomology objects.

  5. (5)

HK. Our center of interest will be (projective) hyper-Kähler varieties. Such a
variety will be called a HK variety or simply a HK.

  6. (6)

Motives. We will work with the category ${\mathcal{M}ot}$ of homological
motives defined over $\mathbb{C}$ and with rational coefficients. The category
comes equipped with a realization functor

$R:{\mathcal{M}ot}\longrightarrow\mathrm{Vect}.$

to the category of rational vector spaces. (The latter are endowed with a
Hodge structure, which we will not need.) For generalities see [And04, §4].

  7. (7)

Tate twists. We will make the brutal choice of fixing an isomorphism
$\mathbb{Q}\cong\mathbb{Q}(1)$ in cohomology, hence ignoring the Tate twists.
This is done so that the operation of cup-producing with a given algebraic
class is an endomorphism of the cohomology ring. See the discussion in [And04,
5.2.3].

However, we will keep track of the Tate twists for motives. Hence the
realization of the motive $\mathbbm{1}(1)$ is the complex $\mathbb{Q}(1)[2]$,
which by our convention is identified with the constant complex $\mathbb{Q}$
in degree $-2$.

  8. (8)

Relative motives. Following Corti–Hanamura [CH00] (cf. also [MNP13, Chapter
8]), there is a category $\mathrm{CHM}(B)$ of relative Chow motives over a
base $B$. Whenever $B$ is projective, there is a commutative square of
functors

${\mathrm{CHM}(B)}$${D^{b}(B)}$${\mathrm{CHM}(\mathbb{C})}$${\mathrm{Vect}.}$$\scriptstyle{R_{B}}$$\scriptstyle{H^{*}}$$\scriptstyle{R}$

Note that the horizontal functors are not faithful. To fix this, we work up to
homological equivalence. This means we can define ${\mathcal{M}ot}(B)$ as (the
pseudo-abelian envelop of) the quotient of the motivic categories with respect
to the tensor ideal given by the kernel of the realization functors. By
construction, this gives the commutative diagram

${{\mathcal{M}ot}(B)}$${D^{b}(B)}$${{\mathcal{M}ot}}$${\mathrm{Vect}.}$$\scriptstyle{R_{B}}$$\scriptstyle{H^{*}}$$\scriptstyle{R}$

where the horizontal functors are now faithful by construction.

##  3\. Statement of the main result

In this section we state our main result, Theorem 3.1, and recall the main
definitions which are needed. The result will be restated and proven as
Theorem 8.2 in Section 8, once the necessary ingredients will have been
developed in Sections 4 to 8.

######  Theorem 3.1.

Let $X$ be a hyper-Kähler variety admitting a Lagrangian fibration
$f:X\rightarrow B$. Assume that:

  1. (1)

The Picard rank of $X$ is $2$.

  2. (2)

The SYZ conjecture (Conjecture 3.8) holds for $X$.

  3. (3)

The Lagrangian fibrations on any birational model of $X$ are Ngô fibrations
(cf. Definition 6.3).

  4. (4)

The fibers of these fibrations are irreducible.

Then the Lefschetz standard conjecture holds for $X$ (and hence all the
standard conjectures hold for $X$).

This is Theorem 1.3 in the introduction; Theorem 1.1 then follows as a special
case. Let us recall some of the concepts appearing in Theorem 3.1.

######  Definition 3.2.

A smooth projective variety $Y$ is called hyper-Kähler (HK) if it is simply-
connected and

$H^{0}(Y,\Omega^{2}_{Y})=\mathbb{C}[\sigma_{Y}]\ ,$

where $\sigma_{Y}$ is a nowhere-degenerate two-form.

######  Remark 3.3.

In every even complex dimension greater or equal to $4$, there are two series
of known examples: those of K3[n]-type, which are deformation equivalent to
the Hilbert schemes of $n$ points on a K3 surface, and those of generalized
Kummer type, which are deformation equivalent to generalized Kummer varieties.
In addition to these, there are two other deformation classes in dimensions
$6$ and $10$. These deformation classes are referred to as OG$6$, respectively
OG$10$, and hyper-Kähler manifolds in these deformation classes are called of
OG6-type, respectively of OG10-type.

######  Definition 3.4.

Let $Y$ be a HK variety. A Lagrangian fibration is a surjective morphism with
connected fibers $f\colon Y\to B$ to a normal variety such that $0<\dim B<\dim
X$.

######  Remark 3.5.

By [Mat99], the smooth fibers of the fibration are Lagrangian (i.e. they are
maximal isotropic submanifolds of $Y$), in particular, $\dim B={1\over 2}\dim
X$. In fact, more is true, namely that every fiber is Lagrangian in the sense
that the smooth locus of the reduced underlying variety is a Lagrangian
submanifold [Mat00]. In particular, $f$ is equidimensional. A long-standing
expectation is that the base of a Lagrangian fibration is a projective space;
this is known to be true when $\dim Y=4$ [HX22] and also known to hold under
the condition that $B$ is smooth [Hwa08].

######  Remark 3.6.

Recall that birational HK varieties are non separated points of the moduli
spaces, in the sense that they appear as central fibers of one parameter
deformations that are isomorphic away from the central fiber; the choice of
such families induces an isomorphism of their cohomology rings and motives
[Huy06, Rie14]. As a consequence, the Lefschetz standard conjecture for one HK
manifold implies the Lefschetz standard conjecture for all of its birational
HK models. Recall also that birational HK manifolds are isomorphic in
codimension $2$ and thus have canonically isomorphic second cohomology groups.

The previous remark is implicitly used in the following definition.

######  Definition 3.7.

Let $Y$ be a HK and $q$ be the Beauville–Bogomolov quadratic form on
$H^{2}(Y)$. A divisor class $l$ on $Y$ is said to be Lagrangian if there
exists a birational HK $Y^{\prime}$ on which $l$ is semi-ample. This means
that $Y^{\prime}$ admits a Lagrangian fibration $Y^{\prime}\to B^{\prime}$ and
that, up to the identification $H^{2}(Y)\cong H^{2}(Y^{\prime})$, $l$ is the
pullback of an ample class on $B^{\prime}$.

The class $l$ is said to be potentially Lagrangian if $q(l)=0$ and $l$ is in
the boundary of the birational Kähler cone.

######  Conjecture 3.8.

(Conjecture SYZ) All potentially Lagrangian classes are actually Lagrangian.

This conjecture holds true for all known examples of HK varieties:

######  Proposition 3.9.

Let $Y$ be a HK variety which is either of K3[n]-type, or of generalized
Kummer type, or of OG6-type, or of OG10-type. Then Conjecture 3.8 is true for
$Y$.

###### Proof.

For HK varieties of K3[n]-type or of generalized Kummer type, the statement is
proven in [Mat17, Corollary 1.1]. For HK varieties of OG6 resp. OG10-type, the
statement is proven in [MR21, Theorem 7.2] resp. [MO22, Theorem 2.2]. ∎

##  4\. The relative Lefschetz standard conjecture

Throughout this section we fix a surjective projective morphism
$f:X\longrightarrow B$ of algebraic varieties with $X$ smooth and connected.
We will denote by $C_{X}$ the constant perverse sheaf on $X$. We also fix a
relatively ample divisor $\eta$.

The goal of the section is to formulate an analogue of the Lefschetz standard
conjecture in this relative setting, and to give some basic properties.

Recall that in this geometric context an algebraic cycle in $X\times_{B}X$
acts on the relative cohomology [CH00], [MNP13, Chapter 8]. Seen in this way
it is called a relative correspondence.

######  Definition 4.1.

We say that the pair $(f,\eta)$ verifies the relative Lefschetz standard
conjecture (or simply that $X$ does) if for all cohomological degrees
$n\in\mathbb{N}$ there exists a (non-unique) relative correspondence
$\gamma_{n}(f)$ inducing the inverse of the relative hard Lefschetz
isomorphism

$\cup\eta^{n}\colon\ \ ^{p}\\!R^{-n}f_{*}C_{X}\ \cong\
^{p}\\!R^{+n}f_{\ast}C_{X}[2n]$

in degree $-n$ (without any prescribed property in the other degrees).

######  Proposition 4.2.

Assume that:

  1. (1)

The fiber over the generic point of $B$ verifies the Lefschetz standard
conjecture,

  2. (2)

For all $n$ the perverse sheaf ${}^{p}\\!R^{n}f_{\ast}C_{X}$ is the
intersection complex of a local system.

Then the pair $(f,\eta)$ verifies the relative Lefschetz standard conjecture
(Definition 4.1).

###### Proof.

By the first hypothesis, there exist correspondences $\gamma_{n}$ on the
square of the generic fiber acting as the inverse of the Lefschetz operators.
For each $n$ we define $\gamma_{n}(f)$ to be any relative correspondence
extending $\gamma_{n}$ (for example the closure of $\gamma_{n}$ in
$X\times_{B}X$). We claim that such a $\gamma_{n}(f)$ satisfies the properties
of Definition 4.1.

We need to check the equalities

$\begin{split}\bigl{(}\gamma_{n}(f)\circ\cup\eta^{n}\bigr{)}|_{{}^{p}\\!R^{-n}f_{*}C_{X}}&=\operatorname{\mathsf{id}}|_{{}^{p}\\!R^{-n}f_{*}C_{X}}\
,\\\
\bigl{(}\cup\eta^{n}\circ\gamma_{n}(f)\bigr{)}|_{{}^{p}\\!R^{n}f_{*}C_{X}}&=\operatorname{\mathsf{id}}|_{{}^{p}\\!R^{n}f_{*}C_{X}}\
.\\\ \end{split}$

Because of the second hypothesis, both equalities live in an algebra of the
form $\operatorname{End}(IC(\mathcal{L}))$, with $\mathcal{L}$ a semisimple
local system on an open $U$ of the base $B$. Since intersection complexes of a
simple local system are simple [BBD82, Théorème 4.3.1], it follows from
Schur’s lemma that restriction from $B$ to $U$ induces an injective map

$\operatorname{End}(IC(\mathcal{L}))\ \hookrightarrow\
\operatorname{End}(\mathcal{L})\ ,$

and so the equality can be checked after restriction to $U$. Finally, since
$\mathcal{L}$ is a local system, the equalities can be checked after
restriction to any fiber and in particular to the generic fiber. ∎

######  Proposition 4.3.

The pair $(f,\eta)$ verifies the relative Lefschetz standard conjecture if and
only if the motive $\mathfrak{h}(X/B)\in{\mathcal{M}ot}(B)$ admits a
decomposition

(4.1) $\displaystyle\mathfrak{h}(X/B)=\bigoplus_{n}\mathfrak{h}^{n}(X/B)\ ,$

satisfying

  1. (1)

The realization of $\mathfrak{h}^{n}(X/B)$ is concentrated in perverse degree
$n$,

  2. (2)

There is an isomorphism $\mathfrak{h}^{n}(X/B)\cong\mathfrak{h}^{-n}(X/B)(-n)$
for all $n$.

###### Proof.

Let us suppose that $(f,\eta)$ verifies the relative Lefschetz standard
conjecture. We show by descending induction on $N\geq 0$ that there is a
motivic decomposition

(4.2) $\displaystyle\mathfrak{h}(X/B)=\mathfrak{h}^{-N\geq n\geq
N}(X/B)\oplus\bigoplus_{n<-N\,\textrm{or}\,n>N}\mathfrak{h}^{n}(X/B)\ ,$

such that properties (1) and (2) of the statement are satisfied and
$R_{B}(\mathfrak{h}^{-N\geq n\geq N}(X/B))$ is concentrated in degrees between
$-N$ and $N$. To show the inductive step, let us decompose further in degree
$-N$ and $+N$. Consider the algebraic correspondences $\gamma_{N}(f)$ and
$\cup\eta^{N}$ from Definition 4.1 and consider the projector $p$ defining
$\mathfrak{h}^{-N\geq n\geq N}(X/B)$ inside $\mathfrak{h}(X/B)$ through the
decomposition (4.2). Consider the endomorphism $q=p\circ\gamma_{N}(f)\circ
p\circ(\cup\eta^{N})\circ p$ of $\mathfrak{h}^{-N\geq n\geq N}(X/B)$. Let us
look at its realization $R_{B}(q)$, which we write as a matrix with respect to
the perverse degrees (for a fixed non canonical decomposition in the derived
category). By construction its image is concentrated in degrees between $-N$
and $+N$. The axioms of a t-structure imply that $R_{B}(q)$ cannot send a
given degree to a higher one so the matrix associated with $R_{B}(q)$ is upper
triangular. Let us look at the diagonal. In degree $-N$ it is the identity by
Definition 4.1 and in higher degrees the diagonal is zero as already
$p\circ(\cup\eta^{N})\circ p$ is so. In conclusion $R_{B}(q)$ is an upper
triangular matrix whose diagonal is the identity in degree $-N$ and zero
otherwise. Hence the powers $R_{B}(q)^{r}$ stabilize for $r$ big enough
($r\geq 2N$), becoming projectors. We deduce that $\pi_{-N}=q^{2N}$ is a
projector whose realization is concentrated in degree $-N$ and coincides with
${}^{p}\\!R^{-N}f_{\ast}C_{X}$.

Arguing similarly we deduce that $\pi_{+N}=(p\circ(\cup\eta^{N})\circ
p\circ\gamma_{N}(f)\circ p)^{2N}$ is a projector whose realization is
concentrated in degree $-N$ and coincides with ${}^{p}\\!R^{+N}f_{\ast}C_{X}$.
Now a subtlety is that the two projectors $\pi_{-N}$ and $\pi_{+N}$ are not
orthogonal a priori. The composition $\pi_{+N}\circ\pi_{-N}$ is a morphism
from lower degrees to upper degrees hence vanishes. However, the other
composition can be nonzero. By direct computation one shows that $\pi_{-N}$
and $\pi_{+N}-\pi_{-N}\circ\pi_{+N}$ are orthogonal projectors and that
$\pi_{+N}-\pi_{-N}\circ\pi_{+N}$ has again ${}^{p}\\!R^{+N}f_{\ast}C_{X}$ as
realization. Let $\mathfrak{h}^{+N}(X/B)$ and $\mathfrak{h}^{-N}(X/B)$ be the
motives defined by these orthogonal projectors. The isomorphism
$\mathfrak{h}^{N}(X/B)\cong\mathfrak{h}^{-N}(X/B)(-N)$ is induced by
$\cup\eta^{N}$ and $\gamma_{N}(f)$. This concludes one implication.

The other implication is easier. The map $\cup\eta^{N}$ induces through the
decomposition $\mathfrak{h}(X/B)=\bigoplus_{n}\mathfrak{h}^{n}(X/B)$ a map
from $\mathfrak{h}^{-n}(X/B)(-n)$ to $\mathfrak{h}^{n}(X/B)$. The goal is to
invert such a map. As these two motives are isomorphic we can see this map as
an endomorphism $F$ of $\mathfrak{h}^{n}(X/B)$. The realization $R_{B}(F)$ is
an isomorphism, so by Cayley–Hamilton its inverse can be written as a
polynomial in $R_{B}(F)$. Hence we can find an algebraic inverse to $F$ in the
algebra $\mathbb{Q}[F]$. ∎

######  Corollary 4.4.

Let $X$ be a smooth projective variety endowed with a morphism $f:X\to B$
verifying the relative Lefschetz standard conjecture. Then its absolute motive
$\mathfrak{h}(X)\in{\mathcal{M}ot}$ admits a decomposition

(4.3)
$\displaystyle\mathfrak{h}(X)=\bigoplus_{n}\mathfrak{h}(B,\mathfrak{h}^{n}(X/B))\
,$

such that the realization of $\mathfrak{h}(B,\mathfrak{h}^{n}(X/B))$ is
$H(B,^{p}\\!R^{n}f_{*}C_{X})$ and the motives verifies
$\mathfrak{h}(B,\mathfrak{h}^{n}(X/B))\cong\mathfrak{h}(B,\mathfrak{h}^{-n}(X/B))(-n).$

###### Proof.

The statement follows directly from Proposition 4.3 by applying the functor
$f_{*}:{\mathcal{M}ot}(B)\rightarrow{\mathcal{M}ot}$. ∎

######  Remark 4.5.

The decompositions (4.1) and (4.3) are not unique in general, since their
cohomological counterparts

(4.4) $\displaystyle Rf_{*}C_{X}=\bigoplus_{n}{}^{p}\\!R^{n}f_{*}C_{X}$ (4.5)
$\displaystyle H(X)=\bigoplus_{n}H(B,{}^{p}\\!R^{n}f_{*}C_{X})$

are not. Let us stress that the above statements do not prove that _any_
decomposition as in (4.4) (or as in (4.5)) is of motivic origin. Proposition
4.3 only shows that there exists one decomposition as in (4.4) of motivic
origin.

Deligne constructed three different decompositions of the form (4.4) which are
better than the others, in the sense that each of them can be characterized as
the unique decomposition satisfying some additional property (for example some
behavior with respect to duality) [Del94]. We do not know if any of these
three is of motivic origin in general.

Based on Deligne’s work, de Cataldo [dC13] showed that there are five
decompositions as in 4.5 with remarkable properties. Again we do not know if
those are motivic in general. The goal of the next section is to show that,
under some special hypotheses which are always verified by Lagrangian
fibrations, it is possible to show that de Cataldo’s decompositions are
motivic. This will be important in the end of Section 8 (proof of Theorem
8.2), where we will use the unicity of such a decomposition.

##  5\. Good decomposition: from relative to absolute

In this section we recall the notion of good decomposition (Definition 5.1)
which is due to de Cataldo [dC13]. We show that the relative Lefschetz
standard conjecture implies that a good decomposition is motivic (cf. Remark
4.5). Good decompositions do not exist in general. However, as we will see in
Section 7, they do exist for lagrangian fibrations.

We keep the same notation as the previous section: we fix a projective
surjective morphism $f:X\longrightarrow B$ of algebraic varieties, with $X$
smooth and connected, and a relatively ample divisor $\eta$.

######  Definition 5.1.

(De Cataldo [dC13]) The pair $(f,\eta)$ admits a good decomposition if there
exists a decomposition

(5.1) $\displaystyle
H^{*}(X)=\bigoplus_{n}H^{*}(B,^{p}\\!\\!R^{n}f_{\ast}C_{X})$

splitting the perverse filtration and such that the map induced by the cup-
product with $\eta$ is homogeneous of degree two (i.e., it is 2-graded). In
this case the induced relative primitive decomposition of $H^{*}(X)$ is called
a good primitive decomposition.

######  Remark 5.2.

In general a good decomposition does not exist. If a good decomposition exists
then it is unique, and it coincides with all the splittings constructed by
Deligne and de Cataldo. In particular, it is self-dual (i.e. Poincaré duality
preserves the direct sum decomposition). The unicity and autoduality hold also
for the induced good primitive decomposition.

The notion of good decomposition is crucial in that it allows to pass from
relative Lefschetz to absolute Lefschetz. This is attested by the following
proposition:

######  Proposition 5.3.

Let $X$ be a smooth projective variety with a morphism $f\colon X\to B$, and
let $\eta$ be a relatively ample divisor. Assume that:

  1. (1)

The pair $(f,\eta)$ verifies the relative Lefschetz standard conjecture
(Definition 4.1),

  2. (2)

The pair $(f,\eta)$ admits a good decomposition (Definition 5.1).

Then the good decomposition is motivic and each motive in the decomposition is
isomorphic to its dual (up to a Tate twist). Moreover, the same statement
holds true also for the good primitive decomposition.

###### Proof.

The proof is similar (and actually easier) to the one of Proposition 4.3 and
its corollary. In this proof, we will use the action of a relative
correspondence on absolute cohomology induced by the functor
${\mathcal{M}ot}(B)\to{\mathcal{M}ot}$.

We construct the motives in the good decomposition by increasing induction on
the degree. By the autoduality of a good decomposition (see Remark 5.2), once
the motive $\mathfrak{h}(B,^{p}\\!\\!R^{-n}f_{\ast}C_{X})$ is constructed the
factor $\mathfrak{h}(B,^{p}\\!\\!R^{+n}f_{\ast}C_{X})$ of the decomposition is
determined by taking the dual (after an appropriate Tate twist). The
isomorphism between these two motives is then deduced by the first assumption.

We can therefore suppose that we have constructed the motive
$\mathfrak{h}(B,^{p}\\!\\!R^{i}f_{\ast}C_{X})$ for every cohomological degree
$i$ strictly smaller than $-n$ and strictly larger than $+n$. Let $p_{i}$
denote the corresponding projectors. The goal is now to construct such a
projector in degree $-n$. To do so, consider $\gamma_{n}(f)$ as in Definition
4.1 and define

$p:=\operatorname{\mathsf{id}}_{\mathfrak{h}(X)}-\sum p_{i}$

to be the projector that acts on $\mathfrak{h}(X)$ by killing all motives
already constructed (i.e., those of cohomological degree strictly smaller than
$-n$ and strictly larger than $+n$). Then, by the very definition of good
decomposition, the correspondance $p\circ\cup\eta^{n}\circ p$ induces an
isomorphism between degree $-n$ and degree $+n$ and zero elsewhere, hence

$p_{n}:=\gamma_{n}(f)\circ p\circ\cup\eta^{n}\circ p$

is the desired projector.

This concludes the proof for the good decomposition, and it is standard to
deduce from it the case of the good primitive decomposition. ∎

##  6\. Ngô fibrations and relative Lefschetz

We keep the notation of the previous section. In order to exhibit morphisms
$f:X\rightarrow B$ verifying the assumptions of Proposition 4.2, we now
introduce the concept of Ngô fibrations. An Ngô fibration is a special case of
a _weak abelian fibration_ , as defined by Ngô [Ngô10], [Ngô11]. Before giving
the definitions, let us recall some terminology concerning commutative group
schemes.

######  Definition 6.1.

Let $g:P\rightarrow B$ be a smooth commutative group $B$-scheme, with identity
component $g^{o}:P^{o}\rightarrow B$. For any point $b\in B$, there exists a
canonical exact sequence of connected commutative group schemes over $k(b)$,
called the _Chevalley dévissage_ :

$0\rightarrow L_{b}\rightarrow P_{b}^{o}\rightarrow A_{b}\rightarrow 0\ ,$

where $L_{b}$ is affine and connected, and maximal with these properties, and
where $A_{b}$ is an abelian variety.

The $\delta$-_loci_ are then the locally closed subvarieties $B_{i}$ of $B$
defined by

$B_{i}:=\\{b\in B|\dim_{k(b)}(L_{b})=i\\}$

Fix a prime $\ell$ and write $\overline{\mathbb{Q}}_{\ell,S}$ for the constant
étale $\overline{\mathbb{Q}}_{\ell}$-adic sheaf over a base $S$ and
$\overline{\mathbb{Q}}_{\ell,S}(n)$ for its $n$-th Tate twist. Let
$d:=\dim_{B}P$. The _Tate module_ of $P/B$ is the constructible $\ell$-adic
sheaf $T_{\ell}(P):=R^{2d-1}g_{!}\overline{\mathbb{Q}}_{\ell,P}(d)$ over $B$.

We say that $P$ is _polarizable_ if there exists a pairing

$T_{\ell}(P)\times T_{\ell}(P)\ \rightarrow\
\overline{\mathbb{Q}}_{\ell,B}(1)$

such that, for any $b\in B$, the null-space of the pairing at $b$ is precisely
$T_{\ell}(L_{b})$.

######  Definition 6.2.

A weak abelian fibration is a pair of morphisms

$(f:X\rightarrow B,g:P\rightarrow B)$

such that:

  1. (1)

$f$ is proper;

  2. (2)

$P$ is a smooth commutative group $B$-scheme;

  3. (3)

there is an action $a:P\times_{B}X\rightarrow X$ of $P$ on $X$ over $B$;

  4. (4)

$f$ and $g$ have the same pure relative dimension, denoted by $d$;

  5. (5)

the action has affine stabilizers at every point $x\in X$;

  6. (6)

the Tate module $T_{\ell}(P)$ of $P/B$ is polarizable.

When the morphisms defining the structure of a weakly abelian fibration on
$X,P,B$ as above are clear from the context, we will write $(X,B,P)$ to denote
the weakly abelian fibration.

We propose the following definition:

######  Definition 6.3.

An Ngô fibration is a weak abelian fibration $(X,B,P)$ with the following
additional property:

$\operatorname{codim}B_{i}\geq i\ \ \forall i\ \in\mathbb{Z}_{\geq 0}$

(NB: in the litterature, an Ngô fibration is called a “$\delta$-regular weak
abelian fibration”).

######  Proposition 6.4.

Let $f:X\longrightarrow B$ be an Ngô fibration with irreducible fibers. Then
it verifies the relative Lefschetz standard conjecture (Definition 4.1).

###### Proof.

We want to apply Proposition 4.2 to this setting. The first assumption is
satisfied because the generic fiber is (possibly after a finite extension) an
abelian variety and the Lefschetz standard conjecture is known for abelian
varieties [Lie68] (and is stable under finite extensions). The second
assumption is satisfied since by Ngô support theorem [Ngô10] an Ngô fibration
with irreducible fibers has only full supports in the decomposition theorem. ∎

######  Corollary 6.5.

Let $f\colon X\to B$ be a Beauville–Mukai integrable system, i.e. the relative
compactified Jacobian of a linear system of curves on a K3 surface [Muk84],
[Bea91]. Assume that $f$ has irreducible fibers, i.e. that all the curves in
the linear system are integral. Then $f\colon X\to B$ verifies the relative
Lefschetz standard conjecture.

###### Proof.

Since $f$ is a Lagrangian fibration, it satisfies all the requirements in the
definition of Ngô fibration except possibly the polarizability (cf. the proof
of Proposition 9.5 below). The polarizability is proven by Ngô for
compactified relative Jacobians of curves [Ngô10, page 76] (cf. also [dC17,
proof of Theorem 3.3.1]). It follows that $f$ is an Ngô fibration and
Proposition 6.4 applies. ∎

In addition to the above corollary, Proposition 6.4 applies also to LSV
tenfolds, as we will explain in Section 9. Both applications can be extended
through the following result.

######  Proposition 6.6.

Consider a family of Lagrangian fibrations parametrized by an irreducible
base. Suppose that one of the fibrations of the family is an Ngô fibration
with irreducible fibers. Then all fibrations in the family satisfy the
relative Lefschetz standard conjecture.

###### Proof.

Let $f:X\rightarrow B$ be the generic fibration and $g:X^{\prime}\rightarrow
B$ be the Ngô fibration with irreducible fibers. By specialization, it is
enough to show that the relative Lefschetz standard conjecture holds for $f$.

The generic fiber of $f$ is an abelian variety $A$, which hence satisfies the
Lefschetz standard conjecture. We can then consider the different Lefschetz
operators with their inverses (for different cohomological degrees) as
correspondences in $A\times A$ and consider their Zariski closures in
$X\times_{B}X\subset X\times X$. These correspondences specialize to
correspondences of $X^{\prime}$. Consider the algebra $\mathcal{A}_{X}$ they
generate inside the correspondences of $X$. It specializes to an algebra
$\mathcal{A}_{X^{\prime}}$ of correspondences of $X^{\prime}$, as their
generators do.

Now Proposition 6.4 and especially the proof of Proposition 4.2 imply not only
that $g$ verifies the relative Lefschetz standard conjecture but also that the
algebra $\mathcal{A}_{X^{\prime}}$ contains projectors inducing the good
decomposition for $g$ and the isomorphisms between dual factors. Hence the
liftings in $\mathcal{A}_{X}$ of those projectors and isomorphisms are again
projectors and isomorphisms for the good decomposition associated with $f$. ∎

##  7\. The Shen–Yin decomposition is good

The object of the present section is to recall a remarkable result of Shen and
Yin concerning Lagrangian fibrations, which implies in particular the
existence of a good decomposition (see Definition 5.1).

Throughout this section, we fix a Lagrangian fibration $f:X\longrightarrow B$.
The dimension of $X$ will be denoted by $\dim X=2r$. Let $q$ be the
Beauville–Bogomolov form, let $\beta$ be the Lagrangian class (i.e. the pull-
back of an ample divisor on $B$) and let $\eta$ be a relatively ample divisor.
Note that $q(\beta)=0$, and that we can (and will) modify $\eta$ in a unique
way by adding a $\mathbb{Q}$-multiple of $\beta$ in order to have the
vanishing $q(\eta)=0$ [SY, (22)]. Up to clearing denominators, we can further
assume that $\eta$ is an integral class. Then, as first observed by Verbitsky,
the vanishing $\eta^{r+1}=\beta^{r+1}=0$ hold.

######  Theorem 7.1.

(Shen–Yin [SY]) Let the notation be as above. Then the following holds:

  1. (1)

The pair $(f,\eta)$ admits a good primitive decomposition (see Definition 5.1)

$H^{*}(X)=\bigoplus_{0\leq i\leq r}\mathbb{Q}[\eta]/(\eta^{r-i+1})\otimes
W^{i}\ .$

  2. (2)

As $\mathbb{Q}[\eta,\beta]$-module, $H^{*}(X)$ can be written as

$H^{*}(X)=\bigoplus_{0\leq i,j\leq
r}\mathbb{Q}[\eta]/(\eta^{r-i+1})\otimes\mathbb{Q}[\beta]/(\beta^{r-j+1})\otimes
V^{i,j}\ $

for certain subspaces $V^{i,j}\subset H^{i+j}:=H^{i+j}(X)$,

  3. (3)

There is an equality $W^{i}=\bigoplus_{0\leq j\leq
r}\mathbb{Q}[\beta]/(\beta^{r-j+1})\otimes V^{i,j}$.

###### Proof.

The vanishings are explained in [SY, (22)]. The existence of $V_{i,j}$ is
explained in [SY, Theorem 3.1 and its proof]. The relation with the good
primitive decomposition is in [SY, Theorem 3.1 and its proof]. ∎

######  Remark 7.2.

The spaces $V_{i,j}$ as in Theorem 7.1 (2) are unique. Indeed, one has

$V_{i,j}=H^{i+j}\cap\operatorname{Ker}(\cup\eta^{r-i+1})\cap\operatorname{Ker}(\cup\beta^{r-j+1})\
.$

######  Proposition 7.3.

Keep the notation as above and suppose that the fibration $f$ is an Ngô
fibration whose fibers are all irreducible. Then the good primitive
decomposition in Theorem 7.1(1) is motivic and each motive in the
decomposition is isomorphic to its dual (up to a Tate twist).

###### Proof.

By Proposition 6.4 such a Lagrangian fibration verifies the relative Lefschetz
standard conjecture. We can then apply Proposition 5.3. ∎

Recall that birational HK varieties have (non-canonically) isomorphic
cohomology rings and motives [Huy06, Rie14], see Remark 3.6. This is implicit
in the following statement.

######  Proposition 7.4.

Let $\eta$ and $\beta$ be as above. Suppose that:

  1. (1)

There exists a HK variety $Y$ birational to $X$ together with a lagrangian
fibration $g:Y\rightarrow B^{\prime}$ such that $\beta$ is relatively ample
for $g$ and $\eta$ is a lagrangian class for $g$,

  2. (2)

Both $f$ and $g$ are Ngô fibrations with irreducible fibers (or can be
deformed to fibrations with this property).

Then the decomposition in Theorem 7.1(2) is motivic and each motive in the
decomposition is isomorphic to its dual (up to a Tate twist). In particular
the standard conjectures hold true for $X$.

###### Proof.

The Shen–Yin decomposition from Theorem 7.1(2) is independent of the choice of
$f$ and $g$, and only depends on $\eta$ and $\beta$ (see Remark 7.2).

By Theorem 7.1(1) if we forget the action of $\beta$ then we get the good
decomposition with respect to $(f,\eta)$ and the latter is motivic by
Proposition 7.3. Similarly we can forget the action of $\eta$ and deduce that
it is motivic as it is the good decomposition with respect to $(g,\beta)$.

If we now put these two decompositions together we deduce that the
decomposition in Theorem 7.1(2) is motivic. Moreover, as each factor of the
two decompositions is selfdual we deduce that the factors in the Shen–Yin
decomposition from Theorem 7.1(2) are so. On the other hand the Shen–Yin
decomposition refines the Künneth decomposition. Hence we have shown that the
Künneth decomposition is motivic and that each of its factors is selfdual.
This is one of the equivalent formulations of the Lefschetz standard
conjecture. ∎

##  8\. Voisin’s trick and the end of the proof

We keep the notation from the previous section. In particular $X$ is a HK
admitting a Lagrangian fibration and $\beta$ and $\eta$ are divisor classes on
$X$, respectively Lagrangian and relatively ample, such that
$q(\beta)=q(\eta)=0$.

We have seen in Proposition 7.3 a criterion to show that the decomposition
with respect to the action of $\eta$ on the cohomology of $X$ is motivic, but
a priori this does not apply to $\beta$, since $\beta$ needs to satisfy the
assumptions of Proposition 7.4. The goal of this section is to recall an
ingenious idea of Voisin [Voi22] which allows, roughly speaking, to invert the
role of $\eta$ and $\beta$ (in a way similar to Proposition 7.4). This will
allow to conclude the proof of Theorem 3.1.

######  Remark 8.1.

(Voisin [Voi22]) Suppose that the Picard rank of $X$ is two (so that $\eta$
and $\beta$ are unique up to multiplication by a positive scalar). Then there
are two cases:

  1. (1)

Both $\eta$ and $\beta$ are potentially Lagrangian (in the sense of Definition
3.7),

  2. (2)

Only $\beta$ is potentially Lagrangian.

In the first case, if we assume 3.8, then there is a second Lagrangian
fibration with respect to which the roles of $\eta$ and $\beta$ are inverted.

In the second case, (without assuming 3.8) the work of Huybrechts and Riess
[Huy06, Rie14] implies the existence of an automorphism of the motive
$\mathfrak{h}(X)$ which commutes with the cup-product and which inverts $\eta$
and $\beta$ (up to scalars).

By work of Boucksom and Huybrechts, the second case happens precisely when
there is a prime exceptional divisor, i.e. a reduced and irreducible divisor
$\Theta$ with $q(\Theta)<0$.

######  Theorem 8.2.

Assume the hypothesis of Theorem 3.1. Then the Shen–Yin decomposition from
Theorem 7.1(2) is motivic and each motive in the decomposition is isomorphic
to its dual (up to Tate twist). In particular the Lefschetz standard
conjecture holds true for $X$.

###### Proof.

We start with Voisin’s idea explained in Remark 8.1 and consider the two cases
listed there. In the first case, one is precisely in the setting of
Proposition 7.4. Hence, let us treat the second case.

Let $\gamma$ be the automorphism of the motive $\mathfrak{h}(X)$ as in Remark
8.1. The conjugation by $\gamma$ sends a projector to a projector hence it
sends the decomposition in Theorem 7.1(2) to a new decomposition.

Moreover, as $\gamma$ commutes with the cup-product and inverts $\eta$ and
$\beta$ (up to scalar) one has the following identities (up to scalar)

(8.1) $\displaystyle\gamma^{-1}\circ\cup\eta\circ\gamma=\cup\beta,\hskip
28.45274pt\gamma^{-1}\circ\cup\beta\circ\gamma=\cup\eta.$

Hence the new decomposition obtained after conjugation is

$H^{*}(X)=\bigoplus_{0\leq i,j\leq
r}\mathbb{Q}[\beta]/(\beta^{r-i+1})\otimes\mathbb{Q}[\eta]/(\eta^{r-j+1})\otimes(\gamma^{-1}V^{i,j}).$

By the unicity of the decomposition (see Remark 7.2) one gets

(8.2) $\displaystyle\gamma^{-1}V^{i,j}=V^{j,i}.$

Now, let us consider the decomposition in Theorem 7.1(2). We want to show that
this decomposition is motivic and that each motive in the decomposition is
selfdual. Arguing as in the proof of Proposition 7.4 it is enough to consider
the two induced decompositions (obtained by forgetting the action of $\beta$
respectively $\eta$) and show that they have the same properties.

The first one is achieved, as in the proof of Proposition 7.4, by combining
Theorem 7.1(1) with Proposition 7.3. The second one is deduced from the first
one because it is obtained from the first one by conjugating by $\gamma$ (as
is shown by the relations (8.1) and (8.2)).

We have shown that the Shen–Yin decomposition is motivic. This implies the
Lefschetz standard conjecture as we explained in the end of the proof of
Proposition 7.4. ∎

######  Remark 8.3.

The main application of the above theorem is given in Section 9 and proves the
Lefschetz standard conjecture for LSV tenfolds. It should be possible to
extend this result to some of their deformations through Proposition 6.6.

##  9\. An application to LSV tenfolds

In this section we prove Theorem 1.2, as a concrete application of Theorem
3.1. We actually prove an extension of it, namely Theorem 9.12 below.

The section is divided in three subsections. The first subsection recalls the
geometry of a general LSV tenfold and identifies the abelian group scheme
acting on it; the second subsection proves the polarizability of this group
scheme; the third subsection gives the proof of the Lefschetz standard
conjecture for all LSV tenfolds (Theorem 9.12) and their twisted analogues.

Let us recall what we mean by _LSV tenfolds_. Let $M$ be a smooth cubic
fourfold and let $Y\rightarrow B$ be the universal family of cubic threefolds
over $B:=(\mathbb{P}^{5})^{\vee}$ whose fibers are the hyperplane sections of
$M$. Let $U\subset B$ be the locus parametrizing smooth hyperplane sections
and let $J_{U}\rightarrow U$ be the intermediate Jacobian fibration. By
Donagi-Markman the total space of this fibration has a holomorphic symplectic
form. By [LSV17, Sac22], there exists a projective HK manifold $X$ of
OG$10$-type which compactifies $J_{U}$, and which is equipped with a
Lagrangian fibration $X\to B$ extending the intermediate Jacobian fibration.
In [LSV17], this is done for a _general_ cubic fourfold $M$ using an explicit
construction that will be recalled below. In [Sac22] this is extended to any
cubic fourfold $M$ (smooth or mildly singular). These Lagrangian fibered HK
manifolds compactifying $J_{U}\to U$ will be referred to as LSV tenfolds. If
$M$ is general, we will refer to the corresponding HK as a general LSV
tenfold. Note that the HK manifolds constructed in [Sac22] are specializations
of general LSV tenfolds. While for general $M$ a Lagrangian fibered HK
compactification of $J_{U}$ is unique, for non-general $M$ there may be
several non isomorphic compactifications. Note, however, that they are all
birational. In [Voi18], Voisin constructs a Lagrangian fibered HK
compactification of the so-called _twisted_ fibration $J^{t}_{U}\to U$
associated to a general $M$. This is a non-trivial torsor over $J_{U}$, and
the fibers parametrize one-cycles of degree one on the hyperplane sections,
modulo rational equivalence. In [Sac22] this is extended to arbitary $M$.

We show that the criterion established in Theorem 3.1 applies to general LSV
tenfolds, as well as to their twisted analogues. From this, Theorem 9.12
follows by a specialization argument.

###  9.1. LSV tenfolds and their Ngô fibration

Let $M$ be a general cubic fourfold. Let us briefly recall how the LSV
tenfolds associated to $M$ and their associated Ngô fibration are constructed.
We keep the notation introduced above and let $f:X\to B$ denote the Lagrangian
fibration on the HK compactification of $J_{U}$ constructed in [LSV17].

For each $b\in B$, one considers the family of lines contained in the fiber
$Y_{b}$ of $Y\rightarrow B$, thus obtaining a relative Fano surface
$\mathcal{F}\rightarrow B$. In [LSV17], an open subscheme
$\mathcal{F}^{0}\rightarrow B$ of $\mathcal{F}$, called subscheme of _very
good lines_ , is defined, via _loc. cit._ , Definition 2.9. We obtain the
smooth commutative group scheme $g:P\rightarrow B$ acting on $X\rightarrow B$
by descent to $B$ of a relative Prym variety over $\mathcal{F}^{0}$, i.e. the
relative Prym attached to a certain étale double cover
$\tilde{\mathcal{C}}\rightarrow\mathcal{C}$ of relative curves over
$\mathcal{F}^{0}$. Let us describe this more precisely.

If one fixes a very good line $\ell\in\mathcal{F}^{0}$ over $b\in B$,
projection from $\ell$ realizes the cubic threefold $\mathcal{Y}_{b}$ as a
conic bundle over $\mathbb{P}^{2}$ with discriminant a quintic curve
$\mathcal{C}_{\ell}$: the latter is precisely the fiber of $\mathcal{C}$ over
$\ell$. The double cover
$\tilde{\mathcal{C}}_{\ell}\rightarrow\mathcal{C}_{\ell}$ is defined as the
curve parametrizing the lines in $\mathcal{Y}_{b}$ incident to $\ell$. By
definition, if the line is very good then for each $\ell$, both
$\tilde{\mathcal{C}}_{\ell}$ and $\mathcal{C}_{\ell}$ are reduced and
irreducible.

We have thus an étale double cover

(9.1) $p:\tilde{\mathcal{C}}\rightarrow\mathcal{C}$

of relative curves over $\mathcal{F}^{0}$ at our disposal.

######  Definition 9.1.

We will denote by $\tilde{\mathcal{J}}$, $\mathcal{J}$ the identity components
of the relative Picard schemes of $\tilde{\mathcal{C}}$ and $\mathcal{C}$ over
$\mathcal{F}^{0}$.

######  Remark 9.2.

Since $p$ is étale and the coherent sheaf
$p_{*}\mathcal{O}_{\tilde{\mathcal{C}}}$ is locally free, we can define a
_relative norm map_ ([Gro61, 6.5])

(9.2) $\mbox{Nm}_{p}:\tilde{\mathcal{J}}\rightarrow\mathcal{J}$

which verifies in particular the following property ([Gro61, Prop. 6.5.8]).
For each closed point $s$ in $\mathcal{F}^{0}$, consider the commutative
diagram

(9.3)
${\tilde{\mathcal{C}}_{s}}$${\mathcal{C}_{s}}$${\tilde{D}_{s}}$${D_{s}}$$\scriptstyle{p_{s}}$$\scriptstyle{\tilde{n}}$$\scriptstyle{\epsilon}$$\scriptstyle{n}$

where $n$, $\tilde{n}$ are normalization morphisms and $\epsilon$ is again an
étale double cover. Observe that the diagram is in fact cartesian, because
normalization commutes with smooth base change. Then the associated norm
morphisms
$\mbox{Nm}_{p_{s}}:\tilde{\mathcal{J}}_{s}\rightarrow\mathcal{J}_{s}$ and
$\mbox{Nm}_{\epsilon}:J_{\tilde{D}_{s}}\rightarrow J_{D_{s}}$ are compatible,
in the sense that

(9.4) $n^{*}\circ\mbox{Nm}_{p_{s}}=\mbox{Nm}_{\epsilon}\circ\tilde{n}^{*}\ .$

We can now give the following:

######  Definition 9.3.

The relative Prym $\mathcal{P}\rightarrow\mathcal{F}^{0}$ associated to the
étale double cover $p$ (9.1) is the identity component of the kernel
$\ker(\mbox{Nm}_{p})$ of the norm map (9.2).

The proof of [LSV17, Lemma 4.9] shows that the above relative Prym can be
equivalently defined as the identity component of the fixed locus of the
involution $\tau$ on $\tilde{\mathcal{J}}$ defined as

$\tau:=-\iota^{\ast}\colon\ \ \tilde{\mathcal{J}}\ \to\ \tilde{\mathcal{J}}\
,$

where $\iota$ is the involution of $\tilde{\mathcal{C}}$ defined by the double
cover $\tilde{\mathcal{C}}\to{\mathcal{C}}$.

######  Lemma 9.4.

The group $\mathcal{F}^{0}$-scheme $\mathcal{P}\rightarrow\mathcal{F}^{0}$
descends along the projection $\mathcal{F}^{0}\rightarrow B$ to a smooth,
commutative group $B$-scheme $g:P\rightarrow B$ with an action
$P\times_{B}X\rightarrow X$ over $B$. Moreover, $P$ is isomorphic over $B$ to
the smooth locus $X^{\mbox{\tiny{sm}}}$ of $X\rightarrow B$.

###### Proof.

The fibers of the morphism $f:X\rightarrow B$ are integral, since each such
fiber can be identified with the closure in $X$ of a (smooth, connected) Prym
variety (cfr. [LSV17, Proposition 4.10 (1), proof of Corollary 4.17]). The
morphism $f$ is then part of an _integrable system_ in the sense of [AF16,
Definition 2.6], and by applying [AF16, Theorem 2], we obtain a smooth,
commutative $B$-group scheme $P\rightarrow B$ with an action
$P\times_{B}X\rightarrow X$ over $B$, such that the smooth locus
$X^{\mbox{\tiny{sm}}}$ of $f$ in $X$ is a torsor over $P$. Since by [Sac22,
Corollary 4.8] $X^{\mbox{\tiny{sm}}}\to B$ has a zero section, it follows that
$P\simeq X^{\mbox{\tiny{sm}}}$. Now remember that by construction (cfr.
[LSV17, Section 5]), $f:X\rightarrow B$ is part of a cartesian square

${X^{\prime}}$${\mathcal{F}^{0}}$${X}$${B}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

where $X^{\prime}$ is a compactification of $\mathcal{P}$ over
$\mathcal{F}^{0}$, and more precisely, $\mathcal{P}$ coincides with the smooth
locus of $f^{\prime}$ in $X^{\prime}$. This tells us that we have a cartesian
diagram

${\mathcal{P}}$${\mathcal{F}^{0}}$${X^{\mbox{\tiny{sm}}}}$${B}$

hence concluding the proof. ∎

Since we want to apply our Theorem 3.1 to an LSV tenfold $X$, it becomes our
task to show:

######  Proposition 9.5.

The data $(X,B,P)$ associated as above to a general LSV tenfold $X$ define an
Ngô fibration (i.e. a $\delta$-regular weak abelian fibration, see Definition
6.2) with integral fibers.

###### Proof.

By construction, our data $(X,B,P)$ satisfy the first four conditions in
Definition 6.2. Now, since $P$ has been constructed in the proof of Lemma 9.4
by applying [AF16, Theorem 2], we have that $P\rightarrow B$ is automatically
part of the data defining a _degenerate abelian scheme_ in the sense of
Definition 2.1 of _loc. cit._. In particular, by Definition 2.1(vi) of _loc.
cit._ , for every schematic point $b\in B$, there is a point $x\in X_{b}$ such
that the stabilizer of $x$ in $P\times_{B}x$ is finite; but then, by _loc.
cit._ , Lemma 5.16, the stabilizer of every other point in the fiber over $b$
is affine, so that condition (5) in Definition 6.2 is satisfied. Moreover,
Definition 2.1(iv) of _loc. cit._ implies that the $\delta$-regularity
condition required by Definition 6.3 is satisfied, too (alternatively, the
$\delta$-regularity also follows from the existence of a symplectic form on
$X$, as explained in [Ngô11, Section 2]). It remains to check that condition
(6) in Definition 6.2 holds. This is proven in the following subsection (Lemma
9.7). ∎

######  Remark 9.6.

Given any general LSV tenfold, the same group scheme $P$ as in the above
proposition acts, by construction, on its twisted version, in such a way that
the first four conditions in Definition 6.2 are satisfied. Then, the above
proof applies to show that the data $(X,B,P)$ define an Ngô fibration with
integral fibers even when $X$ is a twisted general LSV tenfold.

###  9.2. Polarizability of the descended relative Prym

This subsection is devoted to providing the last ingredient needed in the
proof of Proposition 9.5, namely:

######  Lemma 9.7.

Let $P\rightarrow B$ be the commutative group scheme associated to a general
LSV tenfold $X$ as in the previous subsection. Then the Tate module
$T_{\ell}(P)$ is polarizable.

In order to give the proof, we will start by constructing a polarization on
the Tate module of the relative Prym $\mathcal{P}$ of Definition 9.3. This
will be done in the next couple of lemmas.

######  Lemma 9.8.

Let $T_{\ell}$ be the Tate module of the relative Jacobian
$\tilde{\mathcal{J}}$ associated with the relative curve
$\tilde{\mathcal{C}}\rightarrow\mathcal{F}^{0}$, see Definition 9.1. Then
$T_{\ell}$ is polarizable.

###### Proof.

The statement follows from the completely general construction of [Ngô10, page
76] (cf. also [dC17, proof of Theorem 3.3.1]), which defines a polarization
$W_{C}$ on the Tate module of the relative Picard scheme $J_{C}$ of a proper,
flat relative curve $C$ with geometrically reduced fibers over a base $S$. The
polarization is induced by a _Weil pairing_

$\langle\cdot,\cdot\rangle_{C}:J_{C}\times J_{C}\rightarrow\mbox{Pic}^{0}(S)$

constructed via the theory of the determinant of the cohomology. (The
construction is first given for strictly henselian bases $S$; by applying it
to the strict henselianization of the local ring of any point of our base $B$,
we can then obtain a polarization over $B$). The vanishing of $W_{C}$ on the
linear part of $T_{\ell}(J_{C})$ comes from the conjunction of the following
two facts:

(i) By [Ngô10, Lemme 4.12.2], for any geometric point $s\in S$, denoting by
$\nu:D_{s}\rightarrow C_{s}$ the normalization morphism, and for any pair
$L,L^{\prime}$ of degree-zero line bundles on $C_{s}$, we have the
identification of $k(s)$-vector spaces

(9.5) $\langle
L,L^{\prime}\rangle_{C_{s}}=\langle\nu^{*}L,\nu^{*}L^{\prime}\rangle_{D_{s}}\
.$

(ii) By [BLR90, §9.2, Corollary 11], the projection $J_{C_{s}}\rightarrow
J^{ab}_{C_{s}}$ on the abelian part of $J_{C_{s}}$, whose kernel is the linear
part $J^{aff}_{C_{s}}$ of $J_{C_{s}}$, is identified with the morphism
$J_{C_{s}}\rightarrow J_{D_{s}}$ induced by the normalization.

The non-degeneracy of the polarization on $T_{\ell}(J^{ab}_{C_{s}})$ comes
from its identification with cup product on the first étale $\ell$-adic
cohomology group of $D_{s}$ ([Mil80, Chapter V, Remark 2.4(f)]). ∎

######  Lemma 9.9.

Let $T^{\prime}_{\ell}$ be the Tate module of the relative Prym $\mathcal{P}$
of Definition 9.3 and consider it as a sub-object of the Tate module
$T_{\ell}$ of the relative Jacobian $\tilde{\mathcal{J}}$. Then the
restriction to $T^{\prime}_{\ell}$ of the polarization on $T_{\ell}$ provided
by Lemma 9.9 is again a polarization.

###### Proof.

We use the notation for relative Jacobians introduced in Definition 9.1 and
the norm maps associated with the diagram (9.4). Fix a closed point $s$ in
$\mathcal{F}^{0}$ and consider the commutative diagram of exact sequences,
whose rows are given by the Chevalley dévissages,

(9.6)
${0}$${\mathcal{J}^{aff}_{s}}$${\mathcal{J}_{s}}$${\mathcal{J}_{s}^{ab}}$${0}$${0}$${\tilde{\mathcal{J}}^{aff}_{s}}$${\tilde{\mathcal{J}}_{s}}$${\tilde{\mathcal{J}}_{s}^{ab}}$${0}$${0}$${\mathcal{P}^{aff}_{s}}$${\mathcal{P}_{s}}$${\mathcal{P}^{ab}_{s}}$${0}$$\scriptstyle{n^{*}}$$\scriptstyle{\tilde{n}^{*}}$$\scriptstyle{\mbox{Nm}_{p_{s}}}$$\scriptstyle{\mbox{Nm}_{\epsilon}}$$\scriptstyle{u}$$\scriptstyle{v}$$\scriptstyle{w}$

and where the commutativity of the upper-right square comes from (9.4). The
arrow $v$ is the natural inclusion, and $u$ is an inclusion, too: it is
induced by the fact that $\mathcal{P}^{aff}_{s}$ has to map to
$\tilde{\mathcal{J}}^{aff}_{s}$, since the latter is the largest affine
subgroup scheme of $\tilde{\mathcal{J}}_{s}$. The arrow $w$ is then induced by
the commutativity of the lower-left square. Moreover, $\ker(w)$ is finite,
since it is simultaneously a subgroup of the abelian variety
$\mathcal{P}_{s}^{ab}$ and, by the snake lemma, of the affine
$\mbox{coker}(u)$. This means that passing to the corresponding Tate modules,
the lower rectangle yields a commutative diagram of exact sequences

${0}$${T^{aff}_{\ell,s}}$${T_{\ell,s}}$${T^{ab}_{\ell,s}}$${0}$${0}$${T^{\prime,aff}_{\ell,s}}$${T^{\prime}_{\ell,s}}$${T^{\prime,ab}_{\ell,s}}$${0}$

where the vertical arrows are inclusions.

Now look at the restriction to $T^{\prime}_{\ell}$ of the polarization
$W_{\tilde{\mathcal{C}}}$ on $T_{\ell}$ obtained in Lemma 9.8. The above
diagram shows that $W_{\tilde{\mathcal{C}}_{s}}$ vanishes on
$T^{\prime,aff}_{\ell,s}$, since it vanishes on $T^{aff}_{\ell,s}$. Hence, we
need to show that the induced pairing $W^{ab}_{\tilde{\mathcal{C}}_{s}}$
remains non-degenerate on $T^{\prime,ab}_{\ell,s}$. In order to do so, observe
that with notation as in diagram (9.3), we have

$T^{ab}_{\ell,s}\simeq T_{\ell}(J_{\tilde{D}_{s}})\ ,$

and by formula (9.5), the induced pairing $W^{ab}_{\tilde{\mathcal{C}}_{s}}$
coincides with the pairing $W_{\tilde{D}_{s}}$. Now because of the classical
relation

$\mbox{Nm}_{\epsilon}\circ\epsilon^{*}=2\cdot\mbox{Id}_{J_{D_{s}}}\ ,$

we have

(9.7)
$T_{\ell}(J_{\tilde{D}_{s}})\simeq\operatorname{Ker}(T_{\ell}(\mbox{Nm}_{\epsilon}))\oplus\mbox{Im}(T_{\ell}(\epsilon^{*}))\
.$

Moreover, the morphisms $\mbox{Nm}_{\epsilon}$ and $\epsilon^{*}$ are adjoint
with respect to the bilinear forms $W_{\tilde{D}_{s}}$ and $W_{D_{s}}$
([Mum70, page 186, equation (I)], [BL04, page 331, equation (2)]), so that the
direct sum (9.7) is orthogonal with respect to $W_{\tilde{D}_{s}}$. The
restriction of $W_{\tilde{D}_{b}}$ to
$\operatorname{Ker}(T_{\ell}(\mbox{Nm}_{\epsilon}))$ thus remains non-
degenerate. To get what we want, it will be enough to prove that
$\operatorname{Ker}(T_{\ell}(\mbox{Nm}_{\epsilon}))$ coincides with
$T^{\prime,ab}_{\ell,s}$. It is clear by the commutativity of (9.6) that
$T^{\prime,ab}_{\ell,s}$ is included in
$\operatorname{Ker}(T_{\ell}(\mbox{Nm}_{\epsilon}))$, and we will conclude by
showing that the group scheme
$\ker(\mbox{Nm}_{\epsilon})/(\mathcal{P}_{s}^{ab}/\ker(w))$ is finite.

For this, notice first that the upper-right square of (9.6) provides a
surjection

$\ker(\mbox{Nm}_{\epsilon}\circ\tilde{n}^{*})\twoheadrightarrow\ker(\mbox{Nm}_{\epsilon})$

hence, using the commutativity of that same square, a surjection

$\ker(n^{*}\circ\mbox{Nm}_{p_{s}})\twoheadrightarrow\ker(\mbox{Nm}_{\epsilon})/(\mathcal{P}_{s}^{ab}/\ker(w))$

which vanishes on $\ker(\mbox{Nm}_{p_{s}})$, thus yielding a surjection

$\ker(n^{*}\circ\mbox{Nm}_{p_{s}})/\ker(\mbox{Nm}_{p_{s}})\twoheadrightarrow\ker(\mbox{Nm}_{\epsilon})/(\mathcal{P}_{s}^{ab}/\ker(w))\
.$

On the other hand, by definition, the map $n^{*}$ vanishes on the image of the
restriction of $\mbox{Nm}_{p_{s}}$ to $\ker(n^{*}\circ\mbox{Nm}_{p_{s}})$. By
exactness of the upper row of (9.6), $\mbox{Nm}_{p_{s}}$ thus induces a
monomorphism

$\ker(n^{*}\circ\mbox{Nm}_{p_{s}})/\ker(\mbox{Nm}_{p_{s}})\hookrightarrow\mathcal{J}_{s}^{aff}\
,$

which shows that the group scheme
$\ker(n^{*}\circ\mbox{Nm}_{p_{s}})/\ker(\mbox{Nm}_{p_{s}})$ is affine. Since
it surjects onto $\ker(\mbox{Nm}_{\epsilon})/(\mathcal{P}_{s}^{ab}/\ker(w))$,
the latter cannot have non-zero dimension, because otherwise we would get a
non-trivial homomorphism from an affine group scheme to an abelian variety. ∎

######  Remark 9.10.

  1. (1)

The use of the relations between the morphisms $\mbox{Nm}_{\epsilon}$ and
$\epsilon^{*}$ in the above proof is inspired by the strategy of the
polarizability proof of [dC17, Lemma 4.7.1 and Thm. 4.7.2], in the context of
the Hitchin fibration.

  2. (2)

The final diagram chase in the previous proof, showing that the group scheme
$\ker(\mbox{Nm}_{\epsilon})/(\mathcal{P}_{s}^{ab}/\ker(w))$ is finite, could
be replaced by a more direct argument, using the involution $\iota$ on
$\tilde{\mathcal{C}}$ and using that $\ker\mbox{Nm}_{p}$ coincides with the
identity component $Fix(-\iota^{*})^{\circ}$. In fact, the short exact
sequence
$0\to\mathcal{J}^{aff}_{s}\to\mathcal{J}_{s}\to\mathcal{J}_{s}^{ab}\to 0$ is
compatible with the involution $\tau=-i^{*}$ and the corresponding involution
on $\widetilde{D}$. Using the fact that abelian varieties are divisible
groups, one can easily check that the short exact sequence induces a short
exact sequence of identity component of fixed loci. In particular,
$\mathcal{P}_{s}^{ab}=\ker\mbox{Nm}_{\epsilon}$ and $\ker w=0$.

We will deduce Lemma 9.7 from Lemma 9.9 by showing that we can descend to $B$
the polarization over $\mathcal{F}^{0}$ constructed above. For this, we will
need a classical fact whose proof we recall for the convenience of the reader.

######  Lemma 9.11.

Let $\pi:Z\rightarrow B$ be a smooth surjective morphism with connected
fibers. Then the functor $\pi^{*}$ on constructible $\ell$-adic sheaves is
fully faithful.

###### Proof.

We have canonical isomorphisms

(9.8)
$\operatorname{Hom}_{Z}(\pi^{*}F,\pi^{*}G)\simeq\operatorname{Hom}_{Z}(R\pi^{!}F,R\pi^{!}G)\simeq\operatorname{Hom}_{B}(R\pi_{!}R\pi^{!}F,G)\
,$

where the first isomorphism follows from the smoothness of $\pi$, and the
second one is the adjunction isomorphism. Now, for any point
$i_{b}:b\hookrightarrow B$, by looking at the cartesian square

${Z}$${Z_{b}}$${B}$${b}$$\scriptstyle{\pi}$$\scriptstyle{i^{\prime}_{b}}$$\scriptstyle{\pi_{b}}$$\scriptstyle{i_{b}}$

and by applying the base change isomorphism, we have

(9.9) $i_{b}^{*}R\pi_{!}R\pi^{!}F\simeq R\pi_{b,!}\pi_{b}^{!}F_{b}\simeq
R\pi_{b,!}\pi_{b}^{*}F_{b}(\delta)[2\delta]\ ,$

where $F_{b}$ denotes the stalk of $F$ at $b$, the second isomorphism comes
from the smoothness of $\pi_{b}$, and $\delta$ is the relative dimension of
$\pi$. Observe that

(9.10) $R\pi_{b,!}\pi_{b}^{*}F_{b}\simeq R\Gamma_{c}(Z_{b},F_{b})$

where $\Gamma_{c}$ is the compactly supported global sections functor, and by
our hypotheses, $Z_{b}$ is smooth and connected of dimension $\delta$, for any
$b$. As a consequence, the complex $i_{b}^{*}R\pi_{!}R\pi^{!}F$ is
concentrated in degrees $[-2\delta,\dots,0]$ for any $b$, hence so is
$R\pi_{!}R\pi^{!}F$. Thus, as $G$ is concentrated in degree zero,

(9.11)
$\operatorname{Hom}_{B}(R\pi_{!}R\pi^{!}F,G)\simeq\operatorname{Hom}_{B}(H^{0}(R\pi_{!}R\pi^{!}F),G)\
.$

Then, our conclusion will follow as soon as we will prove that the adjunction
morphism

(9.12) $H^{0}(R\pi_{!}R\pi^{!}F)\rightarrow F$

is an isomorphism (for then, (9.8) and (9.11) will yield an isomorphism

$\operatorname{Hom}_{Z}(\pi^{*}F,\pi^{*}G)\simeq\operatorname{Hom}_{B}(F,G)$

which provides an inverse of the morphism induced by $\pi^{*}$). For any $b$,
the morphism (9.12) induces a morphism between stalks at $b$, that by (9.9)
and (9.10) takes the form

$H^{0}\left(R\Gamma_{c}(Z_{b},F_{b}(\delta))[2\delta]\right)=H_{c}^{2\delta}(Z_{b},F_{b})(\delta)\rightarrow
F_{b}$

By [SGA73, Exp. XVIII, Thm. 3.2.5], this morphism coincides with the trace
morphism on top-dimensional compactly supported cohomology. Since by our
hypotheses on $\pi$, the fiber $Z_{b}$ is smooth, connected and non-empty for
all $b$, the trace morphism is an isomorphism for all $b$. ∎

###### Proof of Lemma 9.7.

The group scheme $P$ is a descent to $B$ of the relative Prym $\mathcal{P}$
over $\mathcal{F}^{0}$. We will show that the polarization defined in Lemma
9.9 descends to a polarization of the Tate module of $P\rightarrow B$.

Denote by $W^{\prime}$ the polarization on the Tate module
$T_{\ell}(\mathcal{P})$ of $\mathcal{P}$ constructed in Lemma 9.9. Consider
the cartesian square

${\mathcal{P}}$${\mathcal{F}^{0}}$${P}$${B}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{g}$

If $S$ is a base scheme and $F,G$ are two constructible $\ell$-adic complexes
on $S$, denote
$\operatorname{Hom}_{S}(F,G):=\operatorname{Hom}_{D^{b}(S)}(F,G)$. Observe
then that by the definition of $T_{\ell}(\mathcal{P})$, the pairing
$W^{\prime}$ is an element of

$\operatorname{Hom}_{\mathcal{F}^{0}}\bigl{(}R^{2d-1}g^{\prime}_{!}\overline{\mathbb{Q}}_{\ell,\mathcal{P}}(d)\otimes
R^{2d-1}g^{\prime}_{!}\overline{\mathbb{Q}}_{\ell,\mathcal{P}}(d),\overline{\mathbb{Q}}_{\ell,\mathcal{F}^{0}}(1)\bigr{)}\
.$

By applying the base change isomorphism and the monoidality of pullback, we
see that the above space of morphisms actually coincides with

$\operatorname{Hom}_{\mathcal{F}^{0}}\bigl{(}\pi^{*}\left(R^{2d-1}g_{!}\overline{\mathbb{Q}}_{\ell,P}(d)\otimes
R^{2d-1}g_{!}\overline{\mathbb{Q}}_{\ell,P}(d)\right),\pi^{*}\overline{\mathbb{Q}}_{\ell,B}(1)\bigr{)}\
.$

But since the morphism $\pi$ is surjective and smooth ([LSV17, Section 5, page
110]) and has connected fibers (as follows from [LSV17, Lemma 2.5 (ii)]),
Lemma 9.11 implies that the functor $\pi^{*}$ between constructible sheaves is
fully faithful. In other words, for any two constructible sheaves $F$ and $G$
on $B$, the functor $\pi^{*}$ induces an isomorphism

(9.13)
$\operatorname{Hom}_{B}(F,G)\stackrel{{\scriptstyle\pi^{*}}}{{\simeq}}\operatorname{Hom}_{\mathcal{F}^{0}}(\pi^{*}F,\pi^{*}G)\
.$

This tells us that $W^{\prime}$ arises from a pairing

$W\in\operatorname{Hom}_{B}\bigl{(}R^{2d-1}g_{!}\overline{\mathbb{Q}}_{\ell,P}(d)\otimes
R^{2d-1}g_{!}\overline{\mathbb{Q}}_{\ell,P}(d),\overline{\mathbb{Q}}_{\ell,B}(1)\bigr{)}\
.$

It remains to check that $W$ is a polarization. Take $b\in B$ a closed point
and $s\in\mathcal{F}^{0}$ a closed point in the fiber $\mathcal{F}^{0}_{b}$
over $b$. We already know that the null space of the pairing

$W_{s}^{\prime}\in\operatorname{Hom}(T_{\ell}(\mathcal{P})_{s}\otimes
T_{\ell}(\mathcal{P})_{s},\overline{\mathbb{Q}}_{\ell}(1))$

is precisely the Tate module $T_{\ell}(P^{\prime,aff})_{s}$ of the linear part
of $P_{s}^{\prime}$, and we want to deduce the same for

$W_{b}\in\operatorname{Hom}\bigl{(}T_{\ell}(P)_{b}\otimes
T_{\ell}(P)_{b},\overline{\mathbb{Q}}_{\ell}(1)\bigr{)}\ .$

But observe that the natural isomorphism of fibers
$P_{b}\simeq\mathcal{P}_{s}$ induces a natural isomorphism

$\operatorname{Hom}\bigl{(}T_{\ell}(P)_{b}\otimes
T_{\ell}(P)_{b},\overline{\mathbb{Q}}_{\ell}(1)\bigr{)}\stackrel{{\scriptstyle
I}}{{\simeq}}\operatorname{Hom}\bigl{(}T_{\ell}(\mathcal{P})_{s}\otimes
T_{\ell}(\mathcal{P})_{s},\overline{\mathbb{Q}}_{\ell}(1)\bigr{)}\ ,$

and we claim that $W_{b}$ maps to $W_{s}^{\prime}$ under this isomorphism,
thus concluding the proof.

In fact, for any two constructible sheaves $F,G$ on $B$, the cartesian square

${\mathcal{F}^{0}}$${\mathcal{F}^{0}_{b}}$${B}$${b}$$\scriptstyle{\pi}$$\scriptstyle{i^{\prime}_{b}}$$\scriptstyle{\pi_{b}}$$\scriptstyle{i_{b}}$

yields a commutative diagram, whose subdiagrams all commute,

${\operatorname{Hom}_{B}(F,G)}$${\operatorname{Hom}(F_{b},G_{b})}$${\operatorname{Hom}_{\mathcal{F}^{0}}(\pi^{*}F,\pi^{*}G)}$${\operatorname{Hom}_{\mathcal{F}_{b}^{0}}(\pi_{b}^{*}F_{b},\pi_{b}^{*}G_{b})}$${\operatorname{Hom}((\pi^{*}F)_{s},(\pi^{*}G)_{s})}$$\scriptstyle{\pi^{*}}$$\scriptstyle{i^{*}_{b}}$$\scriptstyle{\pi^{*}_{b}}$$\scriptstyle{I}$$\scriptstyle{i_{b}^{\prime,*}}$$\scriptstyle{\iota}$

Here the vertical arrows are isomorphisms by fully faithfulness, and $\iota$
is pullback to $s\in\mathcal{F}_{b}^{0}$, also an isomorphism since the
sheaves $\pi_{b}^{*}F_{b}$ and $\pi_{b}^{*}G_{b}$ are constant of stalk
$F_{b}\simeq(\pi^{*}F)_{s}$ resp. $G_{b}\simeq(\pi^{*}G)_{s}$ on the fiber
$\mathcal{F}_{b}^{0}$. Now, setting $F=T_{\ell}(P)\otimes T_{\ell}(P)$ and
$G=\overline{\mathbb{Q}}_{\ell,B}$, this implies that $I$ maps $W_{b}$ to the
image under $\iota\circ i^{\prime,*}_{b}$ of $W^{\prime}=\pi^{*}W$. The latter
being precisely $W^{\prime}_{s}$, this proves our claim. ∎

###  9.3. Application of Theorem 3.1 to LSV tenfolds

In this subsection we show Theorem 9.12, which is an extension of Theorem 1.2
from the Introduction.

######  Theorem 9.12.

Let $X$ be a LSV tenfold, respectively $X^{t}$ its twisted version. Then the
standard conjectures hold true for $X$, respectively for $X^{t}$.

###### Proof.

Any given LSV tenfold $X$ is part of a family, where the general member is a
general LSV tenfold $X^{\prime}$ with Picard number 2 (this is just because
the cubic fourfold $M$ defining $X$ can be deformed to a cubic fourfold
$M^{\prime}$ which is general in the sense of [LSV17] and with
$h^{2,2}(M^{\prime})=1$). Via a specialization argument, one is then reduced
to proving the theorem for $X^{\prime}$. To this end, let us check that the
assumptions of Theorem 3.1 are satisfied by $X^{\prime}$. (The analogous
specialization argument holds for the twisted version.)

Point (1) of Theorem 3.1 is clear, and point (2) follows from Proposition 3.9.
As for point (3), it is known that $X^{\prime}$ has only one primitive
Lagrangian class ([Sac22, Corollary 3.10], which follows from the existence of
an effective divisor $\Theta$ with $q(\Theta)<0$), and so we are in the second
case of Remark 8.1. The same holds for the twisted version, using the fact
that by [LPZ22, Theorem 1.3], $X^{t}$ is birational to a divisorial symplectic
resolution, hence the second ray of the birational Kähler cone determines a
divisorial contraction and not a Lagrangian fibration. But then, in both
cases, point (3) reduces to checking that the Lagrangian fibration
$X^{\prime}\to B$ described in [LSV17] (or its twisted analogue) is an Ngô
fibration; this is the content of Proposition 9.5 and Remark 9.6. ∎

######  Remark 9.13.

As explained in the proof, an LSV tenfold $X$ with Picard number 2 falls into
the second case of Remark 8.1, so that there is only one Lagrangian fibration
to study, namely the one naturally provided by the construction of $X$.

######  Remark 9.14.

The following more direct approach is possible to prove the standard
conjectures for twisted intermediate Jacobian fibrations.

Let $Y=X^{t}$ be the twisted intermediate Jacobian fibration associated to a
very general cubic fourfold $Z\subset\mathbb{P}^{5}$. As mentioned earlier, it
is shown in [LPZ22, Theorem 1.3 and Remark 6.9] that $Y$ is birational to the
HK variety $\tilde{M}$ obtained as a symplectic resolution of a certain moduli
space of semi-stable objects in the Kuznetsov category of $Z$. Applying
[FFZ21, Theorem 1.8], the Chow motive of $\tilde{M}$ is in the subcategory
generated by the motive of $Z$. As birational HK varieties have isomorphic
Chow motives, it follows that there is an inclusion of (Chow and hence also of
homological) motives

(9.14) $\mathfrak{h}(Y)\cong\mathfrak{h}(\tilde{M})\ \hookrightarrow\
\bigoplus\mathfrak{h}(Z^{r})(\ast)\ \ \hbox{in}\ {\mathcal{M}ot}\ .$

In particular, $Y$ verifies the standard conjectures.

The same remark applies to the HK varieties of OG6 type constructed as
symplectic resolutions of moduli spaces of semi-stable sheaves on an abelian
surface $A$ in [O’G03]. As shown in [Wu21], these HK varieties $Y$ admit a
Lagrangian fibration that is an Ngô fibration (with possibly reducible
fibers), and so perhaps a stratified version of our argument might apply.
However, it is known that the Chow motive of $Y$ is in the subcategory
generated by the motive of $A$ [Flo22], and so in particular $Y$ verifies the
standard conjectures.

On the other hand, we observe that this line of reasoning does not apply to
the LSV tenfolds. Indeed, it is known that the very general LSV tenfold is not
birational to a moduli space of stable objects in the Kuznetsov category of
$Z$ [Sac22, Corollary 4.2] (cf. also [Sac22, Remark 4.3]). Nevertheless, one
may wonder whether the inclusion (9.14) might still be true for the general
LSV tenfold.

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Giuseppe Ancona, IRMA, Strasbourg, France
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Mattia Cavicchi, IMO, Université Paris-Saclay, France
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Robert Laterveer, IRMA, Strasbourg, France
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Giulia Saccà, Columbia University, US
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