Sorbonne Université
École doctorale de sciences mathématiques de Paris centre

THÈSE DE DOCTORAT
Discipline : Mathématiques
présentée par
Chenyu BAI

Hodge Theory, Algebraic Cycles of Hyper-Kähler Manifolds

dirigée par Claire VOISIN
Soutenance le 14 juin 2024 devant le jury composé de:
Mme Katia AMERIK PR Université Paris-Saclay / NRU HSE Rapporteure M. Olivier
DEBARRE PR Université Paris-Cité Président du jury M. Radu LAZA PR Stony Brook
University Examinateur M. Emanuele MACRÌ PR Université Paris-Saclay
Examinateur M. Gianluca PACIENZA PR IECL Examinateur Mme Claire VOISIN DR CNRS
/ IMJ-PRG Directrice

## Remerciements

Je suis extrêmement honoré de pouvoir remercier ma directrice de thèse, Claire
Voisin. Depuis le M1, son enthousiasme pour les mathématiques, son engagement
envers la vérité, son attitude rigoureuse dans la recherche scientifique, et
sa générosité dans toutes ses interactions, ont profondément influencé mon
propre parcours. Je ne saurais mesurer combien j’ai appris d’elle, mais je
peux dire avec certitude que ma compréhension actuelle et ma vision des
mathématiques ont été fortement impactées par son influence. Cette thèse
reflète également, à bien des égards, l’empreinte de son école de pensée. Je
n’oublierai jamais notre première rencontre un après-midi dans son bureau au
Collège de France, où elle m’a présenté le théorème d’Ehresmann et la théorie
des déformations. Rétrospectivement, je vois le temps qui s’est envolé ; ces
quatre années sont passées en un clin d’œil. Sous sa tutelle, j’ai parcouru un
chemin aussi long que semé d’embûches, compréhensible seulement par ceux qui
ont partagé cette route. Sans les encouragements constants de Claire et
l’inspiration de nos nombreuses rencontres, je n’aurais certainement pas
ouvert la porte du monde des cycles algébriques et de la géométrie algébrique
complexe. Je me souviendrai toujours de cet après-midi, après ma soutenance de
M2, quand elle m’a remis son livre couleur citrouilles et m’a encouragé à le
lire - un excellent texte sur la théorie des cycles, confirmant que je pouvais
commencer mon voyage à partir de là - et ainsi est née cette thèse, imprégnée
de l’esprit et de la méthodologie de son livre. Des milliers de mots ne
pourraient suffire à exprimer ma gratitude envers elle, et c’est avec un cœur
plein d’espoir que je dédie cette thèse à Claire Voisin.

Tout au long de ce voyage, l’aide de Sébastien Boucksom a été inestimable,
tant sur le plan académique que professionnel. Sa bonté et sa sincérité dans
la recherche scientifique m’ont profondément ému. Merci à Bertrand Rémy, dont
la compagnie et l’assistance pendant mon temps à l’École Polytechnique de
Paris m’ont aidé à façonné la personne que je suis aujourd’hui.

Je tiens à remercier mon superviseur de licence, Baohua Fu, qui m’a initié au
monde de la géométrie algébrique avec les 27 droites. La dernière partie de
cette thèse, qui étudie les variétés de Calabi-Yau construites à partir de
droites, peut être vue comme une continuation à la théorie des 27 droites.

Merci à Maxime, pour ces trois années de camaraderie, les discussions
mathématiques avec lui ont toujours été si sereines, et sa bonne humeur a été
un moteur constant dans mon avancée. Merci à Elvaz, toujours en train de
travailler dur dans la salle d’à côté, nous avons été témoins de notre
croissance respective.

Merci à Pietro Beri, Corinne Bedussa et Mauro Varesco, explorer avec eux le
sujet des surfaces d’Enriques a été une des plus belles périodes de ma
carrière scientifique. J’espère qu’ils apprécieront la section sur les
surfaces d’Enriques dans cette thèse. Merci à Matteo, mon petit frère de
thèse, les discussions avec lui ont toujours été productives. Merci également
à Shengxuan pour les discussions mathématiques et philosophiques. Merci à Mai,
Shi, Ruichuan pour les discussions précieuses sur les cubiques.

Je suis reconnaissant envers mes professeurs : Olivier Debarre, Daniel
Huybrechts, Emanuele Macrì. J’ai toujours su que je pouvais toujours apprendre
davantage d’eux, et c’est ce qui s’est avéré. C’est un honneur d’avoir reçu
leurs conseils précieux tout au long de mon parcours de recherche. Merci aussi
à Laurent Manivel, que j’ai eu la chance de rencontrer très tôt et de
bénéficier de ses conseils. Merci à Zhi Jiang pour m’avoir invité à Shanghai
et pour m’avoir donné l’opportunité de parler de mes travaux scientifiques.

Merci à Enrico, Eva, Haohao, Long, Lucas, Mattias, Sacha, Tangi, Tristan,
Xiaohan pour leurs conseils, leurs discussions mathématiques et personnels, et
les bons moments passés ensemble.

Merci à Chenqin, Godard et Yicheng pour ces trois années passées ensemble près
du parc de Sceaux. Un grand merci à Yun-Xiao pour son soutien et ses
encouragements, sans lesquels mon parcours scientifique aurait été bien plus
difficile. Merci également à Shengkai et Peter Yi, mes amis les plus proches
aux États-Unis, et à Yunsong pour nos échanges enrichissants. Je voudrais
remercier Zhuo Pan pour m’avoir toujours encouragé. Un sincère merci à Louise
et à sa famille pour m’avoir encouragé à explorer de nouvelles possibilités
dans la vie et pour avoir enrichi mon expérience.

Un grand merci à Eric Xi Chen pour son calme et son enthousiasme contagieux
pour les mathématiques. Je suis toujours inspiré par nos échanges. Merci à
Jingyi pour nos discussions scientifiques toujours enrichissantes. Et un
spécial remerciement à Ziyi pour m’avoir aidé à découvrir ma passion pour
l’escalade, ce qui a profondément changé ma perspective sur la vie.

Pendant mon parcours à Polytechnique, je tiens à exprimer ma reconnaissance à
plusieurs personnes. En particulier, un grand merci à Mien, qui a été un
soutien moral essentiel pendant le confinement lié au Covid-19. Je lui dois
encore des excuses et un profond remerciement. Je souhaite également remercier
Daniel, mon compagnon de tennis, qui m’a chaleureusement accueilli lors de mes
séjours à Londres.

Un grand merci à Shikang et Yuchen pour m’avoir ouvert la porte vers une autre
possibilité professionnelle. Je suis également reconnaissant envers
BearishSentiment, Daniel, Kaitong, Tianyuan et Xiao pour leur précieuse aide
durant ces procédures.

Et enfin, un immense merci à mes parents, à ma petite sœur et à Anne, vous
êtes mon soutien éternel.

## Résumé

Cette thèse est consacrée à l’étude des cycles algébriques dans les variétés
hyper-Kähleriennes projectives et les variétés de Calabi-Yau strictes. Elle
contribue à la compréhension des conjectures de Beauville et de Voisin sur les
anneaux de Chow des variétés hyper-kählériennes projectives et des variétés de
Calabi-Yau strictes. Elle étudie également certains invariants birationnels
des variétés hyper-kählériennes projectives.

La première partie de la thèse, parue dans Mathematische Zeitschrift [Bai23]
et présentée dans le chapitre 2, étudie si les sous-variétés lagrangiennes
dans une variété hyper-kählérienne partageant la même classe cohomologique ont
également la même classe de Chow. Nous étudions la notion de familles
lagrangiennes et ses applications aux applications d’Abel-Jacobi associées.
Nous adoptons une approche infinitésimale pour donner un critère de trivialité
de l’application d’Abel-Jacobi d’une famille lagrangienne, et utilisons ce
critère pour donner une réponse négative à la question précédente, ajoutant
aux subtilités d’une conjecture de Voisin. Nous explorons également comment la
maximalité de la variation des structures de Hodge sur la cohomologie de degré
$1$ de la famille lagrangienne implique la trivialité de l’application d’Abel-
Jacobi.

La deuxième partie de la thèse, parue dans International Mathematics Research
Notices [Bai24] et présentée dans le chapitre 3, étudie le degré
d’irrationalité, la gonalité fibrante et le genre fibrant des variétés hyper-
kählériennes projectives. Nous commençons par donner une légère amélioration
d’un résultat de Voisin sur la borne inférieure du degré d’irrationalité des
variétés hyper-kählériennes générales de Mumford-Tate. Nous étudions ensuite
la relation entre les trois invariants birationnels susmentionnés pour les
surfaces K3 projectives de nombre de Picard $1$, rajoutant la compréhension
sur une conjecture de Bastianelli, De Poi, Ein, Lazarsfeld, Ullery sur le
comportement asymptotique du degré d’irrationalité des surfaces K3 projectives
très générales.

La troisième partie de la thèse, parue dans ArXiv [Bai24II], présentée dans le
chapitre 4, étudie les applications de Voisin de dimension supérieure sur les
variétés de Calabi-Yau strictes. Voisin a construit des applications auto-
rationnelles de variétés de Calabi-Yau obtenues comme des variétés de
$r$-plans dans des hypersurfaces cubiques de dimension adéquate. Cette
application a été largement étudiée dans le cas $r=1$, qui est le cas de
Beauville-Donagi. Dans les cas de dimensions supérieures, nous étudions
d’abord l’action de l’application de Voisin sur les formes holomorphes. Nous
demontrons ensuite la conjecture de Bloch généralisée pour l’action des
applications de Voisin sur les groupes de Chow dans le cas de $r=2$. Enfin,
via l’étude de l’application de Voisin, nous apportons des éléments de preuve
à une conjecture de Voisin sur l’existence d’un $0$-cycle spécial sur les
variétés de Calabi-Yau strictes.

Mots-clés : Variétés hyper-kählériennes, Variétés de Calabi-Yau strictes,
Conjectures de Voisin, Familles lagrangiennes, Degré d’irrationalité,
Applications de Voisin

## Abstract

This thesis is devoted to the study of algebraic cycles in projective hyper-
Kähler manifolds and strict Calabi-Yau manifolds. It contributes to the
understanding of Beauville’s and Voisin’s conjectures on the Chow rings of
projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It also
studies some birational invariants of projective hyper-Kähler manifolds.

The first part of the thesis, appeared in Mathematische Zeitschrift [Bai23]
and presented in Chapter 2, studies whether the Lagrangian subvarieties in a
hyper-Kähler manifold sharing the same cohomological class have the same Chow
class as well. We study the notion of Lagrangian families and its associated
Abel-Jacobi maps. We take an infinitesimal approach to give a criterion for
the triviality of the Abel-Jacobi map of a Lagrangian family, and use this
criterion to give a negative answer to the above question, adding to the
subtleties of a conjecture of Voisin. We also explore how the maximality of
the variation of the Hodge structures on the degree $1$ cohomology the
Lagrangian family implies the triviality of the Abel-Jacobi map.

The second part of the thesis, appeared in International Mathematics Research
Notices [Bai24] and presented in Chapter 3, studies the degree of
irrationality, the fibering gonality and the fibering genus of projective
hyper-Kähler manifolds, with emphasis on the K3 surfaces case, en mettant
l’accent sur le cas des surfaces K3. We first give a slight improvement of a
result of Voisin on the lower bound of the degree of irrationality of Mumford-
Tate general hyper-Kähler manifolds. We then study the relation of the above
three birational invariants for projective K3 surfaces of Picard number $1$,
adding the understandinf of a conjecture of Bastianelli, De Poi, Ein,
Lazarsfeld, Ullery on the asymptotic behavior of the degree of irrationality
of very general projective K3 surfaces.

The third part of the thesis, appeared in ArXiv [Bai24II], presented in
Chapter 4, studies the higher dimensional Voisin maps on strict Calabi-Yau
manifolds. Voisin constructed self-rational maps of Calabi-Yau manifolds
obtained as varieties of $r$-planes in cubic hypersurfaces of adequate
dimension. This map has been thoroughly studied in the case $r=1$, which is
the Beauville-Donagi case. For higher dimensional cases, we first study the
action of the Voisin map on the holomorphic forms. We then prove the
generalized Bloch conjecture for the action of the Voisin maps on Chow groups
for the case of $r=2$. Finally, via the study of the Voisin map, we provide
evidence for a conjecture of Voisin on the existence of a special $0$-cycle on
strict Calabi-Yau manifolds.

Keywords: Hyper-Kähler manifolds, Strict Calabi-Yau manifolds, Voisin
conjectures, Lagrangian families, Degree of irrationality, Voisin maps

###### Contents

  1. 1 Introduction
    1. 1.1 Chow groups
    2. 1.2 Interaction between Hodge structures and Chow groups
    3. 1.3 Some measures of irrationality
    4. 1.4 Hyper-Kähler Manifolds
    5. 1.5 Beauville’s splitting conjecture and Voisin’s conjectures
  2. 2 On Abel-Jacobi Maps of Lagrangian Families
    1. 2.1 Background
    2. 2.2 A criterion
    3. 2.3 Lagrangian fibrations
    4. 2.4 Maximal variations
    5. 2.5 Example of a Lagrangian family with nontrivial Abel-Jacobi map
  3. 3 On Some Birational Invariants of Hyper-Kähler Manifolds
    1. 3.1 Background and Introduction
    2. 3.2 Fibering genus of very general hyper-Kähler manifolds
    3. 3.3 Relations between birational invariants of $K3$ surfaces
    4. 3.4 Some inequalities about Picard lattices of hyper-Kähler manifolds
    5. 3.5 Related results
  4. 4 On the Geometry of the Higer Dimension Voisin Maps
    1. 4.1 Introduction
    2. 4.2 Action of the Voisin map on top degree holomorphic forms
    3. 4.3 Action of the Voisin map on $CH_{0}(X)_{hom}$
    4. 4.4 Indeterminacy Locus of the Voisin Map
  5. A Computational Verification in Chapter 4
    1. A.1 Computational Verification in Section

## Chapter 1 Introduction

###  1.1 Chow groups

We closely follow the presentations in [Ful84], [Voi03, Chapter 17], and
[Voi14, Chapter 2]. Let $X$ be an algebraic variety of dimension $n$, defined
over a field which will be in this thesis the field of complex numbers. An
algebraic cycle $Z\subset X$ of dimension $k$ is defined as a finite formal
sum $Z=\sum_{i}n_{i}Z_{i}$ with $n_{i}\in\mathbb{Z}$, and $Z_{i}$ is a closed
irreducible subvariety of $X$ of dimension $k$. The cycle group
$\mathcal{Z}_{k}(X)$ is the abelian group of all algebraic cycles of dimension
$k$ in $X$. If $\phi:X\to Y$ is a proper morphism, there is an induced map
$\phi_{*}:\mathcal{Z}_{k}(X)\to\mathcal{Z}_{k}(Y)$, called the push-forward
map, defined as follows: for an irreducible subvariety $Z\subset X$ of
dimension $k$ with $\phi|_{Z}:Z\to Z^{\prime}$ being a generically finite map
of degree $d$, we define $\phi_{*}Z=dZ^{\prime}$. In other cases,
$\phi_{*}Z=0$. Finally, we extend the definition of $\phi_{*}$ linearly to the
entire group $\mathcal{Z}_{k}(X)$.

######  Definition 1.1.1.

Two algebraic cycles $Z_{1},Z_{2}\subset X$ of dimension $k$ are called
rationally equivalent if there exist irreducible subvarieties
$W_{1},\ldots,W_{l}$ of dimension $k+1$ in $X$, nonzero rational functions
$f_{i}$ on $\tilde{W}_{i}$ where $\tau_{i}:\tilde{W}_{i}\to W_{i}$ is the
normalization of $W_{i}$, such that

$Z_{1}-Z_{2}=\sum_{i=1}^{l}\tau_{i*}(\mathrm{div}(f_{i})).$

######  Notation 1.1.2.

Throughout this thesis, we use the following notation. Let $X$ be an algebraic
variety of dimension $n$.

  * •

The Chow group $CH_{k}(X)=\mathcal{Z}_{k}(X)/\equiv_{rat}$ is the quotient
group of $\mathcal{Z}_{k}(X)$ modulo rational equivalence $\equiv_{rat}$.

  * •

$\mathcal{Z}^{k}(X)=\mathcal{Z}_{n-k}(X)$.

  * •

$CH^{k}(X)=CH_{n-k}(X)$.

  * •

$CH_{k}(X)_{\mathbb{Q}}=CH_{k}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$.

  * •

$CH^{k}(X)_{\mathbb{Q}}=CH^{k}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$.

In [Ful84, Chapter 6], an intersection product is constructed for smooth
algebraic varieties.

######  Theorem 1.1.3 (Fulton).

Let $X$ be a smooth quasi-projective variety. There exists a unique product
structure on $CH^{*}(X)$ satisfying the following condition: if two
subvarieties $A,B$ are transverse to each other, then $[A].[B]=[A\cap B]\in
CH^{*}(X)$. This product structure makes $CH^{*}(X)$ a graded ring.

####  1.1.1 Push-forward and pull-back maps

Let $p:X\to Y$ be a flat morphism of relative dimension $l$. Then, there is a
naturally defined map $\phi^{*}:\mathcal{Z}_{k}(Y)\to\mathcal{Z}_{k+l}(X)$,
called the pull-back map, defined by taking the preimage. A fundamental fact
is the following (See Lemma 2.3 and 2.5 in [Voi14]).

######  Lemma 1.1.4.

The push-forward map and the pull-back map, defined at the level of cycle
groups, preserve rational equivalence.

Therefore, the push-forward map of a proper morphism and the pull-back map of
a flat morphism can be defined at the level of Chow groups. Let $p:X\to Y$ be
a morphism between smooth varieties (in fact, we only need $Y$ to be smooth).
We can also define the pull-back map $p^{*}:CH^{k}(Y)\to CH^{k}(X)$ without
flatness conditions (see [Voi03, 17.2]). We have the following functoriality
result.

######  Proposition 1.1.5.

Let $p:X\to Y$ be a morphism between smooth varieties.
(a) (Projection formula) For $Z\in CH(Y)$ and $Z^{\prime}\in CH(X)$, we have

$p_{*}(p^{*}Z.Z^{\prime})=Z.p_{*}(Z^{\prime})\in CH(X).$

(b) For $Z,Z^{\prime}\in CH(Y)$, we have

$p^{*}(Z.Z^{\prime})=p^{*}Z.p^{*}Z^{\prime}\in CH(X).$

###### Proof.

See [Ful84, 8.1]. ∎

A correpondence between $X$ and $Y$ is an element $Z$ in $CH(X\times Y)$. Let
$p_{X}:X\times Y\to X$ and $p_{Y}:X\times Y\to Y$ be the two projection maps.
A correspondence $Z$ between smooth varieties $X$ and $Y$ induces two natural
maps $Z_{*}:CH(X)\to CH(Y)$ and $Z^{*}:CH(Y)\to CH(X)$ in the following way.
For any $z\in CH(X)$,

$Z_{*}x:=p_{Y,*}(p_{X}^{*}x.Z)\in CH(Y),$

and for any $w\in CH(Y)$,

$Z^{*}w:=p_{X,*}(p_{Y}^{*}w.Z)\in CH(X).$

####  1.1.2 Localization exact sequence

Let $X$ be a quasi-projective variety, and let $j:Y\hookrightarrow X$ be a
closed algebraic subset. Let $i:U:=X-Y\hookrightarrow X$ be the open embedding
of the complement of $Y$.

######  Proposition 1.1.6.

We have the following exact sequence:

$CH_{k}(Y)\stackrel{{\scriptstyle
j_{*}}}{{\to}}CH_{k}(X)\stackrel{{\scriptstyle i^{*}}}{{\to}}CH_{k}(U)\to 0.$

###### Proof.

See [Voi03, Lemma 17.12]. ∎

####  1.1.3 Cycle class map

Let $X$ be a smooth complex algebraic variety. To each irreducible subvariety
$Z\subset X$ of codimension $k$, we can associate its cohomological class in
$H^{2k}(X,\mathbb{Z})$. Extending linearly, we get the cycle class map:

$cl:\mathcal{Z}^{k}(X)\to H^{2k}(X,\mathbb{Z}).$

######  Lemma 1.1.7.

Let $Z_{1},Z_{2}\in\mathcal{Z}^{k}(X)$. If $Z_{1}$ and $Z_{2}$ are rationally
equivalent, then $cl(Z_{1})=cl(Z_{2})$ in $H^{2k}(X,\mathbb{Z})$.

Thus, the cycle class map descends to the Chow group level:

$cl:CH^{k}(X)\to H^{2k}(X,\mathbb{Z}),$

and we still call it the cycle class map.

We have the following compatibility results of the cycle class map (see
[Voi03, 17.2.4]).

######  Proposition 1.1.8.

The cycle class map $cl:CH^{*}(X)\to H^{2k}(X,\mathbb{Z})$ is a ring
homomorphism.

######  Proposition 1.1.9.

Let $p:X\to Y$ be a morphism between smooth varieties.
(a) For any $Z\in CH^{k}(Y)$, we have

$p^{*}cl(Z)=cl(p^{*}Z)\in H^{2k}(X,\mathbb{Z}).$

(b) If $p$ is proper, then for any $Z\in CH_{k}(X)$, we have

$p_{*}cl(Z)=cl(p_{*}Z)\in H^{2\dim Y-2k}(Y,\mathbb{Z}).$

######  Notation 1.1.10.

The kernel of the cycle class map $cl:CH^{k}(X)\to H^{2k}(X,\mathbb{Z})$ is
denoted by $CH^{k}(X)_{hom}$. The subscript "hom" signifies "homologous to 0".
Similarly, $CH^{k}(X)_{\mathbb{Q},hom}$ denotes the kernel of
$cl:CH^{k}(X)_{\mathbb{Q}}\to H^{2k}(X,\mathbb{Q})$.

####  1.1.4 Constant cycle subvarieties

The notion of constant cycle subvarieties is introduced and developed in
[Huy14, Voi16], especially in the context of algebraic hyper-Kähler manifold.

######  Definition 1.1.11 ([Huy14, Voi16]).

Let $X$ be an algebraic variety. A subvariety $i:Z\hookrightarrow X$ is called
a constant cycle subvariety if every two points $z_{1},z_{2}\in Z$ are
rationally equivalent in $X$. In other words, the image of the push-forward
map

$i_{*}:CH_{0}(Z)\to CH_{0}(X)$

is $\mathbb{Z}$.

######  Remark 1.1.12.

The study of rational curves in an algebraic variety is a powerful method to
understand the algebraic variety itself ([Kol96]). Constant cycle subvarieties
are a big generalization of rationally connected subvarieties. The existence
of such subvarieties often gives many interesting geometric implications
(e.g., [Voi16, Lin16, Baz17]). On the other hand, constant cycle subvarieties
share similar properties to rationally connected subvarieties. For example, it
is classical that there are no rational curves in abelian varieties. One can
also prove that there are no constant cycle subvarieties of positive dimension
in abelian varieties either.

Constant cycle subvarieties will appear in Chapter 4 where we prove the
following theorem (see Theorem 4.3.26).

######  Theorem 1.1.13.

Let $Y\subset\mathbb{P}^{9}$ be a general cubic eightfold. Let $X=F_{2}(Y)$ be
its Fano variety of planes. Let $F\subset X$ be the closure of the set of the
points $x\in X$ parametrizing the planes $P_{x}\subset Y$ such that there
exists a unique linear subspace $H$ of dimension $3$ such that $H\cap
Y=3P_{x}$. Then $F\subset X$ is a constant cycle subvariety of $X$ of
codimension $3$.

This theorem serves as a crucial ingredient for our understanding of the
Voisin maps in Chapter 4.

####  1.1.5 Bloch-Beilinson filtration

It has been conjectured by Bloch and Beilinson that there exists a decreasing
filtration $F^{i}CH_{k}(X)_{\mathbb{Q}}$ on the Chow groups with rational
coefficients of any smooth complex projective variety $X$, satisfying several
axioms. The precise statements are as follows. We follow closely the
presentation in [Voi04II] and [Voi14, Conjecture 2.19] for the statements of
the Bloch-Beilinson conjectures.

######  Conjecture 1.1.14 (Bloch-Beilinson Conjecture).

For any smooth projective variety $X$, there exists a decreasing filtration
$F^{\bullet}$ on $CH^{i}(X)_{\mathbb{Q}}$, with the following properties:
(i) (Non-Triviality) $F^{0}CH^{i}(X)_{\mathbb{Q}}=CH^{i}(X)_{\mathbb{Q}}$ and
$F^{1}CH^{i}(X)_{\mathbb{Q}}=CH^{i}(X)_{\mathbb{Q},hom}$.
(ii) (Functoriality) If $Z\in CH^{k}(X\times Y)_{\mathbb{Q}}$, then
$Z_{*}(F^{i}CH^{l}(X)_{\mathbb{Q}})\subset F^{i}CH^{l+k-n}(X)_{\mathbb{Q}}$,
where $n=\dim X$.
(iii) (Graded) The induced map $Z_{*}:Gr_{F}^{i}CH^{l}(X)_{\mathbb{Q}}\to
Gr_{F}^{i}CH^{l+k-n}(Y)_{\mathbb{Q}}$ vanishes for any $i$ if $[Z]=0$ in
$H^{2k}(X\times Y,\mathbb{Q})$.
(iv) (Finiteness) One has $F^{k+1}CH^{k}(X)_{\mathbb{Q}}=0$ for any $X$ and
$k$.

There is also a strengthened version of the Bloch-Beilinson conjecture where
(iii) above is replace by

(iii)’ (Graded) For a fixed $i$, the induced map
$Z_{*}:Gr_{F}^{i}CH^{l}(X)_{\mathbb{Q}}\to
Gr_{F}^{i}CH^{l+k-n}(Y)_{\mathbb{Q}}$ vanishes if the following induced map

$[Z]^{*}:H^{2m-2k-2l+2n+i}(Y,\mathbb{Q})\to H^{2n-2l+i}(X,\mathbb{Q})$

vanishes, where $m=\dim Y$ and $n=\dim X$.

###  1.2 Interaction between Hodge structures and Chow groups

####  1.2.1 Hodge Structures and Coniveau

######  Definition 1.2.1.

A weight $k$ Hodge structure (respectively, a rational Hodge structure) on $H$
consists of a free abelian group $H_{\mathbb{Z}}$ (respectively, a
$\mathbb{Q}$-vector space $H_{\mathbb{Q}}$) and a decomposition

$H_{\mathbb{C}}:=H\otimes\mathbb{C}=\bigoplus_{p+q=k}H^{p,q},$

satisfying the Hodge symmetry condition

$\overline{H^{p,q}}=H^{q,p}.$

Hodge structures naturally arise from the Betti cohomology groups of compact
Kähler manifolds [Voi03, 7.1].

######  Theorem 1.2.2 (Hodge Decomposition Theorem [Voi03]).

Let $X$ be a compact Kähler manifold. Then the $k$-th Betti cohomology group
$H^{k}(X,\mathbb{Z})$ of $X$ possesses a weight $k$ Hodge structure.
Specifically, there is a decomposition

$H^{k}(X,\mathbb{C})=\bigoplus_{p+q=k}H^{p,q}(X),$

such that $\overline{H^{p,q}(X)}=H^{q,p}(X)$. Moreover, there is a canonical
isomorphism $H^{q}(X,\Omega_{X}^{p})\cong H^{p,q}(X)$.

######  Definition 1.2.3.

A (rational) Hodge class in a (rational) Hodge structure $H$ of degree $2k$ is
an element in $H_{\mathbb{Z}}\cap H^{k,k}$ (respectively, $H_{\mathbb{Q}}\cap
H^{k,k}$).

######  Definition 1.2.4.

The (Hodge) coniveau $c\leq k/2$ of a weight $k$ Hodge structure
$(H_{\mathbb{Z}},H^{p,q})$ is the smallest integer $p$ for which $H^{p,q}\neq
0$.

A multitude of conjectures relate algebraic cycles and Hodge structures.

######  Conjecture 1.2.5 (Hodge Conjecture).

Let $X$ be a complex smooth projective variety. Then any rational Hodge class
in $H^{2k}(X,\mathbb{Q})$ is representable by a rational algebraic cycle of
codimension $k$.

######  Conjecture 1.2.6 (Generalized Hodge Conjecture, Grothendieck [Gro69]).

Let $X$ be a complex smooth projective variety. Suppose $L\subset
H^{k}(X,\mathbb{Q})$ is a rational sub-Hodge structure of Hodge coniveau $\geq
c$. Then there exists a closed algebraic subset $Z\subset X$ of codimension
$c$ such that $L$ vanishes under the restriction map $H^{k}(X,\mathbb{Q})\to
H^{k}(X\setminus Z,\mathbb{Q})$, where $U:=X\setminus Z$.

Conjecture 1.2.5 is the special case of Conjecture 1.2.6 when the coniveau $c$
is half the weight of the Hodge structure. The significance and recent
advancements in the Hodge conjecture and the generalized Hodge conjecture are
detailed in [Voi16II]. The following conjecture is stated by in [Voi13] by
Voisin who also noticed that it is in fact a consequence of the Lefschetz
standard conjecture (see [Voi14, Remark 2.30]).

######  Conjecture 1.2.7 (Voisin [Voi13]).

Let $X$ be a smooth complex projective variety and $Y\subset X$ be a closed
algebraic subset. Suppose $Z\subset X$ is a codimension $k$ algebraic cycle,
and assume that the cohomology class $[Z]\in H^{2k}(X,\mathbb{Q})$ vanishes in
$H^{2k}(X\setminus Y,\mathbb{Q})$. Then there exists a codimension $k$ cycle
$Z^{\prime}$ on $X$ with $\mathbb{Q}$-coefficients, which is supported on $Y$
and such that $[Z^{\prime}]=[Z]$ in $H^{2k}(X,\mathbb{Q})$.

Combining the generalized Hodge conjecture (Conjecture 1.2.6) and the Bloch-
Beilinson conjecture (Conjecture 1.1.14), the following conjecture is
expected.

######  Conjecture 1.2.8.

Let $Z\subset CH^{n}(X\times X)$ be a self-correspondence of a smooth
projective variety $X$ of dimension $n$. If $[Z]^{*}|_{H^{i,0}(X)}=0$ for some
$i$, then $Z_{*}|_{Gr_{F}^{i}CH_{0}(X)}=0$ for the same $i$ as well, where
$F^{\bullet}$ is the Bloch-Beilinson filtration and $Gr_{F}^{\bullet}$ is the
graded part.

Indeed, By the Künneth decomposition theorem and the Poincaré duality, $[Z]\in
H^{2n}(X\times X,\mathbb{Q})$ can be identified as a graded map
$[Z]^{*}:H^{*}(X,\mathbb{Q})\to H^{*}(X,\mathbb{Q})$ that preserves Hodge
structure. Let us fix a polarization on $X$ and let
$\phi:H^{*}(X,\mathbb{Q})\to H^{*}(X,\mathbb{Q})$ be the restriction of
$[Z]^{*}$ onto the subspace $N^{1}H^{i}(X,\mathbb{Q})$ which is the largest
sub-Hodge structure of $H^{i}(X,\mathbb{Q})$ of coniveau at least $1$. To be
precise, $\phi(\alpha):=[Z]^{*}(\alpha)$ if $\alpha\in
N^{1}H^{i}(X,\mathbb{Q})$ and $\phi(\alpha)=0$ if $\alpha\in
N^{1}H^{i}(X,\mathbb{Q})^{\bot}$ or if $\alpha\in H^{k}(X,\mathbb{Q})$ for
$k\neq i$. The map $\phi$ preserves Hodge-structure, and when viewed as an
element of $H^{2n}(X\times X,\mathbb{Q})$, is a Hodge class. By the Hodge
conjecture, there is an algebraic cycle $Z_{1}\in CH^{n}(X\times X)$ that
represents $\phi\in H^{2n}(X\times X,\mathbb{Q})$. By the construction of
$Z_{1}$, the algebraic cycle $Z_{2}:=Z-Z_{1}$ satisfies the condition
$[Z_{2}]^{*}|_{H^{i}(X,\mathbb{Q})}=0$. By the Bloch-Beilinson conjecture
(Conjecture 1.1.14), we have $Z_{2*}|_{Gr^{i}_{F}CH_{0}(X)}=0$. On the other
hand, we have

######  Lemma 1.2.9.

Assuming the generalized Hodge conjecture (Conjecture 1.2.6) and the
generalized Bloch-Beilinson conjecture (Conjecture 1.1.14),
$Z_{1*}|_{Gr_{F}^{k}CH_{0}(X)}=0$ for any $k$.

Admitting Lemma 1.2.9 for the moment, we conclude $Z_{*}=Z_{1*}+Z_{2*}$ acts
as $0$ on $Gr_{F}^{i}CH_{0}(X)$, as desired.

###### Proof of Lemma 1.2.9.

By construction, $[Z_{1}]^{*}|_{H^{k,0}(X)}=0$ for any $k$. This implies that
the sub-Hodge structure $[Z_{1}]^{*}H^{k}(X,\mathbb{Q})\subset
H^{k}(X,\mathbb{Q})$ has Hodge coniveau at least $1$ for any $k$. By the
generalized Hodge conjecture (Conjecture 1.2.6), there is an open dense subset
$U\subset X$ such that $([Z_{1}]^{*}H^{k}(X,\mathbb{Q}))|_{U}=0$ for any $k$.
This means, by the Künneth decomposition theorem, that the cohomology class of
$Z_{1}|_{U\times X}$ is $0$. Therefore, by Conjecture 1.2.7 (which is implied
by Conjecture 1.2.6), there is a cycle $Z^{\prime}\in CH^{n}(X\times X)$,
supported on $D\times X$, such that $[Z_{1}-Z^{\prime}]=0\in H^{2n}(X\times
X,\mathbb{Q})$, where $D\subset X$ is the complement of $U$ in $X$. Since
$Z^{\prime}$ is supported on $D\times X$ and $D$ is a proper closed subset of
$X$, we have $Z^{\prime}_{*}CH_{0}(X)=0$. On the other hand, since
$Z_{1}-Z^{\prime}$ is cohomologeous to $0$, the (generalized) Bloch-Beilinson
conjecture implies that $((Z_{1}-Z^{\prime}))_{*}|_{Gr_{F}^{k}CH^{*}(X)}=0$
for any $k$. Taken together, we find that the action of
$Z_{1}=Z^{\prime}+(Z_{1}-Z^{\prime})$ on $Gr_{F}^{k}CH_{0}(X)$ is $0$, for any
$k$, as desired. ∎

In the literature, Conjecture 1.2.8 is often called the generalized Bloch
conjecture. It extends the classical Bloch conjecture, stated as follows in
[Blo80] for surfaces.

######  Conjecture 1.2.10 (Bloch [Blo80]).

For a correspondence $Z\in CH^{2}(S\times T)_{\mathbb{Q}}$ between surfaces
that induces a null map $[Z]^{*}:H^{2,0}(T)\to H^{2,0}(S)$, the induced
morphism $Z_{*}:F^{2}CH_{0}(S)\to F^{2}CH_{0}(T)$ is identically zero. Here,
$F^{2}CH_{0}(S)$ is defined as the kernel of the Albanese map from
$CH_{0}(S)_{hom}$ to $Alb(S)$, and similarly for $F^{2}CH_{0}(T)$.

If we take $Z=\Delta_{X}$, the diagonal of $X\times X$, Conjecture 1.2.8
predicts that if $H^{i,0}(X)=0$ for all $i>0$, then
$CH_{0}(X)_{\mathbb{Q},hom}=0$. This is part of the following conjecture, also
named the generalized Bloch conjecture [Voi14, Conjecture 1.9].

######  Conjecture 1.2.11 (Generalized Bloch Conjecture).

Let $X$ be a smooth projective variety. Assume that $H^{p,q}(X)=0$ for $p\neq
q$ and $p<c$ (or $q<c$). Then the cycle class map

$cl:CH_{i}(X)_{\mathbb{Q}}\to H^{2m-2i}(X,\mathbb{Q})$

is injective for $i\leq c-1$.

Some non-trivial incidences of Conjecture 1.2.11 are given in Chapter 4. Let
$Y\subset\mathbb{P}^{9}$ be a general cubic eightfold. Its Fano variety of
lines $F_{1}(Y)$ is a Fano manifold of dimension $12$. It has been established
that $H^{p,q}(F_{1}(Y))=0$ for $p\leq 1$ and $p\neq q$ [DM98], so Conjecture
1.2.11 predicts that $CH_{i}(F_{1}(Y))_{\mathbb{Q},hom}=0$ for $i\leq 1$. In
Chapter 4, we prove the following (see Theorem 4.3.6)

######  Theorem 1.2.12.

We have $CH_{i}(F_{1}(Y))_{\mathbb{Q},hom}=0$ for any $i\leq 1$ and for any
general cubic eightfold $Y$.

####  1.2.2 The case of strict Calabi-Yau manifolds

######  Definition 1.2.13.

A strict Calabi-Yau manifold $X$ is a complex projective manifold of dimension
at least $3$ that is simply connected and has trivial canonical bundle, and
such that for each $0<i<\dim X$, there is no nonzero holomorphic forms of
degree $i$ on $X$.

Conjecture 1.2.8 takes the following form for strict Calabi-Yau manifolds, and
more generally for smooth projective varieties $X$ with $h^{i,0}(X)=0$ for
$0<i<n=dimX$ and $h^{n,0}(X)=1$.

######  Conjecture 1.2.14.

Let $X$ be a strict Calabi-Yau manifold of dimension $n$. Let $\omega_{X}\in
H^{0}(X,K_{X})$ be a nowhere zero top degree holomorphic form on $X$. Let
$Z\in CH^{n}(X\times X)_{\mathbb{Q}}$ be a self-correspondence such that
$[Z]^{*}\omega=0$. Then for any $z\in CH_{0}(X)_{\mathbb{Q},hom}$, we have
$Z_{*}z=0$.

Indeed, since $X$ is a strict Calabi-Yau manifold, $H^{k,0}(X)=0$ for $0<k<n$.
Let $\Delta_{X}\in CH^{n}(X\times X)$ be the diagonal of $X\times X$. Then
$[\Delta_{X}]^{*}|_{H^{k,0}(X)}=0$ for $0<k<n$. By Conjecture 1.2.8,
$\Delta_{X*}|_{Gr^{k}_{F}CH_{0}(X)}=0$ for $0<k<n$, which implies that
$Gr^{k}_{F}CH_{0}(X)=0$ for $0<k<n$. Therefore, the Bloch-Beilinson filtration
on $CH_{0}(X)$ degenerates into

$0=F^{n+1}CH_{0}(X)\subset
F^{n}CH_{0}(X)=\ldots=F^{1}CH_{0}(X)=CH_{0}(X)_{hom}\subset
F^{0}CH_{0}(X)=CH_{0}(X).$

By the assumption of $Z$, we have $[Z]^{*}|_{H^{n,0}(X)}=0$. By Conjecture
1.2.8 again, we have $Z_{*}|_{Gr^{n}_{F}CH_{0}(X)}=0$. Hence, $Z_{*}$ acts as
zero on $Gr^{n}_{F}CH_{0}(X)=F^{n}CH_{0}(X)=F^{1}CH_{0}(X)=CH_{0}(X)_{hom}$,
and we get the desired result.

####  1.2.3 Voisin’s examples of Calabi-Yau manifolds

In Chapter 4, we aim to give evidence to Conjecture 1.2.14 for specific
families of $K$-trivial varieties as constructed in [Voi04]. Let
$Y\subset\mathbb{P}^{n}$ be a smooth cubic hypersurface of dimension $n-1$,
and let $r\geq 0$ denote a nonnegative integer. Define $X=F_{r}(Y)$ as the
Hilbert scheme that parametrizes the $r$-dimensional linear subspaces in $Y$.
As proven in [Voi04, (4.41)], for $n+1=\binom{r+3}{2}$ and a general $Y$, the
variety $X$ is a $K$-trivial variety of dimension
$N=(r+1)(n-r)-\binom{r+3}{3}$. Specifically, when $r=0$, $X$ is an elliptic
curve; and as established in [BD85], $X$ is a hyper-Kähler manifold for $r=1$.
It is further shown (see Lemma 4.1.4) that for $r\geq 2$, $X$ is a strict
Calabi-Yau manifold. As is studied in Theorem 4.4.10 in Chapter 4, the small
deformations of $X$ is relatively easy to understand.

######  Theorem 1.2.15.

Assume $r\geq 2$. Let $X=F_{r}(Y)$ be the above strict Calabi-Yau manifold.

  1. (a)

For any small deformation $X^{\prime}$ of $X$, there is a cubic hypersurface
$Y^{\prime}$ such that $X^{\prime}=F_{r}(Y^{\prime})$.

  2. (b)

The dimension of deformation space of $X$ is $\dim
H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(3))-\dim
GL_{n+1}(\mathbb{C})$.

The distinct feature of the above manifolds $X=F_{r}(Y)$ among all $K$-trivial
manifolds revolves around the presence of a self-rational map,
$\Psi:X\dashrightarrow X$, referred to as the Voisin map. This map was
introduced in [Voi04] through the following construction: Consider a general
point $x\in X$, representing an $r$-dimensional linear space $P_{x}$ within
$Y$. As demonstrated in [Voi04, Lemma 8], there exists a unique
$(r+1)$-dimensional linear subspace $H_{x}$ in $\mathbb{P}^{n}$ tangent to $Y$
along $P_{x}$. The intersection $H_{x}\cap Y$ forms a cubic hypersurface
containing $P_{x}$ doubly, leaving a residual linear subspace in $Y$
represented by a point $x^{\prime}\in X$. This process defines the Voisin map
as $\Psi(x)=x^{\prime}$.

In Chapter 4, we prove the following fact about the Voisin maps in Theorem A:

######  Theorem 1.2.16.

Given a nowhere zero top degree holomorphic form $\omega\in H^{0}(X,K_{X})$ on
$X$, we have

$\Psi^{*}\omega=(-2)^{r+1}\omega.$

By taking $Z=\Gamma_{\Psi}-(-2)^{*}\Delta_{X}\in CH(X\times X)$ where
$\Gamma_{\Psi}$ is the graph of $\Psi$ and $\Delta_{X}$ is the diagonal of
$X\times X$, it should be expected, by Conjecture 1.2.14, that for any $z\in
CH_{0}(X)_{\mathbb{Q},hom}$, $\Psi_{*}z=(-2)^{r+1}z$. In the case $r=2$, we
succeed in proving this result in Theorem B in Chapter 4.

######  Theorem 1.2.17.

Let $Y\subset\mathbb{P}^{9}$ be a general cubic $8$-fold. Let $X=F_{2}(Y)$ be
the Fano variety of planes in $Y$ and let $\Psi:X\dashrightarrow X$ be the
Voisin map. Then for any $z\in CH_{0}(X)_{hom}$, we have

$\Psi_{*}z=-8z\,\textrm{ in }\,CH_{0}(X)_{\mathrm{hom}}.$

###  1.3 Some measures of irrationality

Let $X$ be a projective variety over $\mathbb{C}$ of dimension $n$. By
applying a general linear projection to $X\subset\mathbb{P}^{N}$, there exists
a generically finite dominant rational map
$\phi:X\dashrightarrow\mathbb{P}^{n}$. The following invariant, the degree of
irrationality, was proposed and studied in [BDELU17].

######  Definition 1.3.1 ([BDELU17]).

The degree of irrationality of $X$, denoted as $\mathrm{Irr}(X)$, is the
minimal degree of dominant rational maps
$\phi:X\dashrightarrow\mathbb{P}^{n}$. In other words,

$\mathrm{Irr}(X):=\min\\{\deg\phi:\textrm{there is a dominant rational map
$\phi:X\dashrightarrow\mathbb{P}^{n}$}\\}.$

The degree of irrationality measures how far a variety is from being rational,
in the sense that $X$ is rational if and only if $\mathrm{Irr}(X)=1$. If $X$
is a smooth proper curve, the degree of irrationality of $X$ is the gonality
of the curve $X$, which is a classical invariant of a curve.

In their seminal work [BDELU17], Bastianelli, De Poi, Ein, Lazarsfeld, Ullery
propose the following conjecture on the degree of irrationality of K3
surfaces.

######  Conjecture 1.3.2 ([BDELU17]).

Let $\\{(S_{d},L_{d})\\}_{d\in\mathbb{N}}$ be very general polarized $K3$
surfaces such that $L_{d}^{2}=2d-2$. Then

$\limsup_{d\to\infty}\mathrm{Irr}(S_{d})=+\infty.$

In an attempt to determine the degree of irrationality and to better
understand the geometry of the variety, the following two measures of
irrationality are proposed and studied by Voisin in [Voi21].

######  Definition 1.3.3 ([Voi21]).

(i) The fibering gonality of $X$, $\mathrm{Fibgon}(X)$, is the minimal number
$c$ such that there exists a rational dominant map $\pi:X\dashrightarrow B$
such that $\dim B=\dim X-1$ and the general fiber is a connected curve $C$ of
gonality $c$.
(ii) The fibering genus of $X$, $\mathrm{Fibgen}(X)$, is the minimal number
$c$ such that there exists a rational dominant map $\pi:X\dashrightarrow B$
such that $\dim B=\dim X-1$ and the general fiber is a connected curve $C$ of
geometric genus $c$.

The fibering genus of K3 surfaces has already been studied by Ein and
Lazarsfeld in [EL20], under the name _Konno invariant_. They give a good
estimate of the asymptotic behavior of fibering genus of K3 surfaces as the
genus tends to infinity. Here the genus $d$ of a K3 surface $S$ is defined by
the formula $2d-2=c_{1}(L)^{2}$, where $L$ generates $\mathrm{Pic}(S)$.

######  Theorem 1.3.4 (Ein–Lazarsfeld [EL20]).

Let $S_{d}$ be a polarized K3 surface of Picard rank $1$ and genus $d$. Then

$\mathrm{Fibgen}(S_{d})=\Theta(\sqrt{d}),$

which means that there are constants $C_{1}$, $C_{2}>0$ such that
$C_{2}\sqrt{d}<\mathrm{Fibgen}(S_{d})<C_{1}\sqrt{d}$.

######  Corollary 1.3.5.

We have

$\lim_{d\to\infty}\mathrm{Fibgen}(S_{d})=+\infty.$

It is elementary to prove that for any smooth projective variety $X$, the
following inequlities are satisfied.
(i) $\mathrm{Fibgon}(X)\leq\mathrm{Irr}(X)$.
(ii) $\mathrm{Fibgon}(X)\leq\frac{1}{2}(\mathrm{Fibgen}(X)-1)+2$.

In Chapter 3, we get a finer comparison for very general projective K3
surfaces (see Theorem 3.1.14).

######  Theorem 1.3.6.

Let $S$ be a projective K3 surface whose Picard number is $1$. Then one of the
following two cases holds:
(a) $\mathrm{Irr}(S)=\mathrm{Fibgon}(S)$;
(b) $\mathrm{Fibgen}(S)^{2}\leq\mathrm{Fibgon}(S)^{21}$.

######  Corollary 1.3.7.

Let $\\{S_{d}\\}_{d\in\mathbb{N}}$ be projective K3 surfaces such that the
Picard group of $S_{d}$ is generated by a line bundle with self intersection
number $2d-2$. Then

$\limsup_{d\to\infty}\mathrm{Irr}(S_{d})=+\infty\iff\limsup_{d\to\infty}\mathrm{Fibgon}(S_{d})=+\infty.$

Conjecture 1.3.2 predicts that $\limsup_{d}\mathrm{Irr}(S_{d})=+\infty$, so it
should be expected that $\limsup_{d}\mathrm{Fibgon}(S_{d})=+\infty$ as well.

###  1.4 Hyper-Kähler Manifolds

######  Definition 1.4.1.

A hyper-Kähler manifold $X$ is defined as a compact Kähler manifold that is
simply connected and for which $H^{0}(X,\Omega_{X}^{2})$ is generated by an
everywhere non-degenerate holomorphic $2$-form $\sigma_{X}$.

The presence of an everywhere non-degenerate holomorphic $2$-form implies that
$X$ has an even dimension. Hereafter, we denote by $m=2n$ the dimension of $X$
and by $\sigma_{X}$ a non-degenerate holomorphic $2$-form of $X$ (determined
up to a scalar). Notably, a K3 surface is defined as a hyper-Kähler manifold
of dimension $2$. The broader aspects of hyper-Kähler varieties are
extensively discussed in [Bea83, Fuj87, Huy99].

####  1.4.1 Beauville-Bogomolov-Fujiki Form

Hyper-Kähler manifolds exhibit a non-degenerate symmetric integral quadratic
form on $H^{2}(X,\mathbb{Z})$. This form, which extends the Lefschetz
intersection form applicable to K3 surfaces, has been discovered in [Bea83,
Fuj87]. It is referred to as the Beauville-Bogomolov-Fujiki form.

######  Theorem 1.4.2 ([Bea83, Fuj87]).

Consider a hyper-Kähler manifold $X$ of dimension $m=2n$. There exist a unique
integral quadratic form $q_{X}$ on $H^{2}(X,\mathbb{Q})$ and a positive
rational coefficient $\lambda_{X}\in\mathbb{Q}_{>0}$ such that:
(1) for any $\alpha\in H^{2}(X,\mathbb{Q})$,

$\int_{X}\alpha^{m}=\lambda_{X}q(\alpha)^{n};$ (1.1)

(2) $q_{X}$ is indivisible, meaning that for any $k>1$, $q_{X}/k$ is not
integral;
(3) $q_{X}(\alpha)$ is positive for any Kähler class $\alpha$.

####  1.4.2 Lagrangian Subvarieties

Consider a hyper-Kähler manifold $X$ of dimension $2n$ with a holomorphic
$2$-form $\sigma_{X}$.

######  Definition 1.4.3.

A closed subvariety $L\subset X$ is called Lagrangian if:
(i) The restriction of $\sigma_{X}$ to the smooth part $L_{reg}$ of $L$
vanishes, and
(ii) The dimension of $L$ is $n$.

The deformation of _smooth_ Lagrangian subvarieties with $X$ has been
investigated in [Voi92].

######  Theorem 1.4.4 (Voisin [Voi92]).

For a hyper-Kähler manifold $X$ and a smooth Lagrangian subvariety $L\subset
X$, let $\mathcal{X}\to\mathrm{Def}(X)$ represent the Kuranishi family of $X$,
with $o\in\mathrm{Def}(X)$ as the reference point. Also, let
$\mathrm{Def}(X,L)$ be the deformation germ of the pair $(X,L)$. Then,
(1) $\mathrm{Def}(X,L)$ is smooth, and both its fibers and the image of the
natural map

$\pi:\mathrm{Def}(X,L)\to\mathrm{Def}(X)$

are also smooth.
(2) Furthermore,

$\displaystyle\mathrm{Im}\pi$
$\displaystyle=\\{t\in\mathrm{Def}(X):[L]\textrm{ remains a Hodge class in
}H^{2n}(\mathcal{X}_{t},\mathbb{Q})\\}$
$\displaystyle=\\{t\in\mathrm{Def}(X):[\sigma_{X_{t}}]\in\ker(H^{2}(X,\mathbb{C})\to
H^{2}(L,\mathbb{C}))\\}.$

The second equality in (2) implies that small deformations of $X$ with
constant Picard number contain a deformation of $L$. Theorem 1.4.4 crucially
indicates that the deformation of a smooth Lagrangian subvariety within a
hyper-Kähler manifold is unobstructed. It is used in Chapter 2 to justify the
naturality of a certain technical condition.

####  1.4.3 Lagrangian fibrations

Consider a hyper-Kähler manifold $X$. A Lagrangian fibration of $X$ is
characterized as a holomorphic surjection $\pi:X\to B$, where the general
fibers are Lagrangian subvarieties of $X$. Importantly, a Lagrangian fibration
differs from a topological fibration in that it is a topological fibration
solely over an open subset of the base and includes singular fibers.

This section describes the most important classical results on Lagrangian
fibrations.

######  Theorem 1.4.5 (Matsushita [Mat99]).

Let $\pi:X\to B$ be a holomorphic surjection from a hyper-Kähler manifold $X$
of dimension $2n$ to a compact complex variety $B$, assuming $0<\dim B<2n$ and
the fibers are connected. Then $\dim B=n$ and $\pi:X\to B$ defines a
Lagrangian fibration.

######  Theorem 1.4.6 (Matsushita [Mat99], Voisin [Voi92]).

For a Lagrangian fibration $\pi:X\to B$, the general fibers are projective,
hence are abelian varieties.

######  Theorem 1.4.7 (Hwang [Hwa08]).

Given a Lagrangian fibration $\pi:X\to B$ with a smooth base $B$, then
$B\cong\mathbb{P}^{n}$.

Theorem 1.4.7 also holds under the assumption that $X$ is merely smooth
Kähler. This was demonstrated by Greb and Lehn [GL14] using a result on the
deformation of Lagrangian fibrations by Matsushita [Mat16]. It is conjectured
that the same holds when the base $B$ is normal. Supporting evidence for this
conjecture is provided in [Mat15], which discusses the intersection cohomology
of the base. While this conjecture is straightforward in the context of K3
surfaces, it remains unresolved in a broader scope. In the case of four-
dimensional hyper-Kähler manifolds, the conjecture has been proved by
Huybrechts and Xu in [HX22].

####  1.4.4 Lagrangian families

A natural generalization of Lagrangian fibrations is the notion of Lagrangian
families, introduced by Voisin in [Voi22I, Voi22II].

######  Definition 1.4.8 (Voisin).

A Lagrangian family of a hyper-Kähler manifold $X$ is a diagram

${\mathcal{L}}$${X}$${B}$$\scriptstyle{q}$$\scriptstyle{p}$ (1.2)

In this configuration, $p$ is flat and projective, $\mathcal{L}$ and $B$ are
connected quasi-projective manifolds, and $q$ maps the general fiber
$L_{b}:=p^{-1}(b)$, $b\in B$, birationally to a Lagrangian subvariety of $X$.
For practical purposes, we will denote $j_{b}$ as the composition
$L_{b}\hookrightarrow\mathcal{L}\to X$.

As opposed to the Lagrangian fibrations, the existence of Lagrangian families
are expected to be a less restrictive condition. In Chapter 2, we study the
Abel-Jacobi map associated with the Lagrangian families. We briefly review
Griffiths’ theory on Abel-Jacobi maps in Section 2.1.1. In Chapter 2, we prove
the following criterion to determine if the Abel-Jacobi map of a given
Lagrangian family is trivial (see Proposition 2.1.14).

######  Theorem 1.4.9.

Consider a Lagrangian family of a hyper-Kähler manifold $X$ of dimension $2n$,
satisfying the following condition:

> $\clubsuit$ For general $b\in B$, the contraction by $q^{*}\sigma_{X}$ gives
> an isomorphism $\lrcorner
> q^{*}\sigma_{X}:T_{B,b}\stackrel{{\scriptstyle\cong}}{{\to}}H^{0}(L_{b},\Omega_{L_{b}}).$

Then the Abel-Jacobi map (2.3) is trivial if and only if for general $b\in B$,
the restriction map

$j_{b}^{*}:H^{2n-1}(X,\mathbb{Q})\to H^{2n-1}(L_{b},\mathbb{Q})$

is zero.

This criterion gives a topological method to determine the triviality of the
Abel-Jacobi maps of the Lagrangian families.

####  1.4.5 Mumford-Tate Groups

Our discussion closely follows the presentations of Mumford-Tate groups as
found in [Del71, Section 7], [vGV16], and [Voi10, Section 4.2].

Consider a rational Hodge structure $H=(H_{\mathbb{Q}},H^{p,q})_{p+q=r}$ of
weight $r$. An algebraic group action, denoted $\mu$, of $\mathbb{S}^{1}$ on
$H_{\mathbb{R}}$ is defined as follows: For any $v\in H_{\mathbb{R}}$, with
Hodge decomposition $v=\sum_{p+q=r}v^{p,q}$, and for any
$z=e^{i\theta}\in\mathbb{S}^{1}\subset\mathbb{C}^{*}$, we define

$\mu(z).v:=\sum_{p,q}z^{p}\bar{z}^{q}v^{p,q}.$

######  Definition 1.4.10.

The Mumford-Tate group $MT(H)$ for a Hodge structure $H$ is defined as the
smallest algebraic subgroup $G$ of $GL(H_{\mathbb{Q}})$, defined over
$\mathbb{Q}$, such that $G(\mathbb{R})$ contains $\mu(\mathbb{S}^{1})$.

It should be noted that the Mumford-Tate group as defined here is referred to
as the special Mumford-Tate group of $H$ in [Voi10].

Returning to the context of hyper-Kähler manifolds, consider a projective
hyper-Kähler manifold $X$ with dimension $2n$, and let $q$ denote the
Beauville-Bogomolov-Fujiki form on $H^{2}(X,\mathbb{Q})$. The transcendental
part of $H^{2}(X,\mathbb{Q})$, denoted by $H^{2}(X,\mathbb{Q})_{tr}$, is the
minimal sub-Hodge structure containing $H^{2,0}$ and, thanks to the
projectivity of $X$, can also be defined as the orthogonal complement to the
Néron-Severi group $NS(X)_{\mathbb{Q}}$ in $H^{2}(X,\mathbb{Q})$ with respect
to the Beauville-Bogomolov-Fujiki quadratic form. Due to the first Hodge-
Riemann bilinear relation, the Hodge structure of $H^{2}(X,\mathbb{Q})$ is
compatible with the Beauville-Bogomolov-Fujiki form $q$, which implies that
the Mumford-Tate group $MT(H^{2}(X,\mathbb{Q})_{tr})$ is contained in
$SO(H^{2}(X,\mathbb{Q})_{tr},q)$. The following theorem shows that this
inclusion is an equality for a very general polarized hyper-Kähler manifold
(see, e.g. [vGV16, Lemma 9]):

######  Theorem 1.4.11.

Let $X$ be a very general fiber of a complete family of lattice polarized
deformations. Then the Mumford-Tate group $MT(H^{2}(X,\mathbb{Q})_{tr})$ is
equal to $SO(H^{2}(X,\mathbb{Q})_{tr},q)$.

When $X$ meet the criteria of Theorem 1.4.11, it is said that the Mumford-Tate
group of $X$ is maximal, or that $X$ is a Mumford-Tate very general hyper-
Kähler manifold. This is a technical condition that may help us to prove
results for very general projective hyper-Kähler manifolds. Here are two
examples of how the Mumford-Tate maximality condition appears in this thesis.

The first one gives a condition for the triviality of the Abel-Jacobi map for
a Lagrangian family (see Theorem 2.1.15 and Proposition 2.1.16).

######  Theorem 1.4.12.

Consider a Lagrangian family on a Mumford-Tate very general hyper-Kähler
manifold $X$ of dimension $2n$ satisfying the following conditions

  * (i)

At least one of the Lagrangian subvarieties $L_{b}$ in the Lagrangian family
is smooth;

  * (ii)

$h^{1,0}(L_{b})\leq 2^{\lfloor\frac{b_{2}(X)_{tr}-3}{2}\rfloor}$.

Then the Abel-Jacobi map associated with this Lagrangian family is trivial.

The second example consists of the lower bound of the fibering genus of a
projective hyper-Kähler manifold obtained by Voisin [Voi21].

######  Theorem 1.4.13 (Voisin [Voi21]).

Let $X$ be a Mumford-Tate very general hyper-Kähler manifold of dimension $2n$
with $n\geq 3$ and $b_{2,\mathrm{tr}}(X)\geq 5$. Then

$\mathrm{Fibgen}(X)\geq\min\\{n+2,2^{\lfloor\frac{b_{2,\mathrm{tr}}(X)-3}{2}\rfloor}\\}.$

This result is further improved as follows in Chapter 3 (see Theorem 3.1.7).

######  Theorem 1.4.14.

Let $X$ be a Mumford-Tate very general hyper-Kähler manifold of dimension $2n$
and assume $b_{2,\mathrm{tr}}(X)\geq 5$. Then

$\mathrm{Fibgen}(X)\geq\min\left\\{n+\left\lceil\frac{-1+\sqrt{8n-7}}{2}\right\rceil,2^{\lfloor\frac{b_{2,\mathrm{tr}}(X)-3}{2}\rfloor}\right\\}.$

###  1.5 Beauville’s splitting conjecture and Voisin’s conjectures

We follow closely the article [Voi16]. In this section, the Chow rings are
considered with rational coefficients. Let $X$ be a projective hyper-Kähler
manifold. The splitting conjecture of Beauville [Bea07] predicts that the
Bloch-Beilinson filtration on the Chow ring of $X$ has a natural
multiplicative splitting. As is explained in the Introduction of [Voi16], this
implies the following conjecture, now often called the “weak splitting
conjecture”.

######  Conjecture 1.5.1 (Beauville’s weak splitting conjecture [Bea07]).

Let $X$ be a projective hyper-Kähler manifold. Then the cycle class map is
injective on the subalgebra of $CH^{*}(X)$ generated by divisors.

Beauville’s weak splitting conjecture has been enlarged in [Voi08] as follows.

######  Conjecture 1.5.2 (Voisin [Voi08]).

Let $X$ be a projective hyper-Kähler manifold. Let $C^{*}$ be the subalgebra
of $CH^{*}(X)$ generated by divisors and Chern classes. Then the cycle class
map is injective on $C^{*}$.

It has been shown by Beauville [Bea07, Examples 1.7] that the analogies of
Conjecture 4.1.8 and Conjecture 1.5.2 for strict Calabi-Yau manifolds are
false in general. Beauville constructs in _loc. cit._ a strict Calabi-Yau
threefold $X$ such that the cycle-class map $cl:CH_{1}(X)_{\mathbb{Q}}\to
H^{4}(X,\mathbb{Q})$ is _not_ injective on the subgroup generated by
intersections of divisors. However, it is expected that the analogy of
Conjecture 1.5.2 to strict Calabi-Yau manifolds still holds true for
$0$-cycles.

######  Conjecture 1.5.3 (Voisin).

Let $X$ be a strict Calabi-Yau manifold of dimension $n$. Let $C\subset
CH_{0}(X)_{\mathbb{Q}}$ be the degree $n$ part of the subring of $CH^{*}(X)$
generated by the intersections of divisors and of Chern classes of $X$. Then
the cycle class map

$cl:CH_{0}(X)_{\mathbb{Q}}\to H^{2n}(X,\mathbb{Q})$

is injective on $C$.

We give a conditional result on Conjecture 1.5.3 for the strict Calabi-Yau
$11$-folds $F_{2}(Y)$ constructed by Voisin [Voi04] (see Theorem C in Chapter
4) and studied in this thesis.

######  Theorem 1.5.4.

Let $X=F_{2}(Y)$ be the Fano variety of planes of a general cubic $8$-fold
$Y\subset\mathbb{P}^{9}$. Suppose the indeterminacy locus of the Voisin map is
a constant cycle subvariety. Then $X$ satisfies Conjecture 1.5.3.

In [Voi16], a filtration, now often called the “Voisin’s filtration”, is
proposed for the Chow group of $0$-cycles of $X$. As is defined in _loc. cit._
, for $x\in X$, let $O_{x}\subset X$ be the rational equivalence orbit of $x$,
namely, $O_{x}$ is the set of points in $X$ that are rationally equivalent to
$x$. Standard argument [Voi16, p. 2] shows that $O_{x}$ is a countable union
of closed algebraic subsets of $X$, which makes it possible to define the
dimension of $O_{x}$ as the maximal dimension among the irreducible components
of $O_{x}$.

######  Definition 1.5.5 (Voisin [Voi03]).

We define $S_{i}X\subset X$ to be the set of points in $X$ whose orbit under
rational equivalence has dimension $\geq i$. The filtration $S_{\bullet}$ is
then defined by letting $S_{i}CH_{0}(X)$ be the subgroup of $CH_{0}(X)$
generated by classes of points $x\in S_{i}X$.

Highly motivated by the theory developed in _loc. cit._ , Conjecture 1.5.2 is
enlarged in [Voi16]. Let $2n$ be the dimension of $X$. In the case of $i=n$,
the conjectures in [Voi16] predict that two _constant cycle_ subvarieties of
dimension $n$, upon sharing an identical cohomological class, ought to share
the same Chow class as well. Motivated by this conjecture, in Chapter 2, we
study the Chow classes of Lagrangian subvarieties, raising the question of
whether two Lagrangian subvarieties within the same hyper-Kähler manifold,
sharing identical cohomological classes, also possess the same Chow class.
Given that a constant-cycle subvariety of dimension $n$ is easily seen to be
Lagrangian, see [Voi16, Theorem 0.7], our question emerges as a natural
extension of Voisin’s conjecture. In Chapter 2, we give negative answer to
this question by presenting a counter-example, thereby uncovering the
intricacies and challenges associated with Voisin’s conjecture.

We construct our counter-examples within the framework of the generalized
Kummer varieties as introduced in [Bea83]. Let us remind the construction.
Consider an abelian surface $A$ and its Hilbert scheme $A^{[n+1]}$ of length
$n+1$ subschemes. The morphism $\mathrm{alb}:A^{[n+1]}\to A$ results from
combining the Hilbert-Chow morphism with the summation map as follows:

$A^{[n+1]}\to A^{(n+1)}\to A.$

It is important to note that $\mathrm{alb}$ constitutes an isotrivial
fibration. The generalized Kummer variety, denoted as $K_{n}(A)$, is defined
as the fiber of $\mathrm{alb}$ over $0\in A$. As demonstrated in [Bea83],
$K_{n}(A)$ is a hyper-Kähler manifold of dimension $2n$. We then proceed to
construct Lagrangian subvarieties in $K_{n}(A)$. For any element $x\in A$, we
identify a subvariety $Z_{x}$ within $K_{n}(A)$ that consists of Artinian
subschemes of $A$ of length $n+1$, supported on $x$ and $-nx$, with
multiplicities $n$ and $1$, respectively. For any curve $C\subset A$, we
define $Z_{C}=\bigcup_{x\in C}Z_{x}$. In Section 2.5, we establish the
following theorem:

######  Theorem 1.5.6.

  1. (a)

The subvariety $Z_{C}$ is a Lagrangian subvariety in $K_{n}(A)$.

  2. (b)

For numerically equivalent very ample curves $C$ and $C^{\prime}$ in $A$,
$Z_{C}$ and $Z_{C^{\prime}}$ share the same cohomological class in
$H^{2n}(K_{n}(A),\mathbb{Z})$ but possess distinct Chow classes in
$CH^{n}(K_{n}(A))$.

## Chapter 2 On Abel-Jacobi Maps of Lagrangian Families

We study in this chapter the cohomological properties of Lagrangian families
on projective hyper-Kähler manifolds. First, we give a criterion for the
vanishing of Abel-Jacobi maps of Lagrangian families. Using this criterion, we
show that under a natural condition, if the moduli map for the fibers of the
Lagrangian family is maximal, its Abel-Jacobi map is trivial. We also
construct Lagrangian families on generalized Kummer varieties whose Abel-
Jacobi map is not trivial, showing that our criterion is optimal.

This chapter presents the main result of [Bai23]. The structure is as follows.
In Section 2.1, we give a panorama of known results in this area. Section 2.2,
Section 2.3 and Section 2.4 are the content of the article [Bai23]. In Section
2.2, we prove Proposition 2.1.14. In Section 2.3, we construct a Lagrangian
fibration structure on the relative Albanese variety and use it to prove
Theorem 2.1.15. In Section 2.4, we discuss the condition on the maximality of
the variation of Hodge structures. In Section 2.5, we construct Lagrangian
families satisfying a maximal condition whose Abel-Jacobi map is nontrivial,
showing that Theorem 2.1.15 is optimal.

###  2.1 Background

The primary objective of this chapter is to address the following question:

######  Question 2.1.1.

Suppose $X$ is a hyper-Kähler manifold of dimension $2n$, and $L_{1}$ and
$L_{2}$ are two Lagrangian subvarieties, both sharing the same cohomological
class in $H^{2n}(X,\mathbb{Z})$. Are $L_{1}$ and $L_{2}$ necessarily
rationally equivalent to each other?

The example presented in Section 2.5 gives a negative answer to Question
2.1.1.

####  2.1.1 Abel-Jacobi Maps

We begin by revisiting the Abel–Jacobi invariant and the Abel–Jacobi map
introduced by Griffiths [Gri68], following the framework outlined in [Voi03,
Chapter 12]. Throughout this section, $X$ is a compact Kähler manifold of
dimension $n$.

##### Intermediate Jacobian

######  Definition 2.1.2 (Griffiths).

The $k$-th intermediate Jacobian, denoted $J^{2k-1}(X)$, is the complex torus
defined as:

$J^{2k-1}(X)=H^{2k-1}(X,\mathbb{C})/(F^{k}H^{2k-1}(X)\oplus
H^{2k-1}(X,\mathbb{Z})_{\mathrm{tf}}).$

By applying Poincaré duality, it follows that:

$J^{2k-1}(X)\cong
F^{n-k}H^{2n-2k+1}(X,\mathbb{C})^{*}/H_{2n-2k+1}(X,\mathbb{Z})_{\mathrm{tf}}.$

Here the subscript “tf” signifies the “torsion free part” of the integral
cohomology.

######  Example 2.1.3.

(i) When $k=1$, the intermediate Jacobian $J^{1}(X)=\mathrm{Pic}^{0}(X)$,
representing the degree $0$ part of the Picard group of $X$.
(ii) When $k=n$, the intermediate Jacobian $J^{2n-1}(X)=\mathrm{Alb}(X)$ is
the Albanese variety of $X$.

##### Abel-Jacobi Invariant

Let $\mathcal{Z}^{k}(X)$ denote the free abelian group generated by
irreducible subvarieties of codimension $k$ in $X$. The cycle class map,

$cl:\mathcal{Z}^{k}(X)\to H^{2k}(X,\mathbb{Z})$

maps the class of a subvariety of codimension $k$ to its cohomological class
in $H^{2k}(X,\mathbb{Z})$. Let $\mathcal{Z}^{k}(X)_{hom}$ represent the kernel
of the cycle class map, where the subscript "hom" signifies "homologous to 0".

Associated with any $k$-cocycle $Z\in\mathcal{Z}^{k}(X)_{hom}$ that is
homologous to $0$ is an element $\alpha_{Z}\in J^{2k-1}(X)$, termed the _Abel-
Jacobi invariant of $Z$_, in the intermediate Jacobian of the corresponding
degree. The construction of $\alpha_{Z}$ can be briefly described as follows
[Gri68], [Voi03, 12.1.2].

Given that $Z$ is homologous to $0$, there exists a chain $\Gamma$ of
codimension $2k-1$ such that $\partial\Gamma=Z$. The integration current along
$\Gamma$,

$\int_{\Gamma}:\omega\mapsto\int_{\Gamma}\omega$

can be interpreted as an element in $F^{n-k}H^{2n-2k-1}(X,\mathbb{C})^{*}$. In
fact, if $\omega$ and $\omega^{\prime}$ yield the same cohomological class in
$F^{n-k}H^{2n-2k-1}(X,\mathbb{C})$, then, by a deep result of Hodge theory,
they differ by an exact form $d\phi$ with $\phi\in
F^{n-k-1}\mathcal{A}^{2n-2k-2}_{\mathbb{C}}(X)$. By Stokes’ theorem,
$\int_{\Gamma}d\phi=\int_{Z}\phi$, which vanishes due to type reasons.
Additionally, if $\Gamma$ and $\Gamma^{\prime}$ are two chains such that
$\partial\Gamma=\partial\Gamma^{\prime}=Z$, then modulo the image of
$H_{2n-2k+1}(X,\mathbb{Z})$ in $F^{n-k}H^{2n-2k+1}(X,\mathbb{C})^{*}$, the
linear forms $\int_{\Gamma}$ and $\int_{\Gamma^{\prime}}$ are the same. Hence,
there exists a uniquely determined element:

$\alpha_{Z}=\left[\int_{\Gamma}\right]\in
F^{n-k}H^{2n-2k-1}(X,\mathbb{C})^{*}/H_{2n-2k+1}(X,\mathbb{Z})=J^{2n-2k+1}(X).$

######  Proposition 2.1.4 ([Voi03], Lemme 21.19).

Let $Z$ be an algebraic cycle that is rationally equivalent to $0$ in $X$.
Then the Abel–Jacobi invariant of $Z$ is $0$.

By Proposition 2.1.4, we are now equipped with a map

$\begin{array}[]{cccc}\Phi^{k}_{X}:&CH^{n}(X)_{hom}&\to&J^{2n-1}(X)\\\
&Z=\partial\Gamma&\mapsto&\int_{\Gamma}\end{array}$

##### Abel-Jacobi Map of a Family

Let $\mathcal{Z}\subset B\times X$ be a flat family of subvarieties of
codimension $k$ in $X$. Specifically, $B$ is a connected complex manifold, and
$\mathcal{Z}$ is a subvariety of codimension $k$ in $B\times X$, flat over
$B$. Denote by $p:\mathcal{Z}\to B$ and $q:\mathcal{Z}\to X$ the projection
maps to the two components. Let $0\in B$ be a reference point.

######  Definition 2.1.5.

The Abel–Jacobi map of the family $\mathcal{Z}\subset B\times X$ with respect
to the reference point $0\in B$ is the map

$\begin{array}[]{cccc}\Psi_{\mathcal{L}}^{AJ}:&B&\to&J^{2k-1}(X)\\\
&b&\mapsto&\Phi_{X}^{k}(q_{*}p^{*}(b)-q_{*}p^{*}(0)).\end{array}$

######  Theorem 2.1.6 (Griffiths [Gri68]).

  1. (i)

The Abel–Jacobi map $\Psi_{\mathcal{L}}^{AJ}:B\to J^{2k-1}(X)$ is holomorphic.

  2. (ii)

The image of the differential of $\Psi_{\mathcal{L}}^{AJ}$ at any point lies
in $H^{k-1,k}(X)\subset H^{2k-1}(X,\mathbb{C})$.

####  2.1.2 Lagrangian Families

Consider $X$ as a projective hyper-Kähler manifold of dimension $2n$. Voisin
introduced the notion of Lagrangian families in her work [Voi22I, Voi22II],
which serves as a generalizations of Lagrangian fibrations.

######  Definition 2.1.7 (Voisin [Voi22I, Voi22II]).

A Lagrangian family of a hyper-Kähler manifold $X$ is a diagram

${\mathcal{L}}$${X}$${B}$$\scriptstyle{q}$$\scriptstyle{p}$ (2.1)

In this configuration, $p$ is flat and projective, $\mathcal{L}$ and $B$ are
connected quasi-projective manifolds, and $q$ maps the general fiber
$L_{b}:=p^{-1}(b)$, $b\in B$, birationally to a Lagrangian subvariety of $X$.
For practical purposes, we will denote by $j_{b}$ the composition
$L_{b}\hookrightarrow\mathcal{L}\to X$.

Let us give some examples of Lagrangian families documented in the literature:

######  Example 2.1.8.

  1. (i)

Any Lagrangian fibration of $X$ naturally leads to a Lagrangian family. Note
that the existence of a Lagrangian fibration necessitates that $X$ has a
Picard number of at least $2$, whereas a very general projective hyper-Kähler
manifold typically has a Picard number of $1$.

  2. (ii)

(Voisin [Voi92]) Let $S$ be a K3 surface and $C$ be a curve in $S$. The curve
$C$ can move in a family $\\{C_{b}\\}_{b\in U}$ where $U$ is an open subset of
the Hilbert scheme of deformations of $C$ in $S$. This constitutes a
Lagrangian family of $S$. Moreover, consider $S^{[n]}$, the Hilbert scheme of
$n$ points of $S$, which is a hyper-Kähler manifold of dimension $2n$. Then
$\\{C_{b}^{(n)}\\}_{b\in U}$ is a Lagrangian family of $S^{[n]}$.

  3. (iii)

(Voisin [Voi92]) Let $Y\subset\mathbb{P}^{5}$ be a smooth cubic fourfold, and
$H\subset\mathbb{P}^{5}$ be a general hyperplane. Then the Fano surface
$F_{1}(Y\cap H)$ of lines of $Y\cap H$, is a Lagrangian subvariety of the Fano
variety $F_{1}(Y)$ of lines of $Y$, which is a hyper-Kähler fourfold by
[BD85]. The general hyperplanes $H\subset\mathbb{P}^{5}$ are parameterized by
an open subset $U$ of $(\mathbb{P}^{5})^{*}$. The family $\\{F_{1}(Y\cap
H)\\}_{H\in U}$ provides a Lagrangian family of $F_{1}(Y)$.

  4. (iv)

(Iliev-Manivel [IM08]) Let us consider a linear inclusion
$\mathbb{P}^{5}\subset\mathbb{P}^{6}$ and a smooth cubic fourfold
$Y\subset\mathbb{P}^{5}$. Now, let $Z\subset\mathbb{P}^{6}$ be a general
smooth cubic fivefold containing $Y$. Let $S_{Z}:=F_{2}(Z)$ denote the Fano
surface of planes in $Z$. The map

$\begin{array}[]{cccc}j_{Z}:&S_{Z}&\to&F_{1}(Y)\\\ &P&\mapsto&P\cap
H\end{array}$

is generically 1:1 on its image and the image is a Lagrangian subvariety.
Consequently, $\\{S_{Z}\\}_{Z\in U}$, where $U$ is an open subset of all cubic
fivefolds containing $Y$, provides a Lagrangian family of $F_{1}(Y)$.

####  2.1.3 Voisin’s Conjectures Revisited

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. Recall the
definition of Voisin’s filtration [Voi16] previously mentioned in definition
1.5.5. The subset $S_{n}X$ contains the points whose rational equivalence
orbit has a dimension of $n$.

######  Proposition 2.1.9 ([Voi16], Theorem 0.7).

An irreducible subvariety of dimension $n$, denoted as $L\subset X$, is a
constant-cycle subvariety if and only if $L\subset S_{n}X$.

Based on this proposition, the following conjecture is posed in [Voi16]:

######  Conjecture 2.1.10 ([Voi16]).

Let $L$ and $L^{\prime}$ be two $n$-dimensional _constant cycle_ subvarieties
of a hyper-Kähler manifold $X$. If the cohomological classes
$[L]=[L^{\prime}]$ in $H^{2n}(X,\mathbb{Q})$, then $L$ is rationally
equivalent to $L^{\prime}$ as algebraic cycles in $X$.

It is established [Voi16, Theorem 0.7] that an $n$-dimensional constant cycle
subvariety $L$ of $X$ is Lagrangian. Question 2.1.1 is the question whether,
in Conjecture 2.1.10, the condition "constant cycle subvarieties" can be
substituted with "Lagrangian subvarieties".

Lagrangian families serve as an essential source of Lagrangian subvarieties
sharing the same cohomological class. In view of Question 2.1.1, we are
motived to study the following

######  Problem 2.1.11.

Consider a Lagrangian family on a hyper-Kähler manifold $X$ of dimension $2n$
given by a diagram as in (2.1). What can be said of the map

$\begin{array}[]{cccc}\psi_{\mathcal{L}}:&B&\rightarrow&CH^{n}(X)\\\
&b&\mapsto&q_{*}(L_{b})\end{array}?$ (2.2)

The condition constant cycle is a strong condition for Lagrangian
subvarieties. Notice that in contrast with Lagrangian subvarieties, constant
cycle Lagrangian subvarieties cannot deform into families.

######  Lemma 2.1.12.

Small deformations of constant cycle Lagrangian subvarieties of $X$ are no
longer constant cycle subvarieties.

###### Proof.

Following the notations in [Voi16], let

$S_{n}X:=\\{x\in X:\textrm{ the rational equivalence orbit of }x\textrm{ has
dimension}\geq n\\}.$

As is shown in [Voi16, Theorem 1.3], $S_{n}X$ is a countable union of
irreducible varieties of dimension $\leq n$ and constant cycle Lagrangian
subvarieties of $X$ are exactly irreducible components of $S_{n}X$ of
dimension $n$. Hence, constant cycle Lagrangian subvarieties of $X$ are rigid.
∎

As is described in Section 2.1, a weaker invariant of algebraic cycles in a
projective manifold is the Abel-Jacobi invariant. Problem 2.1.11 thus
motivates the following question.

######  Problem 2.1.13.

Consider a Lagrangian family of a hyper-Kähler manifold $X$ of dimension $2n$
given by a diagram as in (2.1). Let $0\in B$ be a point. Under which
conditions is the Abel-Jacobi map

$\begin{array}[]{cccc}\Psi_{\mathcal{L}}^{AJ}:&B&\rightarrow&J^{2n-1}(X)\\\
&b&\mapsto&\Phi_{X}^{n}(q_{*}(L_{b}-L_{0}))\end{array}$ (2.3)

trivial?

In many instances, Lagrangian families provide affirmative answers to Problem
2.1.11, as highlighted in all examples of Lagrangian families in Example
2.1.8. However, as detailed in Section 2.5, an explicit example has been
constructed that offers a negative solution to Problem 2.1.13. This example is
derived from a Lagrangian family of the generalized Kummer varieties. The
advantage of considering a Lagrangian family over a mere pair of Lagrangian
subvarieties is the utilization of the differential theory of families
developed by Griffiths [Gri68] to explore the triviality of the Abel-Jacobi
map of such a family. Consequently, we have developed a cohomological
criterion in Section 2.2 to determine if a given Lagrangian family possesses a
trivial Abel-Jacobi map. Additionally, an explicit Lagrangian family of
generalized Kummer varieties is constructed and shown to exhibit a nontrivial
Abel-Jacobi map using the criterion we have developed. This implies that any
two Lagrangian subvarieties lacking the same Abel-Jacobi invariant in this
family provide a negative response to Question 2.1.1.

It is important to note that this example does not present counter-examples to
Voisin’s Conjecture 2.1.10. In fact, it adds depth and intricacy to Voisin’s
Conjecture, making it even more engaging and nuanced.

####  2.1.4 Organization of the chapter

In this chapter, we first give a criterion for the vanishing of the Abel-
Jacobi map (2.3) for Lagrangian families of a hyper-Kähler manifold (see also
Proposition 2.2.1).

######  Proposition 2.1.14.

Consider a Lagrangian family on a hyper-Kähler manifold $X$ of dimension $2n$
as in (2.1), satisfying the following condition :

> $\clubsuit$ For general $b\in B$, the contraction by $q^{*}\sigma_{X}$ gives
> an isomorphism $\lrcorner
> q^{*}\sigma_{X}:T_{B,b}\stackrel{{\scriptstyle\cong}}{{\to}}H^{0}(L_{b},\Omega_{L_{b}}).$

Then the Abel-Jacobi map (2.3) is trivial if and only if for general $b\in B$,
the restriction map

$j_{b}^{*}:H^{2n-1}(X,\mathbb{Q})\to H^{2n-1}(L_{b},\mathbb{Q})$

is zero.

The condition $\clubsuit$ is natural. According to [Voi92, Proposition 2.4],
the deformations of a _smooth_ Lagrangian subvariety are non-obstructed, and a
local deformation is still Lagrangian. Therefore, if we take $(B,b)$ to be a
germ of the Hilbert scheme of deformations of a _smooth_ Lagrangian subvariety
$L\subset X$, and $\mathcal{L}\to B$ the corresponding family, then condition
$\clubsuit$ holds since $T_{B,b}\cong H^{0}(L_{b},N_{L_{b}/X})$ by
unobstructedness and $\lrcorner\sigma_{X}:H^{0}(L_{b},N_{L_{b}/X})\to
H^{0}(L_{b},\Omega_{L_{b}})$ is an isomorphism for a smooth Lagrangian
variety.

Using this criterion, we give a response to Problem 2.1.13.

######  Theorem 2.1.15.

Consider a Lagrangian family on a hyper-Kähler manifold $X$ of dimension $2n$
given by a diagram as in (2.1), satisfying condition $\clubsuit$. Assume that
the variation of Hodge structures on the degree $1$ cohomology of the fibers
of $p:\mathcal{L}\to B$ is maximal, i.e., the period map

$\begin{array}[]{cccc}\mathcal{P}:&B&\to&Gr(h^{1,0}(L),H^{1}(L,\mathbb{C}))\\\
&b&\mapsto&H^{1,0}(L_{b})\subset H^{1}(L_{b},\mathbb{C})\cong
H^{1}(L,\mathbb{C}),\end{array}$ (2.4)

where $L$ is a general fiber of $p:\mathcal{L}\to B$, is generically a local
immersion. Then the Abel-Jacobi map (2.3) is trivial.

This response to Problem 2.1.13 is conditional. However, it can be shown that
the condition “maximal variation of Hodge structures” cannot be dropped. In
fact, we construct in Section 2.5 Lagrangian families satisfying $\clubsuit$
for which the Abel-Jacobi map is shown to be nontrivial using Proposition
2.1.14. The variation of weight $1$ Hodge structures of the constructed
Lagrangian families is not maximal.

In Section 2.4, we shall explore under which conditions the variation of
weight $1$ Hodge structures is maximal. Let $H^{2}(X,\mathbb{Q})_{tr}$ be the
orthogonal complement of $NS(X)_{\mathbb{Q}}$ in $H^{2}(X,\mathbb{Q})$ with
respect to the Beauville-Bogomolov-Fujiki form $q$ of $X$ (see [Bea83]) and
let $b_{2}(X)_{tr}$ be the dimension of $H^{2}(X,\mathbb{Q})_{tr}$ .We prove
the following result (see also Proposition 2.4.2):

######  Proposition 2.1.16.

Consider a Lagrangian family on a hyper-Kähler manifold $X$ of dimension $2n$
given by a diagram as in (2.1), satisfying condition $\clubsuit$. Assume that
the Mumford-Tate group of the Hodge structure $H^{2}(X,\mathbb{Q})$ is
maximal, i.e. it is the special orthogonal group of
$(H^{2}(X,\mathbb{Q})_{tr},q)$, and assume that $b_{2}(X)_{tr}\geq 5$. If
$h^{1,0}(L_{b})$ is smaller than
$2^{\lfloor\frac{b_{2}(X)_{tr}-3}{2}\rfloor}$, then the variation of weight
$1$ Hodge structures of $p$ is maximal.

######  Corollary 2.1.17.

Under the same assumptions as in Proposition 2.1.16, the Abel-Jacobi map (2.3)
is trivial.

Let $p:\mathcal{L}\to B$, $q:\mathcal{L}\to X$ be a Lagrangian family. Up to
shrinking $B$, we may assume that the map $p:\mathcal{L}\to B$ is smooth. Let

$\pi:\mathcal{A}:=Alb(\mathcal{L}/B)\to B$

be the relative Albanese variety of $p:\mathcal{L}\to B$. In the proof of
Theorem 2.1.15 and Proposition 2.1.16, we use a similar construction to those
in [LSV17, Voi22I] to get a holomorphic $2$-form $\sigma_{\mathcal{A}}$ on
$\mathcal{A}$. If we assume the condition $\clubsuit$, $\pi:\mathcal{A}\to B$
is a Lagrangian fibration with respect to $\sigma_{\mathcal{A}}$(see Section
2.3). It is interesting to notice that, by this construction, under condition
$\clubsuit$, we can translate the problem concerning Lagrangian families to a
problem concerning Lagrangian fibrations. However, the total space of the
Lagrangian fibration is no longer a hyper-Kähler manifold, but a completely
integrable system over an open subset of the base, as studied for instance in
[DM98, Chapter 7].

###  2.2 A criterion

In this section, we establish a criterion for the vanishing of the Abel-Jacobi
map (2.3). Let $X$ be a hyper-Kähler manifold of dimension $2n$. With the
notation $j_{b}:L_{b}\to X$ as in the introduction, we prove

######  Proposition 2.2.1.

Consider a Lagrangian family of hyper-Kähler manifold $X$ of dimension $2n$
given by a diagram as in (2.1).
(a) If for general $b\in B$, the restriction map

$j_{b}^{*}:H^{2n-1}(X,\mathbb{Q})\to H^{2n-1}(L_{b},\mathbb{Q})$

is zero, then the Abel-Jacobi map (2.3) is trivial.
(b) If condition $\clubsuit$ holds (see Proposition 2.1.14), then the converse
of (a) holds.

######  Remark 2.2.2.

The cohomology group $H^{2n-1}(L_{b},\mathbb{Q})$ has a Hodge structure of
weight $2n-1$ and level $1$. By Hodge symmetry and using the fact
$j_{b}^{*}:H^{2n-1}(X,\mathbb{Q})\to H^{2n-1}(L_{b},\mathbb{Q})$ is a morphism
of Hodge structures, we conclude that $j_{b}^{*}:H^{2n-1}(X,\mathbb{Q})\to
H^{2n-1}(L_{b},\mathbb{Q})$ is zero, if and only if $j_{b}^{*}:H^{n-1,n}(X)\to
H^{n-1,n}(L_{b})$ is zero.

###### Proof.

Let $d\Psi_{\mathcal{L},b}^{AJ}:T_{B,b}\to H^{n}(X,\Omega_{X}^{n+1})$ denote
the differential of the Abel-Jacobi map $\Psi_{\mathcal{L}}^{AJ}$ at point
$b\in B$ (Theorem 2.1.6). Let $j_{b*}:H^{0}(L_{b},\Omega_{L_{b}})\to
H^{n}(X,\Omega_{X}^{n+1})$ be the Gysin map, which is the Serre dual to the
following composition

$j_{b}^{*}:H^{n}(X,\Omega_{X}^{n-1})\to
H^{n}(L_{b},\Omega_{\mathcal{L}|L_{b}}^{n-1})\to
H^{n}(L_{b},\Omega_{L_{b}}^{n-1}).$ (2.5)

By the above remark, the proposition follows from the following lemma and the
fact that

$\cup\sigma_{X}:H^{n}(X,\Omega_{X}^{n-1})\to H^{n}(X,\Omega_{X}^{n+1})$

is an isomorphism since $\wedge\sigma_{X}:\Omega_{X}^{n-1}\to\Omega_{X}^{n+1}$
is a vector bundle isomorphism. ∎

######  Lemma 2.2.3.

The following diagram is commutative:

${T_{B,b}}$${H^{n}(X,\Omega_{X}^{n-1})}$${H^{0}(L_{b},\Omega_{L_{b}})}$${H^{n}(X,\Omega_{X}^{n+1}).}$$\scriptstyle{d\Psi_{\mathcal{L},b}^{AJ}}$$\scriptstyle{\lrcorner
q^{*}\sigma_{X}}$$\scriptstyle{\cup\sigma_{X}}$$\scriptstyle{j_{b*}}$ (2.6)

###### Proof.

We are going to show that the Serre dual of the diagram (2.6) is commutative.

Let $L^{\bullet}\Omega^{i}_{\mathcal{L}|L_{b}}$ be the Leray filtration
[Voi03, Chapter 16] induced on the vector bundle
$\Omega_{\mathcal{L}|L_{b}}^{i}$ by the exact sequence

$0\to p^{*}\Omega_{B,b}\to\Omega_{\mathcal{L}|L_{b}}\to\Omega_{L_{b}}\to 0,$

and defined by
$L^{j}\Omega^{i}_{\mathcal{L}|L_{b}}=p^{*}\Omega_{B,b}^{j}\wedge\Omega_{\mathcal{L}|L_{b}}^{i-j}$.
Since $L_{b}$ is supposed to be Lagrangian, $q^{*}\sigma_{X}\in
H^{0}(L_{b},L^{1}\Omega_{\mathcal{L}|L_{b}}^{2})$ and thus the cup product

$\cup
q^{*}\sigma_{X}:\Omega_{\mathcal{L}|L_{b}}^{\bullet}\to\Omega_{\mathcal{L}|L_{b}}^{\bullet+2}$

sends $L^{k}\Omega_{\mathcal{L}|L_{b}}^{\bullet}$ to
$L^{k+1}\Omega_{\mathcal{L}|L_{b}}^{\bullet+2}$. Denoting
$\overline{q^{*}\sigma_{X}}$ the image of $q^{*}\sigma_{X}$ in
$H^{0}(L_{b},Gr_{L}^{1}\Omega_{\mathcal{L}|L_{b}}^{2})\cong
H^{0}(L_{b},\Omega_{L_{b}})\otimes\Omega_{B,b}$, this implies the existence of
the following commutative diagram

${L^{1}\Omega_{\mathcal{L}|L_{b}}^{n+1}=\Omega_{\mathcal{L}|L_{b}}^{n+1}}$${K_{L_{b}}\otimes
p^{*}\Omega_{B,b}=Gr_{L}^{1}\Omega_{\mathcal{L}|L_{b}}^{n+1}}$${L^{0}\Omega_{\mathcal{L}|L_{b}}^{n-1}=\Omega_{\mathcal{L}|L_{b}}^{n-1}}$${\Omega_{L_{b}}^{n-1}=Gr_{L}^{0}\Omega_{\mathcal{L}|L_{b}}^{n-1},}$$\scriptstyle{\cup
q^{*}\sigma_{X}}$$\scriptstyle{\cup\overline{q^{*}\sigma_{X}}}$ (2.7)

where $K_{L_{b}}$ is the canonical bundle of ${L_{b}}$. Taking the $n$-th
cohomology of (2.7) and combine it with
$q^{*}:H^{n}(X,\Omega_{X}^{\bullet})\to
H^{n}(L_{b},\Omega_{\mathcal{L}|L_{b}}^{\bullet})$, we get the following
commutative ladder

${H^{n}(X,\Omega_{X}^{n+1})}$${H^{n}(L_{b},\Omega_{\mathcal{L}|L_{b}}^{n+1})}$${H^{n}(L_{b},K_{L_{b}}\otimes
p^{*}\Omega_{B,b})\cong\Omega_{B,b}}$${H^{n}(X,\Omega_{X}^{n-1})}$${H^{n}(L_{b},\Omega_{\mathcal{L}|L_{b}}^{n-1})}$${H^{n}(L_{b},\Omega_{L_{b}}^{n-1}).}$$\scriptstyle{q^{*}}$$\scriptstyle{q^{*}}$$\scriptstyle{\cup\sigma_{X}}$$\scriptstyle{\cup
q^{*}\sigma_{X}}$$\scriptstyle{\cup\overline{q^{*}\sigma_{X}}}$ (2.8)

######  Lemma 2.2.4.

The composite in the first row of the diagram (2.8) coincides with the dual of
$d\Psi_{\mathcal{L},b}$.

###### Proof.

Let $\bar{p}:\bar{\mathcal{L}}\to\bar{B}$, $\bar{q}:\bar{\mathcal{L}}\to X$ be
a relative completion of $p:\mathcal{L}\to B$ with respect to the morphism
$q:\mathcal{L}\to X$. More precisely, $\bar{B}$ is a smooth projective
variety, $\bar{p}$ is a flat morphism extending $p$, and $\bar{q}$ is a
morphism extending $q$. The extended family has a Abel–Jacobi map
$\Psi^{AJ}_{\bar{\mathcal{L}}}:\bar{B}\to J^{2n-1}(X)$ that induces a morphism
of complex tori $\psi:\mathrm{Alb}(\bar{B})\to J^{2n-1}(X)$ whose differential
is given by the morphism of Hodge structures ([Voi03, Théorème 12.17])

$\bar{q}_{*}\bar{p}^{*}:H^{2d-1}(\bar{B},\mathbb{Z})\to
H^{2n-1}(X,\mathbb{Z}),$

where $d=\dim B$. It is well-known that the differential of the Albanese map
$\mathrm{alb}:\bar{B}\to\mathrm{Alb}(\bar{B})$ at a point $b\in B$ is given by
the dual of the evaluation map
$ev_{b}:H^{0}(\bar{B},\Omega_{\bar{B}})\to\Omega_{\bar{B},b}$. Hence, the dual
of $d\Psi_{\mathcal{L},b}^{AJ}$ is given by the correspondance
$\bar{p}_{*}\bar{q}^{*}:H^{n}(X,\Omega_{X}^{n+1})\to
H^{0}(\bar{B},\Omega_{\bar{B}})$ composed with the evaluation map
$ev_{b}:H^{0}(\bar{B},\Omega_{\bar{B}})\to\Omega_{\bar{B},b}\cong\Omega_{B,b}$.
The domain $\bar{B}$ can be restricted to $B$ before the evaluation map
$ev_{b}:H^{0}(\bar{B},\Omega_{\bar{B}})\to\Omega_{B,b}$. Therefore, the dual
of $d\Psi_{\mathcal{L},b}^{AJ}$ is given by the correspondance
$p_{*}q^{*}:H^{n}(X,\Omega_{X}^{n+1})\to H^{0}(B,\Omega_{B})$ composed with
the evaluation map $ev_{b}:H^{0}(B,\Omega_{B})\to\Omega_{B,b}$. The Gysin map
$p_{*}:H^{n}(\mathcal{L},\Omega_{\mathcal{L}}^{n+1})\to H^{0}(B,\Omega_{B})$
is given by the Leray filtration. Taken together, the dual of
$d\Psi_{\mathcal{L},b}^{AJ}$ is given by

$H^{n}(X,\Omega_{X}^{n+1})\stackrel{{\scriptstyle
q^{*}}}{{\to}}H^{n}(\mathcal{L},\Omega_{\mathcal{L}}^{n+1})\to
H^{n}(\mathcal{L},K_{\mathcal{L}/B}\otimes p^{*}\Omega_{B})\to
H^{0}(B,R^{n}p_{*}\Omega_{\mathcal{L}/B}\otimes\Omega_{B})\stackrel{{\scriptstyle
ev_{b}}}{{\to}}\Omega_{B,b}.$

Since the restriction to the fiber $L_{b}$ and taking the Leray filtration are
commutative processes, the above composite of maps is equal to the one that
takes the restriction to the fiber $L_{b}$ first and then takes the Leray
filtration. The latter one is exactly the first row of the diagram (2.8). ∎

As in (2.5), the composite in the second row is $j_{b}^{*}$. Taking into
account of Lemma 2.2.4, the diagram 2.8 is indeed the Serre dual of the
diagram (2.6). This concludes the proof of Lemma 2.6. ∎

###  2.3 Lagrangian fibrations

In this section, we associate to a Lagrangian family satisfying condition
$\clubsuit$ on $X$ a fibration on another variety, which turns out to be
holomorphic symplectic in such a way that the fibration is lagrangian. This is
constructed with the help of a construction appeared in [LSV17, Voi22I]. We
use this Lagrangian fibration to prove Theorem 2.1.15.

#### General Constructions

Let (2.1) be a Lagrangian family of a hyper-Kähler manifold $X$ of dimension
$2n$. We fix a relative polarization of $\mathcal{L}\to B$ given by a
hyperplane section of $X$. Let

$\pi:\mathcal{A}:=Alb(\mathcal{L}/B)\to B$

be the relative Albanese variety of $p:\mathcal{L}\to B$.

######  Lemma 2.3.1.

There exist an open dense subset $B_{0}\subset B$ and a finite covering
$B_{0}^{\prime}\to B_{0}$ such that, denoting
$p_{0}^{\prime}:\mathcal{L}_{0}^{\prime}\to B_{0}^{\prime}$ the base change of
$p$ under $B_{0}^{\prime}\to B_{0}\hookrightarrow B$ and
$\pi_{0}^{\prime}:\mathcal{A}_{0}^{\prime}\to B_{0}^{\prime}$ the relative
Albanese variety of $p_{0}^{\prime}$, there is a cycle $Z_{0}\in
CH^{n}(\mathcal{A}_{0}^{\prime}\times_{B_{0}^{\prime}}\mathcal{L}_{0}^{\prime})$
such that

$[Z_{0}]^{*}:p_{0*}^{\prime}\Omega_{\mathcal{L}_{0}^{\prime}/B_{0}^{\prime}}\to\pi_{0*}^{\prime}\Omega_{\mathcal{A}_{0}^{\prime}/B_{0}^{\prime}}$

is an isomorphism.

###### Proof.

Let $B_{0}\subset B$ be the subset of regular points of $p:\mathcal{L}\to B$.
For $b\in B_{0}$, let $j:C_{b}\hookrightarrow L_{b}$ be a complete
intersection curve and $J_{C_{b}}$ the Jacobian variety of $C_{b}$. By
Lefschetz theorem on hyperplane sections, $j_{*}:J_{C_{b}}\to
A_{b}:=Alb(L_{b})$ is surjective. By the semi-simplicity of polarized Hodge
structures, there exists a $\mathbb{Q}$-section $s:A_{b}\to J_{C_{b}}$ of
$j_{*}$, i.e., there exists $N>0$ such that $j_{*}\circ s=N\cdot id_{A_{b}}$.
On $J_{C_{b}}\times C_{b}$, we have the Poincaré divisor $d_{b}\in
CH^{1}(J_{C_{b}}\times C_{b})$ such that
$[d_{b}]^{*}:H^{1}(C_{b},\mathbb{Q})\to H^{1}(J_{C_{b}},\mathbb{Q})$ is an
isomorphism of Hodge structures, exhibiting the inverse of the pull-back of
the Jacobi map $jac_{b}:C_{b}\to J_{C_{b}}$. Let us consider

${A_{b}\times C_{b}}$${J_{C_{b}}\times C_{b}}$${A_{b}\times
L_{b}}$$\scriptstyle{(s,id_{C_{b}})}$$\scriptstyle{(id_{A_{b}},j)}$

and define $Z_{b}:=(id_{A_{b}},j)_{*}(s,id_{C_{b}})^{*}(d_{b})\in
CH^{n}(A_{b}\times L_{b})$. Then $[Z_{b}]^{*}:H^{1}(L_{b},\mathbb{Q})\to
H^{1}(A_{b},\mathbb{Q})$ is given by the composition

$H^{1}(L_{b},\mathbb{Q})\stackrel{{\scriptstyle
j^{*}}}{{\to}}H^{1}(C_{b},\mathbb{Q})\stackrel{{\scriptstyle
d_{b}^{*}}}{{\to}}H^{1}(J_{C_{b}},\mathbb{Q})\stackrel{{\scriptstyle
s^{*}}}{{\to}}H^{1}(A_{b},\mathbb{Q}),$ (2.9)

which is an isomorphism by the definition of $s$. In fact, let
$(j_{*})^{*}:H^{1}(A_{b},\mathbb{Q})\to H^{1}(J_{C_{b}},\mathbb{Q})$ be the
pull-back map of $j_{*}:J_{C_{b}}\to A_{b}=Alb(L_{b})$. Precomposing the left-
hand-side of $(j_{*})^{*}:H^{1}(A_{b},\mathbb{Q})\to
H^{1}(J_{C_{b}},\mathbb{Q})$ with the natural identification
$H^{1}(Alb(L_{b}),\mathbb{Q})\cong H^{1}(L_{b},\mathbb{Q})$, and post-
composing the right hand side with the pull-back of the Jacobi map
$jac_{b}^{*}:H^{1}(J_{C_{b}},\mathbb{Q})\to H^{1}(C_{b},\mathbb{Q})$, we would
get the pull-back map $j^{*}:H^{1}(L_{b},\mathbb{Q})\to
H^{1}(C_{b},\mathbb{Q})$. Since $jac_{b}^{*}:H^{1}(J_{C_{b}},\mathbb{Q})\to
H^{1}(C_{b},\mathbb{Q})$ is the inverse of
$d_{b}^{*}:H^{1}(C_{b},\mathbb{Q})\to H^{1}(J_{C_{b}},\mathbb{Q})$, the
composite $d_{b}^{*}\circ j^{*}:H^{1}(L_{b},\mathbb{Q})\to
H^{1}(J_{C_{b}},\mathbb{Q})$ of the first two maps of the composition (2.9) is
exactly the pull-back map of $j_{*}:J_{C_{b}}\to A_{b}=Alb(L_{b})$ at the
level of cohomology, after the natural identification
$H^{1}(Alb(L_{b}),\mathbb{Q})\cong H^{1}(L_{b},\mathbb{Q})$. Since $j_{*}\circ
s=N\cdot id_{A_{b}}$, we get the desired isomorphism.

The cycles $Z_{b}$ are defined fiberwise, but standard arguments [Voi14,
Chapter 3] show that they can be constructed in family over a smooth
generically finite cover $B_{0}^{\prime}\to B_{0}$. Let us spell out the
standard arguments. By the theory of Hilbert schemes, there are countably many
connected projective $B_{0}$-schemes $H_{1},\ldots,H_{i},\ldots$ together with
the universal families of cycles
$\mathcal{Z}_{1},\ldots,\mathcal{Z}_{i},\ldots$ such that for each
$i\in\mathbb{N}$, every fiber of $\mathcal{Z}_{i}\to H_{i}$ is a cycle
$Z_{b}\in\mathcal{Z}^{1}(A_{b}\times L_{b})$ such that
$[Z_{b}]^{*}:H^{1}(L_{b},\mathbb{Q})\to H^{1}(A_{b},\mathbb{Q})$ is an
isomorphism. By the construction of the previous paragraph, the structure map
$\bigcup_{i\in\mathbb{N}}H_{i}\to B_{0}$ is surjective, thus there is
$i\in\mathbb{N}$, such that $H_{i}\to B_{0}$ is surjective. Since $H_{i}\to
B_{0}$ is projective, we may take a multisection of the map $B_{0}^{\prime}\to
B_{0}$, and the universal family $\mathcal{Z}_{i}$ restricted to $B_{0}$ gives
the desired $1$-cycle. ∎

For the sake of simplicity, we shall note $B_{0}$, $\mathcal{L}_{0}$ and
$\mathcal{A}_{0}$ instead of $B_{0}^{\prime}$, $\mathcal{L}_{0}^{\prime}$ and
$\mathcal{A}_{0}^{\prime}$. We define a holomorphic $2$-form
$\sigma_{\mathcal{A}_{0}}$ on $\mathcal{A}_{0}$ by setting

$\sigma_{\mathcal{A}_{0}}:=[Z_{0}]^{*}q_{0}^{*}\sigma_{X},$ (2.10)

where $q_{0}:\mathcal{L}_{0}\to X$ is the natural map.

######  Proposition 2.3.2.

(a) The $2$-form $\sigma_{\mathcal{A}_{0}}$ is closed.
(b) $\sigma_{\mathcal{A}_{0}}$ vanishes on fibers of
$\pi_{0}:\mathcal{A}_{0}\to B_{0}$.
(c) The composite morphism $\kappa:T_{B_{0}}\stackrel{{\scriptstyle\lrcorner
q_{0}^{*}\sigma_{X}}}{{\longrightarrow}}p_{0*}\Omega_{\mathcal{L}_{0}/B_{0}}\stackrel{{\scriptstyle[Z_{0}]^{*}}}{{\to}}\pi_{0*}\Omega_{\mathcal{A}_{0}/B_{0}}$
is given by the contraction $\lrcorner\sigma_{\mathcal{A}_{0}}$.

###### Proof.

(a) Let $Z_{q}:=(id,q_{0})_{*}Z_{0}\in CH(\mathcal{A}_{0}\times X)$. Then by
the projection formula, $\sigma_{\mathcal{A}_{0}}=[Z_{q}]^{*}\sigma_{X}$. Let
$\mathcal{A}^{\prime}$ be a projective completion of $\mathcal{A}_{0}$. Then
$Z_{q}$ extends to a cycle $\bar{Z}_{q}$ of $\mathcal{A}^{\prime}\times X$.
$\sigma_{\mathcal{A}_{0}}$ extends to a $2$-form
$\sigma_{\mathcal{A}^{\prime}}:=[\bar{Z}_{q}]^{*}\sigma_{X}$ which is
automatically closed since $\mathcal{A}^{\prime}$ is projective. Thus,
$\sigma_{\mathcal{A}_{0}}=\sigma_{\mathcal{A}^{\prime}|\mathcal{A}_{0}}$ is
also closed.

(b) Since $Z_{0}$ is a cycle in
$\mathcal{A}_{0}\times_{B_{0}}\mathcal{L}_{0}\subset\mathcal{A}_{0}\times\mathcal{L}_{0}$,
the morphism $[Z_{0}]^{*}:H^{*}(\mathcal{L}_{0})\to H^{*}(\mathcal{A}_{0})$
preserves the Leray filtrations on both sides. Therefore,
$\sigma_{\mathcal{A}_{0}}\in
H^{0}(\mathcal{A}_{0},\pi_{0}^{*}\Omega_{B_{0}}\wedge\Omega_{\mathcal{A}_{0}})\subset
H^{0}(\mathcal{A}_{0},\Omega_{\mathcal{A}_{0}}^{2})$ since
$q_{0}^{*}\sigma_{X}\in
H^{0}(\mathcal{L}_{0},p_{0}^{*}\Omega_{B_{0}}\wedge\Omega_{\mathcal{L}_{0}})$
by the definition of Lagrangian families. Therefore,
$\sigma_{\mathcal{A}_{0}}$ vanishes on the fibers of
$\pi_{0}:\mathcal{A}_{0}\to B_{0}$.

(c) By Lemma 2.3.1, $[Z_{0}]^{*}$ induces an isomorphism
$H^{0}(B_{0},\Omega_{B_{0}}\otimes p_{0*}\Omega_{\mathcal{L}_{0}/B_{0}})\to
H^{0}(B_{0},\Omega_{B_{0}}\otimes\pi_{0*}\Omega_{\mathcal{A}_{0}/B_{0}})$
which sends $\lrcorner q_{0}^{*}\sigma_{X}$ to
$\lrcorner\sigma_{\mathcal{A}_{0}}$. ∎

By (b) and (c) of the above proposition, we get the following diagram that is
commutative up to a sign:

${0}$${T_{\mathcal{A}_{0}/B_{0}}}$${T_{\mathcal{A}_{0}}}$${\pi_{0}^{*}T_{B_{0}}}$${0}$${0}$${\pi_{0}^{*}\Omega_{B_{0}}}$${\Omega_{\mathcal{A}_{0}}}$${\Omega_{\mathcal{A}_{0}/B_{0}}}$${0.}$$\scriptstyle{(\pi_{0}^{*}\kappa)^{*}}$$\scriptstyle{\lrcorner\sigma_{\mathcal{A}_{0}}}$$\scriptstyle{\pi_{0}^{*}\kappa}$
(2.11)

Here, the commutativity of the second square is dual to Proposition 2.3.2 (c).
Since the dual of
$\lrcorner\sigma_{\mathcal{A}_{0}}:T_{\mathcal{A}_{0}}\to\Omega_{\mathcal{A}_{0}}$
is given by
$-\lrcorner\sigma_{\mathcal{A}_{0}}:T_{\mathcal{A}_{0}}\to\Omega_{\mathcal{A}_{0}}$,
the first square is anti-commutative.

######  Lemma 2.3.3.

If condition $\clubsuit$ (see Proposition 2.1.14) holds for all $b\in B_{0}$,
then $\sigma_{\mathcal{A}_{0}}$ is nowhere degenerate on $\mathcal{A}_{0}$.

###### Proof.

If condition $\clubsuit$ holds, then
$\pi_{0}^{*}\kappa:\pi_{0}^{*}T_{B_{0}}\to\Omega_{\mathcal{A}_{0}/B_{0}}$ is
an isomorphism. By the commutativity of (2.11) and the five lemma,
$\lrcorner\sigma_{\mathcal{A}_{0}}:T_{\mathcal{A}_{0}}\to\Omega_{\mathcal{A}_{0}}$
is an isomorphism, which means that $\sigma_{\mathcal{A}_{0}}$ is nowhere
degenerate. ∎

#### Symmetry

Let (2.1) be a Lagrangian family of a hyper-Kähler manifold $X$ of dimension
$2n$. We fix a relative polarization of $\mathcal{L}\to B$ given by a
hyperplane section of $X$. Let $b\in B$ be a general point. The infinitesimal
variation of Hodge structures on degree $1$ cohomology of the fibers of
$p:\mathcal{L}\to B$ at $b$ is given by (see [Voi03, Lemme 10.19])

$\bar{\nabla}:T_{B,b}\to
Hom(H^{0}(L_{b},\Omega_{L_{b}}),H^{1}(L_{b},\mathcal{O}_{L_{b}})).$

Precomposed with the map $\lrcorner q^{*}\sigma_{X}:T_{B,b}\to
H^{0}(L_{b},\Omega_{L_{b}})$, the map $\bar{\nabla}$ induces a bilinear map

$\begin{array}[]{cccc}S:&T_{B,b}\times
T_{B,b}&\to&H^{1}(L_{b},\mathcal{O}_{L_{b}})\\\
&(u,v)&\mapsto&\bar{\nabla}_{u}(v\lrcorner q^{*}\sigma_{X}).\end{array}$
(2.12)

######  Proposition 2.3.4.

The bilinear map $S$ is symmetric in the sense that $S(u,v)=S(v,u)$ for any
$u,v\in T_{b,B}$.

###### Proof.

By Griffiths’ transversality [Voi03, Chapter 17],
$S(u,v)=\rho(u)\lrcorner(v\lrcorner q^{*}\sigma_{X})$, where $\rho:T_{B,b}\to
H^{1}(L_{b},T_{L_{b}})$ is the Kodaira-Spencer map. Therefore, we need to show
that the following diagram is commutative

${T_{B,b}\otimes T_{B,b}}$${H^{1}(L_{b},T_{L_{b}})\otimes
T_{B,b}}$${H^{0}(L_{b},\Omega_{L_{b}})\otimes
T_{B,b}}$${H^{1}(L_{b},\mathcal{O}_{L_{b}}),}$$\scriptstyle{\rho\otimes
id}$$\scriptstyle{\lrcorner q^{*}\sigma_{X}\otimes
id}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$ (2.13)

where $\alpha(\omega,v)=\rho(v)\lrcorner\omega$ and
$\beta(u,\chi)=u\lrcorner(\chi\lrcorner q^{*}\sigma_{X})$.

To see the commutativity of (2.13), restrict the commutative ladder (2.11) to
$L_{b}$ and apply the cohomology on $L_{b}$, then the commutativity of (2.11)
implies the commutativity of (2.13). Indeed, (2.13) is the connecting
homomorphism of the cohomology of (2.11) tensored by $T_{B,b}$. ∎

######  Remark 2.3.5.

When condition $\clubsuit$ is satisfied, the symmetry of $S$ comes from the
completely integrable system structure on
$(\mathcal{A}_{0},\sigma_{\mathcal{A}_{0}})$. What we proved is in fact the
symmetry of

$\begin{array}[]{cccc}S^{\prime}:&T_{B,b}\times
T_{B,b}&\to&H^{1}(A_{b},\mathcal{O}_{A_{b}})\\\
&(u,v)&\mapsto&\bar{\nabla}_{u}(v\lrcorner\sigma_{\mathcal{A}_{0}}).\end{array}$
(2.14)

Fixing a relative polarisation on $\mathcal{A}_{0}\to B_{0}$, we have natural
isomorphisms (always under $\clubsuit$):
$H^{1}(A_{b},\mathcal{O}_{A_{b}})\cong H^{0}(A_{b},\Omega_{A_{b}})^{*}\cong
T_{B,b}^{*}$ and we can thus view $S^{\prime}$ as an element in
$T_{B,b}^{*}\otimes T_{B,b}^{*}\otimes T_{B,b}^{*}$. If this relative
polarisation is principal, Donagi and Markman proved in [DM98, Lemma 7.2] that
$S^{\prime}$ lies in $Sym^{3}T_{B,b}^{*}$. This result is called “weak cubic
condition” in [DM98, Lemma 7.2]. See also [Voi17, Theorem 4.4]

Now we are ready to prove the main theorem.

###### Proof of Theorem 2.1.15.

Under the assumptions of Theorem 2.1.15, assume by contradiction that the
Abel–Jacobi map (2.3) is not constant. In what follows, we fix a relative
polarization on $p_{0}:\mathcal{L}_{0}\to B_{0}$ induced from a hyperplane
section of $X$, so that $R^{2n-1}p_{0*}\mathbb{Q}\cong R^{1}p_{0*}\mathbb{Q}$.
By Proposition 2.1.14, the morphism

$j^{*}:H^{2n-1}(X,\mathbb{Q})\to R^{2n-1}p_{0*}\mathbb{Q}\cong
R^{1}p_{0*}\mathbb{Q}$

of variations of Hodge structures on an open subset $B_{0}\subset B$
containing $b$ is not zero. Hence, there is a non-zero locally constant sub-
variation of Hodge structures $I:=\mathrm{Im}j^{*}\subset
R^{1}p_{0*}\mathbb{Q}$. Since $I$ is locally constant, for any $\omega\in
I^{1,0}_{b}$ and $u\in T_{B,b}$, $\nabla_{u}(\omega)=0$. Recall that
$\clubsuit$ means that $\lrcorner q^{*}\sigma_{X}:T_{B,b}\to
H^{0}(L_{b},\Omega_{L_{b}})$ is bijective. Let $F:=(\lrcorner
q^{*}\sigma_{X})^{-1}(I^{1,0})\subset T_{B,b}$. Then by the symmetry of $S$
given by Proposition 2.3.4, $F$ lies in the kernel of $\bar{\nabla}$, which
contradicts our assuption that the variation of Hodge structures is maximal.

∎

###  2.4 Maximal variations

In this section, we study under what conditions could the variation of Hodge
structures of a Lagrangian family be maximal. Consider a Lagrangian family of
a hyper-Kähler manifold $X$ of dimension $2n$ satisfying the condition
$\clubsuit$ given by the diagram as in (2.1). Let $U\subset B$ be a simply
connected open subset of $B_{0}\subset B$ and let

$\begin{array}[]{cccc}\mathcal{P}:&U&\to&Gr(h^{1,0}(L),H^{2}(L,\mathbb{C}))\\\
&b&\mapsto&H^{1,0}(L_{b})\subset H^{1}(L_{b},\mathbb{C})\cong
H^{1}(L,\mathbb{C}),\end{array}$ (2.15)

be the local period map of the Lagrangian family.

In what follows, we are going to use the universal property of the Kuga-Satake
construction proved in [vGV16, Proposition 6].

######  Theorem 2.4.1 ([vGV16]).

Let $(H^{2},q)$ be a polarized Hodge structure of hyper-Kähler type of
dimension $\geq 5$. Assume that the Mumford–Tate group of the Hodge structure
on $H^{2}$ is maximal, namely the special orthogonal group of $(H^{2},q)$. Let
$H$ be a simple effective weight-$1$ Hodge structure, such that there exists
an injective morphism of Hodge structures of bidegree $(-1,-1)$

$H^{2}\hookrightarrow Hom(H,A)$

for some weight-$1$ Hodge structure $A$. Then $H$ is a direct summand of the
Kuga–Satake Hodge structure $H^{1}_{KS}(H^{2},q)$. In particular, $\dim H\geq
2^{\lfloor\frac{\dim H^{2}-1}{2}\rfloor}$.

With the same notations as in the introduction, we prove

######  Proposition 2.4.2.

Assume that the Mumford-Tate group of the Hodge structure
$H^{2}(X,\mathbb{Q})$ is maximal, i.e. it is the special orthogonal group of
$(H^{2}(X,\mathbb{Q})_{tr},q)$ and assume $b_{2}(X)_{tr}\geq 5$. If the
dimension of $H^{0,1}(L_{b})$ is smaller than
$2^{\lfloor\frac{b_{2}(X)_{tr}-3}{2}\rfloor}$ for a general fiber $L_{b}$ of
$p:\mathcal{L}\to B$, then the variation of weight $1$ Hodge structure of $p$
is maximal.

###### Proof.

We use the same argument as in [vGV16] where similar results were proved for
Lagrangian fibrations. Assuming that the period map (2.15) is not generically
an immersion, we are going to prove that $\dim H^{0,1}(L_{b})\geq
2^{\lfloor\frac{b_{2}(X)_{tr}-3}{2}\rfloor}$. By assumption, the nonempty
general fibers of $\mathcal{P}$ are of dimension $\geq 1$. Let $b\in U$ be a
general point and let $B_{b}$ the fiber of $\mathcal{P}$ passing through $b$.
Let $U_{b}=B_{b}\cap U$. Then the fibers of
$\pi|_{U_{b}}:\mathcal{A}_{U_{b}}\to U_{b}$ are isomorphic with each other.
Thus, up to a base change by a finite covering of $U_{b}$, we may assume
$\pi|_{U_{b}}:\mathcal{A}_{U_{b}}\to U_{b}$ is trivial, i.e.,
$\mathcal{A}_{U_{b}}=U_{b}\times A_{b}$. Let
$\pi_{F_{b}}:\mathcal{A}_{F_{b}}\to F_{b}$ be a smooth completion of
$\pi|_{U_{b}}$, then $\mathcal{A}_{F_{b}}$ is birational to $F_{b}\times
A_{b}$, which gives a morphism $H^{2}(\mathcal{A}_{F_{b}})\to
H^{2}(F_{b}\times A_{b})$. Recall by Lemma 2.3.1, we get a morphism
$[Z]^{*}:H^{2}(X)\to H^{2}(\mathcal{A})$ that sends $\sigma_{X}$ to a
holomorphic $2$-form which is non-degenerate on $\mathcal{A}_{U}$. Finally,
the rational map $\mathcal{A}_{F_{b}}\dashrightarrow\mathcal{A}$ induces
$H^{2}(\mathcal{A})\to H^{2}(\mathcal{A}_{F_{b}})$. Compositing all these
maps, we get a morphism

$\alpha:H^{2}(X)_{tr}\hookrightarrow H^{2}(X)\to H^{2}(\mathcal{A})\to
H^{2}(\mathcal{A}_{F_{b}})\to H^{2}(F_{b}\times A_{b})\to H^{1}(F_{b})\otimes
H^{1}(A_{b}),$ (2.16)

where the last map is given by the projection in the Künneth decomposition.

######  Lemma 2.4.3.

$\alpha:H^{2}(X)_{tr}\to H^{1}(F_{b})\otimes H^{1}(A_{b})$ is injective.

###### Proof.

Since $h^{2,0}(X)=1$ and $H^{2,0}(X)$ is orthogonal to $NS(X)$ with respect to
the Beauville-Bogomolov-Fujiki form, $H^{2}(X)_{tr}$ is a simple Hodge
structure. Therefore, to show the injectivity of $\alpha$ it suffices to show
that $\alpha$ is not zero. We claim that $\alpha(\sigma_{X})\neq 0$. Indeed,
Since $A_{b}$ is Lagrangian with respect to $\sigma_{\mathcal{A}}$
(Proposition 2.3.2 (b)), in the Künneth’s decomposition of $H^{2}(A_{b}\times
F_{b})$, the image of $\sigma_{X}$ in $H^{0}(F_{b})\otimes H^{2}(A_{b})$ is
zero. If furthermore $\alpha(\sigma_{X})=0$ in $H^{1}(F_{b})\otimes
H^{1}(A_{b})$, then the image of $\sigma_{X}$ on $F_{b}\times A_{b}$ comes
from a $2$-form on $F_{b}$, which has rank $\leq\dim F_{b}$. Therefore, the
rank of $\sigma_{\mathcal{A}}$ has rank $\leq\dim F_{b}$ on
$\mathcal{A}_{U_{b}}$. On the other hand, the codimension of
$\mathcal{A}_{U_{b}}$ in $\mathcal{A}_{U}$ is $\dim B-\dim F_{b}$, and thus
the non-degeneration of $\sigma_{\mathcal{A}_{U}}$ implies that
$\sigma_{\mathcal{A}}$ has rank $\geq 2\dim F_{b}$ on $\mathcal{A}_{U_{b}}$.
This is a contradiction since we are assuming $\dim F_{b}\geq 1$. ∎

We are now in the position to use the universal property of the Kuga-Satake
construction (see Theorem 2.4.1 above). Since $\alpha:H^{2}(X)_{tr}\to
H^{1}(F_{b})\otimes H^{1}(A_{b})$ is nonzero, there is at least one simple
direct factor $A$ of $A_{b}$ such that $H^{2}(X)_{tr}\to H^{1}(F_{b})\otimes
H^{1}(A)$ is nonzero thus injective. Taking $H^{2}$ as $H^{2}(X)_{tr}$, we
conclude by Theorem 2.4.1 that

$\dim H^{0,1}(L_{b})=\dim A_{b}\geq\dim A\geq\frac{1}{2}\times
2^{\lfloor\frac{\dim
H^{2}(X)_{tr}-1}{2}\rfloor}=2^{\lfloor\frac{b_{2}(X)_{tr}-3}{2}\rfloor},$

as desired. ∎

###  2.5 Example of a Lagrangian family with nontrivial Abel-Jacobi map

Recall the construction of generalized Kummer varieties introduced in [Bea83].
Let $A$ be an abelian surface and $A^{[n+1]}$ the Hilbert scheme of length
$n+1$ subschemes of $A$. Let $\mathrm{alb}:A^{[n+1]}\to A$ be the composition
of the Hilbert-Chow morphism and the summation map

$A^{[n+1]}\to A^{(n+1)}\to A.$

Note that $\mathrm{alb}$ is an isotrivial fibration. The generalized Kummer
variety $K_{n}(A)$ is defined to be the fiber of $\mathrm{alb}$ over $0\in A$.
As is shown in [Bea83], $K_{n}(A)$ is a hyper-Kähler manifold of dimension
$2n$.

In this section, we are going to construct Lagrangian families of
$X:=K_{n}(A)$ for $n\geq 2$, satisfying condition $\clubsuit$ and whose Abel-
Jacobi map is _not_ trivial.

For any $x\in A$, one defines a subvariety $Z_{x}$ of $K_{n}(A)$ consisting of
Artinian subschemes of $A$ of length $n+1$ supported on $x$ and $-nx$, with
multiplicities $n$ and $1$, respectively. By [Bri77, Proposition VI.1.1],
$Z_{x}$ is a _rational_ variety of dimension $n-1$ if $x$ is _not_ an
$(n+1)$-torsion point. Let $Z=\bigcup_{x\in A}Z_{x}$ and let $\pi:Z\to A$ send
elements in $Z_{x}$ to $x$. For any curve $C\subset A$, define
$Z_{C}=\bigcup_{x\in C}Z_{x}$.

Now let $B$ be a connected open subset of the Hilbert scheme of deformations
of a smooth curve $C\subset A$ and $\mathcal{C}\to B$ the corresponding
family.

######  Lemma 2.5.1.

$\\{Z_{C}\\}_{C\in B}$ is a Lagrangian family of $K_{n}(A)$ satifying
condition $\clubsuit$.

###### Proof.

Since for general $C$, $Z_{C}$ is a fibration over a curve $C$ whose general
fibers are rational, any holomorphic $2$-form on $Z_{C}$ is $0$. Furthermore,
$\dim Z_{C}=n=\dim K_{n}(A)/2$. These imply that $\\{Z_{C}\\}_{C\in B}$ is a
Lagrangian family. We now show that this family satisfies condition
$\clubsuit$. Denoting $\mathcal{L}$ the total space of the family
$\\{Z_{C}\\}_{C\in B}$ and $L$ a general fiber, and using as before the
following notation

$\leavevmode\hbox to58.38pt{\vbox
to46.31pt{\pgfpicture\makeatletter\hbox{\hskip
29.18918pt\lower-23.15276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.18918pt}{-23.15276pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
7.43054pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.125pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathcal{L}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
7.43054pt\hfil&\hfil\hskip 32.84023pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.53471pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${X}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.84026pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
8.34894pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.0434pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${B}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.34894pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-13.2097pt}{11.50003pt}\pgfsys@lineto{10.9087pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.10869pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.63853pt}{15.21388pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{q}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-20.84024pt}{5.14032pt}\pgfsys@lineto{-20.84024pt}{-12.05977pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-20.84024pt}{-12.25975pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.48747pt}{-4.4861pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{p}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

we need to show that $\lrcorner
q^{*}\sigma_{K_{n}(A)}:H^{0}(L,N_{L/Z})=H^{0}(L,N_{L/\mathcal{L}})\to
H^{0}(L,\Omega_{L})$ is an isomorphism. Since the general fibers of $\pi$ are
rational, $q^{*}\sigma_{K_{n}(A)}=\pi^{*}\sigma_{A}$, where $\sigma_{A}$ is
the unique (up to coefficients) holomorphic $2$-form on $A$. Therefore, we can
conclude by the commutativity of the following diagram

${H^{0}(C,N_{C/A})}$${H^{0}(C,\Omega_{C})}$${H^{0}(L,N_{L/Z})}$${H^{0}(L,\Omega_{L})}$$\scriptstyle{\lrcorner\sigma_{A}}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\lrcorner\pi^{*}\sigma_{A}}$

noting that the two vertical arrows are isomorphims since the fibers of $\pi$
are rational, and that $\lrcorner\sigma_{A}:H^{0}(C,N_{C/A})\to
H^{0}(C,\Omega_{C})$ is an isomorphism since $\sigma_{A}$ is nondegenerate. ∎

######  Proposition 2.5.2.

The Abel-Jacobi map of the Lagrangian family $\\{Z_{C}\\}_{C\in B}$ is _not_
trivial.

###### Proof.

Let $i:C\hookrightarrow A$ be a general curve in the family $\mathcal{C}\to
B$. By Proposition 2.2.1(a), it suffices to show that the restriction map
$H^{2n-1}(X,\mathbb{Q})\to H^{2n-1}(Z_{C},\mathbb{Q})$ is nonzero.

define an injective morphism

$\begin{array}[]{cccc}\beta:&A\times A&\hookrightarrow&A^{(n+1)}\\\
&(x,y)&\mapsto&n\\{x\\}+\\{y\\},\end{array}$

where we use the notation $\\{x\\}\in\mathcal{Z}_{0}(A)$ the $0$-cycle of the
point $x\in A$. Consider the following pull-back diagram definitionning a
subvariety $Z^{\prime}\subset A^{[n+1]}$

$\leavevmode\hbox to90.82pt{\vbox
to51.44pt{\pgfpicture\makeatletter\hbox{\hskip
45.40839pt\lower-25.76926pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.40839pt}{-25.66942pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.84637pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.54083pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${Z^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.84637pt\hfil&\hfil\hskip 38.8695pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.564pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{[n+1]}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
14.86954pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
17.91663pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.61108pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A\times A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
17.91663pt\hfil&\hfil\hskip 39.49174pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.18623pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{(n+1)}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.49178pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{{
{\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@miterjoin\pgfsys@roundcap{{{}} {{}} {}
{{{{}{}{}{}}}{{}{}{}{}}} }
\pgfsys@moveto{0.0pt}{1.95987pt}\pgfsys@curveto{0.6848pt}{1.95987pt}{1.23993pt}{1.52113pt}{1.23993pt}{0.97993pt}\pgfsys@curveto{1.23993pt}{0.43874pt}{0.6848pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-17.00548pt}{11.08337pt}\pgfsys@lineto{14.44711pt}{11.08337pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-17.00548pt}{11.08337pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.6471pt}{11.08337pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.03812pt}{13.43614pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\alpha}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-27.49176pt}{4.72366pt}\pgfsys@lineto{-27.49176pt}{-14.57643pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-27.49176pt}{-14.77641pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.13899pt}{-7.18886pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\pi^{\prime}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{29.91661pt}{4.72366pt}\pgfsys@lineto{29.91661pt}{-12.47643pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{29.91661pt}{-12.6764pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{32.26938pt}{-5.5833pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{c}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-7.93523pt}{-23.16942pt}\pgfsys@lineto{13.82487pt}{-23.16942pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-7.93523pt}{-23.16942pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02486pt}{-23.16942pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.44516pt}{-19.45555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\beta}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $c:A^{[n+1]}\to A^{(n+1)}$ is the Hilbert-Chow morphism. Then
$Z=Z^{\prime}\cap K_{n}(A)\subset A^{[n+1]}$. We have the following
commutative diagram where all three squares are pull-back diagrams

$\leavevmode\hbox to253.94pt{\vbox
to118.91pt{\pgfpicture\makeatletter\hbox{\hskip
126.97084pt\lower-59.50539pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-126.97084pt}{-59.40555pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.07637pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.77083pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${Z}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.07637pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 53.43056pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.12505pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${X=K_{n}(A)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
29.43059pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 32.84634pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.54083pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${Z^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.84637pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 38.8695pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.564pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{[n+1]}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
14.86954pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 41.9166pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.61108pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A\times A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
17.91663pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 39.49174pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.18623pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{(n+1)}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
15.49178pt\hfil&\hfil\hskip 32.05551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-109.17819pt}{46.25282pt}\pgfsys@lineto{-27.58487pt}{46.25282pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-109.17819pt}{46.25282pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-27.38489pt}{46.25282pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-118.89447pt}{39.89311pt}\pgfsys@lineto{-118.89447pt}{-14.05977pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-118.89447pt}{-14.25975pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-116.5417pt}{11.20975pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\pi}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-109.45807pt}{39.31644pt}\pgfsys@lineto{-78.2701pt}{16.38287pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.80563}{0.5924}{0.5924}{0.80563}{-109.45805pt}{39.31644pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.80563}{-0.5924}{0.5924}{0.80563}{-78.10898pt}{16.2644pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{20.32246pt}{36.71553pt}\pgfsys@lineto{55.94557pt}{17.71669pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.88235}{0.47058}{0.47058}{0.88235}{20.32246pt}{36.71552pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.88235}{-0.4706}{0.4706}{0.88235}{56.12202pt}{17.62257pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-58.41522pt}{9.50003pt}\pgfsys@lineto{55.89851pt}{9.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-58.41522pt}{9.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{56.0985pt}{9.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.01729pt}{11.8528pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\alpha}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}