# Fujiki relations and fibrations of
irreducible symplectic varieties

Martin Schwald  Martin Schwald, Universität Duisburg-Essen, Fakultät für
Mathematik, Thea-Leymann-Str. 9, 45127 Essen<EMAIL_ADDRESS>http://www.esaga.uni-due.de/martin.schwald/

###### Abstract.

This paper concerns singular, complex projective, irreducible symplectic
varieties. Generalizing known results, we prove that the generalized
Beauville-Bogomolov form satisfies the Fujiki relations and has rank
$\big{(}3,0,b_{2}(X)-3\big{)}$. As a consequence, we see that these varieties
admit only very special fibrations. Our results give new evidence that the
correct definition of singular analogs of irreducible symplectic manifolds is
[GKP16, 8.16.2].

The author gratefully acknowledges support by the DFG-Graduiertenkolleg GK1821
“Cohomological Methods in Geometry” at the University of Freiburg and partial
support by DFG-SFB/TRR 45.

###### Contents

  1. 1 Introduction
  2. 2 Methods
  3. 3 Generalized Beauville-Bogomolov form
  4. 4 Proofs of the main results

##  1\. Introduction

###  1.1. Symplectic varieties

It is expected that an appropriately defined singular analog of _irreducible
symplectic manifolds_ should play an important role in a possible structure
theory for complex varieties with Kodaira dimension zero, [GKP16, Section 8].
To understand them better, we want to classify their fibrations. Our main
results Theorem 2, 3, 4 show that this new definition, which we call here
_primitive symplectic_ , behaves much like irreducible symplectic manifolds.
In particular Theorem 4 shows that it behaves better than the classical
definition of _irreducible symplectic varieties_.

We briefly recall the relevant definitions required to state our main results.
The different notions in the literature generalizing holomorphic symplectic
manifolds to a singular setting agree in the existence of a _symplectic form_
on the smooth locus, which is best seen as a _reflexive_ $2$-form.

_Reflexive differential forms_ are holomorphic $p$-forms on the smooth locus
$X_{reg}$ of a normal variety $X$. We denote the associated sheaf on $X$ by
$\Omega^{[p]}_{X}$ and get

$\displaystyle\Omega^{[p]}_{X}$ $\displaystyle\cong
i_{*}\Omega^{p}_{X_{reg}}\cong(\Omega^{p}_{X})^{\ast\ast},$
$\displaystyle\textup{H}^{0}(X,\,\Omega^{[p]}_{X})$
$\displaystyle\cong\textup{H}^{0}(X_{reg},\,\Omega_{X_{reg}}^{p}),$

where $i\colon X_{reg}\operatorname{\hookrightarrow}X$ is the inclusion. These
forms satisfy good pull back properties, see Section 2.1. We recommend [Har80]
for a reference on _reflexive sheaves_ and [GKKP11, I–III], [KP16, I] for more
information on reflexive differential forms.

A _quasi-étale_ morphism $\pi\colon X^{\prime}\operatorname{\rightarrow}X$ is
a finite surjective morphism between normal, complex projective varieties that
is étale outside of a subvariety $Z\subset X^{\prime}$ with
$\operatorname{codim}Z\geq 2$.

######  Definition 1 (Symplectic varieties).

Let $X$ be a normal, complex projective variety.

  1. (1)

A _symplectic form_ on $X$ is a reflexive form
$\omega\in\textup{H}^{0}(X,\,\Omega^{[2]}_{X})$ that is not degenerate at any
point $x\in X_{reg}$.

  2. (2)

The pair $(X,\omega)$ is called _symplectic variety_ if for any resolution of
singularities $\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ there is a
holomorphic form
${\widetilde{\omega}\in\textup{H}^{0}(\widetilde{X},\,\Omega^{2}_{\widetilde{X}})}$
that coincides with $\omega$ over $X_{reg}$. We say that $\omega$ _extends_ to
$\widetilde{\omega}$ on $\widetilde{X}$.

  3. (3)

A symplectic variety $(X,\omega)$ is called _irreducible symplectic_ if its
irregularity $\operatorname{h}^{1}(X,\,\mathcal{O}_{X})$ vanishes and if
$\operatorname{h}^{0}(X,\,\Omega^{[2]}_{X})=1$.

  4. (4)

We call an irreducible symplectic variety $(X,\omega)$ _Namikawa symplectic_
if $X$ is $\mathds{Q}$-factorial with
$\operatorname{codim}_{\mathds{C}}X_{sing}\geq 4$.

  5. (5)

We call a symplectic variety _primitive symplectic_ if for all quasi-étale
morphisms $\pi\colon X^{\prime}\operatorname{\rightarrow}X$ the pullback
$f^{*}\omega\in\textup{H}^{0}(X^{\prime},\,\Omega^{[2]}_{X^{\prime}})$, which
exists by Theorem 5, generates the outer algebra of reflexive forms on
$X^{\prime}$, so

$\bigoplus\textup{H}^{0}(X^{\prime},\,\Omega^{[p]}_{X^{\prime}})=\mathds{C}\left[f^{*}\omega\right].$

Section 2.5 reviews properties of these notions and discusses their relations.
Symplectic varieties were introduced by Beauville in [Bea00, Definition 1.1].
Bogomolov and Namikawa showed in [Bog78, Nam01c] that Namikawa symplectic
varieties have a well behaved deformation theory. The codimension property of
the singularities guarantees the extension property of the symplectic form by
[Fle88]. It is conjectured that primitive symplectic varieties are building
blocks in a singular version of the Beauville-Bogomolov decomposition theorem,
[GKP16]. Note that _irreducible symplectic manifolds_ are always primitive
symplectic by [Bea83, Proposition 3].

###  1.2. Main results

On an irreducible symplectic variety $X$, Namikawa and Kirschner constructed
in [Nam01c, Theorem 8 (2)], [Kir15, Definition 3.2.7] an important quadratic
form $q_{X}$ on $\textup{H}^{2}(X,\,\mathds{C})$, the _Beauville-Bogomolov
form_. Its definition will be recalled in Section 3. We will show that it has
the following properties, which are well known in the smooth case, [Fuj87,
Theorem 4.7], [Bea83, Théorème 5(a)]. Matsushita already proved them first for
irreducible symplectic manifolds and then for Namikawa symplectic varieties,
[Mat99, Theorem 3], [Mat01, Theorem 1.2], [Mat15, Proposition 4.1].

######  Theorem 2 (Fujiki relations, index of the Beauville-Bogomolov form).

Let $(X,\omega)$ be a $2n$-dimensional, irreducible symplectic variety. The
Beauville-Bogomolov form $q_{X}$ has the following properties:

  1. (1)

There is a number $c_{X}\in\mathds{R}^{+}$, called _Fujiki constant_ , such
that for all ${\alpha\in\textup{H}^{2}(X,\,\mathds{C})}$ the following, so-
called _Fujiki relation_ , holds,111We will recall integration on singular
cohomology in Section 2.3.

$c_{X}\cdot q_{X}(\alpha)^{n}=\int_{X}\alpha^{2n}.$

In particular for classes of Cartier divisors
$d=c_{1}\big{(}\mathcal{O}_{X}(D)\big{)}$, this relates the Beauville-
Bogomolov form to the intersection product via

$c_{X}\cdot q_{X}(d)^{n}=D^{2n}.$

  2. (2)

The restriction of $q_{X}$ to
$\textup{H}^{2}(X,\,\mathds{R})\operatorname{\rightarrow}\mathds{R}$ is a real
quadratic form of _index_ $\big{(}3,0,b_{2}(X)-3\big{)}$.

Part (2) implies the existence of a $q_{X}$-orthogonal decomposition
${\textup{H}^{2}(X,\,\mathds{R})=V_{+}\oplus V_{-}}$ with
$\dim_{\mathds{R}}V_{+}=3$, such that $q_{X}$ is positive definite on $V_{+}$
and negative definite on $V_{-}$, see Proposition 28.

Matsushita uses his versions of Theorem 2 to show that fibrations of
irreducible symplectic manifolds and Namikawa symplectic varieties have very
special properties, [Mat99, Theorem 2], [Mat00, Corollary 1], [Mat01,
Corollary 1.4], [Mat15, Theorem 1.2, 1.10]. By our Theorems 3 and 4 all these
results hold for primitive symplectic varieties and we show, which of them are
also true for all irreducible symplectic varieties.

######  Theorem 3 (Fibrations of irreducible symplectic varieties).

Let $(X,\omega)$ be an irreducible symplectic variety of complex dimension
$2n$, together with a surjective morphism $f\colon
X\operatorname{\rightarrow}B$ with connected fibers onto a normal, projective
variety $B$ with $0<\dim B<2n$. Then the following properties hold.

  1. (1)

The base variety $B$ is an $n$-dimensional, $\mathds{Q}$-factorial klt variety
with Picard number $\rho(B)=1$.

  2. (2)

The singular locus of $X$ is mapped to a proper closed subset of $B$. In
particular, the general fiber is smooth and entirely contained in $X_{reg}$.

  3. (3)

The general fiber is an $n$-dimensional Abelian variety.

  4. (4)

Any fiber component of $f$ is an $n$-dimensional _Lagrangian subvariety_ 222We
discuss the definition of _Lagrangian subvarieties_ in Section 2.6. of $X$ and
does not lie completely in $X_{sing}$.

######  Theorem 4 (Fibrations of primitive symplectic varieties).

Let $f\colon X\operatorname{\rightarrow}B$ be as in Theorem 3. Then the
following properties do not hold for every irreducible symplectic variety $X$,
but for every primitive symplectic variety $X$.

  1. (1)

The base variety $B$ is _Fano_ , so $-K_{B}$ is ample.

  2. (2)

If $B$ is smooth, then $B\cong\mathds{P}^{n}$.

It is remarkable that both parts of Theorem 4 can fail for irreducible
symplectic varieties, as we will see in Example 29, although they hold for all
irreducible symplectic manifolds, [Mat01, Theorem 2 (3)], [Hwa08, Theorem
1.2], [GL14, Theorem 1.1].

To prove the results on fibrations we work as much as possible on the singular
variety $X$. We hope that this will also allow to tackle further questions on
Lagrangian fibrations. Open problems are for example a classification of the
singular fibers and of the possible base varieties $B$. In general $B$ can be
singular, even if $X$ is primitive symplectic, [Mat15, Theorem 1.9]. However,
we still know little about its possible singularities. It is still unknown if
there is an example where $X$ is smooth and $B$ singular, even in dimension
four, cf. [Ou16, Theorem 1.2].

###  1.3. Outline of the paper

We present in 2.1–2.4 our most important methods to prove our main theorems.
They are the general extension theorem for differential forms from [GKKP11,
Theorem 1.4], the existence of terminal models from [BCHM10, Corollary 1.4.3]
and results on the Hodge theory of klt varieties that we explain in detail in
[Sch16]. In particular we will need the Hodge decomposition of
$\textup{H}^{2}(X,\,\mathds{C})$ and a singular version of the Hodge-Riemann
bilinear relations.

After recapitulating well known properties of symplectic varieties in 2.5, we
prove Propositions 13 and 14 about the structure of their terminal models. In
2.6 we prove in Theorem 16 that singular fibers of Lagrangian fibrations are
automatically also Lagrangian, implying that Lagrangian fibrations are always
equidimensional. This requires a careful revision of the definitions of
Lagrangian subvarieties and a subtle inductive argument. Theorem 16 will be
used to show Theorem 3 (4), but in fact it holds for fibrations of any
symplectic variety. We explain in Section 3 how the Beauville-Bogomolov form
$q_{X}$ is defined in the singular setting in a way that is most suitable for
direct computations in $\textup{H}^{2}(X,\,\mathds{C})$.

In Section 4 we finally prove our main results. We deduce the Fujiki relation
directly from a terminal model using Proposition 13, but we need our Hodge
theoretic methods to calculate the index of $q_{X}$ by hand. Theorem 2 is
essential to prove the rest of Theorem 3 in the spirit of Matsushita’s proofs.
Proving that $B$ is $\mathds{Q}$-factorial becomes particularly involved,
because handling Weil divisors is much harder in the singular case. On the
other hand, we can use a recently proven special case of the Lipman-Zariski
conjecture to prove part (3), which makes the proof even easier and more
geometric. The proof of Theorem 4 uses the branched covering trick,
Proposition 14 and the explicit Example 29.

###  1.4. Conventions

We use the terminology of [Har77] except that we allow _subvarieties_ to be
reducible. We denote the smooth and the singular locus of a variety $X$ by
$X_{reg}$ and $X_{sing}$, respectively. For the terminology of the Minimal
Model Program we refer to [KM98], but we use the sloppy term _klt variety_ for
a variety $X$ that carries an _effective_ Weil divisor $D$, such that the pair
$(X,D)$ is klt according to [KM98, Definition 2.34].

###  1.5. Acknowledgment

The results of this paper are an improved version of results of my PhD thesis,
answering also Questions 6, 8 and Conjecture 7 that were stated in the
introduction of the thesis, [Sch17]. I want to thank my supervisor S. Kebekus,
as well as K. Oguiso, D. Greb, P. Graf, T. Kirschner and B. Taji for their
advice and fruitful discussions.

##  2\. Methods

We recall here the extension theorem, terminal models, integration over
singular cohomology classes and results on the Hodge theory for klt varieties.
We recapitulate well known properties of symplectic varieties that will imply
that terminal models of irreducible or primitive symplectic varieties are
Namikawa or primitive symplectic, respectively. Then we discuss the notions of
Lagrangian subvarieties and fibrations and examine their singular fibers.

###  2.1. Extension Theorem

The extension theorem for differential forms [GKKP11, Theorem 1.4] shows that
on a klt variety $X$ any reflexive form
${\alpha\in\textup{H}^{0}(X,\,\Omega^{[p]}_{X})}$ can be _extended_ to any
resolution of singularities. This allows us to construct pullbacks of
reflexive forms along more general morphisms.

######  Theorem 5 (Pullbacks of reflexive forms, [Keb13, Theorem 1.3]).

Let $f\colon Y\operatorname{\rightarrow}X$ be a morphism between normal,
complex projective varieties where $X$ is a klt variety. Then for any $p$
there is a natural pullback morphism
$f^{*}\Omega^{[p]}_{X}\operatorname{\rightarrow}\Omega^{[p]}_{Y}$, consistent
with the natural pullback of Kähler differentials on $X_{reg}$.∎

The consistency with the natural pullback of Kähler differentials is made
precise in [Keb13, Theorem 5.2]. We will only consider the case when $f\colon
Y\operatorname{\rightarrow}X$ is surjective. Then this consistency means that
the pullback morphism
$f^{*}\Omega^{[p]}_{X}\operatorname{\rightarrow}\Omega^{[p]}_{Y}$ agrees with
the usual pullback of Kähler differentials wherever $X$ and $Y$ are smooth. In
other words, for every $\alpha\in\textup{H}^{0}(X,\,\Omega^{[p]}_{X})$ there
is an $\widetilde{\alpha}\in\textup{H}^{0}(Y,\,\Omega^{[p]}_{Y})$ that
restricts on $Y_{reg}\cap f^{-1}(X_{reg})$ to the natural pullback
$f^{*}({\left.\kern-1.2pt\alpha\vphantom{\big{|}}\right|_{X_{reg}}})$ of
Kähler differentials. We call $\widetilde{\alpha}$ an _extension_ of $\alpha$
to $Y$. Pulling back reflexive forms is contravariant functorial with respect
to $f$.

###  2.2. Terminal models

While not every singular variety admits a crepant resolution, varieties with
klt singularities have always a _terminal model_. For canonical singularities
a terminal model is a _crepant partial resolution_. This is an easy special
case of [BCHM10, Corollary 1.4.3].

######  Theorem 6 (Terminal models, [BCHM10, page 413]).

Let $X$ be a complex projective variety with canonical singularities. Then any
resolution $\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ of $X$ factors
through a $\mathds{Q}$-factorial variety $Y$ with at most terminal
singularities,
${\widetilde{X}\stackrel{{\scriptstyle\widetilde{\pi}}}{{\operatorname{\longrightarrow}}}Y\stackrel{{\scriptstyle\pi}}{{\operatorname{\longrightarrow}}}X}$,
such that $\pi^{*}K_{X}=K_{Y}$. We call $Y$ a _terminal model_ of $X$. ∎

###  2.3. Integration of singular top classes

Recall that we integrate a top singular cohomology class
$\phi\in\textup{H}^{2n}(X,\,\mathds{C})$ over an $n$-dimensional, compact,
complex variety $X$ as
$\int_{X}\phi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[X]\cap\phi$.
Here $[X]\in\textup{H}_{2n}(X,\,\mathds{Z})$ denotes the canonical fundamental
class of $X$, induced by the complex structure on its smooth locus $X_{reg}$.
It exists because $X$ is by [GM80, 1.1] endowed with the structure of a
$2n$-dimensional pseudomanifold and the complex structure on the smooth locus
$X_{reg}$ induces a canonical orientation. Furthermore, the cap product
induces a perfect pairing

$\cap\colon\textup{H}_{2n}(X,\,\mathds{Z})\times\textup{H}^{2n}(X,\,\mathds{Z})$$\textup{H}_{0}(X,\mathds{Z})\cong\mathds{Z}$$(\sigma$,$\phi)$$\sigma\cap\phi,$

so Poincaré duality holds in degree $2n$, see [Sch16, Section 3.4] for
details. This integration commutes with pullbacks by bimeromorphic maps.

######  Lemma 7 (Pullbacks of integrals, [Sch16, Lemma 12]).

If $f\colon X\operatorname{\rightarrow}Y$ is a bimeromorphic morphism of
irreducible, compact, complex varieties, then
$\int_{Y}\alpha=\int_{X}f^{*}\alpha$ for all
$\alpha\in\textup{H}^{2n}(Y,\,\mathds{C})$.∎

###  2.4. Hodge Theory for klt varieties

On Namikawa symplectic varieties, Matsushita and Namikawa consider the Hodge
Frölicher spectral sequence on the smooth locus. As the singularities are of
codimension at least four, the spectral sequence degenerates on the first page
up to degree two, [Ohs87, Theorem 1]. The author does not know if this holds
more generally. However, for klt varieties the Leray spectral sequence and the
extension theorem imply the following decomposition of
$\textup{H}^{1}(X,\,\mathds{C})$ and $\textup{H}^{2}(X,\,\mathds{C})$.

######  Theorem 8 (Structure of $\textup{H}^{1}(X,\,\mathds{C})$ and
$\textup{H}^{2}(X,\,\mathds{C})$, [Sch16, Theorem 7]).

If $X$ is a complex projective klt variety, then for ${p=1,2}$ the mixed Hodge
structure on $\textup{H}^{p}(X,\,\mathds{C})$ is pure of weight $p$.
Furthermore any resolution of singularities
$\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ induces injective pullback
maps

$\nu^{*}_{ab}\colon\textup{H}^{a,b}(X)\to\textup{H}^{a,b}(\widetilde{X})\quad\text{for
}a+b=p,$

where the morphisms $\nu^{*}_{p0}$ and $\nu^{*}_{0p}$ are isomorphisms. There
are canonical isomorphisms
$\kappa_{p0}\colon\textup{H}^{p,0}(X)\stackrel{{\scriptstyle\sim}}{{\operatorname{\longrightarrow}}}\textup{H}^{0}(X,\,\Omega^{[p]}_{X})$
and
$\kappa_{0p}\colon\textup{H}^{0,p}(X)\stackrel{{\scriptstyle\sim}}{{\operatorname{\longrightarrow}}}\textup{H}^{p}(X,\,\mathcal{O}_{X})$
that commute with the natural pullback morphisms and the Dolbeault
isomorphisms $\delta_{p0},\delta_{0p}$.

$\textup{H}^{p,0}(X)$$\textup{H}^{0}(X,\,\Omega^{[p]}_{X})$$\textup{H}^{p,0}(\widetilde{X})$$\textup{H}^{0}(\widetilde{X},\,\Omega^{p}_{\widetilde{X}})$$\sim$$\kappa_{p0}$$\sim$$\delta_{p0}$$\nu^{*}_{p0}$$\wr$$\nu^{*}$$\wr$$\textup{H}^{0,p}(X)$$\textup{H}^{p}(X,\,\mathcal{O}_{X})$$\textup{H}^{0,p}(\widetilde{X})$$\textup{H}^{p}(\widetilde{X},\,\mathcal{O}_{\widetilde{X}})$$\sim$$\kappa_{0p}$$\sim$$\delta_{0p}$$\nu^{*}_{0p}$$\wr$$\nu^{*}$$\wr$

∎

######  Remark 9.

Let $X$ be a complex projective klt variety. Every reflexive two-form on $X$
defines by Theorem 8 a unique cohomology class in
$\textup{H}^{2}(X,\,\mathds{C})$ and we denote this canonical inclusion by:

$\iota_{20}\colon\textup{H}^{0}(X,\,\Omega^{[2]}_{X})$$\textup{H}^{2,0}(X)\subset\textup{H}^{2}(X,\,\mathds{C})$$\alpha$$\underline{\alpha}$$\kappa_{20}$

Due to the first diagram in Theorem 8, the class
$\underline{\alpha}\in\textup{H}^{2}(X,\,\mathds{C})$ has the property that on
any resolution $\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ the
extension of $\alpha$ represents the class $\nu^{*}\underline{\alpha}$ in de
Rham cohomology.

The author explains the Hodge theory of $\textup{H}^{2}(X,\,\mathds{C})$ and
applications in detail in [Sch16]. In this paper, we make use of the following
version of the Hodge-Riemann bilinear relations on klt varieties.

######  Corollary 10 (Bilinear relations on klt varieties, [Sch16, Corollary
16]).

Let $X$ be an $n$-dimensional, complex projective klt variety. Any ample class
${a\in\textup{H}^{2}(X,\,\mathds{R})}$ induces a sesquilinear form
$\psi_{X,a}$, defined by

$\psi_{X,a}\colon\textup{H}^{k}(X,\,\mathds{C})\times\textup{H}^{k}(X,\,\mathds{C})$$\mathds{C}$$(\alpha$,$\beta)$$(-1)^{\frac{k(k-1)}{2}}\cdot\int_{X}\alpha\cup\overline{\beta}\cup
a^{n-k}.$

The so-called _Hodge-Riemann bilinear relation_
$i^{p-q}\cdot\psi_{X,a}(\alpha,\alpha)>0$ holds for any non-zero $\alpha$ with
${\alpha\cup a^{n-k+1}=0}$ that can be written as the cup product of classes
$\alpha_{1},\ldots,\alpha_{r}\in\textup{H}^{1}(X,\,\mathds{C}),\textup{H}^{2}(X,\,\mathds{C})$
.

We write $p\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum
p_{j}$, $q\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum q_{j}$
and $k\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=p+q$ where
$(p_{j},q_{j})$ are such that the $\alpha_{j}$ lie in the
$\textup{H}^{p_{j},q_{j}}(X)$-part of the Hodge decomposition of
$\textup{H}^{1}(X,\,\mathds{C})$ or $\textup{H}^{2}(X,\,\mathds{C})$. ∎

###  2.5. Properties of symplectic varieties

We mention the following properties of symplectic varieties to prove
Propositions 13 and 14, which will be important in the main proofs. The
_canonical sheaf_ on an $n$-dimensional, normal, complex variety $X$ is the
sheaf $\Omega^{[n]}_{X}$ and is often denoted by $\omega_{X}$ or
$\mathcal{O}_{X}(K_{X})$ for the canonical divisor class $K_{X}$ on $X$.

######  Lemma 11 (Properties of symplectic varieties).

  1. (1)

A symplectic variety has canonical singularities and trivial canonical class,
cf. [Bea00, Proposition 1.3].

  2. (2)

A symplectic variety has at most terminal singularities if and only if the
singular locus is at least of codimension four, [Nam01a, Corollary 1].

  3. (3)

An irreducible symplectic variety $X$ is Namikawa symplectic if and only if it
has at most $\mathds{Q}$-factorial, terminal singularities.

#### Proof of (1).

Let $(X,\omega)$ be a $2n$-dimensional symplectic variety with a resolution
$\nu\colon\widetilde{X}\operatorname{\rightarrow}X$. By definition, $\omega$
extends to a regular form $\widetilde{\omega}$ on $\widetilde{X}$. Let
$K_{X},\,K_{\widetilde{X}}$ be the canonical divisor classes on
$X,\,\widetilde{X}$. Due to linear algebra, it is equivalent that $\omega$ is
non-degenerate and that $\omega^{n}\in\textup{H}^{0}(X,\,\Omega^{[2n]}_{X})$
has no zeros on $X_{reg}$. Hence $\operatorname{supp}K_{X}\subset X_{sing}$,
which implies $K_{X}=0$ because $X$ is normal. Thus $\widetilde{\omega}^{n}$
is a global section of $\mathcal{O}_{\widetilde{X}}(K_{\widetilde{X}})$ that
may only have zeros along the set $\mathcal{E}$ of $\nu$-exceptional divisors.
Hence we can write up to linear equivalence

$\displaystyle
0\leq\operatorname{div}(\widetilde{\omega}^{n})=K_{\widetilde{X}}=\underbrace{\nu^{*}\operatorname{div}(\omega^{n})}_{=0}+\sum_{E\in\mathcal{E}}\operatorname{discr}(E)\cdot
E.$

Therefore all discrepancies are non-negative and $X$ has canonical
singularities.

#### Proof of (2) and (3).

Note that in general terminal singularities may be only of codimension three.
However, Namikawa proves in [Nam01a] that the singular locus of any symplectic
variety $X$ has no component of pure codimension three. Furthermore, every
exceptional divisor $E$ over $X$ with vanishing discrepancy has a center of
codimension two in $X$. Namikawa deduces (2) from this in [Nam01a, Corollary
1]. Part (3) follows immediately from (2). ∎

######  Remark 12 (Equivalent definition of symplectic varieties).

Let $X$ be a complex projective variety with a symplectic form
$\omega\in\textup{H}^{0}(X,\,\Omega^{[2]}_{X})$. Then by Lemma 11 $(X,\omega)$
is a symplectic variety if and only if $X$ has canonical singularities. It is
Namikawa symplectic if and only if $X$ has $\mathds{Q}$-factorial terminal
singularities. The form $\omega$ automatically extends to resolutions of $X$
by the extension theorem. This gives definitions that fit better into the
spirit of the _Minimal Model Program_.

We point out that Matsushita uses a slightly different looking definition of
irreducible symplectic varieties. In [Mat01] he uses the condition
$\operatorname{h}^{2}(X,\,\mathcal{O}_{X})=1$ instead of
$\operatorname{h}^{0}(X,\,\Omega^{[2]}_{X})=1$. It follows from Theorem 8 that
these two conditions are equivalent by the Hodge symmetry.

######  Proposition 13.

Every terminal model $\pi\colon Y\operatorname{\rightarrow}X$ of an
irreducible symplectic variety $(X,\omega)$ is a Namikawa symplectic variety
with the extension $\omega^{\prime}\in\textup{H}^{0}(Y,\,\Omega^{[2]}_{Y})$ of
$\omega$ to $Y$ from Theorem 5.

###### Proof.

The form $\omega$ extends to a reflexive form
$\omega^{\prime}\in\textup{H}^{0}(Y,\,\Omega^{[2]}_{Y})$ by Theorem 5. We have
$\pi^{*}K_{X}=K_{Y}$ because $\pi$ is crepant, and Lemma 11 (1) shows that
these canonical classes are trivial. Let $X$ be $2n$-dimensional. As $Y$ is
normal, $(\omega^{\prime})^{n}$ can be extended to a non-trivial section of
$\mathcal{O}_{Y}(K_{Y})$ that has no zeros on $Y_{reg}$. This is equivalent to
$\omega^{\prime}$ being non-degenerate on $Y_{reg}$. The extension theorem
allows us to extend $\omega^{\prime}$ to any resolution
$\widetilde{\pi}\colon\widetilde{X}\operatorname{\rightarrow}Y$ of $Y$, hence
$(Y,\omega^{\prime})$ is a symplectic variety with at most
$\mathds{Q}$-factorial, terminal singularities by construction. As
$\widetilde{X}$ is a common resolution of $X$ and $Y$, Theorem 8 gives the
missing irreducibility conditions of $Y$:

$\displaystyle\operatorname{h}^{1}(Y,\,\mathcal{O}_{Y})$
$\displaystyle=\operatorname{h}^{1}(\widetilde{X},\,\mathcal{O}_{\widetilde{X}})=\operatorname{h}^{1}(X,\,\mathcal{O}_{X})=0,$
$\displaystyle\operatorname{h}^{0}(Y,\,\Omega^{[2]}_{Y})$
$\displaystyle=h^{0}(\widetilde{X},\,\Omega^{2}_{\widetilde{X}})=\operatorname{h}^{0}(X,\,\Omega^{[2]}_{X})=1.$

Hence $(Y,\widetilde{\omega})$ is Namikawa symplectic by Lemma 11. ∎

######  Proposition 14.

Every terminal model $\pi\colon Y\operatorname{\rightarrow}X$ of a primitive
symplectic variety $(X,\omega)$ is primitive symplectic with the extension
$\omega^{\prime}\in\textup{H}^{0}(Y,\,\Omega^{[2]}_{Y})$ of $\omega$ to $Y$.

###### Proof.

Let $g\colon Y^{\prime}\operatorname{\rightarrow}Y$ be a quasi-étale cover of
$Y$. Then the Stein factorization of $\pi\circ g$, [Har77, Corollary
III.11.5], gives a commutative diagram

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where $\pi^{\prime}$ has connected fibers and $h$ is a finite morphism. So
$\pi^{\prime}$ is birational because $\pi\circ g$ is generically finite. If
$g$ is étale outside of $V\subset Y^{\prime}$, then $h$ is étale outside of
$\pi^{\prime}(V)$ because $\pi$ and $\pi^{\prime}$ induce isomorphisms on the
function fields. Thus $h$ is a quasi-étale cover of $X$. We note that quasi-
étale covers of klt varieties are also klt varieties by [KM98, Proposition
5.20]. For a common resolution $\widetilde{X}$ of $X^{\prime}$ and
$Y^{\prime}$ the extension theorem implies
${\textup{H}^{0}(Y^{\prime},\,\Omega_{X^{\prime}}^{[p]})\cong\textup{H}^{0}(\widetilde{X},\,\Omega_{\widetilde{X}}^{p})\cong\textup{H}^{0}(X^{\prime},\,\Omega_{X^{\prime}}^{[p]})}$
for all $p$. As $X$ is primitive symplectic and pulling back reflexive forms
is functorial, the reflexive forms on $Y^{\prime}$ get generated by
$(\pi^{\prime})^{*}h^{*}\omega=g^{*}\omega^{\prime}$. ∎

###  2.6. Lagrangian fibrations

We discuss here the definitions and properties of Lagrangian subvarieties and
fibrations to show in Theorem 18 that Lagrangian fibrations are
equidimensional because also their singular fibers are Lagrangian
subvarieties. Proposition 17 generalizes Matsushita’s proof of [Mat00,
Corollary 1]. However, in the singular case we need an additional inductive
argument to make sure that there are no fiber components in $X_{sing}$, for
which the proof would not work.

######  Definition 15 (Lagrangian subvariety and fibration).

Let $(X,\omega)$ be a symplectic variety. A subvariety $F\subset X$ is called
a _Lagrangian subvariety_ if ${\dim F=\frac{1}{2}\dim X}$ and if on any
embedded resolution $\widetilde{X}$ of $(X,F)$ the extension
$\widetilde{\omega}$ vanishes on the strict transform of $F$. A _Lagrangian
fibration_ of $X$ is a surjective morphism $f\colon
X\operatorname{\rightarrow}B$ with connected fibers onto a normal, complex
projective variety $B$, such that the general fiber is a Lagrangian subvariety
of $X$.

The notion of Lagrangian subvarieties does not depend on the chosen embedded
resolution because for two resolutions we can go over to a common resolution.
We discuss other criteria that can be used to test if a subvariety is
Lagrangian.

######  Lemma 16 (Criteria of Lagrangian subvarieties).

Let $F$ be an irreducible subvariety of a symplectic variety $(X,\omega)$ with
$F\not\subset X_{sing}$ and $\dim F\geq\frac{1}{2}\dim X$. Then the following
implications hold.

  1. (1)

$F$ is Lagrangian if and only if
${\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{X_{reg}\cap F_{reg}}}$
vanishes as a Kähler differential on $X_{reg}\cap F_{reg}$.

  2. (2)

$F$ is Lagrangian if
${i^{*}\underline{\omega}=0\in\textup{H}^{2}(F,\,\mathds{C})}$ for the
inclusion $i\colon F\operatorname{\hookrightarrow}X$. The converse holds if
$F$ is a klt variety.

#### Proof of (1)

Let $\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ be an embedded
resolution of $(X,F)$ with strict transform
$\widetilde{i}\colon\widetilde{F}\operatorname{\hookrightarrow}\widetilde{X}$
of $F$ and extension $\widetilde{\omega}$ of $\omega$ to $\widetilde{X}$. As
$\nu$ is an isomorphism over a dense open subset, the assumption $F\not\subset
X_{sing}$ implies that the conditions
${\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{X_{reg}\cap F_{reg}}}=0$
and
${\left.\kern-1.2pt\widetilde{\omega}\vphantom{\big{|}}\right|_{\widetilde{F}}}=0$
are equivalent. Moreover, linear algebra shows that both latter conditions
imply ${\dim F\leq\frac{1}{2}\dim{X}}$ because $\omega$ is non-degenerate on
$X_{reg}$, [Nam01b, Lemma 1.1]. Together with the assumption $\dim
F\geq\frac{1}{2}\dim{X}$ this implies that the conditions
${\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{X_{reg}\cap F_{reg}}}=0$
and
${\left.\kern-1.2pt\widetilde{\omega}\vphantom{\big{|}}\right|_{\widetilde{F}}}=0$
are both equivalent to $F$ being a Lagrangian subvariety of $X$.

#### Proof of (2)

By Remark 9 we have $\underline{\widetilde{\omega}}=\nu^{*}\underline{\omega}$
and therefore

$\underline{{\left.\kern-1.2pt\widetilde{\omega}\vphantom{\big{|}}\right|_{\widetilde{F}}}}=\widetilde{i}^{*}\nu^{*}\underline{\omega}=\nu|_{\widetilde{F}}^{*}i^{*}\underline{\omega}\in\textup{H}^{2}(\widetilde{F},\,\mathds{C}).$

Hence the condition $i^{*}\underline{\omega}=0$ implies
${\left.\kern-1.2pt\widetilde{\omega}\vphantom{\big{|}}\right|_{\widetilde{F}}}=0$,
which, like in part (1), under the given assumptions is equivalent to $F$
being Lagrangian. The converse implication holds if $F$ is a klt variety
because then
${\nu|_{\widetilde{F}}^{*}\colon\textup{H}^{2}(F,\,\mathds{C})\operatorname{\rightarrow}\textup{H}^{2}(\widetilde{F},\,\mathds{C})}$
is injective by Theorem 8.∎

######  Proposition 17.

Let $f\colon X\operatorname{\rightarrow}B$ be a Lagrangian fibration of a
symplectic variety $(X,\omega)$. Then for any fiber $F$ of $f$ the pullback
$i^{*}\underline{\omega}$ of the symplectic class vanishes in
$\textup{H}^{2}(F,\,\mathds{C})$, where $i\colon F\operatorname{\rightarrow}X$
denotes the inclusion. Thus also
$i^{\prime*}\underline{\omega}=0\in\textup{H}^{2}(F^{\prime},\,\mathds{C})$
for every fiber component $i^{\prime}\colon
F^{\prime}\operatorname{\rightarrow}X$ of $F$.

###### Proof.

The Leray spectral sequence for $f$ and the sheaf $\mathcal{O}_{X}$ gives the
map
$d_{2}\colon\textup{H}^{2}(X,\,\mathcal{O}_{X})\operatorname{\rightarrow}\textup{H}^{0}(B,\,R^{2}f_{*}\mathcal{O}_{X})$.
The class $\underline{\overline{\omega}}$ lies in
$\textup{H}^{2}(X,\,\mathcal{O}_{X})$. Let $i\colon
F\operatorname{\rightarrow}X$ be the fiber over any point $p\in B$, then the
germ of $d_{2}(\underline{\overline{\omega}})$ is given by

$d_{2}(\underline{\overline{\omega}})_{p}=i^{*}\overline{\underline{\omega}}\in\textup{H}^{2}(F,\,\mathcal{O}_{F})\cong(R^{2}f_{*}\mathcal{O}_{X})_{p}.$

As the general fiber is Lagrangian, not completely contained in $X_{sing}$ and
has canonical singularities by Lemma 31, the section
$d_{2}(\underline{\overline{\omega}})$ vanishes at the general point $p\in B$
by Lemma 16. Therefore this section is torsion in $R^{2}f_{*}\mathcal{O}_{X}$.
However, by Lemma 11 the canonical sheaf of $X$ is trivial, so
$\omega_{X}\cong\mathcal{O}_{X}$. Hence the sheaf $R^{2}f_{*}\mathcal{O}_{X}$
is torsion-free by Kollár’s torsion-freeness Theorem, which also holds in the
singular case, see Theorem 33 in the appendix. Thus
${d_{2}(\underline{\overline{\omega}})=0}$, so for any fiber $i\colon
F\operatorname{\rightarrow}X$ the class $i^{*}\underline{\omega}$ vanishes.
Moreover $i^{\prime*}\underline{\omega}=0$ because $i^{\prime}$ factors
through $i$. ∎

######  Theorem 18.

Let $f\colon X\operatorname{\rightarrow}B$ be a Lagrangian fibration of a
symplectic variety $(X,\omega)$. Then every component of any fiber of $f$ is
Lagrangian and does not lie completely in $X_{sing}$. In particular $f$ is
equidimensional.

###### Proof.

We prove this theorem by induction over the dimension of $X$. If $X$ is a
surface, all fibers of $f$ are curves. As $X$ is normal, it has only isolated
singularities and no fiber component can lie in $X_{sing}$. By Proposition 17
the symplectic class vanishes on any fiber component $F^{\prime}$. Hence
$F^{\prime}$ is a Lagrangian subvariety of $X$ by Lemma 16 and the theorem is
proven for $\dim X=2$.

We assume now $2n\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\dim
X>2$ and that the theorem is already proven for all lower dimensional
symplectic varieties. Let $i^{\prime}\colon
F^{\prime}\operatorname{\hookrightarrow}X$ be a component of any fiber $F$ of
$f$. Let $Z$ be any irreducible component $X_{sing}$. We consider the
restriction ${\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{Z}}$ and the lifted
morphism
$\overline{f}\colon\overline{Z}\operatorname{\rightarrow}\overline{f(Z)}$
between the normalizations. Then by [Mat15, Theorem 3.1] the normalization
$\overline{Z}$ is a symplectic variety and $\overline{f}$ is a Lagrangian
fibration. By the induction hypothesis, we know that $\overline{f}$ is
equidimensional, so any fiber of $\overline{f}$ has dimension at most $n-1$.
Hence the dimension of all fibers of
${\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{Z}}$ is also at most $n-1$.
However, by the fiber dimension theorem, [Sha94, Theorem 1.25], we have $\dim
F^{\prime}\geq n$, so $F^{\prime}\not\subset Z$ and $F^{\prime}$ cannot lie
completely in $X_{sing}$. Now Proposition 17 implies
$i^{\prime*}\underline{\omega}=0$, so $F^{\prime}$ is by Lemma 16 an
$n$-dimensional Lagrangian subvariety of $X$. This completes the induction. ∎

##  3\. Generalized Beauville-Bogomolov form

We explain how the Beauville-Bogomolov form is defined in a singular setting,
comparing the equivalent constructions by Kirschner and Namikawa and
clarifying possible notational issues. We hope this also helps to understand
Kirschner’s, Matsushita’s and Namikawa’s results.

On any Riemannian manifold $X$ the Theorem of de Rham allows us to consider
any singular cohomology class as a class of differential forms. Using this,
Beauville defined in [Bea83, page 772] on any compact, Kähler, irreducible
symplectic manifold $(X,\omega)$ the _Beauville-Bogomolov form_ by:

$\displaystyle\begin{aligned}
q_{X,\omega}\colon&\textup{H}^{2}(X,\,\mathds{C})\operatorname{\rightarrow}\mathds{C}\\\
q_{X,\omega}(\alpha)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\;&n\int_{X}(\omega\overline{\omega})^{n-1}\alpha^{2}+(1-2n)\Big{(}\int_{X}\omega^{n}\overline{\omega}^{n-1}\alpha\Big{)}\cdot\Big{(}\int_{X}\omega^{n-1}\overline{\omega}^{n}\alpha\Big{)}.\end{aligned}$

The products and powers denote here the wedge products of the de Rham classes
or equivalently the cup product of their singular cohomology classes.

###  3.1. Namikawa’s generalization

Namikawa defines $q_{X,\omega}$ in the singular setting by pulling back
everything to a resolution of singularities and then calculating Beauville’s
formula. This gives the following definition.

######  Definition 19 (Beauville-Bogomolov form, cf. [Nam01c, Theorem 8 (2)]).

Let $(X,\omega)$ be a $2n$-dimensional, irreducible symplectic variety with a
resolution $\nu\colon\widetilde{X}\operatorname{\rightarrow}X$. The pullbacks
$\nu^{*}\alpha=\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}\widetilde{\alpha}$
of classes $\alpha\in\textup{H}^{2}(X,\,\mathds{C})$ may via de Rham’s Theorem
be regarded as the classes of two-forms. We extend $\omega$ to a form
$\widetilde{\omega}\in\textup{H}^{0}(\widetilde{X},\,\Omega_{\widetilde{X}}^{2})$.
Then the _Beauville-Bogomolov form_ on $X$ is the quadratic form defined by:

$\displaystyle\begin{aligned}
q_{X,\omega}\colon&\textup{H}^{2}(X,\,\mathds{C})\operatorname{\rightarrow}\mathds{C}\\\
q_{X,\omega}(\alpha)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\;&n\int_{\widetilde{X}}(\widetilde{\omega}\overline{\widetilde{\omega}})^{n-1}\widetilde{\alpha}^{2}+(1-2n)\Big{(}\int_{\widetilde{X}}\widetilde{\omega}^{n}\overline{\widetilde{\omega}}^{n-1}\widetilde{\alpha}\Big{)}\cdot\Big{(}\int_{\widetilde{X}}\widetilde{\omega}^{n-1}\overline{\widetilde{\omega}}^{n}\widetilde{\alpha}\Big{)}\end{aligned}$

The products and powers denote here the wedge products of forms on
$\widetilde{X}$.

The Beauville-Bogomolov form does not depend on the chosen resolution. Given
two resolutions $\nu,\,\nu^{\prime}$ of $X$, this can be seen by going over to
a common resolution factoring through $\nu$ and $\nu^{\prime}$.

This definition is well suited for calculations. However, the resolution
$\widetilde{X}$ does not need to be symplectic again. Hence we cannot
trivially deduce all properties of $q_{X,\omega}$ from the smooth case by
considering the resolution $\widetilde{X}$.

###  3.2. Kirschner’s generalization

Kirschner defines the Beauville-Bogomolov form for irreducible symplectic
varieties $X$ by directly integrating singular cohomology classes on the
singular variety $X$ as in Section 2.3. This is similar to Beauville’s formula
and turns out to be equivalent to Namikawa’s Definition 19. It is more
suitable for generalizing properties like the index and the Fujiki relations
of the Beauville-Bogomolov form to irreducible symplectic varieties.

######  Definition 20 (Beauville-Bogomolov form, cf. [Kir15, Definition
3.2.7]).

Let $X$ be a $2n$-dimensional, irreducible, complex projective variety. Then
we can define for any class ${w\in\textup{H}^{2}(X,\,\mathds{C})}$ a quadratic
form

$\displaystyle q_{X,w}\colon$
$\displaystyle\textup{H}^{2}(X,\,\mathds{C})\operatorname{\rightarrow}\mathds{C}$
$\displaystyle
q_{X,w}(v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\;$
$\displaystyle
n\int_{X}(w\overline{w})^{n-1}v^{2}+(1-2n)\Big{(}\int_{X}w^{n}\overline{w}^{n-1}v\Big{)}\cdot\Big{(}\int_{X}w^{n-1}\overline{w}^{n}v\Big{)}.$

Here the products and powers denote the cup product in the cohomology ring
$\textup{H}^{*}(X,\,\mathds{C})$. Using our notation from Remark 9, we can
consider on an irreducible symplectic variety $(X,\omega)$ the class
$\underline{\omega}\in\textup{H}^{2}(X,\,\mathds{C})$. The quadratic form
$q_{X,\underline{\omega}}$ is called _Beauville-Bogomolov form_ on $X$.

We want to relate Namikawa’s and Kirschner’s definitions of the Beauville-
Bogomolov form. For this, we prove in the following lemma that Definition 20
behaves well under pullbacks along birational morphisms.

######  Lemma 21.

Let ${\pi\colon Y\operatorname{\rightarrow}X}$ be a birational morphism from a
normal, complex projective variety $Y$ to a symplectic variety $(X,\omega)$
with an extension $\omega^{\prime}\in\textup{H}^{0}(Y,\,\Omega^{[2]}_{Y})$ of
$\omega$. Then we have
$\pi^{*}\underline{\omega}=\underline{\omega^{\prime}}\in\textup{H}^{2}(Y,\,\mathds{C})$
and therefore

$\displaystyle
q_{Y,\pi^{*}\underline{\omega}}=q_{Y,\underline{\omega^{\prime}}}\quad\text{and}\quad
q_{X,\underline{\omega}}=q_{Y,\pi^{*}\underline{\omega}}\circ\pi^{*}.$

###### Proof.

We take a resolution ${\nu\colon\widetilde{X}\operatorname{\rightarrow}X}$
that factors as
$\widetilde{X}\stackrel{{\scriptstyle\widetilde{\pi}}}{{\rightarrow}}Y\stackrel{{\scriptstyle\pi}}{{\rightarrow}}X$.
Then the extensions of $\omega$ and $\omega^{\prime}$ to $\widetilde{X}$ are a
unique form
$\widetilde{\omega}\in\textup{H}^{0}(\widetilde{X},\,\Omega^{2}_{X})$. Thus
$\widetilde{\pi}^{*}(\pi^{*}\underline{\omega})$ and
$\widetilde{\pi}^{*}(\underline{\omega^{\prime}})$ are both equal to the class
of $\widetilde{\omega}$ in $\textup{H}^{2}(\widetilde{X},\,\mathds{C})$. Due
to Theorem 8, the map $\widetilde{\pi}^{*}$ is bijective on
$\textup{H}^{2,0}(Y)$. Hence we get
$\pi^{*}\underline{\omega}=\underline{\omega^{\prime}}$.

We noted in Lemma 7 that the integration of top cohomology classes is
compatible with pullbacks under birational morphisms. For any
$v,w\in\textup{H}^{2}(X,\,\mathds{C})$ we can apply this to the occurring
integrals in Definition 20 of the Beauville-Bogomolov form, such that we get
$q_{X,w}(v)=q_{Y,\pi^{*}w}(\pi^{*}v)$ for all
$v\in\textup{H}^{2}(X,\,\mathds{C})$. ∎

######  Corollary 22 (Equivalence of the definitions).

On any irreducible symplectic variety $(X,\omega)$ Namikawa’s and Kirschner’s
definitions of the Beauville-Bogomolov form are equivalent in terms of

$q_{X,\omega}=q_{X,\underline{\omega}}.$

###### Proof.

Let $\widetilde{\omega}$ be the extension of $\omega$ to a resolution
$\nu\colon\widetilde{X}\operatorname{\rightarrow}X$. Using the notation of
Kirschner’s Definition 20, we can write Namikawa’s Definition 19 as
${q_{X,\omega}=q_{\widetilde{X},\widetilde{\underline{\omega}}}\circ\nu^{*}}$.
This equals $q_{X,\underline{\omega}}$ by Lemma 21. ∎

###  3.3. Normed Beauville-Bogomolov form

The Beauville-Bogomolov form on an irreducible symplectic variety $(X,\omega)$
depends on the symplectic form. It is convenient to norm $\omega$ in order to
make the Beauville-Bogomolov form on $X$ a unique form $q_{X}$. This is done
in a way, such that we will later obtain the Fujiki relations in Theorem 2. It
can be proven exactly like in the smooth case, [Bea83, Theorem 5a], that the
restriction of $q_{X}$ gives up to a positive real factor an integral form
$\textup{H}^{2}(X,\,\mathds{Z})\operatorname{\rightarrow}\mathds{Z}$. In this
subsection we use the following notation:

######  Notation 23.

On a $2n$-dimensional, irreducible symplectic variety $X$ denote

$\displaystyle
I\colon\textup{H}^{2}(X,\,\mathds{C})\operatorname{\rightarrow}$
$\displaystyle\;\mathds{R}^{+}$ $\displaystyle w\mapsto$
$\displaystyle\int_{X}(w\overline{w})^{n}.$

Furthermore, we write
$I(\omega)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=I(\underline{\omega})$
for any symplectic class $\omega$ on $X$.

######  Lemma 24 (Existence and uniqueness of the norming).

Let $X$ be a $2n$-dimensional, irreducible symplectic variety $X$. Any
symplectic class
${\omega\in\textup{H}^{2}(X,\,\Omega^{[2]}_{X})\setminus\\{0\\}}$ induces a
_normed class_
${\omega^{\prime}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=I(\omega)^{-\frac{1}{2n}}\cdot\omega}$
with $I(\omega^{\prime})=1$. The corresponding Beauville-Bogomolov form
$q_{X,\omega^{\prime}}$ does not depend on the choice of $\omega$.

###### Proof.

As $X$ is irreducible symplectic, any symplectic class
$\alpha\in\textup{H}^{0}(X,\,\Omega^{[2]}_{X})\setminus\\{0\\}$ on $X$ differs
only by a constant factor from the class $\omega$, say $\alpha=c\omega$ for
$c\in\mathds{C}^{\times}$. We see $I(c\omega)=|c|^{2n}I(\omega)$ and therefore

$\displaystyle I(\omega^{\prime})$
$\displaystyle=I\big{(}I(\omega)^{-\frac{1}{2n}}\cdot\omega\big{)}=I(\omega)^{-1}\cdot
I(\omega)=1,$ $\displaystyle(c\omega)^{\prime}$
$\displaystyle=I(c\omega)^{-\frac{1}{2n}}\cdot
c\omega=\frac{c}{|c|}I(\omega)^{-\frac{1}{2n}}\cdot\omega=\frac{c}{|c|}\omega^{\prime}.$

Hence changing $\omega$ merely changes its normed class $\omega^{\prime}$ by a
constant of absolute value one. We see directly from Definition 19 that this
does not affect $q_{X,\omega^{\prime}}$.∎

######  Definition 25 (Normed Beauville-Bogomolov form).

Let $X$ be an irreducible symplectic variety. The _normed Beauville-Bogomolov
form_ is defined as
$q_{X}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=q_{X,\omega}$
for any symplectic form $\omega$ with $I(\omega)=1$.

Lemma 24 shows that Definition 25 and the notation $q_{X}$ make sense, as this
is a unique quadratic form, depending only on $X$. We show that this norming
is compatible with pullbacks like in Lemma 21.

######  Lemma 26 (Pullback of $q_{X}$).

Let ${\pi\colon Y\operatorname{\rightarrow}X}$ be a birational morphism of
irreducible symplectic varieties with normed Beauville-Bogomolov forms
$q_{X},\,q_{Y}$ on $X,\,Y$. If $q_{X}=q_{X,\omega}$, then
$q_{Y}=q_{Y,\omega^{\prime}}$ for the extension $\omega^{\prime}$ of $\omega$
to $Y$ and we have ${q_{X}=q_{Y}\circ\pi^{*}}$.

###### Proof.

We write $q_{X}=q_{X,\omega}$ and
$w\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\underline{\omega}$
for a normed symplectic class
${\omega\in\textup{H}^{0}(X,\,\Omega^{[2]}_{X})}$ with $I(w)=1$. We get
${\int_{Y}\pi^{*}(w\overline{w})^{n}=\int_{X}(w\overline{w})^{n}=1}$ by Lemma
7. Hence $\pi^{*}w$ is also normed in $\textup{H}^{2}(Y,\,\mathds{C})$, so
$q_{Y}=q_{Y,\pi^{*}w}$. Now Lemma 21 gives us
${q_{Y}=q_{Y,\pi^{*}\underline{\omega}}=q_{Y,\omega^{\prime}}}$ and

$q_{X}=q_{X,w}=q_{Y,\pi^{*}w}\circ\pi^{*}=q_{Y}\circ\pi^{*}.\qed$

###  3.4. About the bilinear form and Matsushita’s notation

The Beauville-Bogomolov form $q_{X}$ on an irreducible symplectic variety
$(X,\omega)$ is induced by a symmetric bilinear form. Like Matsushita we also
denote it by $q_{X}$, but with two arguments.

######  Remark 27.

We write the normed Beauville-Bogomolov form on an irreducible symplectic
variety $X$ as $q_{X}=q_{X,w}$ for a normed
$w\in\textup{H}^{2}(X,\,\mathds{C})$. Due to linear algebra we have for all
$a,b\in\textup{H}^{2}(X,\,\mathds{C})\colon$

$\displaystyle q_{X}(a,b)$
$\displaystyle=\frac{1}{2}\big{(}q_{X}(a+b)-q_{X}(a)-q_{X}(b)\big{)}$
$\displaystyle=n\int_{X}(w\overline{w})^{n-1}ab$
$\displaystyle\quad+\frac{1-2n}{2}\Big{(}\int_{X}w^{n}\overline{w}^{n-1}a\int_{X}w^{n-1}\overline{w}^{n}b+\int_{X}w^{n}\overline{w}^{n-1}b\int_{X}w^{n-1}\overline{w}^{n}a\Big{)}.$

In his proofs in [Mat01], Matsushita plugs into the Beauville-Bogomolov form
objects of different types: two-forms derived from the symplectic form
$\omega$, classes in $\textup{H}^{2}(X,\,\mathds{C})$, Cartier divisors
$D\in\operatorname{Div}(X)$ and sums of these types. This should be understood
via the Hodge decomposition of $\textup{H}^{2}(X,\,\mathds{C})$ from Theorem 8
by identifying $\textup{H}^{2,0}(X,\,\mathds{C})$ with the reflexive two-forms
and Cartier divisors with their first Chern classes in $\textup{H}^{1,1}(X)$.
So for example ${q_{X}(\omega+\overline{\omega}+D)}$ in Matsushita’s notation
should be understood as
$q_{X}\Big{(}\underline{\omega}+\underline{\overline{\omega}}+c_{1}\big{(}\mathcal{O}_{X}(D)\big{)}\Big{)}$
in our notation, where $q_{X}$ is the normed Beauville-Bogomolov form from
Definition 25. To avoid confusion, we will only apply $q_{X}$ to classes in
$\textup{H}^{2}(X,\,\mathds{C})$, as we defined it.

##  4\. Proofs of the main results

###  4.1. Fujiki relations and the index of the Beauville-Bogomolov form

We start with the proof of Theorem 2 and show the Fujiki relations for
irreducible symplectic varieties. A natural approach would be to pass to a
resolution of singularities, but this might not be a symplectic variety
anymore. Instead, we pass to a terminal model, which is a Namikawa symplectic
variety by Proposition 13. Part (2) of Theorem 2 will follow directly from the
more precise Proposition 28 about the decomposition of
$\textup{H}^{2}(X,\,\mathds{C})$ into orthogonal subspaces, on which the
Beauville-Bogomolov form is definite. The Hodge theory of
$\textup{H}^{2}(X,\,\mathds{C})$ and the Fujiki relations allow us to
calculate the index by hand.

###### Proof of Theorem 2, part (1).

Let $X$ be a $2n$-dimensional, irreducible symplectic variety with normed
Beauville-Bogomolov form $q_{X}$ on $\textup{H}^{2}(X,\,\mathds{C})$ as in
Definition 25. We need to show that there is a constant
$c_{X}\in\mathds{R}^{+}$, such that for all
$v\in\textup{H}^{2}(X,\,\mathds{C})$ we have the Fujiki relation

$\displaystyle c_{X}\cdot q_{X}(v)^{n}=\int_{X}v^{2n}.$

We write $q_{X}=q_{X,\omega}$ for a normed symplectic class $\omega$ with
${\int_{X}(\underline{\omega\overline{\omega}})^{n}=1}$. By Proposition 13,
every terminal model $\pi\colon Y\operatorname{\rightarrow}X$ of $X$ becomes,
together with an extension
$\omega^{\prime}\in\textup{H}^{2}(Y,\,\Omega_{Y}^{[2]})$ of $\omega$ from
Theorem 5, a Namikawa symplectic variety. We have
$q_{Y}=q_{Y,\omega^{\prime}}$ and $q_{X}=q_{Y}\circ\pi^{*}$ by Lemma 26.

Taking such a Namikawa symplectic terminal model $(Y,\omega^{\prime})$, we can
apply [Mat01, Theorem 1.2] to $Y$ to get a $c_{Y}\in\mathds{R}^{+}$, such that
for all $u\in\textup{H}^{2}(Y,\,\mathds{C})$:

(1) $\displaystyle c_{Y}\cdot q_{Y}(u)^{n}=\int_{Y}u^{2n}.$

Therefore we get for any $v\in\textup{H}^{2}(X,\,\mathds{C})$:

$\displaystyle c_{Y}\cdot q_{X}(v)^{n}$ $\displaystyle=c_{Y}\cdot
q_{Y}(\pi^{*}v)^{n}$ By Lemma 26 $\displaystyle=\int_{Y}(\pi^{*}v)^{2n}$
Equation (1) for
$u\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi^{*}v$
$\displaystyle=\int_{Y}\pi^{*}(v^{2n})$ $\displaystyle=\int_{X}v^{2n}$ By
Lemma 7

Hence we have proved the Fujiki relation for the constant
$c_{X}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=c_{Y}$. ∎

######  Proposition 28 (Index of the Beauville Bogomolov form $q_{X}$).

Let $(X,\omega)$ be a $2n$-dimensional, irreducible symplectic variety with
normed Beauville-Bogomolov form $q_{X}=q_{X,w}$ for
$w\in\textup{H}^{2}(X,\,\mathds{C})$ and an ample class
$a\in\textup{H}^{2}(X,\,\mathds{R})$. Restricting $q_{X}$ gives a real
quadratic form
$\textup{H}^{2}(X,\,\mathds{R})\operatorname{\rightarrow}\mathds{R}$ and
$\textup{H}^{2}(X,\,\mathds{R})$ splits into the following $q_{X}$-orthogonal
subspaces

$\displaystyle V_{+}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\left\langle
w+\overline{w},\,iw-i\overline{w},\,a\right\rangle_{\mathds{R}},$
$\displaystyle V_{-}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\;\\{d\in\textup{H}^{1,1}(X)\cap\textup{H}^{2}(X,\mathds{R})\mid
q_{X}(d,a)=0\\}$ $\displaystyle=$
$\displaystyle\;a^{\perp}\cap\textup{H}^{1,1}(X)\cap\textup{H}^{2}(X,\,\mathds{R}),$

such that $q_{X}$ is positive definite on $V_{+}$, negative definite on
$V_{-}$ and $\dim_{\mathds{R}}V_{+}=3$.

###### Proof that restricting $q_{X}$ gives a real form.

An easy computation for arbitrary ${v\in\textup{H}^{2}(X,\mathds{C})}$ shows
${q_{X,w}\big{(}\overline{v}\big{)}=\overline{q_{X,w}(v)}}$, using that the
integrals are compatible with complex conjugation. Hence restricting gives a
real quadratic form
$\textup{H}^{2}(X,\,\mathds{R})\operatorname{\rightarrow}\mathds{R}$.

#### Proof of $V_{+}\perp V_{-}$ and the positivity of $q_{X}$ on $V_{+}$

We use $\int_{X}(w\overline{w})^{n}=1$ and the formula from Remark 27 to
calculate the Beauville-Bogomolov bilinear form $q_{X}$ on the classes
$w+\overline{w},\,iw-i\overline{w},\,a$. The Hodge decomposition from Theorem
8 allows us to calculate the occurring integrals via the usual type
considerations.

$\displaystyle q_{X}(w+\overline{w})$
$\displaystyle=q_{X}(iw-i\overline{w})=2n+\frac{1-2n}{2}(1+1)=1$
$\displaystyle q_{X}(w+\overline{w},\,iw-i\overline{w})$
$\displaystyle=(-i+i)n+\frac{1-2n}{2}(i-i)=0$ $\displaystyle\forall
v\in\textup{H}^{1,1}(X)\colon q_{X}(w+\overline{w},\,v)$
$\displaystyle=q_{X}(iw-i\overline{w},\,v)=0$ $\displaystyle q_{X}(a)$
$\displaystyle=\int_{X}(w\overline{w})^{n-1}a^{2}$

Therefore $w+\overline{w},\,iw-i\overline{w},\,a$ are orthogonal with respect
to $q_{X}$. The two subspaces $V_{+}$ and $V_{-}$ of
$\textup{H}^{2}(X,\,\mathds{C})$ are likewise orthogonal. The restriction
${\left.\kern-1.2ptq_{X}\vphantom{\big{|}}\right|_{V_{+}}}$ is diagonal with
eigenvalues $1,\,1,\,q_{X}(a)$. As $w^{n-1}\cup a^{3}=0$ by type
considerations, the Hodge-Riemann bilinear relations, Corollary 10, show that
$q_{X}(a)=\psi_{X,a}(w^{n-1},w^{n-1})$ is positive and therefore $q_{X}$
positive definite on $V_{+}$.

#### Proof that $V_{+}\oplus V_{-}=\textup{H}^{2}(X,\,\mathds{R})$.

The classes $w+\overline{w},\,iw-i\overline{w}$ are real because they are
invariant under complex conjugation. The class $a$ is real by the definition
of $c_{1}$. This gives $V_{+}+V_{-}\subset\textup{H}^{2}(X,\,\mathds{R})$.

On the other hand, as $X$ is irreducible symplectic,
$\textup{H}^{2,0}(X),\,\textup{H}^{0,2}(X)$ are spanned over $\mathds{C}$ by
$w,\,\overline{w}$. Hence any real class $v\in\textup{H}^{2}(X,\,\mathds{R})$
can be decomposed as ${v=\alpha w+v^{1,1}+\beta\overline{w}}$ with
$\overline{\alpha}=\beta\in\mathds{C}$ and
${v^{1,1}=\overline{v^{1,1}}\in\textup{H}^{1,1}(X)\cap\textup{H}^{2}(X,\,\mathds{R})}$.
We note ${q_{X}(v^{1,1}-\lambda a,a)=0}$ for
${\lambda\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{q_{X}(v^{1,1},a)}{q_{X}(a)}\in\mathds{R}}$.
This gives the decomposition ${v=v_{+}+v_{-}}$ into the following two parts.

$\displaystyle v_{+}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=Re(\alpha)\cdot(w+\overline{w})+Im(\alpha)\cdot(iw-i\overline{w})+\lambda
a$ $\displaystyle\in V_{+}$ $\displaystyle v_{-}$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=v^{1,1}-\lambda
a$ $\displaystyle\in V_{-}$

Hence we get ${V_{+}+V_{-}=\textup{H}^{2}(X,\,\mathds{R})}$. The directness of
this sum decomposition will follow from the negativity of $q_{X}$ on $V_{-}$.

#### Proof of the negativity of $q_{X}$ on $V_{-}$.

Any class $d$ in the space $V_{-}$ is a real class of type $(1,1)$ with
$q_{X}(d,a)=0$. We need to show $q_{X}(d)<0$ if $d\neq 0$. We use the Fujiki
relations on $X$ to calculate for any $t\in\mathds{R}$:

$\int_{X}(td+a)^{2n}=c_{X}\cdot
q_{X}(td+a)^{n}=c_{X}\cdot\big{(}t^{2}q_{X}(d)+\underbrace{tq_{X}(d,a)}_{=0}+q_{X}(a)\big{)}^{n}.$

Comparing the $t$ and $t^{2}$-terms on both sides yields

$\displaystyle d\cup a^{2n-1}$ $\displaystyle=0,$
$\displaystyle(2n-1)\cdot\int_{X}(d^{2}\cup a^{2n-2})$
$\displaystyle=c_{X}\cdot q_{X}(d)\cdot q_{X}(a)^{n-1}.$

We have $d=\overline{d}$ because $d$ is real. Therefore the Hodge-Riemann
bilinear relations, Corollary 10, imply $\int_{X}(d^{2}\cup
a^{2n-2})=\psi_{X,a}(d,d)<0$. This proves

$q(d)=\frac{2n-1}{c_{X}\cdot q_{X}(a)^{n-1}}\cdot\psi_{X,a}(d,d)\\\ <0.\qed$

###  4.2. Fibrations of irreducible symplectic varieties

Let $(X,\omega)$ be an irreducible symplectic variety of complex dimension
$2n$, together with a surjective morphism $f\colon
X\operatorname{\rightarrow}B$ with connected fibers onto a normal, complex
projective variety $B$ with dimension $0<\dim B<2n$. We subdivide the proof of
Theorem 3 into seven steps, which we will prove in the following order:

  * (1a)

$\dim B=n$.

  * (4a)

The general fiber is a Lagrangian subvariety of $X$.

  * (4b)

Every fiber component of $f$ is an $n$-dimensional Lagrangian subvariety of
$X$ and does not lie completely in $X_{sing}$.

  * (3a)

The general fiber is smooth.

  * (3b)

The general fiber is an Abelian variety.

  * (2)

$X$ is smooth along the general fiber and $f(X_{sing})\subset B$ is a proper
closed subset.

  * (1b)

$B$ is a $\mathds{Q}$-factorial klt variety with $\rho(B)=1$.

###### Proof of Theorem 3.

For the proof we choose very ample divisors $A$ on $X$ and $H$ on $B$. We
denote their first Chern classes by $a\in\textup{H}^{2}(X,\,\mathds{C})$ and
$h\in\textup{H}^{2}(B,\,\mathds{C})$.

#### Proof of ((1a)).

For any class ${d=c_{1}\big{(}\mathcal{O}_{B}(D)\big{)}}$ coming from a
Cartier divisor $D$ on $B$ we get $(f^{*}D)^{2n}=0$. Moreover, the Fujiki
relation $(f^{*}D)^{2n}=c_{X}\cdot q_{X}(f^{*}d)^{n}$ gives

(2) $q_{X}(f^{*}d)=0.$

Furthermore we calculate for any $t\in\mathds{R}^{+}$:

$\displaystyle(A+tf^{*}H)^{2n}$ $\displaystyle=c_{X}\cdot
q_{X}(a+tf^{*}h)^{n}$
$\displaystyle=c_{X}\cdot\big{(}q_{X}(a)+2tq_{X}(a,f^{*}h)+t^{2}\underbrace{q_{X}(f^{*}h)}_{=0}\big{)}^{n}$

We see from the $t$-coefficient that $q_{X}(a,f^{*}h)$ is positive because $A$
is ample:

(3) $q_{X}(a,f^{*}h)=\frac{A^{2n-1}.f^{*}H}{c_{X}\cdot q_{X}(a)^{n-1}}>0.$

Therefore the $t^{k}$-coefficient $\binom{2n}{k}\cdot A^{2n-k}.(f^{*}H)^{k}$
vanishes for $k>n$, but survives for $k=n$:

$\binom{2n}{n}\cdot A^{n}.(f^{*}H)^{n}=2^{n}c_{X}\cdot q_{X}(a,f^{*}h)^{n}>0.$

This shows $\dim B=n$.

#### Proof of ((4a)).

We consider a general fiber $F$ over a smooth point $b\in B$ and the embedding
${i\colon F\operatorname{\hookrightarrow}X}$. By Lemma 31 in the appendix the
general fiber $F$ fulfills ${F_{reg}=F\cap X_{reg}}$ and has at most canonical
singularities. Therefore, considering the symplectic form $\omega$ as a
holomorphic form on the smooth locus $X_{reg}$, we can restrict $\omega$ to
$F\cap X_{reg}$ and obtain a reflexive form on $F$, which we denote by
${{\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{F}}\in\textup{H}^{2}(F,\,\Omega^{[2]}_{F})}$.
By Lemma 16 we need to prove
${\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{F}}=0$ to show that $F$ is
a Lagrangian subvariety of $X$.

We note $\underline{\omega|_{F}}=i^{*}w$ for
$w\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\underline{\omega}$
and that $i^{*}a$ is an ample class on $F$. The idea is now to show

(4) $\int_{F}i^{*}(w\overline{w}a^{n-2})=0.$

This integral equals $\psi_{F,i^{*}a}(i^{*}w,i^{*}w)$ and by the Hodge-Riemann
bilinear relations, Corollary 10, it vanishes if and only if
${\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{F}}=0$, otherwise it is
positive.

As $X,F,B$ are compact, complex varieties, they have canonical fundamental
classes $[X]\in\textup{H}_{4n}(X,\,\mathds{Z})$,
$[F]\in\textup{H}_{2n}(F,\,\mathds{Z})$,
$[B]\in\textup{H}_{2n}(B,\,\mathds{Z})$ with duals in cohomology by section
2.3. By definition, we have $H^{n}=[B]\cap h^{n}$, so the dual to the
fundamental class of $B$ is
${[B]^{\vee}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{H^{n}}h^{n}\in\textup{H}^{2n}(B,\,\mathds{Z})}$.
A calculation with the projection formula, see Lemma 32, gives the equality

(5) $[X]\cap f^{*}\big{(}[B]^{\vee}\big{)}=i_{*}[F]$

in $\textup{H}^{2n}(X,\,\mathds{Z})$. We use this to calculate the integral
(4).

$\displaystyle\int_{F}i^{*}(w\overline{w}a^{n-2})$ $\displaystyle=[F]\cap
i^{*}(w\overline{w}a^{n-2})$ $\displaystyle=i_{*}[F]\cap w\overline{w}a^{n-2}$
Projection formula
$\displaystyle=\frac{1}{H^{n}}\big{(}[X]\cap(f^{*}h)^{n}\big{)}\cap
w\overline{w}a^{n-2}$ By equation (5)
$\displaystyle=\frac{1}{H^{n}}[X]\cap\big{(}w\overline{w}a^{n-2}\cup(f^{*}h)^{n}\big{)}$
[Bre97, Theorem VI 5.2.(3)]
$\displaystyle=\frac{1}{H^{n}}\int_{X}w\overline{w}a^{n-2}(f^{*}h)^{n}$

We calculate the latter integral as a term of the following Fujiki relation
for ${s,t\in\mathds{R}^{+}}$.

$\displaystyle\int_{X}(w+\overline{w}+sa+tf^{*}h)^{2n}$
$\displaystyle=c_{X}\cdot q_{X}(w+\overline{w}+sa+tf^{*}h)^{n}$
$\displaystyle=c_{X}\cdot\big{(}q_{X}(w+\overline{w})+s^{2}q_{X}(a)+2stq_{X}(a,f^{*}h)\big{)}^{n}$

In the last step we used the vanishing of $q_{X}(f^{*}h)$ by equation (2) and
the vanishing of $q_{X}(w+\overline{w},a)$ and $q_{X}(w+\overline{w},f^{*}h)$
that follows directly from the definition of the Beauville-Bogomolov form via
type considerations. Comparing the $s^{n-2}t^{n}$-terms gives
$\int_{X}w\overline{w}a^{n-2}(f^{*}h)^{n}=0$ and therefore
${\left.\kern-1.2pt\omega\vphantom{\big{|}}\right|_{F}}=0$.

#### Proof of ((4b)).

By part ((4a)) $f$ is a Lagrangian fibration. Hence part ((4b)) follows from
Theorem 18.

#### Proof of ((3a)).

As the symplectic form $\omega$ is nowhere degenerate, it induces at every
point ${x\in X_{reg}}$ a linear isomorphism

$\displaystyle\phi_{x}\colon\mathcal{T}_{X_{reg},x}\operatorname{\rightarrow}\;$
$\displaystyle\Omega^{1}_{X_{reg},x}$ $\displaystyle t\mapsto\;$
$\displaystyle\omega_{x}(t,\,\cdot\,).$

These isomorphisms glue together to an isomorphism of sheaves
$\phi\colon\mathcal{T}_{X_{reg}}\stackrel{{\scriptstyle\sim}}{{\operatorname{\longrightarrow}}}\Omega^{1}_{X_{reg}}$.

By Lemma 31 the general fiber $F$ is smooth at a point $x\in F$ if and only if
$x\in X_{reg}$. This enables us to use the exact sequence of the tangent
complex, [Har77, page 182],

(6)
$0\operatorname{\rightarrow}\mathcal{T}_{F_{reg}}\operatorname{\rightarrow}\mathcal{T}_{X_{reg}|F_{reg}}\operatorname{\rightarrow}\mathcal{N}_{F_{reg}|X_{reg}}\operatorname{\rightarrow}0.$

On the other hand, the differential $df$ induces a map of vector bundles

$df\colon\mathcal{T}_{X_{reg}|F_{reg}}$$\mathcal{T}_{b}B\times
F_{reg}$$\forall x\in
F_{reg},\,v\in\mathcal{T}_{X,x}\colon(v,x)$$\big{(}d_{x}f(v),x\big{)}$

where we consider ${\mathcal{T}_{b}B\times F_{reg}}$ as a trivial vector
bundle of rank $n$ over $F_{reg}$. The kernel of $df$ is exactly
$\mathcal{T}_{F_{reg}}$ and $df$ is surjective because of ${\dim B+\dim F=\dim
X}$. Using sequence (6) we get
${\mathcal{N}_{F_{reg}|X_{reg}}\cong\nicefrac{{\mathcal{T}_{X_{reg}|F_{reg}}}}{{\mathcal{T}_{F_{reg}}}}\cong\mathcal{T}_{b}B\times
F_{reg}}$, so the normal bundle $\mathcal{N}_{F_{reg}|X_{reg}}$ is also free
of rank $n$.

We restrict $\phi$ to a morphism
${\left.\kern-1.2pt\phi\vphantom{\big{|}}\right|_{F_{reg}}}$ as follows.

${\left.\kern-1.2pt\phi\vphantom{\big{|}}\right|_{F_{reg}}}\colon\mathcal{T}_{X_{reg}|F_{reg}}$$\Omega^{1}_{F_{reg}}$$\forall
s\in\mathcal{T}_{X_{reg}}\colon{\left.\kern-1.2pts\vphantom{\big{|}}\right|_{F_{reg}}}$${\left.\kern-1.2pt\phi(s)\vphantom{\big{|}}\right|_{F_{reg}}}$

The map ${\left.\kern-1.2pt\phi\vphantom{\big{|}}\right|_{F_{reg}}}$ is
surjective because $\phi$ and the restriction map
$\Omega^{1}_{X_{reg}}\operatorname{\rightarrow}\Omega^{1}_{F_{reg}}$ are
surjective by [Har77, Proposition 8.16]. We know about the kernel
${{{\mathcal{T}_{F_{reg}}}}\subset\ker\big{(}{\left.\kern-1.2pt\phi\vphantom{\big{|}}\right|_{F_{reg}}}\big{)}}$
because $F$ is a Lagrangian subvariety. As $\omega$ is non-degenerate, this
inclusion is by dimension reasons an equality. Using
${\mathcal{N}_{F_{reg}|X_{reg}}\cong\nicefrac{{\mathcal{T}_{X_{reg}|F_{reg}}}}{{\mathcal{T}_{F_{reg}}}}}$
again, we see that $\phi$ induces an isomorphism
${\psi\colon\mathcal{N}_{F_{reg}|X_{reg}}\stackrel{{\scriptstyle\sim}}{{\operatorname{\longrightarrow}}}\Omega^{1}_{F_{reg}}}$.
Hence $\Omega^{1}_{F_{reg}}$ is free of rank $n$ as well.

The sheaves $\mathcal{N}_{F_{reg}|X_{reg}}$ and
$\Omega^{1}_{F_{reg}}\cong{\left.\kern-1.2pt\Omega^{[1]}_{F}\vphantom{\big{|}}\right|_{F_{reg}}}$
are reflexive because they are duals of coherent sheaves, [Har80, Corollary
1.2]. As $F$ is normal, the isomorphism $\psi$ extends uniquely over the
singular locus $F_{sing}$ to an isomorphism
$\mathcal{N}_{F|X}\stackrel{{\scriptstyle\sim}}{{\operatorname{\longrightarrow}}}\Omega^{[1]}_{F}$
of globally free sheaves of rank $n$ on $F$, [Har80, Corollary 1.6].

Therefore the tangent sheaf
$\mathcal{T}_{F}=\big{(}\Omega^{[1]}_{F}\big{)}^{\ast}$ is globally free and
$F$ is smooth by the Lipman-Zariski conjecture, which is already proven for
the singularities of the minimal model program, [GKKP11, Theorem 6.1], [GK14,
Corollary 1.3], [Dru14, Theorem 1.1].

#### Proof of ((3b)).

As we showed in part ((3a)) that the general fiber $F$ of $f$ is a compact
Kähler manifold with globally free tangent sheaf, we may directly conclude by
[Wan54, Corollary 2] that $F$ is an Abelian variety. A more algebraic argument
is to use that we showed that $\Omega^{1}_{F}$ is globally free, so
$\operatorname{h}^{0}(F,\,\Omega^{1}_{F})=\dim F$. The Kodaira dimension
$\kappa(F)$ is zero because $K_{F}=0$. Then by [Kaw81, Corollary 2] the
Albanese morphism $\alpha\colon
F\operatorname{\rightarrow}\operatorname{Alb}(F)$ is a birational morphism to
the Abelian variety $\operatorname{Alb}(F)$. In fact $\alpha$ is an
isomorphism because $K_{F}$ is numerically effective.

#### Proof of ((2)).

As $X_{sing}$ is closed and $f$ is proper, $f(X_{sing})$ is closed too. By
part ((3a)) the general fiber $F$ of $f$ is smooth. It is completely contained
in $X_{reg}$ by Lemma 30 in the appendix. Thus $f(X_{sing})$ is a proper
closed subset of $B$.

#### Proof of ((1b)).

We may assume here that $X$ is Namikawa symplectic, as for the following
argument we can go over to a terminal model of $X$. Let $D$ be a Weil divisor
on $B$ and
$X_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f^{-1}(B_{reg})$.
As $D$ is Cartier on $B_{reg}$, we can consider the pullback
$f|^{*}_{X_{0}}({\left.\kern-1.2ptD\vphantom{\big{|}}\right|_{B_{reg}}})$.
This induces a divisor $\widetilde{D}$ on $X$ by taking the closures of its
components. Then $m\widetilde{D}$ is Cartier for an $m\in\mathds{N}_{>0}$
because $X$ is $\mathds{Q}$-factorial, and we get a class
$d\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{m}c_{1}\big{(}\mathcal{O}_{X}(m\widetilde{D})\big{)}$
in $\textup{H}^{2}(X,\mathds{C})$.

We have $\operatorname{codim}(X\setminus
X_{0})=\operatorname{codim}B_{sing}\geq 2$ because $f$ is equidimensional by
part ((4b)) and $B$ is normal. Hence no components of $f^{-1}(D)$ or $D$ lie
outside of $X_{0}$ or $B_{reg}$, respectively. This implies
$f_{*}\widetilde{D}=D$ as Weil divisors and if $D$ is Cartier also
${\widetilde{D}=f^{*}D}$ and $d=f^{*}c_{1}\big{(}\mathcal{O}_{B}(D)\big{)}$.

Using $q_{X}(f^{*}h)=0$ we calculate for $t\in\mathds{R}$ the following Fujiki
relation.

$\displaystyle(\widetilde{D}$ $\displaystyle+tf^{*}H+sA)^{2n}$
$\displaystyle=c_{X}\cdot\big{(}q_{X}(d)+s^{2}q_{X}(a)+2tq_{X}(d,f^{*}h)+2sq_{X}(d,a)+2stq_{X}(f^{*}h,a)\big{)}^{n}$

Comparing the coefficients of the constant term, of $t^{n}$ and of
$t^{n-1}s^{n-1}$ gives constants $c_{1},\ldots,c_{4}\in\mathds{Q}^{+}$ with

$\displaystyle\widetilde{D}^{2n}$ $\displaystyle=c_{1}\cdot q_{X}(d)^{n}$
$\displaystyle\widetilde{D}^{n}.(f^{*}H)^{n}$ $\displaystyle=c_{2}\cdot
q_{X}(d,f^{*}h)^{n}$ $\displaystyle\widetilde{D}^{2}.(f^{*}H)^{n-1}.A^{n-1}$
$\displaystyle=c_{3}\cdot q_{X}(d)q_{X}(f^{*}h,a)^{n-1}$
$\displaystyle+c_{4}\cdot q_{X}(d,f^{*}h)q_{X}(d,a)q_{X}(f^{*}h,a)^{n-2}.$

We show that all three intersection products vanish. For this we may
substitute the factors $H$ by general hyperplane sections $H_{i}$ of $B$. By
Bertini’s Theorem $H_{1},\ldots,H_{n-1}$ intersect in a smooth curve $C\subset
B_{reg}$ that meets $D$ transversally in finitely many points in $B_{reg}$,
[Aki51]. Thus the intersection products $\widetilde{D}^{n}.(f^{*}H)^{n}$ and
$\widetilde{D}^{2}.(f^{*}H)^{n-1}.A^{n-1}$ can be calculated over $B_{reg}$.
As $X$ is Cohen Macaulay and $f$ equidimensional, the restriction
${\left.\kern-1.2ptf\vphantom{\big{|}}\right|_{X_{0}}}$ is flat, [Eis95,
Theorem 18.16]. By [Ful98, Proposition 2.5d] we can calculate

$\displaystyle\widetilde{D}^{n}.(f^{*}H)^{n}$
$\displaystyle=f^{*}(D|_{B_{reg}}^{n}.H|_{B_{reg}}^{n})=0,$
$\displaystyle\widetilde{D}^{2}.(f^{*}H)^{n-1}.A^{n-1}$
$\displaystyle=f^{*}(D|_{B_{reg}}^{2}.H|_{B_{reg}}^{n-1}).A|_{X_{0}}^{n-1}=0$

because $\dim B=n$. Hence $q_{X}(d,f^{*}h)=0$, and due to $q_{X}(f^{*}h,a)>0$
also ${q_{X}(d)=0}$, which gives $\widetilde{D}^{2n}=0$.

The vanishing of $q_{X}(d,f^{*}h)$, $q_{X}(d)$, $q_{X}(f^{*}h)$ give for any
$\lambda\in\mathds{R}$ the Fujiki relation $(\widetilde{D}-\lambda
f^{*}H)^{2n}=c_{X}\cdot q_{X}(d-\lambda f^{*}h)^{n}=0$. For
$\lambda\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\widetilde{D}.A^{2n-1}}{f^{*}H.A^{2n-1}}\in\mathds{Q}$
we get also $(\widetilde{D}-\lambda f^{*}H).A^{2n-1}=0$, thus the $t^{2n-1}$
terms of the Fujiki relation

$(\widetilde{D}-\lambda f^{*}H+tA)^{2n}=c_{X}\cdot(2tq_{X}(d-\lambda
f^{*}h,a)+t^{2}q_{X}(a))^{n}$

vanish. So $q_{X}(d-\lambda f^{*}h,a)=0$ and therefore $d-\lambda f^{*}h\in
a^{\perp}\cap\textup{H}^{1,1}(X)\cap\textup{H}^{2}(X,\,\mathds{R})$. The form
$q_{X}$ is by Proposition 28 negative definite on this space, so $d=\lambda
f^{*}h$ in $\textup{H}^{2}(X,\,\mathds{Q})$ and $\widetilde{D}\equiv\lambda
f^{*}H$.

If $D$ is Cartier, we get $D\equiv\lambda H$ by the projection formula, so
$\rho(B)=1$. In the general case we show that $D$ is _numerically
$\mathds{Q}$-Cartier_ in the sense of [HMP15]. This means that we have to
construct for any resolution
$\mu\colon\widetilde{B}\operatorname{\rightarrow}B$ a $\mu$-trivial
$\mathds{Q}$-Cartier divisor $D^{\prime}$ on $\widetilde{B}$ with
$\mu_{*}D^{\prime}=D$.

For any resolution $\mu\colon\widetilde{B}\operatorname{\rightarrow}B$ the
fiber product $X\times_{B}\widetilde{B}$ consists of the exceptional
components of $\mu$ and a component $Y$ that is isomorphic to the graph of $f$
and lies birational over $X$. Hence we can construct a commutative diagram of
resolutions

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by taking $\widetilde{X}$ to be a resolution of $Y$, together with the
morphisms $\nu,\widetilde{f}$ induced by the projections from the fiber
product onto $X,B$. We consider
$D^{\prime}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\widetilde{f}_{*}\nu^{*}\widetilde{D}$.
As
$\nu^{*}\widetilde{D}\equiv\lambda\nu^{*}f^{*}H=\lambda\widetilde{f}^{*}\mu^{*}H$,
we have $D^{\prime}\equiv\lambda\mu^{*}H$, which is $\mu$-trivial with
${\mu_{*}D^{\prime}=\mu_{*}\widetilde{f}_{*}\nu^{*}\widetilde{D}=f_{*}\nu_{*}\nu^{*}\widetilde{D}=D}$,
so $D$ is numerically $\mathds{Q}$-Cartier.

Fujino’s result [Fuj99, Theorem 1.2] shows that $B$ is a klt variety, so it
has rational singularities. Hence by [HMP15, Theorem 5.11] $D$ is
$\mathds{Q}$-Cartier, so $B$ is $\mathds{Q}$-factorial. ∎

###  4.3. Fibrations of primitive symplectic varieties

We give an example showing that both parts of Theorem 4 can fail for an
irreducible symplectic variety. The idea is due to Matsushita and was worked
out by Sawon, [Mat01, p. 7f], [Saw14, Lemma 15].

######  Example 29.

Let $E$ be the elliptic curve $\mathds{C}/\Gamma$ with the lattice
$\Gamma=\mathds{Z}+\mathds{Z}\cdot\zeta_{6}$, where $\zeta_{6}$ is a primitive
sixth root of unity. Then
$T\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=E^{6}$ is a complex
$6$-torus. The maps

$\displaystyle\mathds{C}^{6}\operatorname{\rightarrow}$
$\displaystyle\;\mathds{C}^{6}$ $\displaystyle(z_{1},\ldots,z_{6})\mapsto$
$\displaystyle\;(\zeta_{6}z_{1},\zeta_{6}^{5}z_{2},-z_{3},-z_{4},\zeta_{6}^{2}z_{5},\zeta_{6}^{4}z_{6})$
$\displaystyle(z_{1},\ldots,z_{6})\mapsto$
$\displaystyle\;(z_{4},z_{5},z_{1},z_{2},z_{3},z_{4})$

generate a group of automorphisms of $\mathds{C}^{6}$ preserving $\Gamma$,
which induces a subgroup $G\subset\operatorname{Aut}T$. Then
$X\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T/G$ is an
irreducible symplectic variety with
${\omega\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=dz_{1}\wedge
dz_{2}+dz_{3}\wedge dz_{4}+dz_{5}\wedge dz_{6}}$. However, it is not primitive
symplectic, as it carries holomorphic 3-forms. The group $G$ also acts on the
subtorus $T^{\prime}$ with coordinates $(z_{1},z_{3},z_{5})$ and the quotient
$B\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T^{\prime}/G$ is a
Calabi-Yau manifold because the only holomorphic differential forms on $Y$
come from $dz_{1}\wedge dz_{3}\wedge dz_{5}$. Hence the projection $f\colon
X\operatorname{\rightarrow}B$ is a Lagrangian fibration like in Theorem 3, but
over a smooth base variety $B$ with $K_{B}=0$, so both parts of Theorem 4 fail
for $f$.

###### Proof of Theorem 4.

Let $(X,\omega)$ be a primitive symplectic variety with a morphism $f\colon
X\operatorname{\rightarrow}B$ like in Theorem 4. We have to show the following
two parts.

  1. (1)

The base variety $B$ is Fano.

  2. (2)

If $B$ is smooth, then $B\cong\mathds{P}^{n}$.

#### Proof of (1).

As $B$ is $\mathds{Q}$-factorial with $\rho(B)=1$ by Theorem 3, the canonical
divisor $K_{B}$ is $\mathds{Q}$-Cartier with $K_{B}\equiv tH$ for a
$t\in\mathds{Q}$. By the Iitaka conjecture, which due to Kawamata is known to
hold in this special case, see Theorem 34 in the appendix, we have
$\kappa(B)\leq 0$, so $t\leq 0$. Therefore we only need to exclude the case
$t=0$.

We assume that $K_{B}$ is numerically trivial. Applying [Kaw85, Theorem 8.2]
to a terminal model of $B$ shows that there is a ${d\in\mathds{N}_{>0}}$, such
that $\mathcal{O}_{B}(dK_{B})\cong\mathcal{O}_{B}$ is trivial. As the
canonical sheaf is a reflexive sheaf that is locally free on $X_{reg}$, we can
go over to a ramified cyclic covering like in [KM98, 2.52–2.53]. This is a
finite morphism $p\colon\hat{B}\operatorname{\rightarrow}B$ with
${p^{*}\mathcal{O}_{B}(K_{B})=\mathcal{O}_{\hat{B}}}$ that is étale of degree
$d$ over $X_{reg}$ and possibly branched over $X_{sing}$. Hence $p$ is quasi-
étale because $X$ is normal. The variety $\hat{B}$ is by [KM98, Proposition
5.20] also a klt variety. We consider the fiber product $X\times_{B}\hat{B}$
and take the normalization $\hat{X}$ of any component that lies over
$X_{reg}$. The projections from the fiber product give a lift
$\hat{f}\colon\hat{X}\operatorname{\rightarrow}\hat{B}$ of the morphism $f$
with $K_{\hat{B}}=0$ and a quasi-étale cover
$\pi\colon\hat{X}\operatorname{\rightarrow}X$. There is a non-zero reflexive
form $\alpha\in\textup{H}^{0}(\hat{B},\,\Omega^{[n]}_{\hat{B}})$ and, as
$\hat{B}$ is a klt variety, we can pull it back by Theorem 5 to a non-zero
reflexive form
${\hat{f}^{*}\alpha\in\textup{H}^{0}(\hat{X},\,\Omega^{[n]}_{\hat{X}})}$. As
$\hat{f}^{*}\alpha$ is not zero, $n$ has to be even and $\hat{f}^{*}\alpha$ a
multiple of $\pi^{*}(\omega)^{n/2}$ because $X$ is primitive symplectic. This
is a contradiction because $\hat{f}^{*}\alpha^{2}=0$, but $\pi^{*}\omega^{n}$
has no zeros.

#### Proof of (2).

As $X$ is primitive symplectic, any terminal model $\pi\colon
Y\operatorname{\rightarrow}X$ of $X$ is also primitive symplectic by
Proposition 14. Then
${\operatorname{h}^{p}(Y,\,\mathcal{O}_{Y})=\operatorname{h}^{0}(Y,\,\Omega_{X}^{[p]})}$
for all $p$ by Hodge symmetry, [GKP16, Proposition 6.9], so $Y$ is
_cohomologically irreducible symplectic_ in Matsushita’s terminology, [Mat15,
Definition 1.6]. Applying Matsushita’s result [Mat15, Theorem 1.10] to
$f\circ\pi\colon Y\operatorname{\rightarrow}B$ shows $B\cong\mathds{P}^{n}$ if
$B$ is smooth. ∎

## Appendix

For the reader’s convenience and to have a good reference, we explain here
some probably well known results used in the main proofs above. Lemma 30 can
be seen as a trivial case of [Kol96, Theorem I.6.5]. Theorem 33 and 34 are
easy but useful consequences of results of Kollár and Kawamata.

######  Lemma 30 (Smoothness at general fibers).

Let $f\colon X\operatorname{\rightarrow}Y$ be a surjective morphism of complex
varieties with a general fiber $F$. If $F$ is smooth at a point $x\in F$, then
$X$ is smooth at $x$.

###### Proof.

Let $X,\,Y$ be $n+r$ and $n$-dimensional, respectively. A general fiber $F$
lies over a smooth point $y\in Y$ and has the dimension $r$ by the fiber
dimension theorem. Consider a smooth point $x\in F$ with maximal ideals
${\mathfrak{m}_{X,x}\subset\mathcal{O}_{X,x}}$ and
${\mathfrak{m}_{F,x}\subset\mathcal{O}_{F,x}}$ and the maximal ideal
$\mathfrak{m}_{Y,y}\subset\mathcal{O}_{Y,y}$ of $y$. As $y\in Y$ and $x\in F$
are regular, we can write ${\mathfrak{m}_{F,x}=(\phi_{1},\ldots,\phi_{r})}$
and ${\mathfrak{m}_{Y,y}=(\psi_{1},\ldots,\psi_{n})}$. The ideal sheaf
sequence for $F\subset X$ allows us to extend the $\phi_{i}$ to functions
${\overline{\phi_{i}}\in\mathfrak{m}_{X,x}\subset\mathcal{O}_{X,x}}$. Then it
can easily be checked that we get a short exact sequence

$0\operatorname{\rightarrow}f^{*}\mathfrak{m}_{Y,y}\operatorname{\rightarrow}\mathfrak{m}_{X,x}\operatorname{\rightarrow}\mathfrak{m}_{F,x}\operatorname{\rightarrow}0$

that shows
$\mathfrak{m}_{X,x}=(f^{*}\psi_{1},\ldots,f^{*}\psi_{n},\overline{\phi_{1}},\ldots,\overline{\phi_{r}})$.
Hence $x\in X$ is regular and also smooth because we work over a perfect
field. ∎

######  Lemma 31 (Singularities of general fibers).

Let $X\operatorname{\rightarrow}Y$ be a surjective morphism of complex
varieties with connected fibers, where $X$ is normal and irreducible. Then the
general fiber $F$ is irreducible with ${F_{sing}=F\cap X_{sing}}$. When $X$
has canonical (resp. terminal) singularities the general fiber also has
canonical (resp. terminal) singularities.

###### Proof.

A general fiber $F$ lies over a smooth point of $Y$, so it is a local complete
intersection in $X$. By induction we may assume that $F$ is a general
hypersurface section of $X$. It is normal by [BS95, Theorem 1.7.1.1]. The
strict transform of $F$ under a resolution
$\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ is a general hypersurface
section in $\widetilde{X}$, which is smooth by Bertini’s Theorem, [Aki51]. So
$\nu$ is for general $F$ a log resolution. The adjunction formula lets us
compare the minimal discrepancies as ${discr(F)\geq discr(X)}$, which shows
that $F$ also has canonical (resp. terminal) singularities.

Furthermore Bertini’s Theorem implies that $F$ is irreducible and smooth
outside of the singular locus of $X$, so $F_{sing}\subset F\cap X_{sing}$. On
the other hand, a smooth point of $F$ is by Lemma 30 also a smooth point of
$X$. This gives the equality $F_{sing}=F\cap X_{sing}$.∎

######  Lemma 32.

Let $f\colon X\operatorname{\rightarrow}B$ be a surjective morphism between
complex projective varieties, where $X$ is normal and irreducible. Suppose
${\dim X=m}$, ${\dim B=n}$ and ${i\colon F\operatorname{\hookrightarrow}X}$ is
a general fiber of dimension $r=m-n$. Let
${[X]\in\textup{H}_{2m}(X,\,\mathds{Z})}$,
${[F]\in\textup{H}_{2r}(F,\,\mathds{Z})}$,
${[B]\in\textup{H}_{2n}(B,\,\mathds{Z})}$ be the canonical fundamental classes
of $X$, $F$, $B$ and ${{[B]^{\vee}}\in\textup{H}^{2n}(B,\,\mathds{Z})}$ the
dual of $[B]$ in cohomology by section 2.3. Then

(8) $[X]\cap f^{*}\big{(}[B]^{\vee}\big{)}=i_{*}[F].$

###### Proof.

If $X,F,B$ are smooth, equation (8) follows from Poincaré duality. Therefore
Lemma 32 follows by going over to resolutions of singularities like in diagram
7, pushing forward equation (8) and doing calculations with the projection
formula, [Sch17, Lemma 63]. ∎

######  Theorem 33 (Kollár’s torsion-freeness).

Let $f\colon X\operatorname{\rightarrow}B$ be a surjective morphism with
connected fibers between complex projective varieties, where $X$ has rational
singularities. Then all higher direct image sheaves $R^{i}f_{*}\omega_{X}$ for
$i\geq 0$ are torsion-free sheaves on $B$.

###### Proof.

Let $\nu\colon\widetilde{X}\operatorname{\rightarrow}X$ be a resolution of
singularities. We consider the Grothendieck spectral sequence for the functors
$\nu_{*}$, $f_{*}$ and the canonical sheaf
$\omega_{\widetilde{X}}=\Omega^{n}_{\widetilde{X}}$.

$E^{pq}_{2}=R^{p}f_{*}(R^{q}\nu_{*}\omega_{\widetilde{X}})\operatorname{\Rightarrow}R^{p+q}(f\circ\nu)_{*}\omega_{\widetilde{X}}$

By Grauert-Riemenschneider vanishing all higher direct images
$R^{q}\nu_{*}\omega_{\widetilde{X}}$ vanish for $q>0$, [Laz04, Theorem 4.3.9].
Hence we are only left with the row $E^{p0}_{2}$ and the spectral sequence
degenerates already on $E_{2}$. As $X$ has rational singularities, we have
${\nu_{*}\omega_{\widetilde{X}}=\omega_{X}}$ by [KM98, Lemma 5.12]. This gives
us

$R^{p}f_{*}\omega_{X}\cong R^{p}f_{*}(\nu_{*}\omega_{\widetilde{X}})\cong
R^{p}(f\circ\nu)_{*}\omega_{\widetilde{X}}.$

The latter sheaf is torsion-free by Kollár’s result [Kol86, Theorem 2.1]. ∎

######  Theorem 34 (Reduction of the Iitaka conjecture to the minimal model
program).

Let $f\colon X\operatorname{\rightarrow}B$ be a surjective morphism with
connected fibers between normal, complex projective varieties. If the general
fiber $F$ has a _good minimal model_ , in other words $F$ is birational to a
variety with nef and semiample canonical divisor, then
$\kappa(X)\geq\kappa(F)+\kappa(B)$.

###### Proof.

We construct a commutative diagram like diagram 7 that lifts $f$ to a morphism
$\widetilde{f}$ between resolutions $\widetilde{X},\widetilde{B}$ of $X,B$.
The fibers of $\widetilde{f}$ are connected. The general fiber of
$\widetilde{f}$ is smooth by Lemma 31 and lies over the general fiber of $f$.
As the Kodaira dimension is a birational invariant, Theorem 34 follows now
from the smooth case that was proven by Kawamata, [Kaw85, Corollary 1.2]. ∎

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