Fujiki relations and fibrations of irreducible symplectic varieties

0.29 MB·

by  Martin Schwald

FUJIKI RELATIONS AND FIBRATIONS OF
IRREDUCIBLE SYMPLECTIC VARIETIES
MARTINSCHWALD
Abstract. This paper concerns singular, projective, irreducible symplectic
7
varietiesoverC. GeneralizingresultsofKirschner,MatsushitaandNamikawa,
1
we prove that the generalized Beauville-Bogomolov form satisfies the Fujiki
0
relationsandhasrank 3,0,b2(X)−3 . Asaconsequence, weseethat these
2
varietiesadmitonlyver(cid:0)yspecialfibrati(cid:1)ons.
n
a
J

1
3
Contents

]
G 1. Introduction 1
1.1. Symplectic varieties 1
A
1.2. Main results 2
.
h 1.3. Strategy of the proofs 3
at 1.4. Acknowledgement 3
m 2. Methods 3
2.1. Extension Theorem 3
[
2.2. Terminal models 4

1
2.3. Integration of singular top classes 4
v
2.4. Hodge Theory for klt varieties 4
9
6 2.5. Properties of symplectic varieties 5
0 3. Generalized Beauville-Bogomolovform 6
9 3.1. Namikawa’s generalization 7
0
3.2. Kirschner’s generalization 7
.
1 3.3. Normed Beauville-Bogomolovform 8
0 3.4. About the bilinear form and Matsushita’s notation 9
7
4. Proofs of the main results 9
1
4.1. Fujiki relations and the index of the Beauville-Bogomolovform 9
:
v 4.2. Fibrations of irreducible symplectic varieties 11
i
X Appendix 15
References 16
r
a
1. Introduction
1.1. Symplectic varieties. It is expected that symplectic varieties should play
an important role in a possible structure theory for varieties with Kodaira dimen-
sion zero, [GKP16, Section 8]. To understand them better, we want to classify
morphisms from symplectic varieties onto normal spaces of lower dimension.
Date:February1,2017.
The author gratefully acknowledges support by the DFG-Graduiertenkolleg GK1821 “Coho-
mologicalMethodsinGeometry” attheUniversityofFreiburg.
1
2 MARTINSCHWALD
We briefly recall the relevant definitions required to formulate our main results.
Theliteraturecontainsseveralnotionsthatgeneralizeholomorphicsymplecticman-
ifolds toasingularsetting. Theyagreeinthe existenceofasymplectic form onthe
smooth locus.
Reflexive differential forms are holomorphic p-forms on the smooth locus X
reg
of a variety X. We notate the associated sheaf on X as Ω[p] and get
X
Ω[p] =(Ωp )∗∗ =i Ωp ,
X X ∗ Xreg
H0(X, Ω[Xp])∼=H0(Xreg, ΩpXreg),
wherei: X ֒→X istheinclusion. See[Har80]forareferenceonreflexivesheaves
reg
and[GKKP11,I–III],[KP16,I]formoreinformationonreflexivedifferentialforms.
Definition 1 (Symplectic varieties). Let X be a normal, complex projective vari-
ety. A symplectic form on X is a closed, reflexive form ω ∈ H0(X, Ω[2]) that is
X
not degenerate at any point x ∈ X . We call the pair (X,ω) a symplectic va-
reg
riety if for any resolution of singularities ν: X→X there is a holomorphic form
ω ∈H0(X, Ω2 ) that coincides with ω over X . We say that ω extends to ω on
X reeg
Xe. e e
A symplecteic variety (X,ω) is called irreducible symplectic if its irregularity
e
h1(X, O ) vanishes and if h0(X, Ω[2]) = 1. We call an irreducible symplectic
X X
variety (X,ω) Namikawa symplectic if X is Q-factorial with codimCXsing ≥4.
Section2.5reviewspropertiesofthesenotionsanddiscussestheirrelations. Sym-
plectic varieties were introduced by Beauville in [Bea00, Definition 1.1]. The defi-
nition of irreducible symplectic varieties is inspired by attempts to generalize the
Beauville-BogomolovdecompositionTheorem[Bea83,Theorem2]toasingularset-
ting, [GKP16, Section 8]. Bogomolov and Namikawa showed in [Bog78, Nam01b]
that Namikawa symplectic varieties have a well behaved deformation theory.
1.2. Main results. On an irreducible symplectic variety X, Namikawa and
Kirschner constructed in [Nam01b, Theorem 8 (2)], [Kir15, Definition 3.2.7] an
important quadratic form q on H2(X, C), the Beauville-Bogomolov form. Its
X
definition will be recalled in Section 3. We will show that it has the following
properties,whicharewellknowninthesmoothcase,[Fuj87,Theorem4.7],[Bea83,
Th´eor`eme 5(a)]. Matsushita already proved them for Namikawa symplectic vari-
eties in [Mat15, Proposition 4.1], [Mat01, Theorem 1.2].
Theorem 2 (Fujiki relations,index of the Beauville-Bogomolovform). Let (X,ω)
be a 2n-dimensional, irreducible symplectic variety. The Beauville-Bogomolov form
q has the following properties:
X
(1) There is a number c ∈ R+, called Fujiki constant, such that for all
X
α∈H2(X, C) the following, so-called Fujiki relation, holds,1
c ·q (α)n = α2n.
X X
Z
X
In particular for classes of Cartier divisors d=c O (D) , this relates the
1 X
Beauville-Bogomolov form to the intersection prod(cid:0)uct via(cid:1)
c ·q (d)n =D2n.
X X
(2) The restriction of q to H2(X, R)→R is a real quadratic form of index
X
3,0,b (X)−3 .
2
(cid:0) (cid:1)
1WewillrecallintegrationonsingularcohomologyinSection2.3.
3
Part (2) implies the existence of a q -orthogonal decomposition
X
H2(X, R)=V+⊕V− with dimRV+ = 3, such that qX is positive definite on
V and negative definite on V , see Proposition 22.
+ −
Matsushita deduces from his version of Theorem 2, [Mat01, Theorem 1.2], that
Namikawa symplectic varieties have only very special fibrations, [Mat01, Corol-
lary 1.4]. We will prove the same for fibrations of arbitrary irreducible symplectic
varieties.
Theorem 3 (Fibrations of irreducible symplectic varieties). Let (X,ω) be an ir-
reducible symplectic variety of complex dimension 2n, together with a surjective
morphism f: X→B onto a projective, normal variety B, such that the general
fiber F of f is connected with dimension 0 < dimF < 2n. Then the following
properties hold.
(1) The base variety B is n-dimensional with Picard number ρ(B)=1.
(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) ThegeneralfiberF isann-dimensional, Abelianvariety. Itisa Lagrangian
subvariety, in other words, the restriction ω ∈H0(F, Ω2) vanishes.
F F
(4) Every Weil divisor D 6=0 on X fulfills dimC(cid:12)f(suppD)≥dimCB−1.
(cid:12)
1.3. Strategyoftheproofs. Ourproofsrelyonthegeneralextensiontheoremfor
differentialformsfrom[GKKP11,Theorem1.4]andtheexistenceofterminalmod-
els from [BCHM10, Corollary 1.4.3]. Conveniently, terminal models of irreducible
symplectic varieties turn out to be Namikawa symplectic. These two methods had
not been available when Matsushita and Namikawa proved their results.
We show in Section 4 how to deduce the Fujiki relation and Theorem 3 from a
terminal model. However,the calculation of the index of the generalizedBeauville
Bogomolov form requires results on the structure of H2(X, C) that we explain in
detail in [Sch16]. Calculations with the Fujiki relations, the index and the Hodge
theory of H2(X, C) also give us a direct proof of Theorem 3 in Section 4.2, which
generalizes Matsushita’s proof from [Mat01, Corollary 1.4].
1.4. Acknowledgement. The author wants to thank his supervisor
Prof.Kebekus, as well as Prof.Oguiso and B.Taji for their advice and long,
fruitful discussions.
2. Methods
We recallin 2.1–2.3 the generalextensiontheorem for differential forms, the ex-
istence of terminal models and integration over singular cohomology classes. The
results from [Sch16] on the Hodge theory of H2(X, C) are summarized in 2.4, es-
pecially the Hodge decomposition and a version of the Hodge Riemann bilinear
relations on klt varieties. We recapitulate in 2.5 well known properties of sym-
plectic varieties. They will imply that terminal models of irreducible symplectic
varieties are Namikawa symplectic.
2.1. Extension Theorem. The extension theorem for differential forms
[GKKP11, Theorem 1.4] shows that on a klt variety X any reflexive form
α∈H0(X, Ω[p]) can be extended to any resolution of singularities. This allows
X
us to construct pullbacks of reflexive forms along more general morphisms.
Theorem 4 (Pullbacks of reflexive forms, [Keb13, Theorem 1.3]). Let f: Y →X
be a morphism between normal, complex projective varieties. Assume that there is
a Weil divisor D on X, such that (X,D) is klt. Then there is a natural pullback
morphism f∗Ω[p]→Ω[p], consistent with the natural pullback of Ka¨hler differentials
X Y
on X . (cid:3)
reg
4 MARTINSCHWALD
TheconsistencywiththenaturalpullbackofK¨ahlerdifferentialsismadeprecise
in [Keb13, Theorem 5.2]. We will only consider the case when f: Y →X is sur-
jective, for example a birational morphism. Then this consistency means that the
pullbackmorphismf∗Ω[p]→Ω[p] agreeswiththe usualpullbackofK¨ahlerdifferen-
X Y
tialswhereverX andY aresmooth. Inotherwords,foreveryα∈H0(X, Ω[p])there
X
is an α∈H0(Y, Ω[p]) that coincides with f∗(α ), seen as K¨ahler differential on
Y Xreg
Y ∩f−1(X ). We call α an extension of α(cid:12) to Y. Pulling back reflexive forms
reg e reg (cid:12)
is contravariantfunctorial with respect to f.
e
2.2. Terminal models. While not every singular variety admits a crepant reso-
lution, varieties with klt singularities have always a terminal model. For canonical
singularitiesaterminalmodelisacrepant partial resolution. Thisisaneasyspecial
case of [BCHM10, Corollary 1.4.3].
Theorem 5 (Terminal models, [BCHM10, page 413]). Let X be a complex pro-
jective variety with canonical singularities. Then any resolution ν: X→X of
X factors through a Q-factorial variety Y with at most terminal singularities,
e
X −π→Y −π→X, such that π∗K =K . We call Y a terminal model of X. (cid:3)
X Y
2e.3.eIntegrationofsingulartopclasses. Recallthatweintegrateatopsingular
cohomology class φ∈H2n(X, C) over an n-dimensional, compact, complex variety
X as Xφ ..= [X]∩φ. Here [X] ∈ H2n(X, Z) denotes the canonical fundamental
class oRf X, induced by the complex structure on its smooth locus Xreg. It exists,
because X is by [GM80, 1.1]endowedwith the structure ofa 2n-dimensionalpseu-
domanifoldandthecomplexstructureonthesmoothlocusX inducesacanonical
reg
orientation. Furthermore, the cap product induces a perfect pairing
∩: H2n(X, Z)×H2n(X, Z) H0(X,Z)∼=Z
(σ , φ) σ∩φ,
so Poincar´e duality holds in degree 2n, see [Sch16, Section 3.4] for details. This
integration commutes with pullbacks by bimeromorphic maps.
Lemma 6 (Pullbacks of integrals, [Sch16, Lemma 12]). If f: X→Y is a bimero-
morphic morphism of irreducible, compact, complex varieties, then α= f∗α
Y X
for all α∈H2n(Y, C). R R (cid:3)
2.4. Hodge Theory for klt varieties. On Namikawa symplectic varieties, Mat-
sushita and Namikawa consider the Hodge Fr¨olicher 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 decompo-
sition of H1(X, C) and H2(X, C).
Theorem 7 (StructureofH1(X, C)andH2(X, C),[Sch16,Theorem7]). If X is a
complex projective variety with at most klt singularities, then for p=1,2 the mixed
Hodge structure on Hp(X, C) is pure of weight p. Furthermore any resolution of
singularities ν: X→X induces injective pullback maps
e ν∗ : Ha,b(X)→Ha,b(X) for a+b=p,
ab
where the morphisms νp∗0 and ν0∗p are isomeorphisms. There are canonical isomor-
phisms κ : Hp,0(X)−∼→H0(X, Ω[p]) and κ : H0,p(X)−∼→Hp(X, O ) that com-
p0 X 0p X
mute with the natural pullback morphisms and the Dolbeault isomorphisms δ ,δ .
p0 0p
5
Hp,0(X) ∼ H0(X, Ωp ) H0,p(X) ∼ Hp(X, O )
δp0 X δ0p X
νp∗0 ≀e exte ≀ e ν0∗p ≀e ν∗e ≀ e
Hp,0(X) κ∼p0 H0(X, Ω[Xp]) H0,p(X) κ∼0p Hp(X, OX) (cid:3)
Remark 8. Let X be a complex projective variety with at most klt singularities.
Every reflexive two-form on X defines by Theorem 7 a unique singular cohomology
class and we notate this canonical inclusion by:
ι : H0(X, Ω[2]) κ20 H2,0(X)⊂H2(X, C)
20 X
α α
Due to the first diagram in Theorem 7, the class α∈H2(X, C) has the property
that on any resolution ν: X→X the extension of α represents the class ν∗α in
H2(X, C).
e
Tehe author explains the Hodge theory of H2(X, C) andapplications in detail in
[Sch16]. In this paper, we make use of the following versionof the Hodge-Riemann
bilinear relations on klt varieties.
Corollary 9 (Bilinear relations on klt varieties, [Sch16, Corollary 16]). Let X be
an n-dimensional, complex projective variety with at most klt singularities. Any
ample class a∈H2(X, R) induces a sesquilinear form ψ , defined by
X,a
ψ : Hk(X, C)×Hk(X, C) C
X,a
(α , β) (−1)k(k2−1) · Xα∪β∪an−k.
R
The so-called Hodge Riemann bilinear relation ip−q·ψ (α,α) >0 holds for any
X,a
non-zero α with α∪an−k+1 =0 that can be written as the cup product of classes
α ,...,α ∈H1(X, C)∪H2(X, C) .
1 r
Here we notated p ..= p and q ..= q if the α are in the Hpj,qj(X)-part of
j j j
the Hodge decompostion oPf H1(X, C) orPH2(X, C). (cid:3)
2.5. Propertiesofsymplecticvarieties. Wementionthefollowingpropertiesof
symplecticvarieties. Theywillimplythatterminalmodelsofirreduciblesymplectic
varieties are Namikawa symplectic.
Lemma 10 (Properties of symplectic varieties).
(1) A symplectic variety has canonical singularities and trivial canonical class,
cf.[Bea00, Proposition 1.3].
(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) An irreducible symplectic variety X is Namikawa symplectic if and only if
it has at most Q-factorial, terminal singularities.
Proof of (1). Let (X,ω) be a 2n-dimensional symplectic variety with a resolution
ν: X→X. By definition, ω extends to a regular form ω on X. Let K , K be
X X
the canonical divisor classes on X, X. Due to linear algebra, it is equivalent that
e e e
ω is non-degenerate and that ωn ∈ H0(X, Ω[2n]) has no zeros. Since X is noremal,
e X
ωn extends to a global section of O (K ) with vanishing divisor. Thus ωn is a
X X
globalsectionofO (K )thatmayonlyhavezerosalongthesetE ofν-exceptional
divisors. Hence weXcanXwrite up to linear equivalence e
0≤ediv(eωn)=K =ν∗div(ωn)+ discr(E)·E.
X
EX∈E
=0
e
e | {z }
6 MARTINSCHWALD
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
codimensionthree. However,Namikawaprovesin[Nam01a]thatthesingularlocus
ofanysymplecticvarietyX hasnocomponentofpurecodimensionthree. Further-
more,everyexceptionaldivisorE overX withvanishingdiscrepancyhasacenterof
codimension two in X. Namikawa deduces (2) from this in [Nam01a, Corollary 1].
Part (3) follows immediately from (2). (cid:3)
Remark 11 (Equivalent definition of symplectic varieties). Let X be a complex
projective variety with a symplectic form ω ∈ H0(X, Ω[2]). Then (X,ω) is by
X
Lemma 10 a symplectic variety if and only if X has canonical singularities. It
is Namikawa symplectic if and only if X has Q-factorial terminal singularities.
The form ω 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 ir-
reducible symplectic varieties. In [Mat01] he uses the condition h2(X, O ) = 1
X
instead of h0(X, Ω[2]) = 1. It follows from Theorem 7 that these two conditions
X
are equivalent by the Hodge symmetry.
Proposition12. Foranirreduciblesymplecticvariety(X,ω)everyterminalmodel
(Y,ω) is a Namikawa symplectic variety with ω ∈H0(Y, Ω[2]) the extension of ω to
Y
Y from Theorem 4.
e e
Proof. The form ω extends to a reflexive form ω ∈ H0(Y, Ω[2]) by Theorem 4.
Y
We have π∗K = K , because π is crepant, and Lemma 10 (1) shows that these
X Y
canonical classes are trivial. Let X be 2n-dimenesional. As Y is normal, ωn can
be extended to a non-trivial section of O (K ) that has no zeros on Y . This
Y Y reg
is equivalent to ω being non-degenerate on Y . The extension theorem alleows us
reg
to extend ω to any resolution π: X→Y of Y, hence (Y,ω) is a symplectic variety
e
withatmostQ-factorial,terminalsingularitiesbyconstruction. AsX isacommon
e e e e
resolution of X and Y, Theorem 7 gives the missing irreducibility conditions of Y:
e
h1(Y, O )=h1(X, O )=h1(X, O )=0,
Y X X
h0(Y, Ω[2])=h2(Xe, O )=h0(X, Ω[2])=1.
Y Xe X
Hence (Y,ω) is Namikawa symplecticeby Lemma 10. (cid:3)
e
e 3. Generalized Beauville-Bogomolov form
We explain how the Beauville-Bogomolov form is defined in a singular setting,
comparingthe equivalentconstructions by KirschnerandNamikawaandclarifying
possiblenotationalissues. WehopethismightalsohelptounderstandKirschner’s,
Matsushita’s and Namikawa’s results.
On any Riemannian manifold X the Theorem of de Rham allows us to consider
any singular cohomologyclass as a class of differential forms. Using this, Beauville
defined in [Bea83, page 772] on any compact, K¨ahler, irreducible symplectic mani-
fold (X,ω) the Beauville-Bogomolov form by:
q : H2(X, C)→C
X,ω
qX,ω(α)..=n (ωω)n−1α2+(1−2n) ωnωn−1α · ωn−1ωnα
ZX (cid:16)ZX (cid:17) (cid:16)ZX (cid:17)
The products and powers denote here the wedge products of the de Rham classes
or equivalently the cup product of their singular cohomology classes.
7
3.1. Namikawa’s generalization. Namikawadefinesq inthesingularsetting
X,ω
by pulling back everything to a resolution of singularities and then calculating
Beauville’s formula. This gives the following definition.
Definition 13 (Beauville-Bogomolov form, cf.[Nam01b, Theorem 8 (2)]). Let
(X,ω) be a 2n-dimensional, irreducible symplectic variety with a resolution
ν: X→X. The pullbacks ν∗α =.. α of classes α ∈ H2(X, C) may via de
Rham’s Theorem be regarded as the classes of two-forms. We extend ω to a form
ω ∈eH0(X, Ω2 ). Then the Beauville-Beogomolov form on X is the quadratic form
X
defined by:
e e
q e : H2(X, C)→C
X,ω
q (α)..=n (ωω)n−1α2+(1−2n) ωnωn−1α · ωn−1ωnα
X,ω
ZX (cid:16)ZX (cid:17) (cid:16)ZX (cid:17)
ee e e e e e e e
The products and powers denote here the wedge products of forms on X.
e e e
TheBeauville-Bogomolovformdoesnotdependonthechosenresoluetion. Given
two resolutions ν, ν′ of X, this can be seen by going over to a common resolution,
factoring through ν and ν′.
This definition is well suited for calculations. However, the resolution X does
notneedtobe symplectic again. Hence we cannottriviallydeduce allpropertiesof
e
q from the smooth case by considering the resolution X.
X,ω
3.2. Kirschner’s generalization. Kirschner defines thee Beauville-Bogomolov
formforirreduciblesymplecticvarietiesX bydirectlyintegratingsingularcohomol-
ogyclassesonthesingularvarietyX asinSection2.3. ThisissimilartoBeauville’s
formula and turns out to be equivalent to Namikawa’s Definition 13. It is more
suitable for generalizing properties like the index and the Fujiki relations of the
Beauville-Bogomolovform to irreducible symplectic varieties.
Definition 14 (Beauville-Bogomolovform, cf.[Kir15, Definition 3.2.7]). Let X be
a 2n-dimensional, irreducible, complex projective variety. Then we can define for
any class w ∈H2(X, C) a quadratic form
q : H2(X, C)→C
X,w
q (v)..=n (ww)n−1v2+(1−2n) wnwn−1v · wn−1wnv .
X,w
ZX (cid:16)ZX (cid:17) (cid:16)ZX (cid:17)
Here the products and powers denote the cup product in the cohomology ring
H∗(X, C). Using our notation from Remark 8, we can consider on an irreducible
symplecticvariety(X,ω)theclass ω ∈H2(X, C). Thequadraticformq iscalled
X,ω
Beauville-Bogomolovform 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 14
behaves well under pullbacks along birational morphisms.
Lemma 15. Let π: Y →X be a birational morphism from normal, complex pro-
jective variety Y to a symplectic variety (X,ω) with an extension ω′ ∈H0(Y, Ω[2])
Y
of ω. Then we have π∗ω =ω′ ∈H2(Y, C) and therefore
q =q and q =q ◦π∗.
Y,π∗ω Y,ω′ X,ω Y,π∗ω
π π
Proof. We take a resolution ν: X→X that factors as X → Y → X. Then the
extensions ofω andω′ to X areaeunique formω ∈H0(X,eΩe2X). Thus π∗(π∗ω) and
e e e e
8 MARTINSCHWALD
π∗(ω′) are both equal to the class of ω in H2(X, C). Due to Theorem 7, the map
π∗ is bijective on H2,0(Y). Hence we get π∗ω =ω′.
e e e
We noted in Lemma 6 that the integration of top cohomology classes is compa-
teible with pullbacks under birational morphisms. For any v,w ∈H2(X, C) we can
apply this to the occurring integrals in Definition 14 of the Beauville-Bogomolov
form, such that we get q (v)=q (π∗v) for all v ∈H2(X, C). (cid:3)
X,w Y,π∗w
Corollary16(Equivalenceofthedefinitions). Onanirreduciblesymplecticvariety
(X,ω)Namikawa’sandKirschner’sdefinitionsoftheBeauville-Bogomolov formare
equivalent in terms of
q =q .
X,ω X,ω
Proof. Let ω be the extension of ω to a resolution ν: X→X. Using the no-
tation of Kirschner’s Definition 14, we can write Namikawa’s Definition 13 as
q =q e◦ν∗. This equals q by Lemma 15. e (cid:3)
X,ω X,ω X,ω
3.3. Noremeed Beauville-Bogomolov form. The Beauville-Bogomolov form on
an irreducible symplectic variety (X,ω) depends on the symplectic form. It is
convenient to norm ω in order to make the Beauville-Bogomolov form on X a
unique form q . This is done in a way, such that we will later obtain the Fujiki
X
relations in Theorem 2. It can be proven exactly like in the smooth case, [Bea83,
Theorem5a],thattherestrictionofq givesuptoapositiverealfactoranintegral
X
form H2(X, Z)→Z. In this subsection we use the following notation:
Notation 17. On a 2n-dimensional, irreducible symplectic variety X denote
I: H2(X, C)→R+
w7→ (ww)n.
Z
X
Furthermore, we write I(ω)..=I(ω) for any symplectic class ω on X.
Lemma18(Existenceanduniquenessofthenorming). LetX bea2n-dimensional,
irreducible symplectic variety X. Any symplectic class ω ∈H2(X, Ω[2])\{0} in-
X
duces a normed class ω′ ..=I(ω)−21n ·ω with I(ω′) = 1. The corresponding
Beauville-Bogomolov form q does not depend on the choice of ω.
X,ω′
Proof. As X is irreducible symplectic, any symplectic class α ∈ H0(X, Ω[2])\{0}
X
onX differs only by a constantfactorfromthe classω,sayα=cω for c∈C×. We
see I(cω)=|c|2nI(ω) and therefore
I(ω′)=I I(ω)−21n ·ω =I(ω)−1·I(ω)=1,
(cid:0) (cid:1) c c
(cω)′ =I(cω)−21n ·cω = I(ω)−21n ·ω = ω′.
|c| |c|
Hence changing ω merely changes its normed class ω′ by a constant of absolute
value one. We see directly from Definition 13 that this does not affect q . (cid:3)
X,ω′
Definition 19 (NormedBeauville-Bogomolovform). Let X be an irreducible sym-
plectic variety. The normedBeauville-Bogomolovform is defined as q ..=q for
X X,ω
any symplectic form ω with I(ω)=1.
Lemma 18 shows that Definition 19 and the notation q make sense, as this
X
is a unique quadratic form, depending only on X. We show that this norming is
compatible with pullbacks like in Lemma 15.
9
Lemma 20 (Pullback of q ). Let π: Y →X be a birational morphism of irre-
X
ducible symplectic varieties with normed Beauville-Bogomolov forms q , q on
X Y
X, Y. If q = q , then q = q for the extension ω′ of ω to Y and we
X X,ω Y Y,ω′
have q =q ◦π∗.
X Y
Proof. We write qX = qX,ω and w ..= ω for a normed symplectic class
ω ∈H0(X, Ω[2]) with I(w)=1. We get π∗(ww)n = (ww)n =1 by Lemma 6.
X Y X
Hence π∗w is also normed in H2(Y, C),Rso q = q R. Now Lemma 15 gives us
Y Y,π∗w
q =q =q and
Y Y,π∗ω Y,ω′
q =q =q ◦π∗ =q ◦π∗. (cid:3)
X X,w Y,π∗w Y
3.4. About the bilinear form and Matsushita’s notation. The Beauville-
Bogomolov form q on an irreducible symplectic variety (X,ω) is induced by a
X
symmetric bilinear form. Like Matsushita we also notate it with q , but with two
X
arguments.
Remark 21. We write the normed Beauville-Bogomolov form on an irreducible
symplectic variety X as q = q for a normed w ∈ H2(X, C). Due to linear
X X,w
algebra we have for all a,b∈H2(X, C):
1
q (a,b)= q (a+b)−q (a)−q (b)
X X X X
2
(cid:0) (cid:1)
=n (ww)n−1ab
Z
X
1−2n
+ wnwn−1a wn−1wnb+ wnwn−1b wn−1wna .
2 (cid:16)ZX ZX ZX ZX (cid:17)
In his proofs in [Mat01], Matsushita plugs into the Beauville-Bogomolov form
objects of differenttypes: two-formsderivedfromthe symplectic formω, classesin
H2(X, C), Cartier divisors D ∈ Div(X) and sums of these types. This should be
understood via the Hodge decomposition of H2(X, C) from Theorem 7 by identi-
fying H2,0(X, C) with the reflexive two-forms and Cartier divisors with their first
Chern classes in H1,1(X). So for example q (ω+ω+D) in Matsushita’s notation
X
shouldbe understoodasq ω+ω+c O (D) inour notation,whereq is the
X 1 X X
normed Beauville-Bogomolo(cid:16)vform from(cid:0)Definit(cid:1)io(cid:17)n 19. To avoid confusion, we will
only apply q to classes in H2(X, C), as we defined it.
X
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 sin-
gularities, but this might not be a symplectic variety anymore. Instead, we pass
to a terminal model, which is a Namikawa symplectic variety by Proposition 12.
Part (2) of Theorem 2 will follow directly from the more precise Proposition 22
about the decomposition of H2(X, C) into orthogonal subspaces, on which the
Beauville-Bogomolov is definite. The Hodge theory of H2(X, 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
varietywithnormedBeauville-Bogomolovformq onH2(X, C)asinDefinition19.
X
We need to show that there is a constant c ∈R+, such that for all v ∈H2(X, C)
X
we have the Fujiki relation
c ·q (v)n = v2n.
X X
Z
X
10 MARTINSCHWALD
Wewriteq =q foranormedsymplecticclassω with (ωω)n =1. ByPropo-
X X,ω X
sition12,everyterminalmodelπ: Y →X ofX becomes,toRgetherwithanextension
ω′ ∈ H2(Y, Ω[2]) of ω from Theorem 4, a Namikawa symplectic variety. We have
Y
q =q and q =q ◦π∗ by Lemma 20.
Y Y,ω′ X Y
Taking such a Namikawa symplectic terminal model (Y,ω′), we can apply
[Mat01, Theorem 1.2] to Y to get a c ∈R+, such that for all u∈H2(Y, C):
Y
(1) c ·q (u)n = u2n.
Y Y
Z
Y
Therefore we get for any v ∈H2(X, C):
c ·q (v)n =c ·q (π∗v)n By Lemma 20
Y X Y Y
= (π∗v)2n Equation (1) for u..=π∗v
Z
Y
= π∗(v2n)
Z
Y
= v2n By Lemma 6
Z
X
Hence we have proved the Fujiki relation for the constant cX ..=cY. (cid:3)
Proposition 22 (Index ofthe Beauville Bogomolovformq ). Let (X,ω) be a 2n-
X
dimensional, irreducible symplectic variety with normed Beauville-Bogomolov form
q = q , w ∈ H2(X, C) and an ample class a ∈ H2(X, R). Restricting q
X X,w X
gives a real quadratic form H2(X, R)→R and H2(X, R) splits into the following
q -orthogonal subspaces
X
V ..=hw+w, iw−iw, ai ,
+ R
V ..={d∈H1,1(X)∩H2(X,R)|q (d,a)=0}
− X
=a⊥∩H1,1(X)∩H2(X, R),
such that qX is positive definite on V+, negative definite on V− and dimRV+ =3.
Proof that restricting q gives a real form. An easy computation for arbitrary
X
v ∈H2(X,C)showsq v =q (v),usingthattheintegralsarecompatiblewith
X,w X,w
complex conjugation. He(cid:0)nc(cid:1)e restrictinggives a realquadratic formH2(X, R)→R.
Proof of V ⊥ V and the positivity of q on V . We use (ww)n = 1 and the
+ − X + X
formulafromRemark21 to calculatethe Beauville-BogomolRovbilinear formqX on
the elements w+w, iw−iw, a. The Hodge decomposition from Theorem 7 allows
us to calculate the occurring integrals via the usual type considerations.
1−2n
q (w+w)=q (iw−iw)=2n+ (1+1)=1
X X
2
1−2n
q (w+w, iw−iw)=(−i+i)n+ (i−i)=0
X
2
∀v ∈H1,1(X): q (w+w, v)=q (iw−iw, v)=0
X X
q (a)= (ww)n−1a2
X
Z
X
Thereforew+w, iw−iw, a are orthogonalwith respect to q . The two subspaces
X
V and V of H2(X, C) are likewise orthogonal. The restriction q is diagonal
+ − X V+
with eigenvalues1, 1, q (a). As wn−1∪a3 =0 by type consideratio(cid:12)ns,the Hodge-
X (cid:12)
Riemann bilinear relations, Corollary 9, show that q (a) = ψ (wn−1,wn−1) is
X X,a
positive and therefore q positive definite on V .
X +