FOOTNOTES TO THE BIRATIONAL GEOMETRY OF PRIMITIVE SYMPLECTIC VARIETIES CHRISTIAN LEHN GIOVANNI MONGARDI AND GIANLUCA PACIENZA

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FOOTNOTES TO THE BIRATIONAL GEOMETRY OF PRIMITIVE

SYMPLECTIC VARIETIES

CHRISTIAN LEHN, GIOVANNI MONGARDI, AND GIANLUCA PACIENZA

Abstract. In this note, we extend to the singular case some results on the birational

geometry of irreducible holomorphic symplectic manifolds.

In memoria di Alberto Collino

Contents

1. Introduction 1

2. Preliminaries 4

3. Eﬀective birationality 5

4. Termination of ﬂips 8

5. On the geometry of wall divisors 10

References 13

1. Introduction

Irreducible holomorphic symplectic (IHS) manifolds and their singular generaliza-

tions (the so-called primitive symplectic varieties) form a relatively well-studied class

of varieties. Apart from their intrinsic interest, they oﬀer the possibility to test gen-

eral conjectures in algebraic or complex geometry as on the one hand, they are well-

behaved, and on the other hand, they possess a suﬃciently rich geometry. Their

birational geometry is particularly interesting and provides an instance of this princi-

ple. As a sample of this (and taking the risk of leaving many important results aside),

ﬁbers of the period map are exactly birational IHS manifolds (cf. [Ver13], [Huy12,

Corollary 6.1]), the MMP of moduli spaces of sheaves on a K3 surface has been stud-

ied systematically via wall-crossing with respect to Bridgeland stability conditions (cf.

[BM14]), and termination of ﬂips or the Morrison–Kawamata cone conjecture, which

are open problems in general, are already known for IHS manifolds, cf. [LP16], [AV17].

In this note, we discuss three aspects of the birational geometry of primitive sym-

plectic varieties that have been the object of considerable recent interest: eﬀective

2020 Mathematics Subject Classiﬁcation. 14E30, 14J42 (primary), 14E05, 32Q25, 32S15, 53C26

(secondary).

Key words and phrases. hyperk¨ahler variety, primitive symplectic variety, birational boundedness,

wall divisor, MMP.

arXiv:2210.12451v2 [math.AG] 18 Apr 2023

2 CHRISTIAN LEHN, GIOVANNI MONGARDI, AND GIANLUCA PACIENZA

birationality, termination of ﬂips, and monodromy invariance of wall divisors, which

we present in detail below.

Eﬀective Birationality. First, we concentrate on eﬀective birationality. The prob-

lem of boundedness for the birationality of pluricanonical maps has attracted much

attention in the last 10 years (see the papers [HMX14,Bir23] and the references

therein, for results in the framework of irreducible holomorphic symplectic varieties

see [Cha16]). Here, we are able to provide an eﬀective version of such results in the

most general setting, for primitive symplectic varieties. Recall that by the Bogomolov–

Beauville–Fujiki form, the second cohomology (more precisely, the torsion free part)

of such a variety embeds into its dual. We denote by AX:= H2(X, Z)∨/H2(X, Z) the

discriminant group.

Theorem 1.1. Let X be a projective primitive symplectic variety of dimension 2nand

let L∈Pic(X)be a big line bundle on it. Then for all

(1.1) m≥1

2(2n+ 2)(2n+ 3)[(4Card(AX))ρ(X)−1]!,

the map associated to the linear system |mL|is birational onto its image.

If we replace ρ(X) with h1,1(X) in equation (1.1), we obtain an eﬀective bound that

holds for the whole family of deformations of X. Notice that in particular, for any

integer nand positive constant C, the family of all projective primitive symplectic

varieties of dimension 2nand ﬁxed deformation type, endowed with a big line bundle

of volume at most C, is birationally bounded. Therefore, the theorem can be regarded

an eﬀective version of Birkar’s birational boundedness theorem for pluri log-canonical

maps (cf. [Bir23, Corollary 1.4]) in the setting of primitive symplectic varieties. In

the smooth case, the result was obtained in [KMPP19, Corollary 1.3]. The key for

the generalization to primitive symplectic varieties is the recent study of the numerical

and geometric properties of prime exceptional in this setting, cf. [LMP23,LMP22].

Notice that for deformations of moduli spaces of sheaves, the bound is even more

explicit.

Corollary 1.2. Let X be a projective primitive symplectic variety which is deformation

equivalent to a moduli space of sheaves Mv(S)with respect to some v-generic polariza-

tion Hon a projective K3surface S(respectively to Kv(S)in case Sis abelian), for

a Mukai vector v=aw on Swith w2= 2ksuch that (S, v, H)is an (a, k)–triple in

the sense of [PR20, Deﬁnition 1.3]. Then for any big line bundle L∈Pic(X), the map

associated to the linear system |mL|is birational onto its image for any

m≥1

2(dim(X) + 2)(dim(X) + 3)[(8k)ρ(X)−1]!.

Recall that with the notation of the above theorem dim Mv(S)=2a2k+ 2, where

= 1 (respectively =−1) if Sis a K3 (resp. an abelian) surface (see [KLS06,

Theorem 4.4].

FOOTNOTES TO THE BIRATIONAL GEOMETRY OF PRIMITIVE SYMPLECTIC VARIETIES 3

Termination of Flips. Termination of ﬂips is one major open problem in the MMP. A

well-known general strategy due to Shokurov [Sho04] involving the study of some regu-

larity properties of invariants of the singularities (the so-called minimal log-discrepancies)

has allowed to prove termination for IHS manifolds [LP16]. Due to the generalization

to hyperquotient singularities of the lower semicontinuity of minimal log-discrepancies

obtained in [NS20] and to the advances in the theory of primitive symplectic varieties,

we remark here that Shokurov’s strategy works now in a more general framework.

Theorem 1.3. Let X be a projective primitive symplectic variety with terminal hy-

perquotient singularities. Let ∆be an eﬀective R-divisor on X, such that the pair

(X, ∆) is log-canonical. Then every log-MMP for (X, ∆) terminates in a minimal

model (X0,∆0)where X0is a symplectic variety with (canonical) singularities and ∆0

is an eﬀective, nef R-divisor.

Recall that a variety Yhas hyperquotient singularities if Yis a locally complete

intersection inside an ambient variety which is the quotient of a smooth variety by a

ﬁnite group action (see [NS20, Theorem 1.2] for a slightly more general deﬁnition).

In particular, we immediately deduce the following for an important class of singular

IHS varieties which has received much attention in recent years cf. e.g. [Men20,

MR20a,FM21,MR20b].

Corollary 1.4. Let Xbe an IHS orbifold (in the sense of [Men20, Deﬁnition 3.1].

Let ∆be an eﬀective R-divisor on X, such that the pair (X, ∆) is log-canonical. Then

every log-MMP for (X, ∆) terminates.

Invariance of walls under parallel transport. In the study of the birational ge-

ometry of IHS manifolds, wall divisors play a central role, analogous to the role played

by −2 curves on K3 surfaces. As a ﬁnal contribution, we extend an important prop-

erty of wall divisors to the most general singular setting. The following theorem is the

content of Section 5:

Theorem 1.5. Let Xbe a projective primitive symplectic variety, and let D∈P ic(X)

be a wall divisor on it. Then Dremains a wall divisor under parallel transport, and

there exists ϕ∈Mon2,lt

Hdg(X)such that ϕ(D)is dual to an extremal curve. Conversely,

any divisor dual to an extremal curve is a wall divisor.

Notice that under the additional hypotheses of Q-factoriality and terminality (and

b2≥5) the result above was proved in [LMP22, Proposition 7.4], using, among other

things, the fact that under these hypotheses reﬂections in prime exceptional divisors

are integral. This is trivially false in the non-terminal case [LMP22, Example 3.12].

Hence, here we have to take a diﬀerent path to tackle the general case. It is also

important to notice that the deﬁnition of wall divisors in this larger framework (cf.

Deﬁnition 5.1) includes an extra condition which can be deduced from the Morrison–

Kawamata cone conjecture when b2≥5 and the variety is Q-factorial and terminal,

see Remark 5.2 for a detailed discussion of this point.

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FOOTNOTESTOTHEBIRATIONALGEOMETRYOFPRIMITIVESYMPLECTICVARIETIESCHRISTIANLEHN,GIOVANNIMONGARDI,ANDGIANLUCAPACIENZAAbstract.Inthisnote,weextendtothesingularcasesomeresultsonthebirationalgeometryofirreducibleholomorphicsymplecticmanifolds.InmemoriadiAlbertoCollinoContents1.Introduction12.Preliminaries43...

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FOOTNOTES TO THE BIRATIONAL GEOMETRY OF PRIMITIVE SYMPLECTIC VARIETIES CHRISTIAN LEHN GIOVANNI MONGARDI AND GIANLUCA PACIENZA

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