The geometry of antisymplectic involutions, I
Laure Flapan, Emanuele Macrì, Kieran G. O'Grady, Giulia Sacc?
aa r X i v : . [ m a t h . AG ] F e b THE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I
LAURE FLAPAN, EMANUELE MACR`I, KIERAN G. O’GRADY, AND GIULIA SACC `A
Abstract.
We study fixed loci of antisymplectic involutions on projective hyperk¨ahler man-ifolds. When the involution is induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice, we show that the number of connected components of the fixedlocus is equal to the divisibility of the class, which is either 1 or 2.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Surgery on antisymplectic involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. Birational models of Lagrangian fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. Connected components of the fixed locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265. Proof of the Main Theorem in divisibility 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.
Introduction
An involution of a compact irreducible hyperk¨ahler (HK) manifold is antisymplectic if itacts as ( −
1) on the space of global holomorphic 2-forms. The goal of this paper is to describefixed loci of antisymplectic involutions of HK manifolds of K3 [ n ] -type, namely deformationsof the Hilbert scheme of length n subschemes of a K3 surface. The involutions that we studyhave (+1) eigenspace in H spanned by a class of degree 2. HK’s with such an involutionvary in families of the maximum allowable dimension and appear quite frequently; they arethe simplest antisymplectic involutions that deform in codimension one. We carry out our Mathematics Subject Classification.
Key words and phrases.
Projective hyperk¨ahler manifolds, antisymplectic involutions, Lagrangian fibra-tions, moduli spaces, Bridgeland stability.L.F. was partially supported by the NSF grants DMS-1803082, DMS-1645877, as well as DMS-1440140,while in residence at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California duringthe Spring 2019 semester. E.M. was partially supported by the NSF grant DMS-1700751, by the Institutdes Hautes ´Etudes Scientifiques (IH´ES), by a Poste Rouge CNRS at Universit´e Paris-Sud and by the ERCSynergy Grant ERC-2020-SyG-854361-HyperK. K.O’G. was partially supported by PRIN 2017 “Moduli andLie Theory”. G.S. was partially supported by the NSF grant DMS-1801818. analysis by letting the HK degenerate. As a matter of fact, the degeneration is quite mild– itis the contraction of a (smooth) HK—but the involution on the singular variety is descendedfrom an antisymplectic involution on the smooth HK whose (+1) eigenspace in H has rank 2.In turn the smooth contracted HK is birational to a Lagrangian HK. Wall crossing techniquesallow us to reduce everything to an analysis of fiberwise involutions of Lagrangian HK’s.The motivation for studying these fixed loci comes from two different directions. Thefirst direction is to explore further the relationship between HK manifolds of K3 [ n ] -type andFano manifolds of K3 type, seen in classical constructions [BD82, DV10, LLSvS17, Kuz16]as well as recent works [IM15, FM19, BFM19]. These constructions produce a HK manifoldof K3 [ n ] -type starting from a Fano manifold of K3 type. The results of this paper togetherwith the subsequent paper [FM+21] yield a reverse process in which one starts with a HKmanifold X of K3 [ n ] -type and produces a corresponding Fano manifold arising as a connectedcomponent of the fixed locus of an antisymplectic involution on X .The second motivating direction for studying these fixed loci is to produce coveringfamilies of Lagrangian cycles on HK manifolds. Low-dimensional examples, discussed below,suggest that one component of the fixed locus of an antisymplectic involution of a HK ofK3 [ n ] -type with the maximum number of moduli might, up to a multiple, be contained in acovering family of Lagrangian cycles.1.1. Motivating examples.
The following are examples of antisymplectic involutions ofHK’s of K3 [ n ] -type with the maximum number of moduli:(a) Let ( S, l ) be a polarized K3 surface of degree 2, and let τ ∈ Aut( S ) be the coveringinvolution of the 2 : 1 map S → | l | ∨ ∼ = P .(b) Let f : X → Y be the natural double cover of a general EPW sextic Y ⊂ P , and let τ ∈ Aut( X ) be the covering involution of f ; X is of K3 [2] -type (see [O’G06]).(c) Let Y ⊂ P be a smooth cubic fourfold not containing a plane, and let Z be theassociated LLSvS HK manifold of K3 [4] -type. Recall that Z is a specific rationallyconnected quotient of M ( Y ), the irreducible component of the Hilbert scheme of Y containing smooth twisted cubic rational curves. More precisely there is an open densesubset Z → Z whose points parametrize P -families of (possibly degenerate) twistedcubic curves on a cubic surface in Y . The complement Z \ Z is isomorphic to Y itself,and the point y ∈ Y parametrizes the family of cubic curves with an embedded pointat y . One defines an involution Z → Z by mapping a twisted cubic curve C ⊂ Y tothe equivalence class representing the residual intersection of Y and a quadric surfacein h C i containing C . The involution extends to a regular involution τ and Y is aconnected component of the fixed locus (see [Leh15, LLSvS17], [CCL18, Remark 2.19],and [LPZ18]). HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 3
Let us look at the fixed loci in the above three examples. Of course, the fixed locusof an antisymplectic involution is a Lagrangian submanifold (see e.g., [Bea11]). In (a) thefixed locus is an irreducible curve of general type and it moves in a family of Lagrangianscovering S , namely the elements of | l | . In (b) the fixed locus is an irreducible surface ofgeneral type [Fer11, IM11]. The fixed locus itself does not move, but there is a coveringfamily of Lagrangian cycles containing 2 Fix( τ ). In (c) the picture is even more intriguing.The fixed locus Fix( τ ) has two connected components, one whose points are in one-to-onecorrespondence with Cayley cubic surfaces in Y , and one isomorphic to the variety Y itself.As mentioned above we expect that a multiple of the first component moves in a coveringfamily of Lagrangian subvarieties. On the other hand we expect that no multiple of thesecond component (isomorphic to Y ) moves in Z .We predict that the above examples are instances of a general phenomenon. In partic-ular, the main result of this paper characterizes the number of connected components of thefixed locus of the antisymplectic involution considered. In order to start making sense of sucha statement, we record the behaviour of H ( τ ) in the three cases. The polarization has square2 with respect to the Beauville-Bogomolov-Fujiki (BBF) form ([LPZ18]) and H ( τ ) is equalto the reflection in the polarization (this is why the families have maximal dimension), butthere is a key difference between the polarizations in Cases (a), (b) on one hand, and Case (c)on the other. In the first two cases the polarization has divisibility 1, in the third case it hasdivisibility 2 - we recall that if ( L, ( , )) is a (non degenerate) lattice and 0 = v ∈ L , thenthe divisibility of v is equal to the positive generator of the ideal ( v, L ) ⊂ Z .1.2. Statement of the main result.
First we recall some well-known results on HK’s ofK3 [ n ] -type. Let X be such a HK, let q X be the BBF quadratic form on H ( X ; Z ), and let λ be an ample class on X such that q X ( λ ) = 2. By the Global Torelli Theorem (see [Ver13,Mar10b, Huy12] and [Mar11, Theorem 1.3]), there exists an involution τ ∈ Aut( X ) such that H ( τ ) + = Z λ , and this τ is unique ([Bea83]; also [Deb18, Proposition 4.1]). We recall alsothat polarized ( X, λ ) of K3 [ n ] -type with q X ( λ ) = 2 and div( λ ) = 1 exist for any n , while thosewith div( λ ) = 2 exist if and only if 4 | n (note that div( λ ) divides q X ( λ )). Main Theorem.
Let ( X, λ ) be a polarized HK of K [ n ] -type such that q X ( λ ) = 2 , and let τ ∈ Aut( X ) be the involution associated to λ . Then the number of connected components of Fix( τ ) is equal to the divisibility div( λ ) . In the sequel [FM+21] of the current paper, we study the geometry of these fixed com-ponents. In particular, we examine the case of divisibility 2 in more detail and show that oneconnected component Y of the fixed locus Fix( τ ) is a Fano manifold, generalizing the case ofthe cubic fourfold in (c). The other component in divisibility 2 and the unique component indivisibility 1 are expected to be of general type, more details are in [FM+21]. LAURE FLAPAN, EMANUELE MACR`I, KIERAN G. O’GRADY, AND GIULIA SACC `A
In the case of divisibility 2, we show that the two connected components of the fixedlocus can be distinguished by the behavior of any lift of the involution to the total space ofthe line bundle with class λ . More precisely, over one component the action on the fibers of λ is trivial while on the other component it is multiplication by ( − Idea(s) from the proof of the Main Theorem.
Let ( n, d ) ∈ N + × { , } with d = 1if n X, λ ) of K3 [ n ] -type with q X ( λ ) = 2 and div( λ ) = d . Itfollows that the deformation class of Fix( τ ) is independent of ( X, λ ), where τ ∈ Aut( X ) isthe unique involution such that H ( τ ) + = Z λ . Hence in order to prove our results it sufficesto prove them for a single ( X, λ ). This is more easily said than done. Even in the case ofdouble EPW sextics, where the fixed locus has an explicit embedding in P , the properties ofFix( τ ) are obtained (following Ferretti [Fer11]) by first specializing X to S [2] where S ⊂ P isa quartic surface, so that Fix( τ ) specializes to the surface of bitangent lines to S , and thenby invoking (non trivial) results of Welters [Wel81].On the other hand, there are several explicit constructions of HK’s X of K3 [ n ] -type withan antisymplectic involution τ such that H ( τ ) + has rank 2 (and thus deform in codimension 2in moduli). One source of examples are moduli spaces of stable sheaves (or more generally,Bridgeland stable objects) on a K3 surface S which is a double cover S → P : the coveringinvolution of S induces an involution of the moduli space, provided the Chern character of thesheaves and the stability condition are invariant under the involution of S . Another sourceof examples are Lagrangian fibrations X → P n with a fiberwise involution acting as ( − H of a smooth (abelian) fiber. In several such examples (covering all ( n, d ) as above),by making a suitable “very general” assumption, we may suppose that NS( X ) has rank 2,that it equals H ( X ) + , and that it contains a big class λ of square 2. The class λ is notample. Following [BaMa14b] there is an explicit sequence of flops X X + such that theclass λ + ∈ NS( X + ) corresponding to λ is nef. An explicit divisorial contraction X + → X produces a singular holomorphic symplectic variety X with a Cartier divisor class λ pullingback to λ + , and an involution τ which corresponds to τ via the birational map X X .We prove that a general smoothing of ( X , λ ) is a polarized ( X t , λ t ) with q X t ( λ t ) = 2 anddiv( λ t ) = d , and the specialization of the involution τ t for t → τ . It follows thatthe components of Fix( τ t ) can be described starting from a description of the components ofFix( τ ).A baby example of the specialization X t → X is provided by a family of double covers X t → P ramified over a sextic curve Γ t ⊂ P which is smooth for t = 0 and has a node p for t = 0 (and no other singularity). The central surface X has an ordinary double point.After a base change we get a smooth filling f X → T . The central fiber e X = X + is the HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 5 minimal desingularization of X and is a double cover X + → Bl p ( P ) ∼ = F . Since we are indimension 2 there are no flips, hence τ is the covering involution of the double cover X + → F .Clearly Fix( τ ) = e Γ is the desingularization of the branch curve Γ of X → P , and Fix( τ )is obtained from e Γ by joining two points. This is the simplest example of our specializationin divisibility 1.The simplest example of our specialization in divisibility 2 is provided by the LLSvSeightfold associated to a cubic fourfold Y ⊂ P with an ordinary node and not containinga plane. We sketch the picture and we refer to [Leh18] for more details (the results mostrelevant to us had been developed by Starr in unpublished work). The construction of theLLSvS variety associated to a smooth cubic not containing a plane works also for Y andit produces a singular HK 8-dimensional variety Z . The variety parametrizing lines in Y containing the node is a (smooth) K S ⊂ P , i.e., of genus 4. As shown in [Leh18,Section 6], Z is birational to the (HK) moduli space M of sheaves ι C, ∗ ξ where ι C : C ֒ → S is the inclusion of a (genus 4) hyperplane section of S and ξ is a torsion free sheaf on C ofdegree 0 (in [Leh18] the degree is 6, i.e., deg ω C ; ours is simply a different normalization).Notice that there is a Lagrangian fibration M → ( P ) ∨ defined by associating to ι C, ∗ thecurve C . The antisymplectic involution on Z defines a regular antisymplectic involution of M (a priori it is only birational), namely the fiberwise involution τ mapping ι C, ∗ ξ to ι C, ∗ ξ ∨ if ξ is locally free (if Y is very general with a node there is a single antisymplectic birationalinvolution of M ). The upshot is that we are reduced to studying the involution τ on M andthe birational map between M and Z . We notice that in this case the birational map is thecomposition of two flops and a divisorial contraction, see Example 3.27. As the dimensionincreases the birational map becomes more and more complex.1.4. Higher divisibility.
There are other examples of antisymplectic involutions deformingin codimension one. For instance, let X be a projective HK manifold of K3 [ n ] -type and λ anample class on X such that q X ( λ ) = 2( n −
1) and div( λ ) = n −
1. Then again by [Mar11,Section 9.1.1] there exists a unique antisymplectic involution τ of X such that H ( τ ) + = Z λ .However, to study these involutions it seems likely one should chose different degenerationsof the involution than those we consider here. Acknowledgements.
The paper benefited from many useful discussions with the followingpeople which we gratefully acknowledge: Enrico Arbarello, Arend Bayer, Marcello Bernardara,Olivier Debarre, Tommaso de Fernex, Enrico Fatighenti, Alexander Kuznetsov, Giovanni Mon-gardi, Alexander Perry, Paolo Stellari. Parts of the paper were written while the authors werevisiting several institutions. In particular, we would like to thank Coll`ege de France, ´EcoleNormale Sup´erieure, Institut des Hautes ´Etudes Scientifiques, Universit´e Paris-Saclay, CIRMin Luminy and MSRI in Berkeley for the excellent working conditions.
LAURE FLAPAN, EMANUELE MACR`I, KIERAN G. O’GRADY, AND GIULIA SACC `A Surgery on antisymplectic involutions
This is a general section on HK manifolds: we use results by Markman and Namikawato allow for deformations/smoothings of pairs of HK manifolds (or holomorphic symplecticvarieties) together with an involution.Let X be a compact smooth irreducible hyperk¨ahler manifold (in short, HK manifold )and let τ ∈ Aut( X ) be an involution satisfying the following requirements:(a) The eigenspace decomposition of the action of τ on H ( X ; Q ) is(1) H ( τ ) + = h λ, δ i , H ( τ ) − = h λ, δ i ⊥ . where λ, δ ∈ NS( X ) are linearly independent.(b) There is a divisorial contraction ϕ : X → X of an irreducible divisor ∆ representingthe class δ . (c) We have λ = ϕ ∗ ( λ ) for the class λ ∈ H ( X ) of a Cartier divisor on X .(d) The involution τ descends to an involution τ ∈ Aut( X ).By [Nam01], it follows that Def( X ) is smooth and there is a natural branched covering m : Def( X ) → Def( X ) with branch divisor B ( X ) ⊂ Def( X ). By [Mar10a, Section 3], since∆ is irreducible, the general fiber of ∆ → ϕ (∆) is either a P or two copies of P inter-secting transversally; in our applications, only the first case will occur. Hence, by [Mar10a,Theorem 1.4] the map m is a double cover. Moreover m − ( B ( X )) = Def( X, δ ) , where the right hand side parametrizes deformations of X which keep δ a Hodge class. Infact, by [Mar13, Lemma 5.1] the smooth codimension one subvariety Def( X, δ ) parametrizesdeformations of the pair ( X, ∆). In particular, X t and X m ( t ) ( X t and X u are the varietiesparametrized by t ∈ Def( X ) and u ∈ Def( X ) respectively) are isomorphic for t outside m − ( B ( X )), since ϕ is a divisorial contraction.Let Def( X, τ ) ⊂ Def( X ) be the locus parametrizing deformations of the pair ( X, τ ). Proposition 2.1.
Keep notation and hypotheses as above. Then
Def(
X, τ ) is smooth ofcodimension in Def( X ) , and is not contained in B ( X ) .Proof. The involution τ induces an involution on Def( X ) and on the universal family. Inparticular, Def( X, τ ) is the fixed locus of the action on Def( X ) and there is an inducedrelative involution on the restriction of the universal family to Def( X, τ ). Since Def( X ) issmooth it follows that Def( X, τ ) is smooth as well. By divisorial contraction we mean that X is normal and the relative N´eron-Severi has rank 1. HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 7
If Def(
X, τ ) = Def( X ), we would have an antisymplectic involution on a general defor-mation of X . That is impossible, because such an involution would act as multiplication by( −
1) on the whole of H ( X ). Hence Def( X, τ ) has codimension at least 1 in Def( X ).We prove the proposition by comparing Def( X ) and Def( X ). The eigenspaces of H ( τ )are given by (1). Since τ is antisymplectic it follows that the fixed locus of the action of τ on H ( X, Θ X ) ∼ = H ( X, Ω X ) is equal to h λ, δ i ⊥ . Hence Def( X, τ ) ⊂ Def( X ) is a (smooth)subspace of codimension 2. Since m (Def( X, τ )) ⊂ Def(
X, τ ) ∩ B ( X ) , we get that Def( X, τ ) ∩ B ( X ) has codimension at most 1 in B ( X ).Hence in order to finish the proof it suffices to show that Def( X, τ ) is not contained in B ( X ). There is a local equation g of Def( X, δ ) which is a ( − τ on the germ of Def( X ). Since m is a double cover with ramification divisor Def( X, δ ), itfollows that g is the pull-back of an element h of the germ of Def( X ) generating the ideal of B ( X ). By compatibility of τ and τ , h is a (+1)-eigenvalue for the action of τ . This provesthat Def( X, τ ) is not contained in B ( X ), and hence it has codimension 1 in Def( X ). (cid:3) For t ∈ Def(
X, τ ) we let τ t ∈ Aut( X t ) be the antisymplectic involution extending τ . If t ∈ Def( X ) \ B ( X ), then the pair ( X t , τ t ) is a smoothing of ( X, τ ). Corollary 2.2.
There is a line bundle L on the family Y → Def(
X, τ ) with the followingproperties: • c ( L ) = λ . • If t ∈ Def(
X, τ ) , then L t is ample and H ( τ t ) + = Q · c ( L t ) . • For t / ∈ B ( X ) , q X t (c ( L t )) = q X ( λ ) and div(c ( L t )) = div( λ ) .Proof. In general, let X be a (smooth) HK manifold and let ψ ∈ Aut( X ) be an antisymplecticinvolution. Then H ( ψ ) + is contained in H , ( X ) and is spanned by its intersection with H ( X ; Q ). Since it contains K¨ahler classes (if ω is a K¨ahler class on X then ω + ψ ∗ ( ω ) is aK¨ahler class in H ( ψ ) + ), it follows that H ( ψ ) + contains ample classes. If moreover Def( X, ψ )has codimension 1 in Def( X ), then H ( ψ ) + has dimension 1 and hence it is generated by anample primitive class. Applying this to τ t , for t ∈ Def(
X, τ ) \ B ( X ), we get the corollary. (cid:3) Remark 2.3.
Notice that the same proof gives similar results also in the case of floppingcontractions and compatible involutions. More precisely, we still start with a HK manifold X , but we now assume that τ is only a birational involution. We further assume that:(a) H ( τ ) + = h λ i , H ( τ ) − = h λ i ⊥ .(b) There is a birational contraction ϕ : X → X onto a normal variety and a class λ ∈ H ( X ) of an ample Cartier divisor such that λ = ϕ ∗ ( λ ).(c) The rational involution τ descends to a regular involution τ ∈ Aut( X ). LAURE FLAPAN, EMANUELE MACR`I, KIERAN G. O’GRADY, AND GIULIA SACC `A
In this case, again by using [Mar10a, Theorem 1.4], the morphism m gives an isomorphism m : Def( X ) ∼ = −→ Def( X ). Then we can deduce directly that Def( X, τ ) is smooth of codimen-sion 1 in Def( X ), together with the existence of a line bundle L on the universal family withthe same properties as in Corollary 2.2. We will expand on this in [FM+21]. Example 2.4.
Let π : S → P be the double cover ramified over a smooth sextic curve andlet ι ∈ Aut( S ) be the covering involution of π . Then S is a K3 surface and ι is antisymplectic.Let X := S [ n ] and let τ ∈ Aut( X ) be the involution induced by ι , i.e., τ ([ Z ]) := [ ι ( Z )] for[ Z ] ∈ S [ n ] . Then ( X, τ ) satisfies the requirements (a)–(d) above. In fact, let Y := S ( n ) (the n -th symmetric power of S ) and let ϕ : X → X be the Hilbert-Chow map. Then ϕ isthe contraction of the divisor ∆ ⊂ S [ n ] parametrizing non-reduced subschemes. Moreover λ = h ( n ) where h := π ∗ c ( O P (1)) and h ( n ) ∈ H ( S ( n ) ; Z ) is the “symmetrization” of h . Theinvolution τ ∈ Aut( X ) is defined by τ ( P j m j x j ) = P j m j ι ( x j ). By Proposition 2.1 andCorollary 2.2 the involution τ deforms to involutions τ t ∈ Aut( X t ) where X t is smooth and H ( τ t ) + = Q · c ( L t ) where c ( L t ) is a deformation of h ( n ) (and hence q X t ( L t ) = 2). ThusFix( τ ) is a specialization of Fix( τ t ), and we may hope to determine discrete invariants ofFix( τ t ) for general t by examining of Fix( τ ). The latter is a union of irreducible componentseach isomorphic to the product of two factors, a symmetric power of P and a symmetricpower of the branch curve of π . From this description it is not clear that Fix( τ t ) is irreduciblefor general t , nor it is easy to determine its birational invariants. The moral is that one shouldapply the results in this section to carefully chosen examples. On the other hand, we will usea variant of this example to study the Fano component in the divisibility 2 case in [FM+21].3. Birational models of Lagrangian fibrations
In this section we study the nef and movable cone of certain Lagrangian fibrations withinvolution in order to study degenerations of involutions induced by ample classes of degree2 and to apply the surgery criterion of Section 2. We introduce the relevant examples inSection 3.1 and we present a very short review of Bridgeland’s theory of stability conditionsin Section 3.2. The two main results are in Section 3.3 (divisibility 1) and Section 3.4 (divisi-bility 2).Throughout this section S is a smooth projective K3 surface. We let D : D b ( S ) op ∼ = −−→ D b ( S ) D ( ) := R H om S ( , O S )[1]be the anti-autoequivalence given by the derived dual composed with the shift functor [1].3.1. The examples.
Let h be the class of an ample divisor H on S and let h = 2 g −
2. Weassume that all the curves in the linear system | O S ( H ) | are integral; for example, this is thecase if NS( S ) = Z · h . We consider the moduli spaces M h (0 , h,
0) and M h (0 , h, − g ) HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 9 of h -semistable pure sheaves on S with Mukai vector v equal to (0 , h,
0) and (0 , h, − g )respectively. An element of M h (0 , h,
0) is the isomorphism classe of a sheaf i C, ∗ ( ξ ), where i C : C ֒ → S is the inclusion of a curve C ∈ | O S ( H ) | and ξ is a torsion free sheaf of degree g − C . Similarly, an element of M h (0 , h, − g ) is the isomorphism classes of a sheaf i C, ∗ ( ξ )where ξ is a torsion free sheaf of degree 0 on C . In both cases we have a fibration(2) M h ( v ) π −→ | O S ( H ) | ∼ = P g [ F ] Supp det ( F ) , where Supp det ( F ) is the Fitting support (see [LP93]). Of course the fiber of π over a curve C is the compactified Jacobian Pic g − ( C ) and Pic ( C ), depending on whether v = (0 , h, v = (0 , h, − g ). We have the involutions, induced by the anti-autoequivalences D and O S ( − H ) ⊗ D , respectively:(3) M h (0 , h, τ −→ M h (0 , h, F ] [ E xt ( F, O S )] M h (0 , h, − g ) τ −→ M h (0 , h, − g )[ F ] [ E xt ( F, O S ( − H ))](see [ASF15, Lemmas 3.7 & 3.8]). Equivalently, when C is a smooth curve, if [ i C, ∗ ( ξ )] ∈ M h (0 , h,
0) then τ ([ i C, ∗ ( ξ )]) = i C, ∗ ( ω C ⊗ ξ ∨ ), where we set ξ ∨ := H om ( ξ, O C ), while if[ i C, ∗ ( ξ )] ∈ M h (0 , h,
0) then τ ([ i C, ∗ ( ξ )]) = i C, ∗ ( ξ ∨ ).Let v denote one of the two Mukai vectors considered above. Under our assumptionsevery sheaf parametrized by M h ( v ) is stable, hence M h ( v ) is smooth. In fact M h ( v ) is aHK manifold of K3 [ g ] -type (see [Muk84, Huy97, O’G97, Yos01b]) and the fibration in (2) isLagrangian. Let ϑ : v ⊥ ∼ = −−→ H ( M h ( v ); Z )be Mukai’s isomorphism, as defined in [Yos01b, (1.6)]. We recall that ϑ is an isomorphismof lattices, where v ⊥ is equipped with the restriction of Mukai’s pairing on H ∗ ( S ; Z ) andof course H ( M h ( v ); Z ) is equipped with the Beauville-Bogomolov-Fujiki symmetric bilinearform. In addition ϑ is an isomorphism of Hodge structures ( v ⊥ is a sub-Hodge structure of H ∗ ( S ; Z ) with Mukai’s Hodge structure). In particular, it induces an isometry between classesin v ⊥ alg := v ⊥ ∩ H ∗ alg ( S ; Z ) and the N´eron-Severi group NS( M h ( v )). Here H ∗ alg ( S ; Z ) denotesthe group of algebraic classes, or equivalently those of type (1 , H ∗ ( S ; Z ).The result below describes the action of τ on H ( M h ( v ); Z ) in terms of ϑ . Proposition 3.1.
Let v be one of the two Mukai vectors considered above and let τ ∈ Aut( M h ( v )) be the involution in (3) . Suppose that all the curves in the linear system | O S ( H ) | are integral. Then the H ( τ ) -eigenspace decomposition is (4) H ( M h ( v ); Z ) + = ϑ ( Z (1 , ,
0) + Z (0 , , if v = (0 , h, , ϑ ( Z (2 , − h,
0) + Z (0 , , if v = (0 , h, − g ) , and (5) H ( M h ( v ); Z ) − = ϑ ( H ( S ; Z ) pr ) . Proof.
The involution is antisymplectic because it preserves the fibers of the Lagrangian fi-bration π and it acts as ( −
1) on the H of a smooth fiber of π . Since the decomposition into H ( τ )-eigenspaces is independent of ( S, h ), we may assume that NS( S ) = Z · h and hence(6) NS( M h ( v )) = ϑ ( v ⊥ ∩ ( Z (1 , ,
0) + Z (0 , h,
0) + Z (0 , , . Since NS( M h ( v )) ⊥ is an irreducible Hodge structure containing H , ( M h ( v )), and τ is anti-symplectic, it follows that(7) ϑ ( H ( S ; Z ) pr ) ⊂ H ( M h ( v ); Z ) − . On the other hand τ sends π ∗ c ( O P g (1)) to itself i.e., π ∗ c ( O P g (1)) ∈ H ( M h ( v ); Z ) + . If α is anample class on M h ( v ), then α + τ ∗ ( α ) is an invariant ample class on M h ( v ). It follows that (7)is an equality, i.e., (5) holds. We also get that the (+1)-eigenspace has rank at least 2, andtherefore rank 2 by (7). Since it is orthogonal to H ( M h ( v ); Z ) − it is equal to the right-handside of (6). Computing, one gets (4). (cid:3) Our next task is to describe the classes of some key divisors on M h ( v ) in terms of Mukai’smap ϑ . First of all, by [LP93, Section 2.3], we have that f := c ( π ∗ O P g (1)) = ϑ (0 , , − v = (0 , h, v = (0 , h, − g ) then thedivisor is the proper transform (on a HK manifold birational to M h ( v )) to the contracteddivisor. Proposition 3.2.
Let v = (0 , h, and let D be the determinant line bundle on M h ( v ) withfiber D | [ F ] = ^ r H ( S, F ) ∨ ⊗ ^ r H ( S, F ) over [ F ] , where r := h ( S, F ) = h ( S, F ) . The natural section of F is non zero and its divisor ∆ has the following properties: supp ∆ = { [ F ] ∈ M h ( v ) | h ( S, F ) > } , (8) cl(∆) = ϑ (1 , , . (9) Proof.
Equation (8) is the zero locus of the natural section of D and it is not equal to thewhole of M h ( v ) because if C ∈ | O S ( H ) | is smooth its restriction to π − ( C ) = Pic g − ( C ) is thenatural Θ divisor. Equation (9) follows from the Grothendieck-Riemann-Roch Theorem. (cid:3) HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 11
Remark 3.3.
The divisor ∆ defined above is birational to the relative ( g − | O S ( H ) | . Since dim | O S ( H ) | = g , it is birationalto a P -bundle over S [ g − with generic fiber ruling the pencil of curves through g − g is even. The analogue of ∆ on M h (0 , h, − g ) is defined starting fromthe unique stable vector bundle A on S with Mukai vector v ( A ) := ch( A ) . p td S = (cid:16) , − h, g (cid:17) . The existence and uniqueness of A are proved in [Muk87]. Explicitly, A is a Lazarsfeld–Mukaibundle obtained as elementary modification(10) 0 → A → O ⊕ S → i C, ∗ ( ξ ) → , where C ∈ | O S ( H ) | is a smooth curve and ξ is a line bundle of degree g/ h ( C, ξ ) = 2.
Proposition 3.4.
Suppose that
NS( S ) = Z h , g is even, and g ≥ . Let v = (0 , h, − g ) andlet D be the determinant line bundle on M h ( v ) with fiber D | [ F ] = ^ r H ( S, A ∨ ⊗ F ) ∨ ⊗ ^ r H ( S, A ∨ ⊗ F ) over [ F ] , where r := h ( S, A ∨ ⊗ F ) = h ( S, A ∨ ⊗ F ) . The natural section of D is non zeroand its divisor ∆ has the following properties: supp ∆ = { [ F ] ∈ M h ( v ) | h ( S, A ∨ ⊗ F ) > } , (11) cl(∆) = ϑ (cid:16) , − h, g (cid:17) . (12) Moreover if C ∈ | O S ( H ) | is smooth the restriction of ∆ to π − ( C ) = Pic ( C ) is twice theprincipal polarization.Proof. Equation (11) is the zero locus of the natural section of D . The fact that it is not equalto the whole of M h ( v ) follows from the more general results we will prove in Section 3.4 (andlater on in Section 5, we will also need a more precise vanishing result). We give here in anycase a sketch of the classical argument, for completeness. Let C ∈ | O S ( H ) | be smooth. Thezero locus of the restriction of the section to π − ( C ) = Pic ( C ) is equal to(13) { [ ξ ] ∈ Pic ( C ) | h ( C, A ∨| C ⊗ ξ ) > } . It suffices to prove that the above set is not the whole of Pic ( C ). By [Ray82] this holds if(and only if) the restriction of A ∨ to C is semistable. This can be directly proved as follows.Suppose that A ∨| C is not semistable and let A ∨| C → Q be a destabilizing quotient. Let E be the locally free sheaf on S fitting into the exact sequence0 −→ E −→ A ∨ −→ i C, ∗ ( Q ) −→ . We can compute easily that 4 c ( E ) − c ( E ) ≤ − g + 4 <
0, and so by the Bogomolovinequality it follows that E is slope unstable. Since c ( E ) = 0 and NS( S ) = Z h there exists anon zero map O S ( mH ) → E for some m >
0. Composing with the inclusion E ⊂ A ∨ we geta non zero map O S ( mH ) → A ∨ contradicting the stability of A ∨ .This proves that the set in (13) is not the whole of Pic ( C ). Actually the set in (13) isthe support of a cycle which is well-known to be twice the principal polarization of Pic ( C ),and which is the restriction of ∆ to Pic ( C ). Lastly Equation (12) follows again from theGrothendieck-Riemann-Roch Theorem. (cid:3) Example 3.5.
We recall that the LLvSS variety of a cubic fourfold with a node is birationalto M h (0 , h, −
3) (see the discussion at the end of Section 1.3). An explicit description of anopen non empty subset of ∆ ⊂ M h (0 , h, −
3) can be given as follows. Let C ∈ | O S ( H ) | be ageneral smooth curve. There are two distinct g ’s on C , say D , D , and one has the equalityof divisors on C :∆ | Pic ( C ) = { [ ξ ] ∈ Pic ( C ) | h ( C, D ⊗ ξ ) > } + { [ ξ ] ∈ Pic ( C ) | h ( C, D ⊗ ξ ) > } . For general g we will not have a similar explicit description in terms of linear series, which isthe reason we need to look at higher rank vector bundles and thus using Bridgeland stabilityto deal with them on S .3.2. Review of Bridgeland stability conditions.
Stability conditions on derived cate-gories were defined by Bridgeland in [Bri07]; the definition was extended to include existenceof moduli spaces and to also work in families in [KS08] and [BL+19]. We briefly recall herethe definition given in [MS20, Definition 2.1 & Remark 2.2] in the case of K3 surfaces.
Definition 3.6. A Bridgeland stability condition on D b ( S ) is a pair σ = ( Z, P ) where • Z : H ∗ alg ( S, Z ) → C is a group homomorphism, called central charge , and • P = ∪ ϕ ∈ R P ( ϕ ) is a collection of full additive subcategories P ( ϕ ) ⊂ D b ( S )satisfying the following conditions:(a) for all nonzero E ∈ P ( ϕ ) we have Z ( v ( E )) ∈ R > · e iπϕ ;(b) for all ϕ ∈ R we have P ( ϕ + 1) = P ( ϕ )[1];(c) if ϕ > ϕ and E j ∈ P ( ϕ j ), then Hom( E , E ) = 0;(d) (Harder–Narasimhan filtrations) for all nonzero E ∈ D b ( S ) there exists a finite se-quence of morphisms 0 = E s −→ E s −→ . . . s m −→ E m = E such that the cone of s j is in P ( ϕ j ) for some sequence ϕ > ϕ > · · · > ϕ m of realnumbers;(e) (support property) there exists a quadratic form Q on the vector space H ∗ alg ( S ; R ) suchthat HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 13 • the kernel of Z is negative definite with respect to Q , and • for all E ∈ P ( ϕ ) for any ϕ we have Q ( v ( E )) ≥ T and every T -perfect complex E ∈ D T -perf ( S × T ) the set { t ∈ T : E t ∈ P ( ϕ ) } is open;(g) (boundedness) for any v ∈ Λ and ϕ ∈ R such that Z ( v ) ∈ R > · e iπϕ the functor T M σ ( v, ϕ )( T ) := { E ∈ D T -perf ( S × T ) : E t ∈ P ( ϕ ) and v ( E t ) = v, for all t ∈ T } is bounded.The objects in P ( ϕ ) are called σ -semistable of phase ϕ . The simple objects in theabelian category P ( ϕ ) are called σ -stable ; Jordan-H¨older filtrations exist and S -equivalenceclasses of σ -semistable objects are then defined accordingly. The phases of the first and lastfactor in the Harder–Narasimhan filtration of an object E are denoted ϕ + ( E ) and ϕ − ( E ).Definition 3.6 can be reformulated also in terms of slope. More precisely, the extension-closed category A := P ((0 , ,
1] is an abelian category, which is the heart of a bounded t-structure on D b ( S ). The realand imaginary parts of the central charge Z behave like a degree and rank function on A :for a nonzero object E ∈ A , Im Z ( E ) ≥ Z ( E ) = 0, then Re Z ( E ) <
0. Anobject E ∈ A is σ -semistable if and only if it is slope-semistable with respect to the slope µ σ := − Re Z Im Z . Then by property (b) all objects in P are shifts of semistable objects in A .The converse is also true. Let Z be a central charge on the heart of a bounded t-structure A satisfying the above numerical properties. We define (semi)stable objects to be the slope-(semi)stable objects in A , together with their shifts in D b ( S ). We obtain a stability conditionin D b ( S ) once Harder–Narasimhan filtrations exist in A and the remaining properties (e),(f), (g) are satisfied. When we want to stress the category A in the definition of stability weuse the notation σ = ( Z, A ).By Bridgeland’s Deformation Theorem, the set Stab(D b ( S )) of stability conditions is acomplex manifold, when endowed with the coarsest topology such that the functions E ϕ + ( E ) , ϕ − ( E ), for E ∈ D b ( S ), and Z : Stab(D b ( S )) → Hom( H ∗ alg ( S ; C ) , C ) defined by( Z, P ) Z are continuous. We denote by Stab † (D b ( S )) the connected component of thespace of stability conditions on D b ( S ) described in [Bri08]. The map Z on Stab † (D b ( S )) is atopological cover of its image, and the latter is explicitly described in [Bri08, Theorem 1.1]. Example 3.7.
Let us assume that (
S, h ) is a polarized K3 surface with NS( S ) = Z · h . Inthis case, by [BB17, Theorem 1.3], Stab † (D b ( S )) is simply connected and the map Z is theuniversal cover of its image. An open subsetStab geom (D b ( S )) ⊂ Stab † (D b ( S )) can be explicitly defined as follows. Let α, β ∈ R , α >
0. We let σ α,β := ( Z α,β , coh β ( S )),where Z α,β : H ∗ alg ( S, Z ) → C , v ( e ( β + iα ) h , v )and coh β ( S ) := h T β , F β [1] i = E ∈ D b ( S ) : H i ( E ) = 0 , for all i = 0 , − H − ( E ) ∈ F β H ( E ) ∈ T β , where, for a complex E ∈ D b ( S ), H i ( E ) denotes its i -th cohomology sheaf and T β := { E ∈ coh( S ) : all quotients E ։ Q satisfy µ ( Q ) > β } , F β := { E ∈ coh( S ) : all non-trivial subobjects K ֒ → E satisfy µ ( K ) ≤ β } . By [Bri08, Lemma 6.2] and [Tod08, Theorem 1.4], the pair σ α,β gives a stability conditionon D b ( S ) if and only if Z α,β ( δ ) / ∈ R ≤ , for all δ ∈ H ∗ alg ( S, Z ) with δ = −
2. The open subsetStab geom (D b ( S )) consists of all the orbits of the stability conditions σ α,β by the action ofthe group f GL +2 ( R ), the universal cover of the group of 2 × b ( S )) consisting of thosestability conditions where the skyscraper sheaves of length one are all stable of the same phase.The whole component Stab † (D b ( S )) is the union of all the orbits of the closure Stab geom (D b ( S ))with respect to the action of the group of autoequivalences of D b ( S ).Let v ∈ H ∗ alg ( S, Z ) be a primitive Mukai vector. For a stability condition σ = ( Z, P ) ∈ Stab † (D b ( S )), we denote by M σ ( v, ϕ ) the moduli space parametrizing S -equivalence classes of σ -semistable objects in P ( ϕ ) with Mukai vector v . As mentioned above, if the phase is fixed,by shifting we can always assume that all σ -semistable objects lie in A and have Mukai vector ± v . Hence, by abuse of notation, we will write M σ ( v ) and forget the phase. We denote by M st σ ( v ) the open subspace parametrizing σ -stable objects.By [BaMa14a], there is a naturally defined divisor class ℓ σ on M σ ( v ); for the stabilityconditions σ α,β in Example 3.7, the class of this divisor has been computed in [BaMa14a,Lemma 9.2].We have the following result, summarized in [BL+19, Theorem 21.24 & Theorem 21.25],and based on previous work in [Lie06, AP06, Tod08, BaMa14a, AH-LH18]: Theorem 3.8.
Let v ∈ H ∗ alg ( S, Z ) be a Mukai vector with v ≥ − . Then for any σ ∈ Stab † (D b ( S )) the moduli space M σ ( v ) is a non-empty proper algebraic space and the divisorclass ℓ σ is strictly nef. If v is primitive and σ is generic with respect to v , then M σ ( v ) = M st σ ( v ) is smooth projective integral hyperk¨ahler manifold of dimension v + 2 deformation equivalentto the Hilbert scheme of points on a K3 surface and the class ℓ σ is ample. Equivalently, we could have defined M σ ( v ) to denote the moduli space parametrizing S -equivalence classesof σ -semistable objects in A with Mukai vector ± v . HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 15
Mukai’s isomorphism ϑ : v ⊥ ∼ = −→ H ( M σ ( v ); Z ) holds more generally for moduli spaces M σ ( v ), when v is primitive and σ generic with respect to v . By abuse of notation, we willsometimes directly identify v ⊥ with H ( M σ ( v ); Z ), forgetting the map ϑ .We will also need the main result of [BaMa14b] in order to study nef and movable conesof moduli spaces (the case of movable cones, Part (ii) below, was proved earlier and in greatergenerality in [Mar11, Lemma 6.22]). Theorem 3.9.
Let v ∈ H ∗ alg ( S, Z ) be a primitive Mukai vector with v ≥ and let σ ∈ Stab † (D b ( S )) be a generic stability condition with respect to v . Let M := M σ ( v ) .(i) All smooth projective HK manifolds which are birational to M are isomorphic to M σ ( v ) ,for some σ ∈ Stab † (D b ( S )) generic with respect to v , and conversely each M σ ( v ) isbirational to M .(ii) The interior of the movable cone of M is the connected component of Pos( M ) \ [ a ∈ H ∗ alg ( S ; Z ) s.t.either a = − a,v )=0 , or a =0 and ( a,v )=1 , ϑ ( a ⊥ ) , containing an ample divisor, where Pos( M ) = { D ∈ NS( M ) R : q M ( D ) > } .(iii) The ample cones of each birational model M σ ( v ) can be identified to the connectedcomponent of Pos( M ) \ [ a ∈ H ∗ alg ( S ; Z ) s.t. a ≥− ≤ ( a,v ) ≤ v / ϑ ( a ⊥ ) containing an ample divisor on M σ ( v ) . The connection between Part (i) and Parts (ii) and (iii) in Theorem 3.9 is via the map ℓ . More precisely, this map glues to a piece-wise analytic continuous map ℓ : Stab † (D b ( S )) → NS( M ) R . Then any wall in the positive cone of M, namely the codimension 1 boundarycomponents of the above chamber decomposition, is given by a wall in Stab † (D b ( S )), namelythe codimension 1 components of the locus where semistable objects of Mukai vector v change.Conversely, a wall W in Stab † (D b ( S )) gives a wall in the positive cone if and only if, for σ ∈ W , if we write ℓ σ = ϑ ( w σ ), then w σ is orthogonal to a ∈ H ∗ alg ( S ; Z ) with a ≥ − ≤ ( a, v ) ≤ v / Example 3.10 ([BaMa14a, Example 9.7]) . Let (
S, h ) be a polarized K3 surface with NS( S ) = Z · h , h = 2 g −
2. Let v ∈ H ∗ alg ( S, Z ) be a primitive Mukai vector, v = ( r , c · h, s ), with r , c , s ∈ Z , r ≥
0. We assume there exists x, y ∈ Z , x >
0, such that xc − yr = 1. Wedefine α > α := x √ g − , if ( g − y +1 x ∈ Z , otherwise.and we consider the stability conditions σ α,y/x in Stab geom (D b ( S )), for α > α .Then there is no wall for M σ α,y/x ( v ) for all α > α . Indeed, the imaginary part of Z α,y/x has the property that1 α · Im Z α,y/x ( r, c · h, s ) = (2 g − xc − yrx ∈ (2 g −
2) 1 x · Z and (1 /α ) Im Z α,y/x ( v ) = (2 g − /x is the smallest possible positive value.As a consequence, ℓ α,y/x is ample on M σ α,y/x ( v ) for all α > α . In particular, if we denotethe constant moduli space M := M σ α,y/x ( v ), we have that h ℓ α,y/x : α > α i ⊂ Amp( M ) . Example 3.11.
In the notation of Example 3.10, we assume now there exists x, y ∈ Z , x > xc − yr = 2. In this case, there may be walls, but since (1 /α ) Im Z α,y/x ( v ) =2(2 g − /x , if we denote by v the Mukai vector of a destabilizing subobject or quotient,we can only have (1 /α ) Im Z α,y/x ( v ) = (2 g − /x . Hence, if there exists α > α such that E ∈ M σ α ,y/x ( v ) is not stable, we must have an exact sequence0 → F → E → Q → /α ) Im Z α,y/x ( F ) = (2 g − /x and (1 /α ) Im Z α,y/x ( Q ) = (2 g − /x . Thus, F and Q arestable for all α > α , by Example 3.10. In particular, the birational transformation inducedby such a wall at α is a Mukai flop whose exceptional locus is the projective bundle (in theanalytic topology) P Z ( V ), where Z = M σ α ,y/x ( v ( F )) × M σ α ,y/x ( v ( Q ))and V is the twisted vector bundle of rank ( v ( F ) , v ( Q )) induced by the relative Ext-sheaf. Finally, we recall that, under the assumptions of Example 3.10 and Example 3.11, themoduli space M σ α,y/x ( v ) is isomorphic to a moduli space of h -semistable sheaves, for α ≫ α sufficiently large, by [Bri08, Proposition 14.2] and [Tod08, Section 6.2]. V is an actual vector bundle when universal families exist on M σ α ,y/x ( v ( F )) and M σ α ,y/x ( v ( Q )). HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 17
Divisibility 1.
In this section we follow mostly [Bay18], but see also [Mar01, Yos99,Yos01a, AB13]. Let (
S, h ) be a polarized K3 surface of genus g with NS( S ) = Z · h .We let v = (0 , h,
0) and M := M h ( v ). Let f = ϑ (0 , , − , δ = ϑ (1 , ,
1) = cl(∆) , λ = ϑ (1 , , −
1) = 2 f + δ, where ∆ is the divisor defined in Proposition 3.2. Notice that λ has square 2. Recall fromSection 3.1 that the class f induces the Lagrangian fibration π : M → | O S ( H ) | ∼ = P g , and is thus one of the two rays of the nef cone. We can determine the other ray of the nefcone as well. Lemma 3.12.
The nef and movable cone coincide for M : Nef( M ) = Mov( M ) = h f, λ i . Proof.
We want to apply Theorem 3.9. This is an explicit computation with Pell’s equations,but it can also be easily done by using Example 3.10: indeed v = (0 , h,
0) satisfies theassumptions, with x = 1 and y = 0. By [BaMa14a, Lemma 9.2], the associated divisorclass ℓ α, on M satisfies ℓ α, → f, for α → ∞ ℓ α, → λ, for α → √ g − , up to positive multiplicative constants. HenceNef( M ) ⊃ h f, λ i . But δ = − δ, λ ) = 0. Hence, by Theorem 3.9, λ is not in the interior of the movablecone of M and Mov( M ) ⊂ h f, λ i , finishing the proof. (cid:3) Lemma 3.12 gives that the effective cone of M is h f, δ i . Since δ is indivisible and we areassuming that NS( S ) = Z · h , we notice the following consequence about the divisor ∆. Corollary 3.13.
The divisor ∆ on M is prime i.e., reduced and irreducible. Notice that the proof of Lemma 3.12 gives a result slightly stronger than the thesis,namely that all objects in M are stable for all stability conditions σ α, , α > √ g − . Next, wedescribe the divisorial contraction induced by λ (see [Bay18, Corollary 6.7]). Lemma 3.14.
The class λ induces a divisorial contraction ϕ : M → M with exceptional divisor ∆ .Proof. We use the notation from [BaMa14b, Section 5]. The rank 2 primitive hyperboliclattice H ⊂ H ∗ alg ( S, Z ) associated to the class λ is given by H = Z δ + Z v. By Theorem 3.9 (see also [BaMa14b, Theorem 5.7]), we are looking for classes a ∈ H suchthat a ≥ − ≤ ( a, v ) ≤ g −
1. It is immediate to see that the only possibility is when a = ± δ . Hence, the exceptional locus of the divisorial contraction ϕ is ∆. (cid:3) Next we examine a stratifcation of ∆. Given k ≥
1, set∆( k ) := (cid:8) F ∈ M : h ( S, F ) = k (cid:9) . Thus ∆ is the disjoint union of the ∆( k )’s. A more precise result is the following. Lemma 3.15.
We have (14) ∆ = G ≤ k ≤⌊√ g ⌋ ∆( k ) , ∆( i ) = G i ≤ k ≤⌊√ g ⌋ ∆( k ) . Moreover let σ ∈ Stab geom (D b ( S )) ⊂ Stab † (D b ( S )) be a stability condition such that ℓ σ = λ .Then each ∆( k ) is isomorphic to a locally trivial (in the analytic topology) Grassmannianbundle Gr( k, U k ) over the (2 g − k ) –dimensional moduli space M st σ ( b k ) of stable objectswhere b k = − ( k, − h, k ) and where U k is a twisted vector bundle of rank k induced by therelative Ext-sheaf.Proof. We describe each ∆( k ) as a Grassmannian bundle over M st σ ( b k ). Explicitly, if we choose σ = σ α,β with ( α, β ) sufficiently close to (1 / √ g − ,
0) and β <
0, then F ∈ ∆( k ) if and onlyif its Jordan–H¨older filtration with respect to σ is(15) O S ⊗ W ∨ → F → F k , where F k ∈ M st σ ( b k ), dim W = k , U k := Hom( F k , O S [1]), and where W is viewed as a subspaceof U k by applying Hom( , O S ) to the triangle above and using the fact that Hom( F, O S ) = 0.The twisted vector bundle U k has fiber equal to U k over the point F k ∈ M st σ ( b k ). The upperbound k ≤ ⌊√ g ⌋ follows because the moduli space has dimension (2 g − k ), and the secondequation in (14) holds because each stratum has the expected codimension, as determinantalsubvariety. (cid:3) HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 19
Finally, we look at the involution τ on M ; recall from Section 3.1 that it is inducedby the functor D and acts fiberwise with respect to the Lagrangian fibration π . Recall thatST O S : D b ( S ) ∼ = −→ D b ( S ) denotes the spherical twist at O S . Lemma 3.16.
We keep the notation of Lemma 3.14. The involution τ on M maps each ∆( k ) to itself and is compatible with the Grassmannian bundle ∆( k ) → M st σ ( b k ) , with the involutionon M st σ ( b k ) induced by the anti-autoequivalence Φ : D b ( S ) op ∼ = −→ D b ( S ) given by Φ := ST O S ◦ D .In particular τ induces a regular involution τ : M ∼ = −→ M .Proof. Let F ∈ ∆( k ). Since χ ( O S , F ) = 0, the dimension of hom( O S , F ) and hom( O S , D ( F ))are the same. Hence, by definition, τ maps each ∆( k ) to itself.To describe explicitly the involution induced on M st σ ( b k ) in terms of the functor Φ, wesimply apply the functors D and Φ to (15):(16) O S ⊗ K / / ev (cid:15) (cid:15) O S ⊗ U / / ev (cid:15) (cid:15) O S ⊗ Q ev (cid:15) (cid:15) O S ⊗ W / / (cid:15) (cid:15) D ( F k ) / / (cid:15) (cid:15) D ( F ) (cid:15) (cid:15) Φ( O S ⊗ W ∨ [1]) / / Φ( F k ) / / Φ( F )for the graded vector spaces K := R Hom( O S , O S ) ⊗ W, U := R Hom( O S , D ( F k )) , Q := R Hom( O S , D ( F )) . By using the identification U ∼ = U k , we can identify the first line of (16) with O S ⊗ ( W ⊕ W [ − −→ O S ⊗ U k −→ O S ⊗ ( U k /W ⊕ W [ − , and so we deduce the exact triangle O S ⊗ ( U k /W ) → D ( F ) → Φ( F k ) , as we wanted. Notice that this is nothing but the Mukai involution in divisibility 1, studiedin [Bea83, O’G05]. (cid:3) Remark 3.17.
Given an element F k ∈ M st σ ( b k ) such that Φ( F k ) = F k , we will describe in[FM+21] the fixed locus of the involution τ on the Grassmannian Gr( k, U k ), where U k =Hom( F k , O S [1]): it is irreducible and isomorphic to the Lagrangian Grassmannian LGr( k, U k ). Remark 3.18.
Observe that the moduli space M , the involution τ , and the contraction ϕ : M → M satisfy the requirements of Section 2: (a) is Proposition 3.1, (b) and (c) areby construction, and (d) follows from Lemma 3.16. Hence in the divisibility 1 case, thespecialization we consider for the proof of the Main Theorem is the pair ( M , λ ), where λ denotes the ample Cartier divisor class induced by λ on M , equipped with the involution τ . Divisibility 2.
Let (
S, h ) be a polarized K3 surface of genus g with NS( S ) = Z · h .Throughout the present subsection we assume that the genus is divisible by 4. Let v =(0 , h, − g ) and let M := M h ( v ).Let f = ϑ (0 , , − , δ = ϑ (cid:16)(cid:16) , − h, g (cid:17)(cid:17) = cl(∆) , λ = ϑ (cid:16)(cid:16) , − h, g − (cid:17)(cid:17) = f + δ, where ∆ is the divisor defined in Proposition 3.4. Notice that λ has square 2. Recall fromSection 3.1 that the class f induces the Lagrangian fibration π : M → | O S ( H ) | ∼ = P g , and is thus one of the two rays of the nef cone.In order to describe the wall and chamber decomposition for the moduli spaces M , weintroduce two integers c, d ∈ Z with c ≥ d ≥ −
1, and satisfying the following two conditions:(a) 4 d + (2 c + 1) ≤ g − ( g − c − d c +1 ∈ Z .Then we set µ c,d := g − − d − (2 c + 1) c + 1) ≥ a c,d := (cid:18) c + 1 , − c · h, ( g − c − d c + 1 (cid:19)e a c,d := λ + µ c,d f. Notice that by definition d = a c,d /
2, ( e a c,d , a c,d ) = 0, and by (b) a c,d ∈ H ∗ alg ( S, Z ). We definean ordering on the pairs ( c, d ) as above by using the slope µ c,d :( c, d ) (cid:22) ( c ′ , d ′ ) if and only if µ c,d ≤ µ c ′ ,d ′ . With respect to this ordering, the first values are:(0 , − (cid:23) (0 , (cid:23) . . . The Mori chamber decomposition of the movable cone is more interesting than in thedivisibility 1 case discussed in the previous subsection.
Lemma 3.19.
The movable cone for M is Mov( M ) = h f, λ i . The ordered rays generated by ϑ ( e a c,d ) , where c, d ∈ Z , c ≥ , d ≥ − , satisfy (a) and (b), givethe stable base locus decomposition of Mov( M ) . HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 21
Proof.
This is an immediate calculation from Theorem 3.9. Indeed, we look for Mukai vectors a ∈ H ∗ alg ( S, Z ) satisfying a ≥ −
2, 0 ≤ ( a, v ) ≤ g −
1. There are only two possibilities for( a, v ): either ( a, v ) = 0 (and so, a = − a, v ) = g −
1. The first case gives λ . For thesecond case, if we write a = ( r, − c · h, s ), then the condition ( a, v ) = g − r = 2 c + 1and, by setting d := a /
2, we can directly express s as a function of c and d ; hence, these a areexactly the vectors a c,d , whose corresponding ray in the positive cone Pos( M ) is exactly theone generated by ϑ ( a c,d ). The fact that this is inside Mov( M ) corresponds to the positivityof its slope µ c,d , which is guaranteed by (a). (cid:3) The birational maps f c,d corresponding to the vectors a c,d in Lemma 3.19 can be explicitlydescribed and have a very simple form: they are all Mukai flops over products of moduli spacesof stable objects. Lemma 3.20.
For c, d ∈ Z as above, the divisor class e a c,d induces a Mukai flop whose excep-tional locus is the projective bundle, in the analytic topology, P c,d := P Z c,d ( V c,d ) where Z c,d = M σ c,d ( a c,d ) × M σ c,d ( v − a c,d ) is a product of smooth HK manifolds of dimension d +2 , where V c,d is the twisted vector bundleof rank ( a c,d , v − a c,d ) induced by the relative Ext-sheaf, and where σ c,d ∈ Stab geom (D b ( S )) ⊂ Stab † (D b ( S )) is a stability condition associated to e a c,d .Proof. We use Example 3.11. Indeed v = (0 , h, − g ) satisfies all the assumptions with x = 2and y = −
1. By [BaMa14a, Lemma 9.2], the associated divisor class ℓ α, − / on M satisfies ℓ α, − / → f, for α → ∞ ℓ α, − / → λ, for α → √ g − , up to multiplicative constants. Hence, all walls in the movable cone will come from a stabilitycondition in this segment σ α, − / , for α > √ g − , thus finishing the proof. (cid:3) Remark 3.21.
The above proof allows us to say a bit more about birational models of M and the structure of stable objects.(a) We first notice that there are no totally semistable walls (in the sense of [BaMa14b,Definition 2.20]) for the Mukai vector v on the segment σ α, − / , α > √ g − ; concretely, thismeans that the general object E in M is σ α, − / -stable, for all α > √ g − . Indeed, by using[BaMa14b, Theorem 5.7] we see that in our case such a totally semistable wall exists if andonly if there exist α > √ g − and a σ α , − / -stable spherical object R in coh − / ( S ) whichsatisfies ( v ( R ) , v ) < E in M on σ α , − / . Example 3.11 then forces ( v ( R ) , v ) = −
1, which is impossible, being a multiple of 1 − g with4 | g . (b) Given c, d ∈ Z as above, we let α ( c,d ) > √ g − ℓ α ( c,d ) , − / is equal to the class ϑ ( e a c,d ), up to a positive constant(namely, we can choose σ c,d as σ α ( c,d ) , − / ). We define for 0 < ǫ ≪ M c,d := M σ α ( c,d )+ ǫ, − / ( v ) ,M ′ c,d := M σ α ( c,d ) − ǫ, − / ( v ) ,M last := M σ √ g − ǫ, − / ( v ) . Notice that M last is equal to M ′ c ,d where ( c , d ) is the minimum ( c, d ) for the total orderingthat we have defined above. Moreover M , − = M and M ′ , − = M , ,ϑ ( e a c,d ) is nef on M c,d , and ϑ ( λ ) is nef on M last .For E ∈ M c,d , we have that E is in the projective bundle P c,d if and only if its Jordan-H¨older filtration with respect to σ α ( c,d ) , − / has only two terms of the form(17) R → E → R ′ with R ∈ M σ α ( c,d ) , − / ( a c,d ) and R ′ ∈ M σ α ( c,d ) , − / ( v − a c,d ). The twisted vector bundle V c,d thenhas fiber Ext ( R ′ , R ) over the point ( R, R ′ ). Example 3.22.
We can describe the first flop, induced by e a , − as follows. The exceptionallocus of the flop consists of those E = O C , for C ∈ | O S ( H ) | , and thus can be identifiedwith the image of the zero section ˇ P g := P ( S, H ( O S ( H ))). The Jordan-H¨older filtration withrespect to σ α (0 , − , − / is given by O S → E → O S ( − H )[1] . On the other side of the flop, the objects are given as non-trivial extensions of the form O S ( − H )[1] → E ′ → O S , which are then parametrized by P ( S, H ( O S ( − H ))) = P ( H ( S, O S ( H )) ∨ ) = P g .As in the divisibility 1 case, we have an explicit description of the divisorial contractioninduced by λ . Let A be the (unique) spherical stable vector bundle on S with Mukai vector v ( A ) = (2 , − h, g/ c, d ∈ Z as above and k ≥
1, we let∆ c,d := { F ∈ M c,d : hom S ( A, F ) > } , ∆ c,d ( k ) := { F ∈ M c,d : hom S ( A, F ) = k } , (18) HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 23 and similarly on M last ∆ last := { F ∈ M last : hom S ( A, F ) > } , ∆ last ( k ) := { F ∈ M last : hom S ( A, F ) = k } . Notice that ∆ c,d , respectively ∆ last , is the strict transform of ∆ on M c,d , respectively M last .An argument analogous to the one given for Lemma 3.14 and Lemma 3.15, gives thefollowing two results. Lemma 3.23.
The class λ induces a divisorial contraction ϕ : M last → M with exceptional divisor ∆ last . Lemma 3.24.
The divisor ∆ last is prime and (19) ∆ last = G ≤ k ≤⌊√ g ⌋ ∆ last ( k ) , ∆ last ( i ) = G i ≤ k ≤⌊√ g ⌋ ∆ last ( k ) . Moreover let σ ∈ Stab geom (D b ( S )) ⊂ Stab † (D b ( S )) be a stability condition such that ℓ σ = ϑ ( λ ) .Then each ∆ last ( k ) is isomorphic to a locally trivial (in the analytic topology) Grassmannianbundle Gr( k, U k ) over the (2 g − k ) –dimensional moduli space M st σ ( b k ) of stable objects,where b k = − (cid:18) k, (1 − k ) h, (cid:18) − k (cid:19) g − (cid:19) and U k is a twisted vector bundle of rank k induced by the relative Ext-sheaf.Proof. We prove that ∆ last ( k ) is a Grassmannian bundle as stated. Explicitly, choose σ = σ α,β with ( α, β ) sufficiently close to (1 / √ g − ,
0) and β < − /
2. Then F ∈ ∆ last ( k ) if and onlyif its Jordan–H¨older filtration with respect to σ is A ⊗ W ∨ → F → F k , where F k ∈ M st σ ( b k ), dim W = k , U k := Hom( F k , A [1]), and the twisted vector bundle U k has fiber equal to U k over the point F k ∈ M st σ ( b k ). By irreducibilty of moduli spaces of stableobjects on K last is irreducible. Since its class δ is idivisible ∆ last isa prime divisor. The equalities in (19) follow as in the proof of Lemma 3.15. (cid:3) Recall from Section 3.1 that the moduli space M is equipped with an involution τ induced by the functor Ψ := O S ( − H ) ⊗ D : D b ( S ) op ∼ = −−→ D b ( S )and that τ acts fiberwise with respect to the Lagrangian fibration π . We now verify that τ ismoreover compatible with the Mukai flops f c,d and divisorial contraction ϕ introduced above. Lemma 3.25.
We adopt the notation of Lemma 3.20 and Lemma 3.23.(i) The involution τ on M is compatible with the Mukai flops f c,d . More precisely, thefunctor Ψ induces an isomorphism M σ c,d ( a c,d ) ∼ = −→ M σ c,d ( v − a c,d ) so that the inducedaction on M σ c,d ( a c,d ) × M σ c,d ( v − a c,d ) exchanges the two factors and given a fixed point ( R c,d , Ψ( R c,d )) the induced actions on the fibers P ( V c,d | ( Rc,d, Ψ( Rc,d )) ) and P ( V ∨ c,d | ( Rc,d, Ψ( Rc,d )) ) are naturally dual to each other.(ii) The involution τ on M maps each ∆ last ( k ) to itself and is compatible with the Grass-mannian bundle ∆ last ( k ) → M st σ ( b k ) ; the corresponding involution on M st σ ( b k ) is in-duced by the anti-autoequivalence Φ : D b ( S ) op ∼ = −→ D b ( S ) given by Φ := ST A ◦ ( O S ( − H ) ⊗ D ) . In particular, τ induces a regular involution τ : M ∼ = −→ M .Proof. Part (i) follows immediately from Lemma 3.20 and the following observations. Let E be an object in M c,d and consider its Jordan-H¨older filtration with respect to σ c,d = σ α ( c,d ) , − / given in (17): R c,d → E → R ′ c,d By applying the functor Ψ, we get a triangle:Ψ( R ′ c,d ) → Ψ( E ) → Ψ( R c,d )By using Example 3.10, Ψ( R c,d ) (respectively Ψ( R ′ c,d )) is also σ c,d -stable with Mukaivector v − a c,d (respectively a c,d ). Hence, the involution τ is compatible with the Mukai flop andit induces via the functor Ψ, an isomorphism M σ c,d ( a c,d ) ∼ = −→ M σ c,d ( v − a c,d ). The identificationof τ with Ψ thus shows that the action of τ on the product M σ c,d ( a c,d ) × M σ c,d ( v − a c,d )exchanges the two factors.It follows that there is an induced natural action on the universal relative Ext-(twisted)sheaf V c,d on the product M σ c,d ( a c,d ) × M σ c,d ( v − a c,d ) and thus on the projective bundle P c,d = P Z c,d ( V c,d ). The induced action on the fibers P ( V c,d | ( Rc,d, Ψ( Rc,d )) ) and P ( V ∨ c,d | ( Rc,d, Ψ( Rc,d )) )are then indeed naturally dual to each other.For part (ii), the argument is analogous to the proof of Lemma 3.16. (cid:3) Remark 3.26.
In the divisibility 2 case, the fixed locus of the action of τ on the GrassmannianGr( k, U k ) over a given element F k ∈ M st σ ( b k ) with Φ( F k ) = F k is more complicated than inthe divisibility 1 case (see Remark 3.17). In divisibility 2, this fixed locus may have either oneirreducible component (isomorphic to a Lagrangian Grassmannian) or two components (eachisomorphic to an orthogonal Grassmannian). An explicit description in both the divisibility 1and divisibility 2 cases will be given in [FM+21]. HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 25
Example 3.27.
In low genera the walls and Mukai flops can be easily described.(a) g = 4. In such a case we must have c = 0 and the only vectors allowed are a , − =(1 , ,
1) and a , = (1 , , M = M , − π (cid:15) (cid:15) & & ▼▼▼▼▼▼▼▼▼▼▼ o o f , − / / ❴❴❴❴❴❴❴❴❴❴ M , { { ✇✇✇✇✇✇✇✇ " " ❊❊❊❊❊❊❊❊ o o f , / / ❴❴❴❴❴❴❴❴❴ M last | | ①①①①①①①① ϕ (cid:15) (cid:15) | O S ( H ) | ∼ = P M , − M , M . where f , − is the Mukai flop at the 0-section of π described in Example 3.22 and f , is theMukai flop at the P -bundle in the Zariski topology over the product S × S . The destabilizingexact sequences are O S → E → O S ( − H )[1]for f , − (as already observed), and I p → E → D ( I q ) ⊗ O S ( − H )for f , , where p, q ∈ S . The induced action of τ is trivial on P (Hom( O S ( − H ) , O S )) and it isof type (1 ,
2) on P (Ext ( D ( I p ) ⊗ O S ( − H ) , I p )).The above example gives an explicit decomposition of the birational map between theLLvSS variety of a cubic fourfold with a node and M h (0 , h, − g = 8. Also in this case we must have c = 0 and the vectors allowed are a , − =(1 , , a , = (1 , , a , = (1 , , − M = M , − π (cid:15) (cid:15) & & ▼▼▼▼▼▼▼▼▼▼▼ o o f , − / / ❴❴❴❴❴❴❴❴❴❴ M , { { ✇✇✇✇✇✇✇✇ " " ❊❊❊❊❊❊❊❊ o o f , / / ❴❴❴❴❴❴❴❴❴ M , | | ②②②②②②②② " " ❊❊❊❊❊❊❊❊ o o f , / / ❴❴❴❴❴❴❴❴❴ M last | | ①①①①①①①① ϕ (cid:15) (cid:15) | O S ( H ) | ∼ = P M , − M , M , M . where f , − is the Mukai flop at the 0-section, f , the Mukai flop at the P -bundle in theZariski topology over the product S × S , and f , the Mukai flop at the P -bundle in theZariski topology over the product S [2] × S [2] . Destabilizing sequences for f , − and f , areanalogous to the g = 4 case, while the one for f , is I Γ → E → D ( I Γ ′ ) ⊗ O S ( − H )where Γ , Γ ′ ⊂ S are 0-dimensional closed subschemes of length 2. The induced action ofthe involution τ is trivial on P (Hom( O S ( − H ) , O S )), it is of type (1 ,
6) on P (Ext ( D ( I p ) ⊗ O S ( − H ) , I p )), and of type (2 ,
3) on P (Ext ( D ( I Γ ) ⊗ O S ( − H ) , I Γ )). (c) The first higher rank flop ( c >
0) starts at g = 12, with the vector a , − = (3 , − h, ±
3, and the geometric intuitionis less clear.
Remark 3.28.
As in the divisibility 1 case, the moduli space M , the involution τ , and thecontraction ϕ : M → M satisfy the requirements of Section 2 via Proposition 3.1, Lemma 3.23and Lemma 3.25. Hence, analogously to the divisibility 1 case, in the divisibility 2 case, thespecialization we consider for the proof of the Main Theorem is the pair ( M , λ ), where λ denotes the ample Cartier divisor class induced by λ on M , equipped with the involution τ .4. Connected components of the fixed locus
The goal of this section is to study the connected components of the fixed locus of theinvolution τ on the Lagrangian fibrations introduced in Section 3, for a general polarized K3surface. This study is easier on these models since the fixed locus here correspond either to(the closure of the) locus of 2–torsion points or of theta-characteristics on the linear sectionsof the K3 (see Section 4.1). For a general K3 surface, we compute the monodromy group andthe rule out the existence of exotic components fully supported over singular curves. Thisallows us to compute the number of connected components.As an immediate application, in Section 4.2 we use the results in Section 2 to prove theMain Theorem in the divisibility 1 case. The proof of the Main Theorem in the divisibility 2case requires a more in-depth analysis of how the connected components of the fixed locus of τ behave under wall-crossing. This will be done in Section 5.4.1. Components of the fixed locus on Lagrangian fibrations.
Let (
S, h ) be a polarizedK3 surface of genus g for which every curve in the linear system | O S ( H ) | is integral. Weconsider the two moduli spaces of Section 3.1 M h (0 , h, − g ) and M h (0 , h, O S ( − H ) ⊗ D on M h (0 , h, − g ) and D on M h (0 , h, τ .We recall that since τ is antisymplectic, the fixed locus Fix( τ ) on each of these modulispaces is a smooth Lagrangian submanifold. The two key results of this section are thefollowing propositions. Proposition 4.1.
The irreducible components of
Fix( τ ) on M h (0 , h, − g ) are (20) Fix( τ ) = Σ ⊔ Ω , where Σ is the image of the zero section (and thus parametrizes the structure sheaves of thecurves in | O S ( H ) | ) and where Ω is the closure of the locus parametrizing sheaves i C, ∗ ( ξ ) where HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 27 C ∈ | O S ( H ) | is a smooth curve and ξ is a non-trivial square root of O C (and thus is theclosure of -torsion points). Proposition 4.2.
The irreducible components of
Fix( τ ) on M h (0 , h, are (21) Fix( τ ) = S + ⊔ S − , where S + , respectively S − , is the closure of the locus parametrizing sheaves i C, ∗ ( ξ ) where C ∈| O S ( H ) | a smooth curve and ξ is an even, respectively odd, theta-characteristic. Remark 4.3. If | O S ( H ) | contains non integral curves then Fix( τ ) may very well have morethan two irreducible components; this is possible because M h ( v ) is singular in such cases. Forexample, if ( S, h ) is hyperelliptic covering a rational scroll F or F (depending on the parityof g ) then Fix( τ ) ⊂ M h (0 , h, − g ) has ⌊ g +32 ⌋ irreducible components dominating | O S ( H ) | . Infact this follows from the explicit description of theta-characteristics on a hyperelliptic curve:the irreducible components dominating | O S ( H ) | are indexed by the l (including l = − S, h ), we denote by | O S ( H ) | ⊂ | O S ( H ) | the open subsetparametrizing smooth curves. Notice that | O S ( H ) | is dense unless ( S, h ) is unigonal.
Proposition 4.4.
Let ( S, h ) be a general polarized K3 surface of genus g ≥ . Let C ∈| O S ( H ) | and let h , i be the intersection product on H ( C , Z ) . Then the natural represen-tation π ( | O S ( H ) | , C ) −→ Sp( H ( C , Z ) , h , i ) is surjective.Proof. According to Theorem 3 in [Bea86] (based on [Jan83, Jan85]), and the discussionthereafter, it suffices to exhibit a polarized K3 surface (
S, h ) such that(i) there exists C ∈ | O S ( H ) | with a singularity of Type E , and(ii) there exists ( C ′ + C ′′ ) ∈ | O S ( H ) | such that C ′ , C ′′ are reduced, they intersect trans-versely, and C ′ · C ′′ is odd.Trigonal K3 surfaces provide such examples. We get trigonal polarized K3’s of genus g by oneof the following constructions:(a) g ≡ S ⊂ P be a smooth quartic surface containing a plane cubic curve E . Let D ∈ | O S (1) | . Then H := D + ( g − E is an ample divisor with H · H = 2 g − S, h ) is trigonal because H · E = 3. (b) g ≡ S ⊂ P be the transverse intersection of a quadric cone Q andcubic hypersurface. Let D ∈ | O S (1) | . The projection from the vertex of Q defines afinite map S → P × P . Composing with one of the two projections P × P → P we get an elliptic fibration S → P ; let E be an element of the fibration. Then H := D + g − E is an ample divisor with H · H = 2 g −
2, and (
S, h ) is trigonal because H · E = 3.(c) g ≡ g ≥
5: Let S ⊂ P × P be a smooth element of | O P (3) ⊠ O P (2) | .Let p i be the projections of S to P and P respectively, and let D ∈ | p ∗ O P (1) | , E ∈ | p ∗ O P (1) | . Then H := D + g − E is an ample divisor with H · H = 2 g −
2, and(
S, h ) is trigonal because H · E = 3.For any such trigonal ( S, h ) as above Item (ii) holds. In fact we can take C ′ ∈ | E | and C ′′ ∈ | H − E | (which has no base locus). We will show that in each of (a), (b), (c), onecan choose ( S, h ) such that Item (i) holds as well. In each of (a), (b), H ≡ D + mE with m ≥
0. Since E has no base locus, it suffices to show that there exists a choice of ( S, h ) witha D ∈ | O S (1) | with a singularity of Type E . In (c), H ≡ D + mE with m ≥
1: hence itsuffices to show that there exists a choice of (
S, h ) with a C ∈ | O S ( D + E ) | with a singularityof Type E .In case (a), we let [ x, y, z, t ] be homogeneous coordinates on P and S be x + y t − z + zt = 0 . As is easily checked S is smooth, the curve S ∩ V ( z ) has an E singularity at [0 , , , S contains the line R := V ( t, x − z ), and hence it contains the plane cubic cuves inthe pencil | H − R | .In case (b), let [ x , . . . , x ] be homogeneous coordinates on P and let S be x x − x x = x + f ( x , . . . , x ) = 0 . The vertex of the quadric cone containing S is p = [0 , . . . , , p defines afinite map S → T , where T ⊂ P is the smooth quadric x x − x x = 0. If the curve B ⊂ P defined by x x − x x = f ( x , . . . , x ) = 0 is smooth, then S is smooth. If Λ ⊂ P is aplane such that there exists a point q ∈ Λ ∩ B with mult q (Λ · B ) = 4, then the hyperplane h p, Λ i ⊂ P intersects S in a point of Type E . One easily writes down such examples.In case (c), first we define a curve C ⊂ P × P with an E singularity, and then weargue that there exists a smooth S as above containing C as a divisor in | O S ( D + E ) | . Let[ x, y, z ], [ s, t ] be homogeneous coordinates on P and P respectively, and let C be defined by x st + y s + y t = sx − tz = 0 . The curve C has an E singularity at ([1 , , [0 , , S ⊂ P × P be defined by x st + y s + y t + ϕ ( x, y, z ; s, t )( sx − tz ) = 0 . HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 29 where ϕ ( x, y, z ; s, t ) is a bi-homogeneous polynomial of degree 3 in x, y, z and degree 2 in s, t .Then C ⊂ S and C ∈ | O S ( D + E ) | . It remains to prove that there exists ϕ ( x, y, z ; s, t ) suchthat S is smooth. This follows from Bertini’s theorem and easy considerations. (cid:3) As another preliminary result, we need to exclude the existence of connected componentsin the fixed locus entirely contained over the locus parametrizing singular curves.
Proposition 4.5.
Let ( S, h ) be a polarized K3 surface of genus g such that every curve inthe linear system | O S ( H ) | is integral. Then every irreducible component of Fix( τ ) dominates | O S ( H ) | .Proof. Over a smooth curve C , the fiber π − ( C ) ∩ Fix( τ ) of the induced morphism Fix( τ ) →| O S ( H ) | is finite. Since every component of Fix( τ ) is of dimension g , it is enough to givea dimension estimate of the fibers over the locus in | O S ( H ) | parametrizing singular curves.Since every curve in the linear system | O S ( H ) | is integral, the two Lagrangian fibrations for v = (0 , h,
0) and v = (0 , h, − g ) are fiberwise isomorphic and one can furthermore choose theisomorphisms to be compatible with the involutions on the two moduli spaces. It is thereforeenough to prove the statement for the case v = (0 , h, − g ).We claim that if C is a curve of cogenus δ > C is singular), then dim π − ( C ) ∩ Fix( τ ) < δ . Since by [dCRS19, Proposition 2.4.2], the locally closed subset of | O S ( H ) | parametrizing curves with cogenus δ has codimension ≥ δ , the proposition follows from theclaim.To prove the claim we use some well-known facts about compactified Jacobian of integrallocally planar curves. Standard references are [Reg80, Coo93, Bea99]. Let C ∈ | O S ( H ) | beas above, i.e., an integral curve of arithmetic genus g and cogenus δ >
0. Since C has locallyplanar singularities, the degree zero compactified Jacobian π − ( C ) = J( C ), parametrizingtorsion free sheaves of rank 1 and degree 0 on C , is irreducible of dimension g and con-tains Pic ( C ), the locus parametrizing locally free sheaves, as a dense open subset. SinceFix( τ ) ∩ Pic ( C ) is 0-dimensional, we only have to worry about non locally free sheaves. Thecomplement of Pic ( C ) in J( C ) is non empty because C is singular and it can be describedin the following way. For every non locally free sheaf F ∈ J( C ) there exists a unique partialnormalization n ′ : C ′ → C such that n ′∗ O C ′ = E nd ( F ) and a unique torsion free sheaf F ′ on C ′ of rank 1 such that F = n ′∗ F ′ (see [Bea99, Section 2]). The degree 0 Picard varietyPic ( C ) acts on J( C ) by tensorization and for F = n ′∗ F ′ as above, L ⊗ F ∼ = F if and onlyif L ∈ ker[ n ′∗ : Pic ( C ) → Pic ( C ′ )] (see [Bea99, Lemma 2.1]). The involution τ commuteswith the action of Pic ( C ).The possible dimensions of the stabilizers of the Pic ( C ) action on J( C ) go from 0 to δ .For every 0 ≤ δ ′ ≤ δ , let Z δ ′ ⊂ J( C ) be the locally closed subset of points whose stabilizer has The cogenus of a reduced curve is the difference between the arithmetic genus and geometric genus. dimension equal to δ ′ . This defines a finite τ -invariant and Pic ( C )-invariant stratification onJ( C ) with Z = Pic ( C ). Now set g ′ = g − δ ′ , so that dim Z δ ′ = g ′ + ǫ g ′ for some ǫ g ′ ≥
0. For δ ′ > Z δ ′ is strictly contained in J( C ), so(22) g ′ + ǫ g ′ < g and ǫ g ′ < δ ′ ≤ δ. We want to show that codim Z δ ′ ( Z δ ′ ∩ Fix( τ )) ≥ g ′ By (22), this implies that dim Z δ ′ ∩ Fix( τ ) ≤ ǫ g ′ < δ and hence the claim.For every partial normalization n ′ : C ′ → C , set g ′ := g ( C ′ ) and δ ′ := g − g ′ . Then0 < δ ′ ≤ δ . For any integer d , let J d ( C ′ ) denote the moduli space of degree d ′ torsion freesheaves of rank 1 on C ′ . The pushforward map n ′∗ : J − δ ′ ( C ′ ) → J( C ) is a closed embedding(see [Bea99, Lemma 3.1]) and the image is clearly preserved by the action of Pic ( C ). Inparticular there is an induced action of Pic ( C ′ ) on n ′∗ (J − δ ′ ( C ′ )), and this action is free onthe locus, denoted by n ′∗ (J − δ ′ ( C ′ ) ◦ ), of sheaves F with n ′∗ O C ′ = E nd ( F ). In particular, n ′∗ (J − δ ′ ( C ′ ) ◦ ) ⊂ Z δ ′ . By relative duality,(23) H om C ( n ′∗ F ′ , O C ) ∼ = n ∗ H om C ′ ( F ′ , ω n ′ ) , where ω n ′ is the relative dualizing sheaf of the finite morphism n ′ : C ′ → C (see [HLS19,Section 3]). The involution τ acts on J( C ) preserving n ′∗ (J − δ ′ ( C ′ ) ◦ ) and its Pic ( C ′ )-orbits.Consider a sheaf F = n ′∗ F ′ in n ′∗ (J − δ ′ ( C ′ ) ◦ ) and let G be another sheaf in the same Pic ( C ′ )-orbit. Write G = n ′∗ G ′ with G ′ = F ′ ⊗ L ′ and L ′ ∈ Pic ( C ′ ). Suppose both F and G are τ -invariant. Then by (23) and the fact that n ′∗ is an embedding, L ′⊗ = O C ′ . The conclusionis that for each Pic ( C ′ )-orbit in n ′∗ (J − δ ′ ( C ′ ) ◦ ) its intersection with Fix( τ ) is a torsor over the2-torsion line bundles on C ′ and is thus zero dimensional. It follows that every irreduciblecomponent of the intersection of Fix( τ ) with Z δ ′ has codimension at least g ′ . (cid:3) Proof of Proposition 4.1.
By Proposition 4.5 the equality in (20) holds. Since Σ is isomorphicto | O C ( H ) | ∼ = P g it is irreducible.It remains to prove that Ω is irreducible. Let Ω := π − ( | O S ( H ) | ). The restriction of π defines a map π : Ω −→ | O S ( H ) | which is proper and a local homeomorphism. It suffices to prove that Ω is connected. If g ≥
3, we can deform and assume that (
S, h ) is general, and then this follows at once fromProposition 4.4.If g = 2 we argue as follows. The map f : S → | O S ( H ) | ∨ ∼ = P is a double cover ramifiedover a smooth sextic curve. Let D ⊂ S be the ramification divisor. Let C ∈ | O S ( H ) | be asmooth curve. Then C intersects transversely D in 6 points x , . . . , x , andPic ( C )[2] \ { } = { x i − x j } ≤ i Hence it suffices to prove that the action of π ( | O S ( H ) | , C ) on the set { x , . . . , x } is thefull symmetric group. This holds by the Uniform Position Theorem applied to the hyperplane(i.e., line) sections of the branch curve f ( D ) ⊂ P . (cid:3) Proof of Proposition 4.2. By Proposition 4.5 the equality in (21) holds. It remains to provethat S ± is irreducible. Let ( S ± ) := π − ( | O S ( H ) | ). The restriction of π defines maps( π ± ) : ( S ± ) −→ | O S ( H ) | . which are proper and local homeomorphisms. It suffices to prove that ( S ± ) is connected. If g ≥ g = 2 one argues as in the analogous case in the proof of Proposition 4.1. (cid:3) Recall that an irreducible closed subset Z of a variety X is called constant cycle ifall points of Z (or all points of a dense open subset of Z ) are rationally equivalent in X .By [Lin20, Lemma 3.2], if X is a HK manifold then this is equivalent to the cycle mapCH ( Z ) Q → CH ( X ) Q being constant. Proposition 4.6. Let M be M h (0 , h, − g ) , respectively M h (0 , h, . The irreducible subvari-eties Σ , Ω ⊂ M , respectively S − , S + ⊂ M , are constant cycle subvarieties. In particular, theyhave no non trivial holomorphic forms.Proof. The second statement follows from the first using the Mumford-Roitman Theorem[Mum68, Roi80], [Voi03, Theorem 10.17]. The first statement follows from arguments in[Lin20], which we now recall for the reader’s sake. Let H be an ample τ -invariant line bundleon M . For a general fiber M t of π , take for the origin one of the fixed points of τ . Therestriction H t is a symmetric line bundle with respect to this group structure on the abelianvariety M t and the points of Fix( τ ) ∩ M t are precisely the two torsion points of M t . By [Lin20,Lemma 3.4], for any 2-torsion x ∈ M t , the class D [ x ] is rationally equivalent to H gt , where D = deg( H gt ). Since all fibers M t are rationally equivalent in M , it follows that all points ofan irreducible component of Fix( τ ) are rationally equivalent in M . (cid:3) Proof of the Main Theorem, divisibility 1. As an immediate application of theresults in the previous section, we can prove the Main Theorem, in the easier case of divisi-bility 1. We just need two preliminary observations. Proposition 4.7. Let ( S, h ) be a polarized K3 surface of genus g with NS( S ) = Z · h . Wekeep the notation of Sections 3.3 and 4.1.(i) The irreducible subvariety S − is contained in ∆ , it is uniruled, and the ruling is inducedby the ruling of ∆ .(ii) The irreducible subvariety S + is not contained in ∆ . Proof. The first statement of Part (i) is clear by the definitions of S − and ∆; see Remark 3.3.The second statement is [FV14, Proposition 2.2]: let η be an odd theta characteristic on asmooth curve C corresponding to a general point in S − . Then h ( C, η ) = 1 and η = O C ( D ) fora unique effective divisor of degree g − C . The member of the ruling of ∆ corresponding tothe pencil of curves in | O S ( H ) | containing D is then clearly contained in S − . The statement inPart (ii) follows from the well known fact that on every smooth curve there exist non-effectivetheta characteristics (see [Har82, Theorem 1.10(ii)] and use the fact that for the general curvethere are no effective even theta characteristics). (cid:3) Corollary 4.8. Let S ± be the image of S ± under the divisorial contraction ϕ : M → M .(i) The general fiber of the induced morphism ϕ : S − → S − has dimension and thus dim ϕ ( S − ) = g − .(ii) The induced morphism ϕ : S + → S + is birational.Proof of the Main Theorem, divisibility 1 case. We keep notation as above, so that M is thesingular model of M under the divisorial contraction of Lemma 3.14, τ is the involution on M of Proposition 3.1 and τ is the induced involution on M .As observed in Remark 3.18, we can apply the results in Section 2. Let M → D be the pullback of the universal family over Def( M ) to a general disc D ⊂ Def( M , τ ). ByProposition 2.1 and Corollary 2.2, M → D is a smoothing of M and there is a global involution τ on M which preserves the fibers M t and is such that, for t = 0, τ t is the involution whosefixed locus we are interested in and for t = 0, τ = τ . Let Fix( τ, M ) be the fixed locus of τ on M . The fiber at zero satisfies Fix( τ, M ) = Fix( τ , M ) which, by Corollary 4.8, is thedisjoint union S + ` S − of a g -dimensional component and a ( g − g − S − is of course contained in the singular locus of M .At a general point of S + , a tangent space computation shows that Fix( τ, M ) is smoothof dimension g + 1 and that the morphism Fix( τ, M ) → D is smooth of relative dimension g . Let M ∗ := M \ M and let F := Fix( τ, M ∗ ) be the closure of the fixed locus on M ∗ .Then F → D is a flat surjective morphism, which is generically smooth along the centralfiber F = S + , and so F t is connected for any t ∈ D . (cid:3) Proof of the Main Theorem in divisibility 2 In this section we complete the proof of the Main Theorem in the divisibility 2 case. Theapproach is similar to the case of divisibility 1, with two extra issues. First, we describe howthe fixed locus of τ in the Lagrangian fibration behaves under the flopping transformationsof Section 3.4 and we verify that no connected component of this fixed locus is created ordestroyed in the process. This, which is the content of Section 5.2, immediately gives thatthere are at most two connected components. The second problem is to show that there are HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 33 exactly two components. This is done in Section 5.3. To deal with this, we notice that the linebundle L last on M last with c ( L last ) = λ has a natural linearization with respect to the actionof the involution τ . Then we study the action of τ on the fibers of L last over the fixed locus:the two connected components correspond to different irreducible representations. This willshow that even in the singular model M the two connected components are disjoint and, sincethey both intersect the smooth locus, they each are limits of a component of the fixed locuswe are interested in.We start with a general section on flops and involutions on HK manifolds, Section 5.1,where we outline the geometric picture of what happens to the fixed loci when we pass througha Mukai flop.5.1. Flops and involutions, general results. The goal of this section is to describe locallythe birational transformation induced on the fixed loci by a Mukai flop.We start with elementary considerations. Let V be a vector space and let τ be aninvolution on P V . We can linearize the action to lift the involution to V and get dual actionson V ∨ and P V ∨ . By abuse of notation we denote all of these actions by τ . The dual actionson V and V ∨ are such that the ( ± V ± and ( V ∨ ) ± satisfy ( V ∨ ) ∓ = Ann( V ± ).There is a natural action of τ on the Euler sequence0 → Ω P V → O P V ( − ⊗ V ∨ → O P V → , and the ( − P V ) − = ( O P V ( − ⊗ V ∨ ) − . Notice that while the action on V depends on the choice of the linearization, the action on O P V ( − ⊗ V ∨ does not. A similar statement also holds true in the relative setting, when weconsider a relative action on a projective bundle over a base.We now consider the following setting. Let ˆ M be an open (in the analytic or Zariskitopology) subset of a HK manifold and ˆ P ⊂ ˆ M be a coisotropic subvariety which is isomorphicto the projectivization of a vector bundle over a smooth manifold ˆ Z . To fix notation, we let V be the vector bundle on ˆ Z such that ˆ P ∼ = P ˆ Z V ρ −→ ˆ Z . The holomorphic symplectic formthen induces an isomorphism N ˆ P / ˆ M ∼ = Ω ˆ P / ˆ Z and the restriction of the symplectic form to ˆ P isthe pullback of a holomorphic symplectic form on ˆ Z . Assume that there is an antisymplecticinvolution τ acting on ˆ M and preserving the projective bundle ˆ P → ˆ Z , with induced anti-symplectic involution on ˆ Z denoted by τ ˆ Z . Let ψ : ˆ M ˆ M ′ be the Mukai flop of ˆ M alongˆ P and denote by ˆ P ′ the exceptional locus of ψ − , so that ˆ P ′ = P ˆ Z V ∨ ρ ′ −→ ˆ Z , where V ∨ is thedual bundle. With these assumptions, there is an induced regular antisymplectic involution τ ′ on ˆ M ′ , restricting to the dual action on ˆ P ′ . We want to describe the general behavior of the restriction of ψ to the connected com-ponents of Fix( τ, ˆ M ). We first examine the case where a connected component is containedin ˆ P . Proposition 5.1. Keep notation and assumptions as above, and suppose that there is a com-ponent ˆ F of the fixed locus Fix( τ, ˆ M ) contained in ˆ P . Then the following hold:(i) There exists a component ˆ Y ⊂ ˆ Z of the fixed locus of τ ˆ Z such that ˆ F = ρ − ( ˆ Y ) .(ii) The subset ( ρ ′ ) − ( ˆ Y ) ⊂ ˆ P ′ is a component of the fixed locus Fix( τ ′ , ˆ M ′ ) .Proof. Part (i) holds by an easy dimension count. By (i), the action of τ on each fiber ofˆ P → ˆ Z over a point of ˆ Y is trivial. Since ρ ′ : ˆ P ′ → ˆ Z is the dual of ˆ P → ˆ Z and the action of τ ′ is the dual action, it follows that the action of τ ′ on each fiber of ˆ P ′ → ˆ Z over a point ofˆ Y is trivial. This proves (ii). (cid:3) For applications in [FM+21], we can also study the case when a connected componentis not contained in ˆ P . Proposition 5.2. We let ˆ F be a component of Fix( τ, ˆ M ) such that ˆ F ˆ P and let ˆ F ′ be theproper transform of ˆ F in ˆ M ′ . Then ˆ F ′ is a connected component of Fix( τ ′ , ˆ M ′ ) and the Mukaiflop ψ restricts to a birational map ϕ : ˆ F ˆ F ′ , which is a finite series of disjoint flips, onefor each connected component ˆΓ of ˆ F ∩ ˆ P . More precisely, we will show that any connected component ˆΓ of ˆ F ∩ ˆ P (with its reducedinduced scheme structure) is identified with a projective bundle(25) p : ˆΓ ∼ = P ˆ W V ˆΓ −→ ˆ W , where ˆ W ⊂ ˆ Z is a connected component of Fix( τ ˆ Z , ˆ Z ) and V ˆΓ is one of the two eigenbundles of V | ˆ W . The normal bundle of ˆΓ in ˆ P satisfies N ˆΓ / ˆ P ∼ = O p ( − ⊗ V ′ ˆΓ , where V ′ ˆΓ := Ann ( V ˆΓ ) ⊂ V ∨| ˆΓ ,and locally around ˆΓ the birational map ϕ is the corresponding standard flip of ˆ F along ˆΓ,namely ˆΓ is replaced by ˆΓ ′ := P ˆ W V ′ ˆΓ . Proof. If ˆΓ is a connected component of ˆ F ∩ ˆ P , then ˆΓ is a connected component of the fixedlocus of τ on ˆ P . Thus ˆΓ is smooth and coincides with a projective bundle P ˆ W V ˆΓ , as in (25).Let ˜ F be the proper transform of ˆ F in the blow up of ˆ M along ˆ P . Then ˜ F is a componentof the fixed locus of an involution and thus is smooth. Hence the birational map ˜ F → ˆ F ,which has connected fibers, has exceptional locus of codimension one and is isomorphic to aprojective bundle over ˆΓ. One can see that ˜ F → ˆ F is, in fact, just the blowup of ˆ F alongˆΓ: by the universal property of blowing up, there exist a birational morphism ˜ F → Bl ˆΓ ˆ F ,which has connected connected fibers and is finite; it is thus an isomorphism because Bl ˆΓ ˆ F is smooth. HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 35 To compute the normal bundle of ˆΓ in ˆ F we look at the natural action of τ (and τ ˆ Z ) onseveral conormal/cotangent bundles sequences and at the corresponding ( ± F with that of ˆΓ in ˆ M and using thefact that action on Ω ˆΓ is trivial and the action on N ∨ ˆ F / ˆ M is ( − N ˆΓ / ˆ F = ( N ˆΓ / ˆ M ) + .Similarly, looking at normal bundle sequence for ˆΓ ⊂ ˆ P ⊂ ˆ M and using the fact that theaction on N ˆΓ / ˆ P is ( − N ˆΓ / ˆ M ) + = ( N ˆ P / ˆ M ) + . Since τ is antisymplectic, theisomorphism N ˆ P / ˆ M ∼ = Ω ˆ P / ˆ Z switches the ( ± ϕ . (cid:3) Flops and involutions in a special case. In this section, we return to our morespecific situation. We let ( S, h ) be a polarized K3 surface of genus g such that 4 | g andNS( S ) = Z · h . We are in the context of Section 3.4 and keep the notation therein: M := M h ( v ),its birational models are denoted M c,d , M last the divisorial contraction associated to λ isdenoted by ϕ : M last → M , and the involution is denoted by τ . We also recall the two fixedcomponents Σ( ∼ = P g ) and Ω of τ in M introduced in Proposition 4.1. The goal is to show thefollowing result. Proposition 5.3. The fixed locus of τ on M last has exactly two connected components: Fix( τ, M last ) = Σ last ⊔ Ω last . Moreover, Σ last is rational while Ω last is birational to Ω and has no non trivial holomorphicforms. We adopt the notation of Remark 3.21. In particular f c,d : M c,d M ′ c,d is the Mukaiflop induced by the divisor class ϑ ( e a c,d ). Recall the definition of ∆ c,d in (18). Lemma 5.4. For all ( c, d ) the indeterminacy locus of the flop f c,d : M c,d M ′ c,d is containedin ∆ c,d .Proof. It suffices to prove that if E ∈ M c,d and Hom S ( A, E ) = 0, then E is σ α, − / -stablefor all α ( c,d ) ≥ α > √ g − . If E is not σ α, − / -stable for some α ( c,d ) ≥ α > √ g − , since byRemark 3.21(a) there are no totally semistable walls on the segment σ α, − / , then we canassume that there exists α ( c ′ ,d ′ ) < α ( c,d ) such E is not σ α ( c ′ ,d ′ ) , − / -stable. Then, we have aJordan-H¨older filtration with respect to σ α ( c ′ ,d ′ ) , − / of the form R c ′ ,d ′ → E → R ′ c ′ ,d ′ with v ( R c ′ ,d ′ ) = a c ′ ,d ′ and v ( R ′ c ′ ,d ′ ) = v − a c ′ ,d ′ .By applying the functor Hom S ( A, ), we get an exact sequence:Hom S ( A, R ′ c ′ ,d ′ [ − → Hom S ( A, R c ′ ,d ′ ) → Hom S ( A, E ) . Since R ′ c ′ ,d ′ ∈ coh − / ( S ) is σ α ( c ′ ,d ′ ) , − / -stable and the slope of A [1] is infinite, we havehom S ( A, R ′ c ′ ,d ′ [ − S ( A [1] , R ′ c ′ ,d ′ ) = 0 . Similarly, by Serre duality, since R c ′ ,d ′ , A [1] ∈ coh − / ( S ), we havehom S ( A, R c ′ ,d ′ [2]) = hom( R c ′ ,d ′ , A ) = hom( R c ′ ,d ′ , A [1][ − . An immediate calculation, by using condition (a) in Section 3.4, shows that( δ, a c ′ ,d ′ ) ≤ − (2 c ′ + 1) < , which gives then 0 = Hom S ( A, R c ′ ,d ′ ) ֒ → Hom S ( A, E ) , a contradiction. (cid:3) In M = M , − the fixed locus Fix( τ, M ) has two connected components, namely Σ andΩ. Motivated by Lemma 5.4 we determine whether Σ or Ω is contained in ∆ = ∆ , − . Firstwe deal with Σ. We need a few preliminary results on the spherical vector bundle A . Lemma 5.5. Let C ∈ | O S ( H ) | be a curve. Then the restriction map H ( S, A ∨ ) −→ H ( C, A ∨| C ) is an isomorphsim and h ( S, A ∨ ) = h ( C, A ∨| C ) = 2 + g . Proof. By the explicit description in (10), we deduce immediately that h ( S, A ) = h ( S, A ) = 0and, by Serre duality, h ( S, A ∨ ) = g/ h ( S, A ∨ ) = h ( S, A ∨ ) = 0.Finally, from the exact sequence0 → A ∨ ( − H ) ∼ = A → A ∨ → A ∨| C → , for C ∈ | O S ( H ) | , we deduce our statement. (cid:3) Let r := h ( S, A ∨ ) / g/ 4. Then r is a positive integer because 4 | g . Lemma 5.6. Let x , . . . , x r ∈ S be general distinct points. Then H ( S, I { x ,...,x r } ⊗ A ∨ ) = 0 . Proof. Since A ∨ is globally generated of rank 2, if x ∈ S then h ( S, I x ⊗ A ∨ ) = h ( S, A ∨ ) − g . Since A ∨ is slope stable with c ( A ) = h and NS( S ) = Z · h , if F ⊂ A ∨ is a rank 1 subsheafthen c ( F ) = mh with m ≤ 0. Thus, if F is globally generated then m = 0, so F ∼ = O S and HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 37 h ( S, F ) = 1. Hence, since g/ > A ∨ generated by H ( S, I x ⊗ A ∨ )must have rank 2. It follows that if x ∈ S is general, then h ( S, I { x ,x } ⊗ A ∨ ) = h ( S, A ∨ ) − g − . Iterating, we get the result. (cid:3) Lemma 5.7. In M , we have Σ ⊂ ∆ and mult Σ (∆) = h ( C, A ∨| C ) = 2 + g . Proof. It suffices to prove that if C ∈ | O S ( H ) | is smooth curve thenmult [ O C ] ∆ | Pic ( C ) = h ( C, A ∨| C ) . Let r = h ( S, A ∨ ) / x , . . . , x r ∈ S be general points. By a dimension count,there exists a smooth curve C ∈ | O S ( H ) | containing x , . . . , x r . By Lemma 5.6, we have that H ( S, I { x ,...,x r } ⊗ A ∨ ) = 0. Since by Lemma 5.5 the restriction map H ( S, A ∨ ) → H ( C, A ∨| C )is an isomorphism, we deduce that H ( C, A ∨| C ( − ( x + · · · + x r )) = 0 . The statement follows now from [C-MTB11, Lemma 6.1] (see also [C-MF05]). (cid:3) Next we deal with Ω. Lemma 5.8. In M , we have Ω ∆ .Proof. We recall that if ( V, Θ) is a principally polarized abelian variety of dimension g and welet η : V → | O V (2Θ) | ∨ ∼ = P g − be the morphism associated to the divisor 2Θ, then the points η ( x ), for x ∈ V [2], span thewhole P g − , i.e., no non zero section of O V (2Θ) vanishes on all points in V [2]. Let C ∈ | O S ( H ) | be a smooth curve. By Proposition 3.4 the restriction of ∆ = ∆ , − to π − ( C ) = Pic ( C )is a non zero section of twice the natural principal polarization. Since ∆ | Pic ( C ) vanishesat [ O C ] ∈ Pic ( C )[2], it follows that there exists [ ξ ] ∈ Pic ( C )[2], such that i C, ∗ ( ξ ) is notcontained in ∆. (cid:3) We notice that the irreducibility of Ω (namely, Proposition 4.1) gives the slightly strongerresult that, for general smooth C ∈ | O S ( H ) | , i C, ∗ ( ξ ) is not contained in ∆, for all ξ = O C inPic ( C )[2].We now focus our attention on the first flop f , − , which is the flop of Σ, see Example 3.22.To simplify notation in what follows, we set: M ′ := M , , f := f , − : M M ′ , ∆ ′ := ∆ , . We moreover let Σ ′ ⊂ M ′ be the “flopped” Σ (i.e., the indeterminacy locus of the inverse f − ), and we let Ω ′ := f (Ω). Clearly, Fix( τ, M ′ ) = Σ ′ ⊔ Ω ′ . Lemma 5.9. In M ′ , we have Ω ′ ∆ ′ and Σ ′ ∆ ′ .Proof. Since Ω is disjoint from the center of the flop f we have Ω ′ ∆ ′ by Lemma 5.8. Itremains to prove that Σ ′ ∆ ′ .Let(26) Ξ (cid:31) (cid:127) / / (cid:127) (cid:127) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ * * f M g ′ ❇❇❇❇❇❇❇❇ g ~ ~ ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ Σ (cid:31) (cid:127) / / M o o f ∼ = / / ❴❴❴❴❴❴❴ M ′ Σ ′ ? _ o o be the resolution of the Mukai flop f , with common exceptional divisor Ξ. Let d := mult Σ ′ ∆ ′ .We only need to show that d = 0.Let us denote by e ∆, respectively e ∆ ′ , the strict transform of ∆, respectively ∆ ′ , on f M .Then e ∆ = e ∆ ′ and we have g ∗ ∆ = e ∆ + (cid:0) g + 2 (cid:1) Ξ g ′∗ ∆ ′ = e ∆ ′ + d Ξ , where we used that mult Σ ∆ = g/ e γ ⊂ Ξ be a lift of a line γ ⊂ Σ. Then g ′ ( e γ ) = pt ∈ M ′ . Thus, we have0 = g ′∗ ∆ ′ · e γ = (cid:16) e ∆ ′ + d Ξ (cid:17) · e γ = e ∆ ′ · e γ − d. Since e ∆ ′ · e γ = e ∆ · e γ , we get − (cid:16) g (cid:17) = ∆ · γ = g ∗ ∆ · e γ = e ∆ · e γ + (cid:16) g (cid:17) Ξ · e γ = d − (cid:16) g (cid:17) , and thus d = 0, as we wanted. (cid:3) Concretely, as explained in Example 3.22, Lemma 5.9 means that for a general non-trivialextension O S ( − H )[1] → E ′ → O S we have that Hom S ( A, E ′ ) = 0. Proof of Proposition 5.3. Recall that the ( c, d ) introduced in Subsection 3.4 are totally or-dered. The maximum is (0 , − , − (cid:23) (0 , (cid:23) . . . (cid:23) last.We know that for M ′ = M , , we have Fix( τ, M ′ ) = Σ ′ ∪ Ω ′ and, by Lemma 5.9, that neitherΣ ′ nor Ω ′ is contained in ∆ ′ . Let (0 , (cid:23) ( c , d ). By Lemma 5.4 and Lemma 5.9 the rationalmap M , M c,d given by composing successive f c,d for (0 , (cid:23) ( c, d ) (cid:23) ( c , d ) is regular atthe generic point of Σ , and of Ω , . We let Σ c ,d , Ω c ,d ⊂ M c ,d be the image of Σ , , Ω , re-spectively. Let us prove by descending induction on the ( c, d ) ′ s that Fix( τ, M c,d ) = Σ c,d ∪ Ω c,d . HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 39 Thus we assume the result for M c,d and we prove it for M ′ c,d . Assume that it does not holdfor M ′ c,d . Then there is a component of Fix( τ, M ′ c,d ) which is contained in the indeterminacylocus of f − c,d . The hypotheses of Proposition 5.1 hold for ˆ M = M ′ c,d and ψ := f − c,d . It followsthat there is a component of Fix( τ, M c,d ) which is contained in the indeterminacy locus of f c,d ,contradicting the inductive hypothesis. (cid:3) As already remarked, as an immediate consequence of Proposition 5.3 we can use thesame argument as in the divisibility 1 case (in Section 4.2) to show that in the divisibility 2case there are at most two connected components.5.3. Linearizations and fixed loci. To complete the proof of the Main Theorem, in thedivisibility 2 case, we only have to show the following two results. We defineΣ := ϕ (Σ last )Ω := ϕ (Ω last ) , where Σ last , Ω last ⊂ M last are the two connected components of the fixed locus of τ on M last .The fact that there are two follows from Proposition 5.3. Proposition 5.10. Neither Σ last nor Ω last is contained in the exceptional divisor ∆ last of thedivisorial contraction ϕ : M last → M . Proposition 5.11. The two irreducible components Σ , Ω ⊂ M do not intersect; in symbols, Σ ∩ Ω = ∅ . By assuming Proposition 5.10 and Proposition 5.11, we can now complete the proof ofthe Main Theorem. Proof of the Main Theorem, divisibility 2 case. Let M → D be the pullback of the universalfamily over Def( M ) to a general disc D ⊂ Def( M , τ ). By Proposition 2.1 and Corollary 2.2, M → D is a smoothing of M and there is a global involution τ on M which preserves thefibers M t and is such that for t = 0 τ t is the involution whose fixed locus we are interested inand for t = 0, τ = τ .By Proposition 5.11, the fiber at zero Fix( τ, M ) of the fixed locus Fix( τ, M ) is thedisjoint union Σ ` Ω. Moreover, by Proposition 5.10, both components are g -dimensional andat a general point of either component the fixed locus Fix( τ, M ) → D is smooth of relativedimension g . By letting M ∗ := M \ M and F := Fix( τ, M ∗ ), we get that any fiber F t hasexactly two components, for all t ∈ D , thus concluding the proof. (cid:3) We are left with the task of proving the two propositions. The key part of the proofis to understand the linearization of the action of τ on the line bundle associated to λ . Let G := { id , τ } ∼ = µ be the group acting on each of the birationals model M c,d and M last of M . We denote by ρ tr and ρ det the two irreducible representations of G , acting respectively by( ± L c,d , respectively L last be the line bundle on M c,d , respectively M last , with firstChern class λ . Theorem 5.12. There is a G -linearization on L last such that • if x ∈ Ω last , then L last | x ∼ = ρ tr ; • if x ∈ Σ last , then L last | x ∼ = ρ det . The above theorem provides the final ingredient to finish the proof of the Main Theoremin divisibility 2, which establishes that the fixed locus of τ consists of two connected compo-nents. Theorem 5.12 moreover shows that one can distinguish between these two connectedcomponents according to the action of τ on the fibers of θ ( λ ) at the fixed points. In particular,one of the two components will always be contained in the base locus of the linear system | λ | . This is exactly what happens in the case of the LLSvS HK manifold associated to cubicfourfolds, where the cubic itself is the base locus of the linear system ([LLSvS17]).The idea of the proof of Theorem 5.12 is to first study the linearization on M and thencompute how this linearization changes on the various flops f c,d . Proposition 5.13. There is a G -linearization on L = L , − such that if x ∈ Ω ∪ Σ , then L | x ∼ = ρ tr .Proof. We can write L = O M (∆) ⊗ π ∗ O P g (1) . Each factor is a G -invariant line bundle. We define a G -linearization on L by considering thetrivial G -linearization on π ∗ O P g (1) and defining an explicit G -linearization on the line bundle O M (∆) as follows. We write locally I ∆ = ( f )and lift the G -action on O M (∆) by mapping the local generator 1 /f as follows:1 f τ ∗ ( f ) . Let x ∈ Ω ∪ Σ, i.e., τ ( x ) = x . We choose local coordinates y , . . . , y g , t , . . . , t g centered at x such that τ ∗ ( y i ) = − y i , y i ’s coordinates “on the fibers of π ” τ ∗ ( t i ) = t i , t i ’s pull-back of coordinates on P g . HE GEOMETRY OF ANTISYMPLECTIC INVOLUTIONS, I 41 Since τ preserves the zero-locus of f , we have that L | x ∼ = ρ det , if f ( − y, t ) = − f ( y, t ) , i.e., if mult (0 , ( f ) is odd ρ tr , if f ( − y, t ) = f ( y, t ) , i.e., if mult (0 , ( f ) is even.The statement now follows immediately from Lemma 5.7 if x ∈ Σ, and from Lemma 5.8 if x ∈ Ω. (cid:3) We now return to our study of the first flop: f : M M ′ which is the Mukai flop atΣ. Then the linearization on L of Proposition 5.13 on M induces a linearization on L ′ = L , on M ′ . In fact, it induces a linearization on each L c,d on M c,d and on L last as well. Proposition 5.14. With respect to the above linearization on L ′ on M ′ we have that if x ∈ Σ ′ ,then L ′| x ∼ = ρ det .Proof. Let m := deg( L | γ ) , where γ ⊂ Σ ∼ = P g is a line. Then, in the notation of Diagram 26 giving the resolution of theMukai flop f := F , − , we have g ∗ L ∼ = g ′∗ L ′ ⊗ O f M ( m Ξ) . In particular, if x ∈ Σ ′ , then L ′| x ∼ = ( − m ρ tr . To prove the statement, we need to show that m is odd.Since L ∼ = O M (∆) ⊗ π ∗ O P g (1), we have that m = 1 + ∆ · γ. By construction, since ϑ ( δ ) = ∆, where δ = (2 , − h, g/ a , − = (1 , , · γ = (cid:16)(cid:16) , − h, g (cid:17) , (1 , , (cid:17) = − (cid:16) g (cid:17) . Since 4 | g , this completes the proof. (cid:3) Proof of Proposition 5.10. By Lemma 5.4, this follows immediately from Lemma 5.8 andLemma 5.9. In fact, let F ∈ Ω \ ∆. By Lemma 5.4, F is not contained in any center ofthe flops f c,d , and thus F ∈ Ω last \ ∆ last , as we wanted. We argue similarly for F ∈ Σ ′ \ ∆ ′ . (cid:3) Proof of Theorem 5.12. This follows by a similar argument as for Proposition 5.10. 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(2001), 817–884. 9 Michigan State UniversityDepartment of Mathematics619 Red Cedar Road, East Lansing, MI 48824, USA Email address : [email protected] Universit´e Paris-SaclayCNRS, Laboratoire de Math´ematiques d’OrsayRue Michel Magat, Bˆat. 307, 91405 Orsay, France Email address : [email protected] Sapienza Universit`a di RomaDipartimento di MatematicaP.le A. Moro 5, 00185 Roma, Italia Email address : [email protected] Columbia UniversityDepartment of Mathematics2990 Broadway, New York, NY 10027, USA Email address ::