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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS EYAL MARKMAN Abstract. Let X be any compact K¨ ahler manifold deformation equiv- alent to the Hilbert scheme of length n subschemes on a K3 surface, n 2. For each point x X we construct a rank 2n - 2 reflexive coher- ent twisted sheaf Ex on X, locally free over X \{x}, with the following properties. (1) Ex is ω-slope-stable with respect to some K¨ ahler class ω on X. (2) Set κ(Ex) := ch(Ex) exp -c 1 (Ex) 2n-2 . It is well defined, even though c1(Ex) is not. The characteristic class κi (Ex) H i,i (X, Q) is monodromy-invariant, up to sign. Furthermore, κi (Ex) can not be expressed in terms of classes of lower degree, if 2 i n/2. (3) The Beauville-Bogomolov class is equal to c2(TX)+2κ2(Ex). Contents 1. Introduction 2 1.1. The main results 2 1.2. Two applications 5 1.3. Notation 7 2. Characteristic classes of projective bundles and twisted sheaves 7 3. The rational Hodge classes κ i (X ) 10 3.1. Relation of Theorem 1.4 with the Hodge conjecture 11 4. Monodromy invariant classes over X × X 12 4.1. Monodromy invariant classes κ i (F ) over M(v) ×M(v) 12 4.2. Proof of Proposition 4.5 14 4.3. Lifting deformations of a moduli space M to deformation of the pair (M, PV ) 17 5. Proof of the monodromy invariance of κ i (X ) and κ i (F ) 18 6. Hyperholomorphic sheaves 21 6.1. Twistor deformations of pairs 21 6.2. Projectively ω-stable-hyperholomorphic sheaves 24 6.3. Deformation of pairs along twistor paths 26 7. Stable hyperholomorphic sheaves of rank 2n - 2 on all manifolds of K3 [n] -type 29 7.1. A very twisted E xt 1 π 13 (π * 12 E * 23 E ) 29 7.2. Polystability of E nd(E) for a very twisted sheaf E 31 7.3. Stability of an untwisted E xt 1 π 13 (π * 12 E * 23 E ) 33 1

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THE BEAUVILLE-BOGOMOLOV CLASS AS A

CHARACTERISTIC CLASS

EYAL MARKMAN

Abstract. Let X be any compact Kahler manifold deformation equiv-alent to the Hilbert scheme of length n subschemes on a K3 surface,n ≥ 2. For each point x ∈ X we construct a rank 2n− 2 reflexive coher-ent twisted sheaf Ex on X, locally free over X \ x, with the followingproperties.(1) Ex is ω-slope-stable with respect to some Kahler class ω on X.

(2) Set κ(Ex) := ch(Ex) exp“

−c1(Ex)2n−2

. It is well defined, even though

c1(Ex) is not. The characteristic class κi(Ex) ∈ Hi,i(X, Q) ismonodromy-invariant, up to sign. Furthermore, κi(Ex) can notbe expressed in terms of classes of lower degree, if 2 ≤ i ≤ n/2.

(3) The Beauville-Bogomolov class is equal to c2(TX) + 2κ2(Ex).

Contents

1. Introduction 21.1. The main results 21.2. Two applications 51.3. Notation 72. Characteristic classes of projective bundles and twisted sheaves 73. The rational Hodge classes κi(X) 103.1. Relation of Theorem 1.4 with the Hodge conjecture 114. Monodromy invariant classes over X ×X 124.1. Monodromy invariant classes κi(F) over M(v) ×M(v) 124.2. Proof of Proposition 4.5 144.3. Lifting deformations of a moduli space M to deformation of

the pair (M,PV ) 175. Proof of the monodromy invariance of κi(X) and κi(F) 186. Hyperholomorphic sheaves 216.1. Twistor deformations of pairs 216.2. Projectively ω-stable-hyperholomorphic sheaves 246.3. Deformation of pairs along twistor paths 267. Stable hyperholomorphic sheaves of rank 2n− 2 on all manifolds

of K3[n]-type 297.1. A very twisted Ext1π13

(π∗12E , π∗23E) 297.2. Polystability of End(E) for a very twisted sheaf E 317.3. Stability of an untwisted Ext1π13

(π∗12E , π∗23E) 331

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2 EYAL MARKMAN

7.4. Proof of the deformability Theorem 1.7 348. Proof of Lemma 1.5 35References 36

1. Introduction

1.1. The main results. An irreducible holomorphic symplectic manifoldis a simply connected compact Kahler manifold X, such that H 0(X,Ω2

X)is generated by an everywhere non-degenerate holomorphic two-form. Anirreducible holomorphic symplectic manifold of real dimension 4n admitsa Riemannian metric with holonomy Sp(n) [Be]. Such a metric is calledhyperkahler.

Let S be a smooth KahlerK3 surface and S [n] the Hilbert scheme of lengthn zero dimensional subschemes of S. S [n] is an irreducible holomorphicsymplectic manifold [Be]. An irreducible holomorphic symplectic manifold

X is said to be of K3[n]-type, if X is deformation equivalent to S [n], for a K3surface S. The moduli space of manifolds of K3[n]-type is 21-dimensional,if n ≥ 2. In particular, a generic manifold of K3[n]-type is not the Hilbertscheme of any K3 surface.

Definition 1.1. Let X be an irreducible holomorphic symplectic manifold.An automorphism g of the cohomology ring H∗(X,Z) is called a monodromyoperator, if there exists a family X → B (which may depend on g) of ir-reducible holomorphic symplectic manifolds, having X as a fiber over apoint b0 ∈ B, and such that g belongs to the image of π1(B, b0) under themonodromy representation. The monodromy group Mon(X) of X is thesubgroup of GL(H∗(X,Z)) generated by all the monodromy operators.

Parallel transport of a class α inH2i(S[n],Q), which isMon(S [n])-invariant,defines a class αX in any X of K3[n]-type. More generally, if spanQα is a

non-trivial Mon(S [n])-character, then we get a well defined unordered pair±αX of a class and its negative. Such a class αX is of Hodge type (i, i), bythe following Lemma.

Lemma 1.2. Let α ∈ H2i(X,C) be a class, which is invariant under afinite-index subgroup of Mon(X). Then αX is of Hodge type (i, i).

Proof. The statement is proven in [Ma4], Proposition 3.8 part 3, which reliesalso on results of Verbitsky [Ve1].

We define in the proposition below certain classes κi(X) in H i,i(X,Q), fori an integer in the range 1 ≤ i ≤ n+2

2 , and for any irreducible holomorphic

symplectic manifold X of K3[n]-type, n ≥ 2. Given a coherent OX -moduleE of rank r > 0, set

κ(E) := ch(E) ∪ exp(−c1(E)/r),

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 3

and let κi(E) ∈ H2i(X,Q) be the summand of κ(E) of degree 2i. Let E be

the ideal sheaf of the universal subscheme in S×S [n], fi the projection fromS × S[n] onto the i-th factor, i = 1, 2, and IZ the ideal sheaf of a length nsubscheme Z ⊂ S. Let

(1.1) EZ := Ext1f2(f∗1 (IZ), E)

be the relative extension sheaf over S [n]. EZ is a torsion free reflexive sheaf ofrank 2n−2, which is locally free away from the point of S [n] corresponding tothe ideal sheaf IZ (Proposition 4.5). Set κi(S

[n]) := κi(EZ) ∈ H2i(S[n],Q).

Proposition 1.3. (Proposition 3.1). Let i be an integer in the range

2 ≤ i ≤ n+22 . The class κi(S

[n]) is monodromy invariant, for even i. The

pair κi(S[n]),−κi(S

[n]) is monodromy invariant, for odd i. Parallel trans-

port of the pair κi(S[n]),−κi(S

[n]) yields a well defined unordered pairκi(X),−κi(X) of classes of type (i, i) on any irreducible holomorphic sym-

plectic manifold X of K3[n]-type, n ≥ 2.

The class κi(X) is non-trivial; it can not be expressed as a polynomial inclasses of degree less than 2i, if i ≤ n

2 ([Ma1], Lemma 10). In contrast, theodd Chern classes c2k+1(TX) vanish, since TX is a holomorphic symplecticvector bundle.

Twisted coherent sheaves and their characteristic classes are reviewed insection 2. For the uninitiated reader it would suffice at this point to note thatthe data of a locally free twisted sheaf is equivalent to that of a projectivebundle. Furthermore, the definition of the characteristic classes κi(E) abovemay be extended to define characteristic classes of both twisted sheaves andprojective bundles.

Theorem 1.4. Let X be any irreducible holomorphic symplectic manifoldof K3[n]-type and i an integer in the range 2 ≤ i ≤ n+2

2 . The class κi(X)is a characteristic class of a (possibly twisted) reflexive coherent sheaf Ex ofrank 2n− 2 on X, which is locally free away from a single point x of X.

Theorem 1.4 is proven below after Theorem 1.7. The cohomology groupH2(X,Z), of an irreducible holomorphic symplectic manifold X, admits acanonical, symmetric, non-degenerate, and primitive bilinear pairing q ∈Sym2H2(X,Z)∗ [Be]. Theorem 1.4 yields an expression of the Beauville-

Bogomolov pairing in terms of characteristic classes, for X of K3[n]-type,n ≥ 2, by the following Lemma. The inverse of q is a class in Sym2H2(X,Q),and we denote by q−1 its image in H4(X,Q) as well.

Lemma 1.5. The following equation holds in H4(X,Q), for any X of K3[n]-type, n ≥ 2.

(1.2) q−1 = c2(TX) + 2κ2(X).

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4 EYAL MARKMAN

The dimension of the subspace1 spanq−1, c2(TX), κ2(X) is 2, for n ≥ 4,and 1, for n = 2, 3.

The Lemma is proven in section 8. The main technical result of thispaper is the following Theorem. Let n be an integer ≥ 2. Set r := 2n− 2.Let S be a K3 surface admitting an ample line bundle H of degree 2r2 + r,which is not the tensor power of another ample line-bundle of lower degree.Let M be the moduli space of Gieseker-Maruyama H-stable sheaves F ofrank r with determinant line-bundle H and χ(F ) = 2r. Let πij be theprojection from M× S ×M onto the product of the i-th and j-th factors.

Theorem 1.6. There exists a K3 surface S with an ample line-bundle Hof degree 2r2 + r as above, such that the moduli space M has the followingproperties.

(1) M is smooth and projective of dimension 2n.(2) (Proposition 4.5 and Theorem 7.10) There exists an untwisted uni-

versal sheaf E over S ×M. The relative extension sheaf

(1.3) E := Ext1π13(π∗12E , π∗23E)

over M×M is reflexive, of rank 2n− 2, locally free away from thediagonal, and OM(1) OM(1)-slope-stable, for some2 ample line-bundle OM(1) over M.

(3) (Proposition 4.2) Let i be an integer satisfying 4 ≤ 2i ≤ n + 2.If i is even, then the class κi(E) in H∗(M ×M,Q) is invariantunder the diagonal action of Mon(M). If i is odd, then the pairκi(E),−κi(E) is Mon(M)-invariant.

(4) [Y1] Let F be a stable sheaf over S with isomorphism class [F ] ∈M.The restriction of the sheaf E to [F ] × M is isomorphic to therelative extension sheaf EF := Ext1f2

(f∗1F, E) over M. Furthermore,

the pair (M, EF ) is deformation equivalent to the pair (S [n], EZ),given in equation (1.1). In particular, κi(EF ) is equal to the classκi(M) of Proposition 1.3.

(5) (Theorem 7.10) The sheaf EF in part 4 is OM(1)-slope-stable, for ageneric such F .

Fix an integer n ≥ 2 and a moduli spaceM as in Theorem 1.6. Associatedto the Kahler class c1(OM(1)) is a twistor deformation X → P1 of M as anirreducible holomorphic symplectic manifold. A reflexive sheaf over M×Mis said to be projectively-hyperholomorphic, if it extends as a twisted reflexivesheaf over the fiber product X ×P1 X , flat over P1. The sheaf E in Theorem

1When n = 3, the relation 4q−1 = 3c2(TM) holds as well. It follows from Chern num-

bers calculations, by comparing two formulas for the Euler characteristic χ(S [n], L) of a

line bundle L on S[n]. One as a binomial coefficient χ(S[n], L) =

q(c1(L),c1(L))2

+ n + 1n

«

[EGL], the other provided by Hirzebruch-Riemann-Roch.2More information about OM(1) is provided in Theorem 7.10.

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 5

1.6 is projectively-hyperholomorphic, by the stability result in part (2), themonodromy-invariance result in part (3) of Theorem 1.6, and a deep resultof Verbitsky ([Ve4] and Corollary 6.12 below). We use Verbitsky’s theoryof hyperholomorphic sheaves in order to deform the pair (M, E) to a pair

(X,E′), for any irreducible holomorphic symplectic variety X of K3[n]-type.More precisely, we prove the following statement.

Theorem 1.7. (Theorem 7.11) Let X be an irreducible holomorphic sym-

plectic variety of K3[n]-type. There exist a reduced, connected, projectivecurve C of arithmetic genus 0, which may be reducible, a smooth and properfamily X → C of irreducible holomorphic symplectic varieties, a torsion-freereflexive coherent twisted3 sheaf G of OX×CX -modules of rank 2n − 2, flatover C, and points s, t, in C, with the following properties. The fiber Xs

of X over s is isomorphic to M, the restriction Gs of G to Xs ×Xs is iso-morphic to the sheaf E in Theorem 1.6, and the fiber Xt is isomorphic toX.

A conjectural alternative approach to the construction of the deforma-tion in the above Theorem, for X a sufficiently small deformation of M, issketched in [Ma5]. This alternative approach is more geometric, but leadsonly to local deformations.

Proof. (Of Theorem 1.4) The pairs ±κi(EF ) and ±κi(M) are equal, by part4 of Theorem 1.6. The pairs (M, E) and (X,Gt) are deformation equivalent,where Gt is the restriction of G to the point t of C in Theorem 1.7. Givena point x ∈ X, denote by Ex the restriction of Gt to x × X. The pairs(M, EF ) and (X,Ex) are then deformation equivalent. The characteristicclass κi(E) is defined in section 2 for any twisted sheaf E. Theorem 1.7 thusimplies that the pair ±κi(Ex) is a parallel transport of the pair ±κi(EF ).We conclude that the pairs ±κi(Ex) and ±κi(X) are equal, by Proposition1.3. This completes the proof of Theorem 1.4.

The fact that the deformations of the sheaf E in Theorem 1.7 are twistedis a blessing, rather than a nuisance. It is key to the proof of its slope-stability. This is due to the following general result.

Proposition 1.8. (Proposition 7.8) Let (X,ω) be a compact Kahler man-ifold, θ ∈ H2

an(X,O∗X ) a class of order r > 0, and E a reflexive, rank r,

θ-twisted sheaf. Then End(E) is ω-slope polystable.

1.2. Two applications.

1.2.1. Application to the Lefschetz standard conjecture for projective irre-ducible holomorphic symplectic manifolds of K3[n]-type. Theorem 1.7 ex-hibits a positive dimensional family of reflexive coherent sheaves over everyirreducible holomorphic symplectic variety X of K3[n]-type, n ≥ 2. Indeed,

3The data of the twisted reflexive sheaf G is equivalent to a projective bundle over thecomplement of the diagonal embedding of X in X ×C X .

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6 EYAL MARKMAN

the rank 2n−2 twisted reflexive sheaf E over X×X, constructed in Theorem1.7, is locally free away from the diagonal, but singular along the diagonal.The restriction Ex, of E to x×X, corresponds to a reflexive twisted sheafon X, locally free on X \ x, and singular at the point x. Thus, the iso-morphism class of Ex determines the point x. The same applies to End(Ex),which is untwisted.

The sheaves End(Ex), x ∈ X, are shown to be hyperholomorphic in theproof of Theorem 1.7. The hyperholomorphicity property is defined in sec-tion 6. Here we point out only that the existence of such a positive dimen-sional family of isomorphism classes of reflexive hyperholomorphic sheavesseems to imply the Lefschetz standard conjecture in degree 2, for projectiveirreducible holomorphic symplectic manifolds of K3[n]-type (see [Ch], The-orem 3). Charles considers locally free hyperholomorphic sheaves, but hisproof seems to go through for reflexive hyperholomorphic sheaves as well.

1.2.2. Non-commutative deformations of the derived category of K3 sur-faces. Let S, M, and E , be as in Theorem 1.6. Let Db(S) be the boundedderived category of coherent sheaves on S. We expect that the hyperkahlerdeformations of M, which do not come from deformations of S, neverthelesscorrespond to “deformations” of Db(S) as follows. Consider the exact func-tor ΦE : Db(S) → Db(M) with the universal sheaf E ∈ Db(S ×M) as itskernel. Set ER := E∨[2] and let ΦER

: Db(M) → Db(S) be the exact functorwith kernel ER. Then ΦER

is the right adjoint of ΦE [Mu4]. Set

Φ := ΦE ΦER: Db(M) → Db(M).

The kernel F ∈ Db(M×M), of the endo-functor Φ, is the convolution of Eand ER. Let πij be the projection from M×S×M onto the product of the

i-th and j-th factors. Then F := Rπ13

(π∗12ER

L⊗ π∗23E

)[2]. Hence, F fits in

an exact triangle E[−1] → F → OM×M → E, where E is given in equation(1.3).

The sheaf E deforms to a twisted sheaf over the self product of everyX of K3[n]-type, n := dim(M)/2, by Theorem 1.7. The kernel F of Φsimilarly deforms to an object in the derived category of twisted sheavesover X × X [MS]. We get a deformation of the endo-functor Φ to endo-functors of bounded derived categories of twisted sheaves Db(X, θ), whereθ ∈ H2(X,µ2n−2) is a monodromy invariant class, up to sign, given inLemma 7.2.

The functor ΦE and the endo-functor Φ are studied in a joint work withS. Mehrotra [MS]. It is shown that ΦE is faithful when M = S [n] and itis expected to be faithful for any M as above. Whenever ΦE is faithful,the category Db(S) can be reconstructed from the co-monad (Φ, ε, δ), whereε : Φ → id is the co-unit for the adjoint pair (ΦE ,ΦER

), η : id → ΦER ΦE

the unit, and δ := ΦEηΦER: Φ → Φ2 the co-action. Associated to the

co-monad (Φ, ε, δ) is the category C(Φ, ε, δ) of co-algebras for the co-monad

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 7

([Mac], section VI). The reconstruction of Db(S) is an application of theBar-Beck Theorem ([Mac], section VI.7, and [MS]), which states that thenatural functor Db(S) → C(Φ, ε, δ) is an equivalence, when ΦE is faithful.We expect the co-monad structure (Φ, ε, δ) to deform along every hyperkahlerdeformation of M, including deformations along which the K3 surface Sdoes not deform. Consequently, the category C(Φ, ε, δ) would deform alongevery hyperkahler deformation of M.

Organization: The note is organized as follows. In section 2 we re-view the definition of the characteristic classes κi of projective bundles andtwisted sheaves. In section 3 we recall the language of the Mukai latticeand repeat the construction of the classes κi(M), for moduli spaces M ofstable sheaves over a K3 surface. The realization of the classes κi(S

[n]), as

characteristic classes, depends on a choice of a point in S [n]. The realizationis canonical, when carried out in a relative setting over S [n]×S[n] in section4. The canonical sheaf E over S [n]×S[n], given in (1.3) above, is shown to betorsion free, reflexive, and its characteristic classes are monodromy-invariantand restrict to κi(S

[n]). The sheaf E is shown to be the direct image of avector bundle V over the blow-up of the diagonal in S [n] × S[n]. In section6 we review Verbitsky’s results about hyperholomorphic sheaves. In section7 we prove the stability Theorem 1.6. We then prove Theorem 1.7 usingVerbitsky’s results on hyperholomorphic sheaves.

1.3. Notation. Let f : X → Y be a proper morphism of complex mani-folds or smooth quasi-projective varieties. We denote by f∗ the push-forwardof coherent sheaves, as well as the Gysin homomorphism in singular coho-mology, while f! is the Gysin homomorphism in K-theory (algebraic, holo-morphic, or topological). We let Ktop(X) be the Grothendieck K-ring ofequivalence classes of formal sums of topological vector bundles.

The pullback homomorphism is denoted by f ∗ for coherent sheaves andin singular cohomology, while f ! is the pull back in K-theory. Given a classα in Heven(X), we denote by αi the graded summand in H2i(X).

Given a Cech 2-cocycle θ of O∗X on a complex variety X, we define the

notion of a θ-twisted coherent sheaf in Definition 2.1. A (coherent) sheafwill always mean an untwisted (coherent) sheaf, unless we explicitly mentionthat it is twisted.

2. Characteristic classes of projective bundles and twisted

sheaves

Let Y be a topological space and y a class in the ring KtopY generatedby classes of complex vector bundles over Y . Assume that the rank r of yis non-zero, set

(2.1) κ(y) := ch(y) ∪ exp(−c1(y)/r),

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8 EYAL MARKMAN

and let κi(y) be the summand of κ(y) in H2i(Y,Q). In terms of the Chern

roots yj, we have chi(E) =∑r

j=1

yij

i! , c1(E) =∑r

j=1 yj, and

κi(y) =

r∑

j=1

[yj −

(Pr

k=1 yk

r

)]i

i!.

The characteristic class κ is multiplicative, κ(y1 ⊗ y2) = κ(y1) ∪ κ(y2),and κ([L]) = 1, for any line bundle L. Given a vector bundle E over Y ,the equality κ(E) = κ(E ⊗ L) thus holds, for any line bundle L. Note theequalities

κi(y∨) = (−1)iκi(y),

κ(−y) = −κ(y).We define next the invariant κ(P), for any holomorphic Pr−1-bundle, r ≥

1. The definition is clear, if P is the projectivization of a vector bundle E,since κ(E) is independent of the choice of E. More generaly, let

θ(P) ∈ H2(Y,O∗Y )

be the obstruction class to lifting P to a holomorphic vector bundle. Theclass θ(P) is the image of the class [P] ∈ H1(Y, PGLr), under the connectinghomomorphism of the short exact sequence of sheaves

0 → O∗Y → GLr(O) → PGLr(O) → 0.

Consider the dual bundle π : P∗ → Y . The pullback π∗P has a tautologicalhyperplane subbundle, hence a divisor, hence a holomorphic line-bundleOπ∗P(1). The obstruction class θ(P) is in the kernel of π∗ : H2(Y,O∗

Y ) →H2(P∗,O∗

P∗) and the projective bundle π∗P over P∗ is the projectivization of

some vector bundle E. The class κ(E) belongs to the image of the injective

homomorphism π∗ : H∗(Y,Q) → H∗(P∗,Q), since κ(E) restricts as 1 to eachfiber of π. Define4

(2.2) κ(P) ∈ H∗(Y,Q)

as the unique class satisfying π∗(κ(P)) = κ(E).The class θ(P) is determined by a topological class, which we now define.

Let µr be the group of r-th roots of unity. Denote the corresponding localsystem by µr as well, and let ι : µr → O∗ be the inclusion. Let

(2.3) θ : H1(Y, PGLr(O)) → H2(Y, µr)

be the connecting homomorphism of the short exact sequence

0 → µr → SLr(O) → PGLr(O) → 0.

4The construction has an analogue for topological complex Pr−1-bundles. Note that thetopological analogue of H2(Y,O∗Y ), for the sheaf of invertible continuous complex valuesfunctions, is isomorphic to H3(Y, Z), via the connecting homomorphism of the exponentialsequence.

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 9

Then the following equality clearly holds.

(2.4) θ(P) = ι[θ(P)].

When P is the projectivization of a vector bundle V over Y , the followingequality holds ([HSc], Lemma 2.5)

(2.5) θ(PV ) = exp

(−2π√−1

rc1(V )

).

Definition 2.1. Let Y be a scheme or a complex analytic space, U :=Uαα∈I a covering, open in the complex or etale topology, and θ ∈ Z 2(U ,O∗

Y )

a Cech 2-cocycle. A θ-twisted sheaf consists of sheaves Eα of OUα -modulesover Uα, for all α ∈ I, and isomorphisms gαβ : (Eβ)|Uαβ

→ (Eα)|Uαβsatisfy-

ing the conditions:(1) gαα = id,(2) gαβ = g−1

βα ,

(3) gαβgβγgγα = θαβγ · id.The θ-twisted sheaf is coherent, if the Eα are.

The abelian categories of θ-twisted and θ ′-twisted coherent sheaves areequivalent, if the cocycles θ and θ′ represent the same cohomology class. Theequivalence is not canonical, but the ambiguity is only up to tensorizationby an untwisted line-bundle [Ca]. Thus, the classes κi of a twisted sheaf,defined below, are preserved under the equivalences. We will often abuseterminology and refer to a θ-twisted sheaf, where θ is a class in H 2(Y,O∗

Y ),meaning the equivalence class of θ-twisted sheaves, for different choices ofCech cocycles θ′, representing the class θ.

Remark 2.2. Observe that the determinant det(E), of a θ-twisted coherentsheaf E of rank r, is a θr-twisted line-bundle. Thus, θr is a coboundary.Consequently, the order of the class [θ], of θ in H 2(Y,O∗

X), divides the rankof every θ-twisted locally free sheaf E.

Assume Y is a complex manifold. A projective Pr−1 bundle P over Ycorresponds to a rank r locally-free twisted coherent sheaf E, with twistingcocycle θ in Z2(U ,O∗

Y ), for some open covering U of Y . The θ-twisted sheafE is unique, up to tensorization by a line-bundle. The characteristic classκ(E) := κ(P) can be generalized for twisted sheaves, which are not locallyfree, as we show next.

Let θ ∈ Z2(U ,O∗Y ) be a two cocycle representing the class of a projective

Pr−1-bundle P and F := (Fα, gαβ) a θ-twisted sheaf of rank ρ > 0. Thecocycle θ pulls back via π : P∗ → Y to a coboundary over P∗. Say π∗θ =δ(ψ), for a cochain ψ in C1(π−1U ,O∗

P∗). Then the product of π∗(gαβ) with

ψ−1 glues (π∗Fα)α∈I to an untwisted coherent sheaf F of rank ρ over P∗.A choice of a different cochain ψ′, satisfying δ(ψ′) = π∗θ, would yield a

tensorization of the previous F by the holomorphic line-bundle associated

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10 EYAL MARKMAN

to the cocycle ψ · ψ′−1. The class κ(F ) is again the pullback of a uniqueclass

(2.6) κ(F ) ∈ H∗(Y,Q).

The class κ(F ) is independent of the choice of the bundle P representing theclass [θ] ∈ H2(Y,O∗

Y ). We obtain the class κ(F ), for any θ-twisted sheaf ofpositive rank, for any class [θ] in the Brauer group of Y . Note: the Cherncharacter ch(F ) of a θ-twisted sheaf, with a topologically trivial class θ, wasdefined in [HSt], depending on a choice of a lift of θ to a class in H 2(Y,Q).Another definition is provided in [Li].

3. The rational Hodge classes κi(X)

Let S be a projective K3 surface and v ∈ KtopS a primitive class of rank> 0 with c1(v) of type (1, 1). Assume that (v, v) ≥ 2. There is a systemof hyperplanes in the ample cone of S, called v-walls, that is countable butlocally finite [HL], Ch. 4C. An ample class is called v-generic, if it does notbelong to any v-wall. Choose a v-generic ample class H. Then the modulispace MH(v) is a projective irreducible holomorphic symplectic manifold,

deformation equivalent to S [n], with n = 1 + (v,v)2 . This result is due to

several people, including Huybrechts, Mukai, O’Grady, and Yoshioka. Itcan be found in its final form in [Y1].

Let f1 and f2 be the projections on the first and second factors of S ×MH(v). Assume further that a universal sheaf E exists over S ×MH(v).Let e : KtopS → KtopMH(v) be the homomorphism given by

(3.1) ex := f2!

(f !1(−x∨)⊗ [E ]

).

The class ex has rank (v, x), in terms of the Mukai pairing

(3.2) (x, y) := −χ(x∨ ⊗ y),

for x, y ∈ KtopS. Let v⊥ be the sublattice of KtopS orthogonal to v.Mukai defines a weight 2 Hodge structure on KtopS ⊗Z C as follows. The

(2, 0) summand is the pull-back of H2,0(S), via the Chern character isomor-phism ch : KtopS → H∗(S,Z), and the pullback of H0(S,Z) and H4(S,Z)are both of Hodge-type (1, 1). Recall that H2(MH(v),Z) is endowed withthe Beauville-Bogomolov pairing. The homomorphism

v⊥ → H2(MH(v),Z),(3.3)

x 7→ c1(ex),

is an isometry and an isomorphism of weight 2 Hodge structures [Y1].The Mukai vector of a class v ∈ KtopS is the class ch(v)

√tdS ∈ H∗(S,Z).

Following Mukai, we write the Mukai vector of v as a triple (r, c1(v), s), wherethe rank r corresponds to the summand in H0(S,Z), while the summand

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 11

in H4(S,Z) corresponds to the integer s times the class Poincare-dual to apoint. The Hirzebruch-Riemann-Roch Theorem yields the equality

(v, v) = c1(v)2 − 2rs.

Proposition 3.1. The class κ(ev) is invariant under a finite-index sub-group5 of Mon(MH(v)). Let i be an integer satisfying 4 ≤ 2i ≤ n + 2. Ifi is even, then κi(ev) is Mon(MH(v))-invariant. If i is odd, then the linespanQκi(ev) in H2i(MH(v),Q) is Mon(MH(v))-invariant.

The proposition is proven in section 5 using results of [Ma2, Ma4]. Propo-sition 3.1 yields a monodromy-invariant pair of a class and its negative,denoted by

(3.4) ±κi(X),

for any irreducible holomorphic symplectic manifold X of K3[n]-type, n ≥ 2,and for 4 ≤ 2i ≤ n+ 2. The class κi(X) is of type (i, i), by Lemma 1.2.

Recall that the first main goal of this note is to express the Hodge-classesκi(X) as characteristic classes of sheaves on X (Theorem 1.4). The genericsuch X does not contain any proper closed positive-dimensional analyticsubvariety [Ve2].

3.1. Relation of Theorem 1.4 with the Hodge conjecture. Let Xbe a compact Kahler manifold. Denote by Hodge(X) the Q-subalgebra ofH∗(X,Q), generated rational (p, p)-classes, and let Chern(X) be the Q-subalgebra of Hodge(X) generated by Chern classes of coherent sheaves onX. Chern(X) is in fact additively generated by Chern classes of coher-ent sheaves on X [Vo]. Chern(X) is equal to the subalgebra of H ∗(X,Q)generated by the classes

f∗(ci(E)),

where f : Y → X is a morphism from another compact Kahler manifold,E is a holomorphic vector bundle on Y , and i is an integer [Vo]. When Xis projective, Chern(X) admits a third characterization; the Q-subalgebragenerated by classes of subvarieties of X.

The classes κi(E), of a θ-twisted sheaf on X of positive rank, all belong toChern(X). This follows from their definition (2.6), in terms of characteristicclasses on a projective bundle over X, and the following general fact. Amorphism f : Y → X, from another compact Kahler manifold, satisfies

(3.5) f∗Chern(Y ) ⊂ Chern(X).

When X is a projective variety, the Hodge Conjecture predicts the equal-ity Chern(X) = Hodge(X). The extension of the Hodge Conjecture toKahler manifolds fails; Voisin proved that there exist four-dimensional com-plex tori X, for which Chern(X) 6= Hodge(X) [Vo].

5Conjecturally, any monodromy operator takes κ(ev) to itself or its dual κ((ev)∨). The

conjecture is proven in [Ma4] Corollary 1.6, if n is congruent to 0 or 1 modulo 4

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12 EYAL MARKMAN

4. Monodromy invariant classes over X ×X

In section 4.1 we construct a reflexive sheaf over MH(v) ×MH(v), sin-gular along the diagonal, which is, roughly, the canonical representative ofa relative version of the class ev . We resolve this sheaf as a locally free sheafV , over the blow-up of the diagonal in MH(v)×MH (v), in sections 4.1 and4.2.

4.1. Monodromy invariant classes κi(F) over M(v)×M(v). SetM :=MH(v). Assume that a universal sheaf E exists over S×M. (This assump-tion will be dropped later). A choice of a stable sheaf E in M yields a liftof the class ev, given in (3.1), to a class in the bounded derived category ofcoherent sheaves Db

Coh(M). Avoiding such a choice, we construct instead anatural class over M×M.

Let πij be the projection from M× S ×M onto the product of the i-th and j-th factors. Consider the following object in the bounded derivedcategory of coherent sheaves over M×M:

(4.1) F := Rπ13∗

[π∗12E∨

L⊗ π∗23E

][1].

Let ιE : M → M×M, be the embedding sending a point [E ′] ∈ M to(E,E′). Then ιE relates the class of F in Ktop(M×M) to ev :

Lemma 4.1. ev = ι!E [F ].

Proof. Denote by ιE : S × M → M × S × M the morphism given by(x,E′) 7→ (E, x,E ′). The Cohomology and Base Change Theorem yieldsthe second equality below:ι!E[−F ] = ι!Eπ13!

(π!12E∨ ⊗ π!

23E) = f2!ι!E(π!

12E∨ ⊗ π!23E) = f2!

(f !1E

∨ ⊗ E) =−ev.

Proposition 4.2. The class κ(F) in H∗(M×M,Q) is invariant under thediagonal action of a finite-index subgroup of Mon(M). Let i be an integersatisfying 4 ≤ 2i ≤ n+2. If i is even, then κi(F) is Mon(M)-invariant. If iis odd, then the line spanQκi(F) in H2i(M×M,Q) is Mon(M)-invariant.

The proposition is proven in section 5 using results of [Ma2, Ma4]. Com-pare also with Theorem 4.4 below.

Lemma 4.3. c1(F) = −π∗1c1(ev) + π∗2c1(ev).

The lemma is proven in section 5. When v is the class of the ideal sheafof a length n subscheme, and E is the universal ideal sheaf, then c1(ev) is

half the class of the big diagonal in S [n] ([Ma4], Lemma 5.9).The object F fits in an exact triangle

Ext1π13(π∗12E , π∗23E) → F → Ext2π13

(π∗12E , π∗23E)[−1] → Ext1π13(π∗12E , π∗23E)[1].

Furthermore, Ext2π13(π∗12E , π∗23E) is isomorphic to the structure sheaf O∆ of

the diagonal ∆ ⊂ [M×M], while Ext1π13(π∗12E , π∗23E) is a reflexive sheaf of

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 13

rank (v, v), and is locally free away from ∆ (Proposition 4.5). The objectF plays a central role in the study of the cohomology of the moduli spaceM. It was used by Mukai to prove that if M is 2-dimensional then it isa K3 surface [Mu2]. In the higher dimensional case properties of F leadto a simple proof of the irreducibility of M [KLS], originally proven by adegeneration argument; via a combined effort of several authors ([O’G], [Y1]Theorem 8.1, and [Y2] Corollary 3.15).

Theorem 4.4. (1) ([Ma1], Theorem 1) The Chern classes of F sat-ify: c2n−1(F) = 0, and c2n(F) is Poincare-Dual to the class [∆] ofthe diagonal. Hence, c2n

(Ext1π13

(π∗12E , π∗23E))

is Poincare-Dual to(1− (2n− 1)!)[∆].

(2) ([Ma3], Theorem 1) Consequently, the Chern classes ci(ex), of theKunneth factors ex ∈ KtopM, x ∈ KtopS, of E, given in (3.1),generate the integral cohomology ring H∗(M,Z).

Generators for the cohomology ring, with rational coefficients, were foundin [LQW, Ma1]. The ring structure was determined in [LS].

Let β : B → [M×M] be the blow-up of M×M along ∆, D := P(T∆)the exceptional divisor, ι : D → B the closed imersion, δ : ∆ → M×Mthe diagonal embedding, p : D → ∆ the bundle map, ` the tautological line-sub-bundle of p∗T∆, and `⊥ the symplectic-orthogonal subbundle of p∗T∆.Let τ be the involution of M×M, interchanging the two factors, and τ theinduced involution of B. Note that τ ∗(F) = F∨, by Grothendieck-Serre’sDuality, and the triviality of the relative canonical line-bundle ωπ13 .

Proposition 4.5. (1) The sheaf E := Ext1π13(π∗12E , π∗23E) is reflexive of

rank (v, v).(2) E restricts to [M ×M] \ ∆ as a locally free sheaf. We have the

following isomorphism:

(4.2) δ∗E ∼=(

2∧ T ∗M

)/OM · σ,

where σ is the symplectic form. For i > 0, we have

(4.3) TorM×Mi (E, δ∗OM) ∼= δ∗

i+2∧ T ∗M.

(3) The quotient

(4.4) V :=[β∗Ext1π13

(π∗12E , π∗23E)](D)/tor,

by the torsion subsheaf, is a locally free sheaf of rank (v, v) over B.(4) β∗(V ) ∼= Ext1π13

(π∗12E , π∗23E) and Riβ∗

(V ) = 0, for i > 0.

(5) τ∗V is isomorphic to V ∗.(6) The restriction V|D is naturally identified with the sub-quotient

(4.5) [`⊥/`].

In particular, V|D is a symplectic vector bundle.

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14 EYAL MARKMAN

4.2. Proof of Proposition 4.5. The following lemma will be used in theproof of Proposition 4.5.

Lemma 4.6. The following natural homomorphism is surjective:

(4.6) p∗p∗([`⊥/`]⊗ `∗

)→ [`⊥/`]⊗ `∗.

Proof. We identify each of the vector bundles T∆ and [`⊥/`] with its dual,via the symplectic forms. We have the short exact sequence

0 → [`⊥/`]⊗ `∗ → [p∗T ∗∆/`]⊗ `∗ → `−2 → 0.

p∗([`⊥/`]⊗ `∗

) ∼= ker[p∗(p∗T ∗∆⊗ `∗/O) → p∗(`−2)

], which is naturally

isomorphic to the quotient [2∧ T ∗∆]/O, by the line-sub-bundle spanned by

the symplectic form. The homomorphism (4.6) is dual to the wedge product

[`⊥/`]⊗ ` → p∗([2∧ T∆]/O), which is clearly injective.

The proof of Proposition 4.5 requires a review of the following construc-tion carried out in [Ma1]. There exists a (non-canonical) complex

(4.7) V−1g−→ V0

f−→ V1,

of locally free sheaves over M×M, representing the object F [Lan]. Thesheaf homomorphism g is injective, since Ext0π13

(π∗12E , π∗23E) vanishes. The

middle cohomology sheaf ker(f)/Im(g) is isomorphic to Ext1π13(π∗12E , π∗23E),

and coker(f) is isomorphic to Ext2π13(π∗12E , π∗23E), and hence also to δ∗O∆.

Furthermore, the dual complex represents the pullback τ ∗(F) of the objectF . In particular, coker(g∗) is also isomorphic to δ∗O∆.

Claim 4.7.

Ext1(Im(f),OM×M) = 0,(4.8)

ker(f)∗ ∼= coker(f ∗),(4.9)

ker(g∗)∗ ∼= coker(g).(4.10)

Proof. Consider the long exact sequence of extension sheaves, obtained byapplying Hom(•,OM×M) to the short exact sequence

0 → Im(f) → V1 → O∆ → 0.

Exti(V1,OM×M) = 0, for i > 0, and Exti(O∆,OM×M) = 0, for 0 ≤ i <dim(M) = 2n, by the Local Duality Theorem. The vanishing (4.8) follows.

Applying Hom(•,OM×M) to the short exact sequence

0 → ker(f) → V0 → Im(f) → 0,

we get the short exact sequence

0 → V ∗1f∗−→ V ∗

0 −→ ker(f)∗ → 0,

by the vanishing (4.8). Equation (4.9) follows.Equation (4.10) is the analogue of Equation (4.9) for the dual of the

complex (4.7).

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 15

Let K be the kernel of g|∆ : (V−1)|∆ → (V0)|∆ and F the image of f|∆ :(V0)|∆ → (V1)|∆ . Then K and (V1)|∆/F are both isomorphic to O∆. LetU−1 be the subsheaf of (β∗V−1)(D), whose sections restrict to D as sectionsof [ι∗(p∗K)](D). We get the short exact sequence:

(4.11) 0 → β∗V−1 → U−1 → [ι∗(p∗K)](D) → 0.

Define U1 ⊂ β∗V1 as the subsheaf, whose sections restrict to D as sectionsof ι∗(p∗F ). It fits in the short exact sequence:

(4.12) 0 → U1 → β∗V1 → ι∗(p∗coker(f)) → 0.

We get the complex of vector bundles over B

U−1g−→ β∗V0

f−→ U1,

where both f and g∗ are surjective. Both U−1 and U1 are locally free OB-modules. Set

(4.13) V := ker(f)/Im(g)

(we no longer regard (4.4) as a definition, but rather as an equality, to beproven below). Then V is locally free as well.

Claim 4.8. (1) β∗(U−1) ∼= V−1, and Riβ∗

(U−1) = 0, for i > 0.

(2) β∗(U1) ∼= Im(f), and Riβ∗

(U1) = 0, for i > 0.

(3) β∗(ker(f)) ∼= ker(f), and Riβ∗

(ker(f)) = 0, for i > 0.

Proof. 1) The higher direct images Rip∗(OD(D)) vanish, for i ≥ 0. This

vanishing implies Part 1, using the long exact sequence of higher directimages via β, associated to the short exact sequence (4.11).

2) The pushforward p∗OD is isomorphic to O∆, and all the higher directimages vanish. Part 2 follows from the long exact sequence of higher directimages via β, associated to the short exact sequence (4.12).

Part 3 follows part 2 using the long exact sequence of higher direct imagesvia β, associated to the short exact sequence

0 → ker(f) → β∗V0f→ U1 → 0.

Proof of Proposition 4.5:Part 1) Applying Hom(•,OM×M) to the short exact sequence

0 → Ext1π13(π∗12E , π∗23E) → coker(g) → Im(f) → 0,

we get the short exact sequence

0 → V ∗1f∗−→ ker(g∗) −→

[Ext1π13

(π∗12E , π∗23E)]∗ → 0,

by the vanishing (4.8) and equation (4.10). Hence,[Ext1π13

(π∗12E , π∗23E)]∗

is the middle sheaf cohomology of the complex dual to (4.7). The dualcomplex represents the object τ ∗F , in the derived category, so the middle

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16 EYAL MARKMAN

sheaf cohomology is the pullback τ ∗Ext1π13(π∗12E , π∗23E). Reflecsivity now

follows, by applying the above argument to the dual complex, since τ 2 = id.Part 4 follows from Claim 4.8 and the long exact sequence of higher direct

images via β, associated to the short exact sequence

0 → U−1g→ ker(f) → V → 0.

Part 6) Let Z be the total space of the vector bundle Hom(V−1, V0),h : Z → X × X the projection, g′ : h∗V−1 → h∗V0 the tautological ho-momorphism, Z1 ⊂ Z the determinantal stratum, where the rank of g ′ isrank(V−1) − 1, and g : X × X → Z the section given in (4.7). Z1 is asmooth locally closed subvariety, whose normal bundle NZ1 is isomorphic to

Hom(ker(g′|Z1

), coker(g′|Z1))

[ACGH]. The diagonal ∆ is the scheme theo-

retic inverse image g−1(Z1). Hence, the homomorphism

dg : N∆ −→ g∗NZ1 = Hom(ker(g|∆), coker(g|∆)

)

is injective at every fiber of N∆. ∆ is also the degeneracy locus of thehomomorphism f given in (4.7), and f g = 0. Thus, the image of dgis contained in Hom

(ker(g|∆), ker(f|∆)/Im(g|∆)

). Now, ker(g|∆) ∼= O∆ and

ker(f|∆)/Im(g|∆) is isomorphic to T∆, by the well known identification of

TM with the relative extension sheaf Ext1f2(E , E). We conclude that dg

factors through a homomorphism

dg : N∆ −→ T∆,

which is fiberwise injective, and hence an isomorphism.Over B we have the tautological line-sub-bundle η : OD(D) → p∗N∆

and the homomorphism d(β∗g) is the composition p∗(dg)η. It follows thatthe image of d(β∗g) is ` ⊂ T∆, by the definition of `. On the other hand,the image of d(β∗g) is precisely

Hom(p∗ ker(g|∆), Im(g|D)/Im(β∗g|D)

).

These two descriptions of the image of d(β∗g) provide a canonical isomor-phism ` ∼= Im(g|D)/Im(β∗g|D). We see that V|D is a sub-bundle of [p∗T∆]/`.

Repeating the above argument, for the dual of the complex (4.7) and forthe homomorphism f ∗, we get that (V ∗

|D) is a subspace of [p∗T∆]/` as well

(under the identification T∆ ∼= T ∗∆, via the symplectic structure). Hence,V|D is isomorphic to `⊥/`.

Part 3) The direct image p∗[(V )|D ] vanishes, by part 6 and the vanish-

ing of p∗[`⊥/`]. Hence, β∗[V (−D)] is isomorphic to β∗V . The natural ho-momorphism β∗β∗[V (−D)] → V (−D) is surjective, by part 6 and Lemma4.6. Part 3 follows from part 4, since the kernel of the homomorphismβ∗β∗[V (−D)] → V (−D) is supported on D, and is hence the torsion sub-sheaf of β∗β∗[V (−D)].

Part 5) If we repeat the construction of the vector bundle (4.13), usingthe dual of the complex (4.7), we obtain the vector bundle V ∗, by a direct

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 17

check. On the other hand, the dual complex represents τ ∗F , and the proofof the equality of the sheaves (4.4) and (4.13) yields the isomorphism

V ∗ ∼= β∗[τ∗

Ext1π13

(π∗12E , π∗23E)]

(D)/tor.

The statement now follows from the equality β∗τ∗ = τ∗β∗. This completesthe proof of Proposition 4.5.

Part 2) Consider the exact triangle

Ea−→ [V−1 → V0 → V1]

b−→ O∆[−1] → E[1].

Restriction to ∆ yields the long exact sequence

TorM×M2 (E,O∆)

a−2−→ 0b−2−→ TorM×M

3 (O∆,O∆)δ−1−→

TorM×M1 (E,O∆)

a−1−→ O∆b−1−→ TorM×M

2 (O∆,O∆)δ0−→

E ⊗O∆a0−→ T∆

b0−→ TorM×M1 (O∆,O∆)

δ1−→0

a1−→ O∆b1−→ O∆ ⊗O∆ → 0.

Note that TorM×Mi (O∆,O∆) is isomorphic to

i∧ T ∗∆. Clearly, δ−i is an

isomorphism, for i ≥ 2. The isomorphism in Equation (4.3) follows. Thehomomorphism b0 is surjective, hence an isomorphism. Thus a0 = 0 and δ0is surjective.

The isomorphism (4.2) would follow, once we prove that b−1 is injective.The proof is by contradiction. Assume that b−1 vanishes. Then δ0 is injectiveand δ0(σ) is a non-zero global section of H0(E ⊗ O∆). Let tor(β∗E) bethe torsion subsheaf of β∗E. The endo-functor Rβ∗Lβ∗ of Db(M ×M)is the identity. Hence, β∗(tor(β∗E)) = 0, since E is torsion free, by part1. In particular, H0(tor(β∗E)) = 0. Now [β∗E/tor(β∗E)]|D

∼= `⊥/`, by

part 6, and H0(`⊥/`) = 0. Thus, H0(D, [β∗E]|D) = 0. Consequently,

H0(E ⊗O∆) = 0. A contradiction.

4.3. Lifting deformations of a moduli space M to deformation of

the pair (M,PV ). Let S ′ be another projective K3 surface, v ′ ∈ KtopS′ a

primitive class satisfying (v′, v′) = 2n− 2, n ≥ 2, and H ′ a v′-generic ampleline bundle H ′. Assume that M′ := MH′(v′) is non-empty. Let B ′ be theblow-up of the diagonal in M′ ×M′ and PV ′ the projective bundle overB′ associated to the (possibly twisted) locally free sheaf (4.4). Yoshiokaproved that the moduli space M′ is an irreducible holomorphic symplecticvariety, deformation equivalent to S [n] [Y1]. His proof implies the existenceof a sequence of families of K3-surfaces Si → Ti, 1 ≤ i ≤ N , over quasi-projective curves Ti, with smooth and proper relative families of such modulispaces MSi/Ti

having the following properties. There exist points t′i ∈ Ti

and t′′i+1 ∈ Ti+1, and an isomorphism φi from the fiber Mt′ionto the fiber

Mt′′i+1. Finally, Mt′1

= MH′(v′), and Mt′′N

= S[n].

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18 EYAL MARKMAN

The isomorphism φi comes in two flavors. One is induced by a Fourier-Mukai transformations between the derived categories of St′i

and St′′i+1map-

ping stable sheaves to stable sheaves. Such Fourier-Mukai transformationsrelate a twisted universal sheaf over St′i

×Mt′ito one over St′′i

×Mt′′i([Mu3],

Theorem 1.6). Let B ′i be the blow-up of the diagonal in Mt′i

×Mt′iand B′′

i+1

the analoge for Mt′′i+1. The isomorphism φi pulls back the bundle PV ′′

i+1

over B′′i+1 to PV ′i over B′

i.The second flavor is induced by the composition, of a Fourier-Mukai trans-

formation, with the functor, which takes an object or a morphism, in thederived category, to its dual. The composite functor relates a twisted uni-versal sheaf over St′i

× Mt′ito the dual of one over St′′i+1

× Mt′′i+1. The

isomorphism φi pulls back the bundle PV ′′i+1 over B′′

i+1 to (PV ′i)∗ over B′i.

The following Lemma thus follows from Yoshioka’s work. It will play acentral role in section 6.

Lemma 4.9. The pair (M′, PV ′, (PV ′)∗) deforms to the pair (M, PV,PV ∗),associated to the vector bundle (4.4).

5. Proof of the monodromy invariance of κi(X) and κi(F)

We prove Propositions 3.1 and 4.2 and Lemma 4.3, after reviewing thenecessary facts about the monodromy group of S [n].

Let S be aK3 surface, v ∈ KtopS a primitive class with c1(v) of type (1, 1),and H a v-generic line bundle. Assume that MH(v) is non-empty (in par-ticular, rank(v) ≥ 0, (v, v) ≥ −2, etc . . . ). Then MH(v) is a projective irre-

ducible holomorphic symplectic manifold of K3[n]-type, with 2n = (v, v)+2.Assume that (v, v) ≥ 2. Then H2(MH(v),Z), endowed with the Beauville-Bogomolov pairing, is Hodge isometric to v⊥ ⊂ KtopS, via Mukai’s isometry(3.3).

We define next the orientation character of O(KtopS). A 4-dimensionalsubspace V of KtopS ⊗Z R is positive definite, if the Mukai pairing restrictsto V as a positive-definite pairing. The positive cone C+ ⊂ KtopS ⊗Z R,given by

C+ := x : (x, x) > 0,is homotopic to the unit 3-sphere in any 4-dimensional positive definite sub-space. Hence H3(C+,Z) is isomorphic to Z and is a natural character

cov : O(KtopS) −→ ±1.of the isometry group. Let O+(KtopS) be the kernel of cov.

Denote by O(KtopS)v the subgroup of isometries of KtopS, stabilizing v.Let g be any isometry in O(KtopS)v . It is not assumed to preserve the Hodgestructure. Denote by

(g ⊗ 1) : KtopS ⊗KtopM(v) −→ KtopS ⊗KtopM(v)

the homomorphism acting via the identity on the second factor. The KunnethTheorem identifies KtopS ⊗KtopM(v) with Ktop[S ×M(v)] ([A], Corollary

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 19

2.7.15). Assume that a universal sheaf Ev exists over S ×M(v) and let [Ev]be its class in Ktop[S ×M(v)]. Let D : KtopS → KtopS be the involution,sending a class x to its dual x∨. Set m := (v, v) + 2. We define a class inthe middle cohomology H2m(M(v) ×M(v),Z):

mon(g) :=

cm

(−π13!

π!

12 [(g ⊗ 1)[Ev1 ]]∨ ∪ π!

23[Ev2 ])

if cov(g) = 1,cm

(−π13!

π!

12 [(Dg ⊗ 1)[Ev1 ]] ∪ π!23[Ev2 ]

)if cov(g) = −1.

Denote by

(5.1) mon(g) : H∗(M(v),Z) −→ H∗(M(v),Z)

the homomorphism obtained from mon(g) using the Kunneth and Poincare-Duality Theorems.

Theorem 5.1. ([Ma2] Theorems 1.2 and 1.6)

(1) The endomorphism mon(g) is an algebra automorphism and a mon-odromy operator.

(2) The assignment

(5.2) mon : O(KtopS)v −→ Mon(M(v)),

sending an isometry g to the operator mon(g), is a group-homomorphism.The homomorphism is injective, if (v, v) ≥ 4, and its kernel is gen-erated by the involution

(5.3) w 7→ −w + (w, v)v,

if (v, v) = 2. The image mon[O(Ktop(S)v] is a normal subgroup offinite index in the monodromy group Mon(MH(v)).

(3) There exists a topological complex line bundle `g on MH(v) satisfyingone of the following equations:

(g ⊗mon(g))[Ev ] = [Ev ]⊗ f∗2 `g, if cov(g) = 1,((D g) ⊗mon(g))[Ev ] = [Ev]

∨ ⊗ f∗2 `g, if cov(g) = −1.

The action of mon(g) on Ktop(M(v)), in part 3 of the Theorem, is con-structed as follows. The Chern character homomorphism ch : KtopM(v) →H∗(M(v),Q) in injective, since KtopM(v) is torsion free [Ma3]. The ho-momorphism ch is monodromy equivariant; hence it maps KtopM(v) to amon(g)-invariant subalgebra, for all g ∈ O(KtopS)v, by part 1 of Theo-rem 5.1. Denote by mong the corresponding monodromy automorphism ofKtopM(v). Part 3 of the Theorem can be rephrased in terms of the homo-morphism e : KtopS → KtopM(v), given in (3.1):

(5.4) mong(eg−1(x)) =

ex ⊗ `g, if cov(g) = 1,

(ex)∨ ⊗ `g, if cov(g) = −1.

Consequently, the line bundle `g is determined by the following formula:

(5.5) c1(`g) =mong(c1(ev))− cov(g) · c1(ev)

(v, v).

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20 EYAL MARKMAN

Let Mon2(M(v)) be the image in O[H2(M(v),Z)] of Mon(M(v)), underthe restriction homorphism from H∗(M(v),Z) to H2(M(v),Z). Let

mon2 : O(KtopS)v →Mon2(M(v))

be the composition of mon with the restriction homomorphism.

Theorem 5.2. ([Ma4], Theorem 1.2 and Lemma 4.2). The homomorphismmon2 : O(KtopS)v → Mon2(M(v)) is surjective. It is an isomorphism, if(v, v) ≥ 4, and its kernel is generated by the involution (5.3), if (v, v) = 2.

Theorem 5.3. ([Ma4] Theorem 1.5) Any monodromy operator, which actsas the identity on H2(M(v),Z), acts as the identity also on Hk(M(v),Z),

0 ≤ k ≤ dimR(M(v))4 + 2.

Proof of Proposition 3.1: Observe first that the pair κ(ev), κ ((ev)∨)

is invariant under the image of O(KtopS)v in Mon(M(v)) via mon. Thisfollows from the reformulation (5.4) of part 3 of Theorem 5.1, and the factthat g(v) = v. We prove next the Mon(M(v))-invariance of spanQκi(ev),for 4 ≤ 2i ≤ n + 2. Let f be an element of Mon(M(v)). There existsan isometry g ∈ O(KtopS)v, such that f and mon(g) have the same imagein Mon2(M(v)), by Theorem 5.2. Theorem 5.3 implies that the actions off and mon(g) on Hk(M(v),Z) agree in the range required. We concludethe f -invariance of spanQκi(ev), from its mon(g) invariance, for i in thatrange.

Proof of Proposition 4.2: It suffices to show that the pair κ(F), κ (F ∨)is invariant under the image of O(KtopS)v in Mon(M(v)) via mon, by theargument used in the proof of Proposition 3.1. Denote by

DM : KtopM(v) → KtopM(v)

the duality involution y 7→ y∨ and by DS the duality involution of KtopS.Note that [Ev]

∨ = (DS ⊗ DM)[Ev]. Caution: while DM commutes withMon(M(v)), DS does not commute with O(KtopS)v . The class [F ] is theimage in KtopM(v) ⊗KtopM(v) of the class (1⊗DM)[Ev] ⊗ [Ev] via thecontraction with the Mukai pairing:

ψ : [KtopS ⊗KtopM(v)] ⊗ [KtopS ⊗KtopM(v)] → KtopM(v)⊗KtopM(v)

x1 ⊗ y1 ⊗ x2 ⊗ y2 7→ −χ(x∨1 ⊗ x2)y1 ⊗ y2

The equality ψ = ψ (g ⊗ 1⊗ g⊗ 1) holds, for any isometry g of the Mukailattice. Hence, the following equality holds:ψ (g ⊗mong DM)[Ev]⊗ (g ⊗mong)[Ev ] =

ψ (1⊗mong DM)[Ev ]⊗ (1⊗mong)[Ev] .The right hand side is (mong ⊗mong)[F ], while the left hand side is equalto

ψ(1⊗DM)([Ev ]⊗ f !

2`g)⊗ [Ev]⊗ f !2`g

, if cov(g) = 1,

ψ([Ev]⊗ f !

2`∨g )⊗ (1⊗DM)([Ev ]⊗ f !

2`∨g )

, if cov(g) = −1,

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 21

by part 3 of Theorem 5.1. The latter contractions simplify to

(5.6) (mong ⊗mong)[F ] =

[F ]⊗ π!

1(`∨g )⊗ π!

2`g, if cov(g) = 1,

([F ])∨ ⊗ π!1(`

∨g )⊗ π!

2`g, if cov(g) = −1,

by the projection formula. We conclude that the pair κ(F), κ(F∨) ismon(g)-invariant.

Proof of Lemma 4.3: For every g ∈ O+KtopS)v , we have:

(mong ⊗mong)(c1(F))(5.6)= c1(F) − (v, v)[π∗1c1(`g)− π∗2c1(`g)]

(5.5)=

c1(F) − π∗1 [mong(c1(ev))− c1(ev)] + π∗2 [mong(c1(ev))− c1(ev)].

Consequently, c1(F) + π∗1(c1(ev)) − π∗2c1(ev) is O+(KtopS)v-invariant. TheO+(KtopS)v-invariant subspace of H2(M(v)×M(v)) vanishes, since the lat-ter is the direct sum of two copies of the non-trivial irreducible O+(KtopS)v-module H2(M(v)).

6. Hyperholomorphic sheaves

We review Verbitsky’s theory of hyperholomorphic reflexive sheaves [Ve4].It plays a central role in the proof of Theorem 1.7.

6.1. Twistor deformations of pairs. LetX be an irreducible holomorphic-symplectic manifold, ω a Kahler class of X, and X → P1

ω the associatedtwistor deformation [HKLR, Hu]. Recall that associated to ω and the com-plex structure I is a hermitian metric g. Furthermore, any two among I, ω,and g, determine the third. The twistor deformation X → P1

ω comes with acanonical differentiable trivialization X ∼= X×P1

ω. The hermitian metric onX is constant with respect to this trivialization, but the complex structureIt and the associated Kahler form ωt vary as we vary t ∈ P1

ω. We denoteby Xt the differentiable manifold X endowed with the complex structure It.We denote by 0 ∈ P1

ω the point corresponding to the complex structure I onX.

Let F be a reflexive sheaf on X and (F )sing the singular locus of F . Then(F )sing has codimension ≥ 3 in X. Set (F )sm := X \ (F )sing. Let gF be ahermitian metric on the restriction of F to (F )sm. Associated to gF and theholomorphic structure ∂ of F is the Chern connection ∇ ([GH], Ch. 0 Sec.5, Lemma page 73). Recall that ∂ is the (0, 1)-part of ∇. The decompositionT ∗CX := T 1,0X⊕T 0,1X, of the complexified cotangent bundle of X, dependson the complex structure I of X.

When the sheaf F is ω-slope-stable, then there exists a unique Hermite-Einstein metric gF , whose curvature form is L2-integrable, on the restrictionof F to (F )sm [BS]. We will refer to gF as the Hermite-Einstein metric ofF and to its Chern connection as the Hermite-Einstein connection of F .Denote by ∂t, t ∈ P1

ω, the (0, 1)-part of ∇ with respect to the complexstructure It. Then ∂2

0 = 0, but ∂2t need not vanish for a general t ∈ P1

ω.

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22 EYAL MARKMAN

Definition 6.1. ([Ve4], Definition 3.15) An ω-slope-stable reflexive sheaf Fover (X,ω) is ω-stable-hyperholomorphic, if ∂2

t = 0, for all t ∈ P1ω. An ω-

slope-polystable reflexive sheaf F is ω-polystable-hyperholomorphic, if eachω-slope-stable direct summand of F is ω-stable-hyperholomorphic.

Definition 6.2. ([Ve4], Definition 2.9) A subvariety Z ofX is ω-tri-analytic,if the canonical differentiable trivialization X ∼= X × P1

ω maps Z × P1ω to a

closed analytic subvariety of X .

Verbitsky proves that the singularity locus (F )sing, of a reflexive hyper-holomorphic sheaf, is supported over a tri-analytic6 subvariety of X ([Ve4],Claim 3.16). The complex structure ∂t on F defines a locally free OXt-module over Xt \ (F )sing. We denote by Ft the reflexive sheaf on Xt cor-responding to the push-forward of the latter via the inclusion into Xt. Inparticular, F0 = F . The following is a fundamental result of Verbitsky.

Theorem 6.3. ([Ve4], Theorem 3.19) Let E be an ω-slope-stable reflexivesheaf on X. Assume that ci(E) is of Hodge-type (i, i), for i = 1, 2, andfor all complex structures parametrized by the twistor line P1

ω. Then E isω-stable hyperholomorphic.

The notion of ω-slope-stability is well defined for twisted sheaves as well.Slope-stability of a torsion-free sheaf E depends on the sheaf End(E) ofLie-algebras and its subsheaves of maximal parabolic subalgebras. Given asubsheaf F of E, the condition slopeω(F ) < slopeω(E) is equivalent to

(6.1) degω(Hom(E,F )) < 0.

The sheaf Hom(E,F ) is untwisted, for every θ-twisted subsheaf F of a θ-twisted sheaf E.

Definition 6.4. Let E be a torsion free θ-twisted sheaf and ω a Kahler classon X. We say that E is ω-slope-stable, if the inequality (6.1) holds, for everynon-zero θ-twisted proper subsheaf F of E. The sheaf E is ω-slope-semi-stable, if the analogue of (6.1), with strict inequality replaced by ≤, holdsfor every such F . The sheaf E is said to be ω-slope-polystable, if away froma locus of codimension 2, it is isomorphic to a direct sum of ω-slope-stablesheaves.

Note that if E is reflexive and ω-slope-polystable, then E is a direct sumof ω-slope-stable sheaves ([HL], Corollary 1.6.11).

Lemma 6.5. ([Ve4], section 3.5) Let F and G be two reflexive ω-polystable-hyperholomorphic sheaves of ω-slope 0. Then the following statements hold.

6Note that Z is tri-analytic if and only if Z is analytic with respect to It, for all t ∈ P1ω.

The ‘only if’ direction is clear. The ‘if’ direction follows from the following fact. Givenpoints t ∈ P1

ω and x ∈ X, we get the direct sum decomposition T(x,t)X = TtP1ω ⊕ TxX, of

the real tangent space, induced by the differentiable trivialization of X . The relevant factis that both summands are complex subspaces, even though the projection X → X is notholomorphic ([HKLR], formula (3.71)).

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 23

(1) Any global section f of F is flat with respect to the Hermite-Einsteinconnection. In particular, f is a holomorphic section with respect toall complex structures ∂t, t ∈ Pω.

(2) There exists a canonical isomorphism of vector spaces Hom(Ft, Gt) →Hom(Fs, Gs), for all s, t ∈ Pω.

(3) If Ft is endowed with an associative multiplication mt : Ft⊗Ft → Ft,or more specifically a structure of an Azumaya OXt-algebra, or a Lie-algebra structure [, ]t : Ft ⊗ Ft → Ft, then Fs is naturally endowedwith such a structure, for all s ∈ Pω.

(4) Any saturated subsheaf F ′ of F of ω-slope zero is reflexive and ω-polystable-hyperholomorphic.

(5) Let ϕ : F → G be a homomorphism. Then ker(ϕ) and the saturationof Im(ϕ) are ω-polystable-hyperholomorphic.

(6) Let F ′t be a saturated subsheaf of Ft, of ωt-slope 0, for some t ∈ Pω.Then F ′t extends to an ω-polystable-hyperholomorphic subsheaf F ′

s ofFs, for all s ∈ Pω.

(7) If Ft has a structure of an Azumaya algebra and the subsheaf F ′t in

part 6 is a maximal parabolic subalgebra, then the subsheaf F ′s is a

maximal parabolic subalgebra, for all s ∈ Pω.

Proof. Parts 1 and 2) See [Ve4], Theorem 3.27.Part 3) The sheafHom (Hom(F ∗t , Ft), Ft) is ωt-polystable-hyperholomorphic

and m (or [, ]) is a global section of this sheaf, hence a flat section with re-spect to the induced Hermite-Einstein connection, hence a holomorphic sec-tion with respect to all induced complex structures, by part 1. The axiomsof the coresponding algebraic structure are easily verified.

Part 4) The ω-slope-stable summands of F are hyperholomorphic, and F ′

is necessarily isomorphic to a direct sum of such summands.Part 5) The kernel and image of ϕ must have ω-slope zero.Part 6) The sheaf F ′t is ωt-slope-polystable-hyperholomorphic, by part 4.

The sheaf Hom(F ′t , Ft)⊗C F∗s is ωs-slope-polystable-hyperholomorphic, it is

canonically isomorphic to Hom(F ′s, Fs)⊗C F∗s , by part 2, and the evaluation

homomorphism from the latter into Fs has a hyperholomorphic image, bypart 5.

Part 7) Assume that F ′t is a saturated ωt-slope 0 Lie subalgebra. Thenits extension, in part 6, consists if Lie subalgebras F ′

s, for all s ∈ Pω, bypart 3. Let F ′′s be the Killing-perp of L′s. Then L′s is a sheaf of maximalparabolic subalgebra over (F )sm, if and only if the following two conditionshold: a) F ′′s is a subsheaf of F ′s, and b) the homomorphism F ′′s ⊗ F ′′s → Fs,given by a⊗ b 7→ ab, vanishes. Now F ′′s is the kernel of the homomorphismF ′s → (F ′s)

∗, induced by the Killing form. Hence F ′′s is saturated of ωs-slope0, and so ωs-polystable-hyperholomorphic, by part 4.

Assume now that F ′t is a saturated ωt-slope 0 maximal parabolic subal-gebra of Ft. We get a flag F ′′ ⊂ F ′ ⊂ F of ω-polystable-hyperholomorphicreflexive sheaves of ω-slope 0. Furtheremore, each of the conditions a) and b)

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24 EYAL MARKMAN

above is expressed in terms of the vanishing of a natural homomorphism be-tween ω-polystable-hyperholomorphic sheaves of slope zero. Hence, if theyboth hold for t, then they both hold for all s ∈ Pω, by part 1.

6.2. Projectively ω-stable-hyperholomorphic sheaves. The theory ofω-slope-stable θ-twisted reflexive sheaves is incomplete, as the Ulenbeck-Yautheorem, about the existence of Hermite-Einstein metric, is not available7

in this generality. Nevertheless, such a metric does exist, when we startwith an untwisted ω-slope-stable reflexive sheaf F of rank r and attempt todeform it as a twisted sheaf.

Definition 6.6. Let F be a reflexive ω-slope-stable (untwisted) sheaf of pos-itive rank over (X,ω). Denote by ∂t the (0, 1)-part of its Hermite-Einsteinconnection. We say that F is projectively ω-stable-hyperholomorphic, if theequation

(6.2) ∂2t = λt ⊗ id

holds, for some (0, 2)-form λt on Xt, for all t ∈ P1ω. A reflexive ω-slope-

polystable sheaf is projectively ω-polystable-hyperholomorphic, if it is a directsum of projectively ω-stable-hyperholomorphic sheaves.

Remark 6.7. The two form λt above is ∂-closed and(

r√−1

)λt, with r =

rank(F ), represents the projection of c1(F ) to H0,2(Xt), since ∂2t is the

(0, 2)-part of the curvature form of ∇ with respect to the complex structureIt of Xt.

Let F be a projectively ω-stable-hyperholomorphic reflexive sheaf of rankr > 0. Let θ ∈ H2(X,µr) be the class exp

(−2π

√−1c1(F )/r

), as in equation

(2.5). Denote by θt the image of θ in H2(Xt,O∗Xt

). Similarly, let θ be the

image of θ in H2(X ,O∗X ) via the composite homomorphism

H2(X,µr) → H2(X , µr) → H2(X ,O∗X ),

where the left homomorphism is the pull-back via the projection X → X,associated to the differentiable trivialization of the twistor deformation.

Construction 6.8. The sheaf F corresponds to a θ-twisted family F ofsheaves over the twistor space X . Following is the construction of such afamily. Denote by Z ⊂ X the image of P1

ω×(F )sing via the differentiable triv-ialization of the twistor deformation. Then Z is an analytic subvariety of X ,since (F )sing is ω-tri-analytic ([Ve4], Claim 3.16, applied to End(F ), whichis ω-polystable-hyperholomorphic by Lemma 6.9 below). Set U := X \ Zand let Ut be the fiber of U over t ∈ P1

ω. Let π1 : U → (F )sm be given by the

differentiable trivialization of U . The curvature Θ, of the pull-back π∗1∇ of

the connection ∇ on (F )sm, is the pullback of the curvature of ∇. Thus Θ

7See however Proposition 7.8 below, which implies the existence of a Hermite-Einsteinmetric on End(E), in the special case of a θ-twisted reflexive sheaf E, such that the orderr of the class θ is equal to rank(E).

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 25

is a section of π∗1

[End(F )⊗

2∧ T ∗C(F )sm

]. Let I be the complex structure of

X . The subspace π∗1(T∗RX) is I invariant (even though the map π1 : X → X

is not holomorphic ([HKLR], formula (3.71)). Fix t ∈ P1ω. We see that

the projection of π∗1(2∧ T ∗CX)|Xt

to T 0,2(Xt) factors through the projection

to the (0, 2) summand of π∗1(2∧ T ∗CX)|Xt

, followed by an isomorphism from

the latter onto T 0,2(Xt). So the (0, 2)-part of Θ is of the form λ ⊗ id, forsome (0, 2)-form λ on U , if and only if the (0, 2)-part of the restriction Θt

of the curvature Θ to Ut is of the form (6.2), for all t. The latter condi-tion holds, by definition. Hence, the pullback π∗1PF is endowed with thestructure of a holomorphic projective bundle over U , and so defines a classin H1

an(U,PGL(r)), which may be lifted to a locally free θ-twisted sheaf Fover U . The push-forward of F , via the inclusion U → X , is a reflexive θ-twisted sheaf over X , since Z is a closed analytic subvariety of codimension≥ 3. Denote this push-forward by by F as well, and let Ft be its restrictionto Xt, t ∈ P1

ω.

Lemma 6.9. Let F be a reflexive ω-slope-stable sheaf of positive rank over(X,ω). Then F is projectively ω-stable-hyperholomorphic, if and only ifEnd(F ) is ω-polystable-hyperholomorphic.

Proof. Assume that F is projectively ω-stable-hyperholomorphic. Let ∇′

be the induced Hermite-Einstein connection on End(F ) and ∂′t its (0, 1)-part with respect to It. The condition (6.2) of Definition 6.6 implies that(∂′t)

2 = 0. We know that End(F ) is ω-slope-polystable. Let E be an ω-slope-stable direct summand of End(F ) and E⊥ its orthogonal complement withrespect to the Hermite-Einstein metric gEnd(F ) induced by that of F . Then

E and E⊥ are ∇′-invariant. Denote by ∇E the restriction of ∇′ to E, andlet ∂E,t be the (0, 1)-part of ∇E with respect to It. The vanishing of (∂′t)

2

implies the vanishing of (∂E,t)2. Hence, E is ω-stable-hyperholomorphic.

Assume next that End(F ) is ω-polystable-hyperholomorphic with respectto some Hermite-Einstein metric g′. Denote by ∇′ its associated Hermite-Einstein connection, and let ∂′t be the (0, 1)-part of ∇′ with respect to It.Then (∂′t)

2 = 0, by assumption. Now the Hermite-Einstein connection ofan ω-slope-polystable sheaf E is unique. Hence, ∇′ is also the Hermite-Einstein connection on End(F ) induced by that of F . Thus, condition (6.2)of Definition 6.6 is satisfied.

Lemma 6.10. Let F be a reflexive projectively ω-polystable-hyperholomorphicsheaf. If the θt-twisted sheaf Ft over Xt, in Construction 6.8, is ωt-slope-stable, for some t, then it is ωt-slope-stable for every t.

Proof. Assume that F ′t is an ωt-slope-stable subsheaf of Ft of rank r′, suchthatHom(Ft, F

′t ) has ωt-slope 0, for some t ∈ Pω. ThenHom(Ft, F

′t ) extends

as an ω-polystable-hyperholomorphic subsheaf P of End(F ), of ω-slope 0,by Lemma 6.5 part 6. The subsheaf Pt := Hom(Ft, F

′t ) is a sheaf of maximal

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26 EYAL MARKMAN

parabolic subalgebras. Its extension is also a subsheaf of maximal parabolicsubalgebras, by Lemma 6.5 part 7. In particular, if 0 < r ′ < rank(F ), thenFt is not ωt-slope-stable, for any t.

Theorem 6.11. ([Ve4], Corollary 3.24) Let F be an ω-slope-polystable re-flexive sheaf on (X,ω), of ω-slope 0, and It an induced complex structuresuch that It 6∈ I,−I. Then

(6.3)

Xκ2(F )ω2n−2 ≥

∣∣∣∣∫

Xκ2(F )ω2n−2

t

∣∣∣∣ ,

and equality holds, if and only if each stable direct summand F ′ of F isω-stable-hyperholomorphic. Furthermore, equality holds in (6.3) if κ2(F ) isof Hodge-type (1, 1), with respect to It, for all t ∈ Pω.

Proof. When F is ω-slope-stable this is precisely ([Ve4], Corollary 3.24). Ithank Misha Verbitsky for pointing out this statement and the fact thatthe statement holds also when F is ω-slope-polystable. Assume that F =

⊕Ni=1Fi, where Fi is ω-slope-stable. Then κ2(F ) =

∑Ni=1 κ2(Fi). We get

Xκ2(F )ω2n−2 =

N∑

i=1

Xκ2(Fi)ω

2n−2 ≥N∑

i=1

∣∣∣∣∫

Xκ2(Fi)ω

2n−2t

∣∣∣∣ ≥∣∣∣∣∫

Xκ2(F )ω2n−2

t

∣∣∣∣ ,

where the first inequality is by [Ve4], Corollary 3.24, and the second by thetriangle inequality. Clearly, equality holds above, if and only if it holds foreach Fi.

If κ2(F ) is of Hodge-type (1, 1) with respect to It, for all t ∈ Pω, thenequality holds in (6.3), by Claim 3.21 in [Ve4].

The following generalization of Theorem 6.3 was explained to me by MishaVerbitsky.

Corollary 6.12. Let E be an ω-slope-stable (untwisted) reflexive sheaf andassume that κ2(E) is Mon(X)-invariant. Then End(E) is ω-polystable-hyperholomorphic and E is projectively ω-stable-hyperholomorphic.

Proof. It would suffice to prove that End(E) is ω-polystable-hyperholomorphic,since Lemma 6.9 would then imply that E is projectively ω-stable-hyperholomorphic.Apply Theorem 6.11 with F := End(E). Equality holds in (6.3), since κ2(E)is of Hodge-type (1, 1), for all t ∈ Pω, by Lemma 1.2.

6.3. Deformation of pairs along twistor paths. A marking of an irre-ducible holomorphic symplectic manifoldX is an isometry φ : H 2(X,Z) → Λwith a fixed lattice Λ. Let MΛ be the moduli space of isomorphism classesof marked irreducible holomorphic symplectic manifolds [Hu]. A twistor-path in MΛ is a sequence of twistor lines, in which each consecutive pair hasnon-trivial intersection in MΛ. If the intersection, of each consecutive pair,contains a point corresponding to a manifold with trivial Picard group, wecall the twistor-path generic.

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 27

Theorem 6.13. ([Ve3], Theorems 3.2 and 5.2.e) Let (Xi, φi), i = 1, 2,be two marked irreducible holomorphic symplectic manifolds, in the sameconnected component of MΛ. Then there exists a generic twistor-path inMΛ connecting (X1, φ1) with (X2, φ2).

We will need the following evident Lemma.

Lemma 6.14. Let X be a compact Kahler manifold with a trivial Picardgroup Pic(X) = OX, ω and ω′ two Kahler classes on X, and E a torsionfree coherent OX -module of rank r. Then E is ω-slope-stable, if and only ifE does not admit any subsheaf of rank r ′, for 0 < r′ < r. In particular, Eis ω-slope-stable, if and only if E is ω ′-slope-stable.

The differential geometric Definition 6.6, of projectively ω-stable-hyperholomorphicsheaves, is convinient along a single twistor line, as the Hermite-Einsteinconnection is constant along this line. When we attempt to deform a reflex-ive sheaf along a twistor path, the language of Azumaya algebras is moreconvenient.

Definition 6.15. (1) A reflexive sheaf of Azumaya8 OX -algebras of rankr over a Kahler manifold X is a sheaf E of reflexive coherent OX -modules, with a global section 1E , and an associative multiplicationm : E ⊗E → E with identity 1E , admitting an open covering Uαof X, and an isomorphism E|Uα

∼= End(Fα) of unital associativealgebras, for some reflexive sheaf Fα of rank r, over each Uα.

(2) Let ω be a Kahler class on X. A reflexive sheaf E of Azumaya OX -algebras is ω-slope-stable, if E is ω-slope-polystable as a sheaf ofOX -modules, and E does not admit a slope zero subsheaf of maximalparabolic subalgebras.

From now on the term a sheaf of Azumaya algebras will mean a reflexivesheaf of Azumaya algebras.

Remark 6.16. (1) Fix a closed analytic subset Z ⊂ X, of codimension≥ 3, and set U := X \Z. A reflexive Azumaya OX -algebra is deter-mined by its restriction to U . Hence, the set of isomorphism classesof reflexive Azumaya OX -algebras E of rank r, which are locallyfree over U , is in natural bijection with H1

an(U,PGL(r)) [Mi]. Sim-ilarly, H1

an(U,PGL(r)) parametrizes equivalence classes of coherentreflexive twisted OX -modules, which are locally free over U .

(2) Let F be a reflexive (untwisted) sheaf of positive rank on X and ω aKahler class. Then F is ω-slope-stable, if and only if End(F ) is an ω-slope-stable Azumaya OX -algebra. Furthermore, F is projectivelyω-stable-hyperholomorphic, if and only if End(F ) is ω-polystable-hyperholomorphic and an ω-slope-stable Azumaya OX -algebra, byLemma 6.9 and Lemma 6.5 part 3.

8Caution: The standard definition of a sheaf of Azumaya OX -algebras assumes that Eis a locally free OX -module, while we assume only that it is reflexive.

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28 EYAL MARKMAN

A parametrized twistor-path γ : C → MΛ consists of a connected reducednodal curve C, of arithmetic genus 0, with an ordering of the irreduciblecomponents, so that two consecutive components meet at a node, and amorphism γ from C to MΛ, mapping the i-th component of C isomorphicallyonto a twistor line. If γ maps each node to a point with a trivial Picardgroup, we call γ a generic parametrized twistor-path. Let γ : C → MΛ be aparametrized twistor-path, X → C the natural twistor deformation, 0 ∈ Ca point of the first component of C, and X0 the fiber of X over 0. Let E bea reflexive twisted sheaf on X0.

Definition 6.17. We say that E can be deformed along γ, if there exists areflexive twisted coherent sheaf over X , flat over C, which restriction to X0

is isomorphic to E. Equivalently, there exists a sheaf of reflexive AzumayaOX -algebras, flat over C, which restriction to X0 is isomorphic to End(E).

Let X be an irreducible holomorphic symplectic manifold and γ : C →MΛ a generic parametrized twistor path, with X0 = X. Let ω0 be a Kahlerclass on X0, such that Pω0 is the first twistor line. Let ωt−i

, 1 ≤ i ≤ N , be a

Kahler class on the i-th node Xti , such that Pωt−

i

is the i-th twistor line, and

ωt+i, 1 ≤ i ≤ N −1, a Kahler class on Xti , such that Pω

t+i

is the i+1 twistor

line. Note that ω0 determines ωt−1, and ωt+i

determines ωt−i+1. At a node

ti ∈ C, the group Pic(Xti) is trivial. Slope-stability is then independent ofthe Kahler class, by Lemma 6.14. We will abuse notation and say that asheaf on Xti is ωti-slope-stable, if it is slope-stable with respect to some,hence any Kahler class.

Proposition 6.18. Let F be an ω0-slope-stable (untwisted) reflexive sheaf.Assume that κ2(F ) is Mon(X)-invariant. Then F deforms along γ, in thesense of definition 6.17.

Proof. The following argument is similar to the proof of Theorem 10.8 in[Ve4]. The proof is by induction on the number N of twistor lines in C.F is projectively ω0-stable-hyperholomorphic and End(F ) is ω0-polystable-hyperholomorphic, by Corollary 6.12. Consequently, F deforms along thefirst twistor line, by Construction 6.8.

Set E0 := End(F ). Assume that E0 deforms, as an ωt-slope-stable sheafof Azumaya algebras, along the first i twistor lines, and i < N . Then Eti isslope-polystable with respect to ωt−i

and hence also with respect to ωt+i, by

Lemma 6.14. Hence, Eti is ωt+ipolystable-hyperholomorphic, by Theorem

6.11. The structure of an Azumaya algebra deforms along the i+ 1 twistorline, by Lemma 6.5 part 3.

The ωt-slope-stability of Et is proven by induction as well. The stabilityfor t in the first twistor line follows from Lemma 6.10. Stability of Et1 forωt+1

follows from that for ωt−1and Lemma 6.14. The proof of the induction

step is similar.

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 29

Remark 6.19. With the exception of Theorem 6.13, Verbitsky proves the re-sults mentioned above for hyperkahler varieties, without assuming the con-dition h2,0 = 1 (the irreducibility condition). In particular, all the definitionsand results in this section hold for X ×X, where X is an irreducible holo-morphic symplectic manifold, provided the twistor deformations of X ×Xwe consider are only fiber-square X ×P1

ωX of twistor deformations of X,

associated to a Kahler class ω on X.

7. Stable hyperholomorphic sheaves of rank 2n− 2 on all

manifolds of K3[n]-type

Definition 7.1. Let X be a complex manifold and E a torsion free θ-twistedcoherent sheaf on X. E is said to be very twisted, if the rank of E is equalto the order of the class of θ in H2

an(X,O∗X).

A very-twisted sheaf does not have any non-trivial subsheaves of lowerrank, so it is trivially slope-stable. In section 7.1 we construct a very twistedversion of the sheaf E in Theorem 1.6. In section 7.2 we show that if F isvery twisted, then the untwisted sheaf End(F ) is ω-slope-polystable with re-spect to every Kahler class ω. Consequently, there exists a Hermite-Einsteinmetric on End(F ). In section 7.3 we carefully degenerate the K3 surface, themoduli space M, and the sheaf E to those in Theorem 1.6, so that the sheafE is untwisted, yet still slope-stable. The stability Theorem 1.6 is proven insection 7.3. In section 7.4 we prove the deformability Theorem 1.7.

7.1. A very twisted Ext1π13(π∗12E , π∗23E). We construct a very twisted re-

flexive sheaf Ext1π13(π∗12E , π∗23E), over the self-product of a suitable choice

of a moduli space M (Theorem 7.4). Recall that an untwisted analogue ofExt1π13

(π∗12E , π∗23E) appears in Theorem 1.6.Let MH(v) be a smooth and projective moduli space of H-stable sheaves

on a projective K3 surface S. Set r := (v, v). Assume, that (v, v) ≥ 2. Letµr be the group of r-th roots of unity.

Lemma 7.2. (1) There exists a unique r(v⊥) coset w in v⊥ of classesw, such that w − v belongs to rKtopS.

(2) Define a class in H2(MH(v), µr), by

(7.1) θ := exp(−2π√−1w/r),

where we identify v⊥ with H2(MH(v),Z) via Mukai’s isometry (3.3).

Then the pair θ, θ−1 is monodromy invariant.

Proof. 1) Uniqueness is clear. When v is the class of the ideal sheaf of alength n subscheme, with Mukai vector (1, 0, 1−n), choose w = (1, 0, n−1).The existence of such a class follows, for an arbitrary primitive class v with(v, v) = 2n−2, since any two such classes belong to the same O(KtopS)-orbit.

2) The class θ is determined by the primitive isometric lattice embeddingH2(MH(v),Z) ∼= v⊥ ⊂ KtopS and the choice of a generator v of the lineorthogonal to the image of H2(MH(v),Z). Any monodromy operator of

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30 EYAL MARKMAN

H2(MH(v),Z) can be extended to an isometry of KtopS, which necessarilymaps v to v or −v, by Theorem 5.2.

Denote by θ the image of θ in H2(MH(v),O∗), via the sheaf inclusionι : µr → O∗. Let β : B →MH(v)×MH(v) be the blow-up of the diagonaland PV the projective bundle over B associated to the twisted locally freesheaf (4.4).

Lemma 7.3. (1) The class θ(PV ) ∈ H2(B,µr), defined in (2.3), satis-fies

(7.2) θ(PV ) = β∗((π∗1 θ)

−1π∗2 θ).

(2) The order of the class θ in H2an(MH(v),O∗) is given by:

gcd(v, x) : x ∈ KtopS and c1(x) is of type (1, 1).Proof. 1) Assume first, that v is the class of the ideal sheaf of a length nsubscheme. Then V is a vector bundle, which restricts to the exceptionaldivisor D as a vector bundle with trivial determinant (Proposition 4.5).Thus, c1(V ) = β∗c1(F) = β∗[−π∗1c1(ev) + π∗2c1(ev)], by Lemma 4.3. When

E is the universal ideal sheaf over S × S [n], then ev = ew, where v hasMukai vector (1, 0, 1 − n), that of w is (1, 0, n − 1), and that of w − v is(2n − 2)(0, 0, 1), by Lemma 5.9 in [Ma4]. The equality (7.2) follows fromequation (2.5).

The general case of equation (7.2) follows, by deformation of the classeson both sides, via a deformation to the Hilbert scheme case, as in Lemma4.9.

2) Set M := MH(v). Consider the short exact sequence

(7.3) 0 → µrι→ O∗ (•)r

→ O∗ → 0.

The connecting homomorphism H1(M,O∗) → H2(M, µr) sends the class ofa line bundle L to exp(2π

√−1c1(L)/r). Let d be a positive integer dividing

(v, v). Then ι(dθ) = 1, if and only if dθ = exp(−2π√−1`/r), for some

` ∈ H1,1(M,Z). Identify H2(M,Z) with v⊥, via Mukai’s Hodge-isometry(3.3). Set ¯ := `+ rv⊥. It suffices to prove that the following are equivalent.

(1) There exists ` ∈ v⊥, with c1(`) of type (1, 1), such that ¯ = dw inv⊥/rv⊥, where w is the coset in Lemma 7.2.

(2) d = (v, x), for some x ∈ KtopS, with c1(x) of type (1, 1).

1⇒ 2: The (1, 1) class x := dv−`r is integral, by the assumption on `, and

satisfies (x, v) = d(v,v)r = d.

2⇒1: Set ` := dv − (v, v)x. Then (`, v) = 0 and `− dv = −rx belongs torKtopS. Thus, ¯= dw in v⊥/rv⊥, by Lemma 7.2 part 1.

Set r := 2n − 2, n ≥ 2. Let S be a projective K3 surface with a cyclicPicard group generated by an ample line-bundle H with c1(H)2 = 2r2 + r.Let v ∈ KtopS be the rank r class with c1(v) = c1(H), and χ(v) = 2r.

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 31

Its Mukai vector ch(v)√tdS is (r,H, r). Then (v, v) = r and (v, x) ≡ 0,

(modulo r), for every class x ∈ KtopS with c1(x) of type (1, 1). The modulispace MH(v) is smooth and projective (see section 3). Let E be the rankr (π∗1 [θ]

−1π∗2[θ])-twisted sheaf Ext1π13(π∗12E , π∗23E), over MH(v)×MH(v). E

is reflexive, by Proposition 4.5.

Theorem 7.4. The (π∗1 [θ]−1π∗2 [θ])-twisted sheaf E is ω-slope-stable (Defini-

tion 6.4) and the untwisted sheaf End(E) is ω-polystable-hyperholomorphic(Definition 6.1), with respect to every Kahler class ω on MH(v)×MH(v).

Proof. The class θ has order r, by lemma 7.3. It follows that E does nothave any non-zero twisted subsheaves of rank < r (see Remark 2.2). Theclass κ2(E) is monodromy-invariant, by Proposition 4.2. The polystabilityof End(E) is proven in the next section (Proposition 7.8). Consequently,End(E) is ω-polystable-hyperholomorphic, by Theorem 6.11.

7.2. Polystability of End(E) for a very twisted sheaf E.

Definition 7.5. Let X be a complex manifold and E a torsion free θ-twistedcoherent sheaf on X. A subsheaf A ⊂ End(E) is said to be degenerate, ifthe generic rank of every local section of A is lower than that of E.

Lemma 7.6. Let (X,ω) be a compact Kahler manifold, θ ∈ H 2an(X,O∗)

a class of order r > 0, and E a reflexive, rank r, θ-twisted sheaf. ThenEnd(E) does not have any non-zero degenerate subsheaf.

Proof. The proof is by contradiction. Let A′ be a non-zero degenerate sub-sheaf of End(E) and A its saturation in End(E). Then A is a reflexivedegenerate subsheaf of End(E). Let U ⊂ X be the open subset, where A islocally free, and set Z := X \ U . Then the codimension of Z in X is ≥ 3.

We have the commutative diagram of exponential sequences

H2(X,Z) → H2an(X,O) → H2

an(X,O∗) → H3(X,Z)∼= ↓ ρ1 ↓ ρ2 ↓ ↓ ∼=

H2(U,Z) → H2an(U,O) → H2

an(U,O∗) → H3(U,Z)

Set n := dimC(X). The left and right vertical homomorphisms are isomor-phisms, by the codimension of Z, Lefschetz DualityH i(U,Z) ∼= H2n−i(X,Z,Z),and the vanishing of H2n−i(Z,Z), for i < 6. The homomorphism ρ1 is in-jective, since the codimension of Z is ≥ 3 [Sch]9. It follows that the homo-morphism ρ2 is injective as well, by a diagram chase. We conclude, that theimage θ′ of θ in H2

an(U,O∗) has order r.Set Y := P[A|U ] and let π : Y → U be the natural morphism. The pull-

back π∗ : H2(U,O∗) → H2(Y,O∗) is injective, by a similar diagram chase,since the homomorphism H3(U,Z) → H3(Y,Z) is injective, and both

H2(U,Z)/c1[Pic(U)] → H2(Y,Z)/c1[Pic(Y )]

9If X is projective, and we consider the Zariski topology instead, the injectivity of ρ1

follows from the vanishing of the cohomology H iZ(X,OX), with support along Z, for i ≤ 2

[H].

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32 EYAL MARKMAN

and H2an(U,O) → H2

an(Y,O) are isomorphisms. Hence, the pull-back θ ′′ :=π∗(θ′) to Y has order r. Consequently, the θ ′′-twisted sheaf π∗E does nothave any non-trivial proper θ′′-twisted subsheaf. Let τ ⊂ π∗A be the tauto-logical line subbundle. The image of the composition

τ ⊗ π∗E → π∗(A⊗E) → π∗([End(E)] ⊗E) → π∗E

is a non-trivial θ′′-twisted proper subsheaf, since τ is a degenerate subsheafof π∗End(E). A contradiction.

Lemma 7.7. Let (X,ω) be a compact Kahler manifold, θ ∈ H 2an(X,O∗)

a class of order r > 0, and E a reflexive, rank r, θ-twisted sheaf. ThenEnd(E) is an ω-slope-semistable sheaf.

Proof. The proof is by contradiction. Assume that End(E) is not semi-stable, and let F be an ω-slope-stable destabilizing subsheaf of End(E) ofmaximal slope. Then (F ⊗ F )/tor is ω-slope-polystable of slope 2µ(F ).The image of F ⊗ F in End(E) must be zero, since otherwise the slope ofthe image is ≥ 2µ(F ), constradicting the assumption that the slope of Fis maximal. We conclude that F is a degenerate subsheaf. We obtain acontradiction, by Lemma 7.6.

Proposition 7.8. Let (X,ω) be a compact Kahler manifold, θ ∈ H 2an(X,O∗)

a class of order r > 0, and E a reflexive, rank r, θ-twisted sheaf. ThenEnd(E) is ω-slope polystable.

Proof. End(E) is ω-slope semistable, by Lemma 7.7. Let F ⊂ End(E) be themaximal polystable subsheaf ([HL], Lemma 1.5.5). Then F is reflexive, andis hence locally free away from a closed analytic subvariety Z of codimension≥ 3 in X. Let F⊥ ⊂ End(E) be the subsheaf orthogonal to F with respectto the trace-pairing on End(E). Set A := F ∩ F⊥.

We show first that A must vanish. Note first that the multiplicationhomomorphismm : End(E)⊗End(E) → End(E) maps F⊗F onto a subsheafof slope 0. We conclude that the image is slope-polystable, and is hencecontained in F . Consequently, F is a sheaf of unital associative subalgebrasof End(E). Let a be a local section of A. Then an is a section of F , forall n ≥ 0. Thus, tr(ak) = tr(ak−1a) = 0, for all k > 0. It follows that ais nilpotent. Hence, the sheaf A is a degenerate subsheaf of End(E). Weconclude that A = 0, by Lemma 7.6.

We may assume that A vanishes. Thus the homomorphism

φ : F ⊕ F⊥ −→ End(E)

is injective. Its degeneracy divisor, in X \ Z, must be trivial, since F andF⊥ must both have ω-slope 0. We conclude that φ is an isomophism, sinceboth its domain and target are reflexive sheaves. If F ⊥ does not vanish,then it contains a stable subsheaf of ω-slope 0, contradicting the maximalityof F . Thus F = End(E).

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 33

7.3. Stability of an untwisted Ext1π13(π∗12E , π∗23E). Let π : M → C be

a non-isotrivial family of irreducible holomorphic-symplectic manifolds overa (connected) Riemann surface C. Fix a positive integer m. Note thatH2(M, µm) is isomorphic to H0(C,R2

π∗µm), since π has simply connected

fibers and H2(C, µm) is trivial. Assume given a global non-zero section θ ofH0(C,R2

π∗µm). Denote by Cθ the subset of C given by

Cθ := t ∈ C : θ maps to the trivial class in H2(Mt,O∗Mt

).Proposition 7.9. [Ogu] Cθ is a dense subset in the classical topology of C.Furtheremore, either Cθ = C, or Cθ is enumerable.

Proof. The class θt ∈ H2(Mt, µm) maps to the trivial class in H2(Mt,O∗Mt

),

if and only if θt belongs to the image of Pic(Mt) under the homomorphism

θ : Pic(Mt) → H2(Mt, µm), given by θ(L) = exp(2π√−1c1(L)/m). Indeed,

we have seen that θ is the connecting homomorphism of the short exactsequence (7.3).

We may assume that the local system R2π∗µm is trivial. Given a global sec-

tion κ of the local system, denote by κt the corresponding class inH2(Mt, µm).For each κ ∈ H0(C,R2

π∗µm), let Cκ be the subset of the curve C, consistingof points t ∈ C, such that κt is the image of a non-trivial class in Pic(Mt).Theorem 1.1 in [Ogu] shows that the finite union D := ∪κ∈H0(C,R2

π∗µm)Cκ is

either equal to C or dense and enumerable. Proposition 7.9 states that eachCκ has this property and is thus a slight generalization of Theorem 1.1 in[Ogu]. Oguiso’s proof is easily seen to establish this generalization.

Next we prove slope-stability of untwisted reflexive sheaves in a familycontaining a very-twisted reflexive sheaf.

Set r := 2n − 2, n ≥ 2. Let π : S → C be a non-isotrivial smoothand projective family of K3 surfaces, admitting a π-ample line-bundle Hon S of fiberwise degree 2r2 + r. Assume, further, that there exists a point0 ∈ C, such that Pic(S0) is generated by H0. There exists a projectivemorphism p : M→ C, whose fiber Mt, t ∈ C, is isomorphic to the modulispace MHt(r,Ht, r) of Ht-semistable sheaves on St of class (r,Ht, r), by

[Sim]. The fiber M0 of p is smooth and connected of K3[n]-type, by ourassumption on Pic(S0). Hence, we may assume that p is smooth and all itsfibers are of K3[n]-type, possibly after restricting to a Zariski dense opensubset of C. Let OM(1) be a p-ample line-bundle on M and denote byOMt(1) its restriction to Mt. Note that c1 (OM(1)) maps to a section ofR2

p∗Z, which is in the image via (3.3) of the trivial local system of Mukaivectors spanned by the Mukai vectors (2r + 1, c1(H), 0) and (1, 0,−1), byour assumption on Pic(S0).

There exists a θ-twisted universal sheaf E over S ×C M, for some classθ ∈ H2

an(M,O∗M). Let Et be the restriction of E to a twisted universal sheaf

over St ×Mt and denote by θt ∈ H2(Mt,O∗Mt

) its Brauer class. Set

Et := Ext1π13(π∗12Et, π

∗23Et).

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34 EYAL MARKMAN

Et is a rank 2n − 2 reflexive sheaf on Mt ×Mt (Proposition 4.5). Givena point m ∈ Mt, denote by Et,m the restriction of Et to Mt × m. LetZDµs ⊂ M be the subset consisting points m, such that Et,m is OMt(1)-slope-stable. Let pi be the projection from Mt ×Mt onto the i-th factor.Set OMt×Mt(1) := p∗1OMt(1)⊗ p∗2OMt(1). Let Σ ⊂ C be the subset given by

Σ := t ∈ C : Et is OMt×Mt(1)-slope-stable and θt is trivial.Theorem 7.10. The subset Σ is a dense countable subset of C. The in-tersection of Σ with the image of ZDµs is a dense countable subset of C aswell.

Proof. Let vt be the Mukai vector (r,Ht, r). Then (vt, vt) = r. The classvt ∈ KtopSt is the restriction of a class in KalgS, since H is a line bundleon the family S. Hence, the order of the class θ divides r, by Lemma 7.3.Consequently, the class θ lifts to a class θ ∈ H2(M, µr). Let Cθ ⊂ C be thesubset consisting of points t ∈ C, such that θt is trivial. We conclude thatCθ is countable and dense, by Proposition 7.9. Let C s be the subset of C,consisting of points t where Et is OMt×Mt(1)-slope-stable. Cs contains thepoint 0, by Theorem 7.4. Cs is a Zariski open subset of C, by [Li], Corollary2.3.2.11. Σ is the intersection Cs∩Cθ, which is a dense and countable subsetof C. Lieblich’s result also establishes that ZDµs is a Zariski open subsetof M. ZDµs contains the whole fiber M0, by Theorem 7.4. Hence, ZDµs isa Zariski dense open subset of M. In particular, the image of ZDµs in Ccontains a Zariski dense open subset of C.

7.4. Proof of the deformability Theorem 1.7. Let n be an integer ≥ 2.Set r := 2n− 2. Let S be a K3 surface admitting an ample line bundle Hof degree 2r2 + r. Set v := (r,H, r) and M := MH(v). Assume that Mis smooth, there exists an untwisted universal sheaf E over S ×M, and thereflexive sheaf

(7.4) E := Ext1π13(π∗12E , π∗23E)

over M ×M is OM×M(1)-slope-stable, where OM×M(1) := p∗1OM(1) ⊗p∗2OM(1) and OM(1) corresponds to a Mukai vector

a(1, 0,−1) + b(2r + 1,H, 0) ∈ v⊥

via the Mukai isomorphism (3.3), for some integers a and b. The existenceof such a pair (S,H) is proven in Theorem 7.10.

Theorem 7.11. Let E be the sheaf given in equation (7.4) and X an irre-

ducible holomorphic symplectic manifold of K3[n]-type. Then there exists aparametrized twistor path connecting MH(v) and X, along which E can bedeformed (in the sense of Definition 6.17).

Proof. The proof is by induction on the length of the twistor path. The classκ2(E) is Mon(MH(v))-invariant, by Proposition 4.2. Let ω ′ be the Kahlerclass of OMH(v)×MH (v)(1). Then E was chosen to be ω′-slope-stable. Hence,

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THE BEAUVILLE-BOGOMOLOV CLASS AS A CHARACTERISTIC CLASS 35

E is ω-slope-stable, for ω in an open neighborhood of ω ′ in the Kahler cone[HL], Ch. 4C. The sheaf E is projectively ω-stable-hyperholomorphic, byCorollary 6.12 and Remark 6.19. We choose ω, so that the hyperplane ω⊥

intersects trivially the lattice H1,1(MH(v),Z). Then Pic(Xt1) is trivial, fora generic t1 ∈ P1

ω. There exists a generic parametrized twistor path fromXt1 to X, by Theorem 6.13. We get a generic parametrized twistor pathfrom MH(v) to X. We conclude that E deforms along the twistor path γ,by Proposition 6.18.

8. Proof of Lemma 1.5

It suffices to prove the Lemma for every smooth and compact modulispace M := MH(v), for all (v, v) ≥ 2. Let

u : KtopS −→ H∗(M,Q)

u(x) := ch(ex) · exp

(−c1(ev)(v, v)

),

where ex is given in (3.1), and u2i : KtopS → H2i(M,Q) the composition ofu with the projection on the degree 2i-summand. Note that u(v) = κ(ev),u0(x) = (v, x),

u2(x) = c1(ex)− (v, x)

(v, v)c1(ev),

u2(v) = 0, and u2 restricts to v⊥ and the standard Mukai isomorphism

(u2)|v⊥

: v⊥∼=−→ H2(M,Z).

Moreover, u is O+(KtopS)v equivariant, by Theorem 5.1 and equation (5.5).Let q ∈ Sym2KtopS be the Mukai pairing. The following equality is a

special case of equation (4.8) in [Ma2]:

(8.1) c2(TM) = (u2 ∪ u2 − 2u4 ∪ u0)(q),

where (u2 ∪ u2 − 2u4 ∪ u0) is the homomorphism from KtopS ⊗ KtopS toH4(M,Q).

The orthogonal decomposition (KtopS)Q = Qv + (v⊥)Q induces the de-

composition q = v⊗v(v,v) + q−1, where we identified v⊥ with H2(M,Z), via u2.

Equation (1.2) follows from (8.1) and the following equations

(u4 ∪ u0)(q−1) = 0,(8.2)

(u2 ∪ u2)(q−1) = q−1,(8.3)

(u2 ∪ u2)(v ⊗ v) = 0,(8.4)

(u4 ∪ u0)

(v ⊗ v

(v, v)

)= u4(v) = κ2(X).(8.5)

Proof of Equation (8.2): u4 ∪ u0 is O+(KtopS)v-equivariant, and thus

sends the O+(KtopS)v-invariant class q−1 in (v⊥ ⊗ v⊥)Q to an O+(KtopS)v-

invariant class in u4(v⊥)Q. But the image u4(v

⊥) either vanishes, or is an

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36 EYAL MARKMAN

irreducible O(KtopS)v-module isomorphic to v⊥. Thus, any invariant class

in u4(v⊥) vanishes.

Equations (8.3) and (8.5) are clear and Equation (8.4) follows from thevanishing of u2(v), observed above.

It remains to calculate the dimension of spanq−1, c2(TX), κ2(X). The

homomorphism Sym2H2(S[n],Q) → H4(S[n],Q) is known to be injective[Ve1]. When n = 2, the homomorphism is surjective, by Gottsche’s formulafor the Betti numbers [Go]. When n = 3, the co-kernel of the homomorphismis an irreducible 23-dimensional representation of Mon(S [3]) [Ma2]. Thus,the monodromy invariant subspace of H4(X,Q) is one domensional, and is

spanned by each of the three classes, for X of K3[n]-type, n ≤ 3.Assume that n ≥ 4. Then the monodromy invariant subspace of the

quotient H4(S[n],Q)/Sym2H2(S[n],Q) is one-dimensional and is spannedby the image of each of κ2(X) and c2(TX) ([Ma2], Lemma 4.9).

Acknowledgements: I would like to thank Misha Verbitsky and DanielHuybrechts for valuable comments. I thank Andrei Caldararu for intro-ducing me to the theroy of twisted coherent sheaves. I thank Jun Li forexplaining to me the conjectural generalization of the Ulenbeck-Yau Theo-rem for slope-stable twisted sheaves. I thank Francois Charles for explainingto me his interesting results in [Ch]. This paper was presented in two work-shops: “Workshop on Moduli spaces of vector bundles”, at the Clay Math.Inst., October 2006, and “Non-linear integral transforms: Fourier-Mukai andNahm” at the Centre de Research Mathematique, Montreal, August 2007. Iapologize to the organizers of the Clay workshop for my failure to completethe write-up of the paper in time to be included in the proceedings.

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