Brill-Noether duality for moduli spaces of sheaves on K3 surfaces Eyal Markman 1 9 Contents 9 9 1 1 Introduction 1 n 2 Stratified Elementary Transformations 10 a 2.1 Dualresolutionsoftheclosureofanilpotentorbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 J 2.2 Constructionofstratifiedtransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8 1 3 Determinantal Varieties - Background 17 3.1 Blowingupdeterminantalideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ] 3.2 Transpositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 G 3.3 ThePetrimap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Blowingupthesmalleststratum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A . 4 Constructionof the dual collection 27 h 4.1 ProofofTheorem6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 t a 4.2 Isomorphismofcohomologyrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 m 5 Brill-Noether duality for moduli spaces of sheaves 34 [ 5.1 Dualizablecollections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Self-dualmodulispaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1 5.3 Stabilitycriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 v 5.4 Coherentsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 5.4.1 Universalproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 5.5 ConstructionoftheBrill-Noetherloci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 0 5.6 Tyurin’sextensionmorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1 5.7 Dualizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 0 5.8 Lazarsfeld’sreflectionisomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 9 5.9 Thecollections aredual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9 / h 1 Introduction t a m Hilbertschemes ofpointsareamongthesimplest modulispacesofsheaves onanalgebraic : v surface S. Compactified relative Picards over a linear system of curves in S may be i X considered as moduli space of sheaves with pure one-dimensional support. The latter r a is complicated in comparison. For example, the punctual Hilbert schemes are always smooth while the compactified relative Picard is rarely smooth. This work was motivated by a desire to understand (resolve) the birational Abel-Jacobi isomorphism S[g] Picg ← ··· → |OS(1)| between the punctual Hilbert scheme S[g] of a symplectic surface with a linear system (1) of curves of genus g and the relative compactified Picard Picg . The latter |OS | |OS(1)| is a completely integrable hamiltonian system [Hur, Mu1]. Of particular interest is the case where S is the cotangent bundle of a smooth algebraic curve and Pic “is” a |OS(1)| 1Partially supported by NSF grant number DMS-9802532 1 Hitchin system. The recursive nature of the geometry involved forced us to consider a more general duality among the moduli spaces of sheaves of arbitrary rank on the symplectic surface. However, due to the complexity of the birational isomorphisms, we consider here only the simplest setup; that of a K3 surface S and the collection of moduli spaces of sheaves whose first Chern class is minimal (Condition 6 Section 5.1). The structure, both local and global, of the birational Abel-Jacobi isomorphism turns out to be surprisingly beautiful. Local analytically, the birational isomorphism is modeled after two dual Springer resolutions T G(t,H) t T G(t,H ) ∗ ∗ ∗ → N ← of the closure of the nilpotent orbit in End(H) of square-zero matrices of rank t. We proceed to describe the global structure (Theorems 1 and 2 below). A projective polarized K3 surface is a simply connected surface S with a trivial canonical bundle and a choice of an ample line bundle (1). There is a sequence of 19- S O dimensional irreducible moduli spaces of polarized K3 surface parametrized by the genus g 2 of a curve in the linear system (1) . The linear system (1) is g dimensional S S ≥ |O | |O | and the line bundle (1) gives rise to a morphism ϕ : S Pg which is an embedding S O → as a surface of degree 2g 2 if g > 2 and a double cover of P2 branched along a sextic if − g = 2. Consider first the generic case of a K3 surface with a cyclic Picard group generated by (1). A theorem of Mukai implies that all moduli spaces of Gieseker-Simpson stable S O sheaves on S, which happen to be compact, are smooth projective hyperkahler varieties (theyadmitanalgebraicsymplecticstructure)[Mu1]. Compactnessofthemodulispaceis automatic for the collection of components parametrizing sheaves whose rank r, determi- nant (d), andEuler characteristic χsatisfy gcd(r,d,χ) = 1 (these correspond to primi- S O tivevectorsintheMukailatticeintroducebelow). Inparticular,compactnessofthestable locus holds for the relative compactified Picards over the linear system (1) (union of S |O | compactified Picards of all curves in the linear system). Here, the fact that all curves in the linear system (1) are reduced and irreducible is crucial. The polarized weight 2 S |O | Hodge structure of a moduli space of stable sheaves is identified as a sub-quotient of the Mukai-lattice. This is the cohomology lattice H(S,Z) := H0(S,Z) H2(S,Z) H4(S,Z) ⊕ ⊕ endowed with the non-degenerate symmetric pairing e (r ,c ,s),(r ,c ,s ) := c c r s r s. h ′ ′1 ′ ′′ ′1′ ′′ i ′1 · ′1′ − ′ ′′ − ′′ ′ The Euler characteristic of a coherent sheaf F on S of rank r and Chern classes c , 1 c is χ(F) = 2r + (c1)2 c . Following Mukai, we associate to F its Mukai vector 2 2 − 2 v(F) := ch(F) Td(S) = (r,c ,s) where s = χ(F) r. Pic(S) acts on the Mukai lattice 1 − by tensorizatiopn (d) (r,c ,s) = r,c +rd,s+ 1 (d) (2c +r (d)) . OS ⊗ 1 1 2OS · 1 OS The dimension of the moduli space (cid:0)(v) of stable sheaves with Mukai vector(cid:1)v is v,v + M h i 2 = (c )2 2rs+2. Conjecturally, the weight 2 Hodge structure of a smooth projective 1 − 2 moduli space (v) isisomorphic to v if dim( (v)) > 2 andto v /Z v if dim( (v)) = ⊥ ⊥ M M · M 2 (see [Mu2, O, Yo] for many known cases). Above, v is the orthogonal hyperplane with ⊥ respect to the Mukai pairing. The collection of non-empty moduli spaces consists of the points in the “hyperboloid” := v = (r, (d),s) 1 dim (v) = 1+d2(g 1) rs 0 V { OS | 2 M − − ≥ } in Z3 satisfying the additional condition that the rank r is 0. is symmetric with ≥ V respect to r and s but the last constraint is not. Let us illustrate this lack of symmetry in the plane c = (1). We get the planar region := (c = 1) := v = 1 S 1 O H V ∩ { (r, (1),s) 1 dim (v) = g rs 0 bounded by a hyperbola (see Figure 1). The OS | 2 M − ≥ } symmetry is restored via Brill-Noether theory. There is a natural Z/2 Z/2 action by a × group of isometries of the Mukai lattice. The hyperbola is invariant and we get three H involutions σ, τ and τ σ where σ and τ are reflections with respect to the hyperplanes ◦ r s=0 and r+s=0. − r M(g,1,-g) M(g,1,0)* *M(g,1,1) * M(g-1,1,0) * H [2g] [g+1] rs=g S S[g+2] S S[g] S[g-1] S[0] * * * * * * s *-1 * * * g J Jg-2 Jg-1 J rs=g σ(r,1,s) = (s,1,r) τ (r,1,s) = (-s,1,-r) τσ(r,1,s) = (-r,1,-s) Figure 1: The region in the Mukai lattice of non-empty moduli spaces with c = (1). 1 S O The Hilbert scheme S[n] of length n zero dimensional subschemes is represented by 3 (1,0,1 n) as well as by any Pic(S)-translate, hence also by (1,1,g n). The M − M − relative compactified Picard of degree i over the linear system (1) is the moduli S |O | space Ji := (0,1,i+1 g). Observe that the birational isomorphism between Jg and M − S[g], realized by the Abel-Jacobi map, is a lift of the involution σ(1,1,0) = (0,1,1). The local Torelli theorem holds for the weight 2 Hodge structure of projective hyperkakler varieties [B1]. Hence, given a universal isometry of the Mukai lattice, it is natural to ask if it “lifts” to a birational transformation on the level of moduli spaces. Tyurin observed that this is indeed the case for σ and moduli spaces parametrized by vectors in the first quadrant of the region (see [Ty3] (4.11) and Theorem 4.1). In [G-H] the reflection H τ(1,1, 2) = (2,1, 1) was lifted to a birational isomorphism S[g+2] (2,1, 1). − − ↔ M − Stratified Mukai elementary transformations: An elementary Mukai transforma- tion is a birational transformation (M,P) (W,P ) between a symplectic variety M ∗ ↔ containing a smooth codimension n subvariety P which is a Pn-bundle P Y. The blow- → up of M along P admits a second ruling, the contraction of which results in a symplectic variety W containing the dual bundle [Mu1]. In Section 2 Theorem 6 we construct a stratified analogue of a Mukai elementary transformation - a birational transformation of a symplectic variety M admitting a stratification with a highly recursive structure, which we call a dualizable stratified collection (see (12) and Definition 5). The birational transformation produces another symplectic variety W and has the affect of “replacing” each stratum in M - a grassmannian bundle - by the dual grassmannian bundle. The base of each grassmannian bundle is itself the dense open stratum in a smaller dualizable collection. The elementary transformationis a duality; when applied twice it recovers the original variety. In Section 2.1 the simplest example is introduced: A Springer resolution of the closure of a square-zero nilpotent (co)adjoint orbit in gl is related to the dual n Springer resolution by a stratified elementary transformation. We use the Brill-Noether stratification (3) of the moduli spaces in , and prove: H Theorem 1 Given a Mukai vector v in with negative rank, define the moduli space H (v) to be identical to (σ τ(v)). Then, the Z/2 Z/2 symmetry of , as a set M M ◦ × H of Mukai vectors, lifts to an action by stratified elementary transformations on the level of moduli spaces. Consequently, we obtain a resolution of the birational isomorphisms (v) (σ(v)) and (v) (τ(v)) as a sequence of blow-ups along smooth sub- M ↔ M M ↔ M varieties, followed by a dual sequence of blow-downs. For a more detailed statement, see Theorem 20. The formal identification (v) = M (σ τ(v)) is well defined because the only vectors in , for which both v and σ τ(v) M ◦ H ◦ have non-negative rank, are Mukai vectors with r = 0 and c = (1). They correspond 1 S O to the compactified Jacobians over (1) and σ τ takes the Mukai vector of Jn to that S |O | ◦ of J2g 2 n. These two moduli spaces are naturally isomorphic (see [Le1] Theorem 5.7). − − Moreconceptually, sinceσ andτ lifttostratifiedelementarytransformationswithrespect to the same (Brill-Noether) stratification, they coincide as birational transformations of (τ(v)). Hence, σ τ[ (v)] = τ τ[ (v)] = (v). When the rank of v is negative, M ◦ M ◦ M M 4 (v) can be also described as a moduli space of (equivalence classes) of complexes of M sheaves. The general definition of a dualizable stratified collection is illustrated in the context of moduli spaces of sheaves. Given an integer t, denote by ~t the Mukai vector (t,0,t) of the trivial rank t bundle. Given a Mukai vector v in , let µ(v), the distance of v from H the boundary of , be H max t : v+~t , t Z , if χ(v) 0 µ(v) := (cid:26) max{t : v ~t ∈ H, t ∈Z≥0}, if χ(v) ≥0. (1) 0 { − ∈ H ∈ ≥ } ≤ Then µ(v)+1 is the length of the Brill-Noether stratification. For example, on the line r=0 of compactified relative Jacobians in , µ(v)+1 is equal to the usual length H of the Brill-Noether stratification of a Petri-generic curve, µ(Jg 1+n) = µ(0,1,n) = − n+ √n2+4g max 0, − ⌈ ⌉ . Choose, forexample, v withnon-negative Euler characteristic { 2 } ∈ H χ(v) = r+s 0. If µ(v) > 0 then µ(v+~1) = µ(v) 1. The square (µ(v)+1) (µ(v)+1) ≥ − × upper triangular matrix (v) (v)1 (v)t (v)µ M ⊃ M ⊃ ··· ⊃ M ··· ⊃ M ↓ (v+~1) (v+~1)1 (v+~1)µ 1 − M ⊃ M ⊃ ··· ⊃ M (v+↓~2) (v+~2)µ 2 (2) − M ⊃ ··· ⊃ M . . . ↓ (v+~µ) M is a stratified dualizable collection. If χ(v) 0, replace (v+~i)t by (v ~i)t in the ≤ M M − matrix (2) to obtain the analogous stratified dualizable collection (here, even if we start with v satisfying rank(v) 0, the Mukai vector v ~i may have negative rank and we ≥ − use the convention of Theorem 1). The diagonal entries are symplectic projective moduli spaces. Each row is the Brill-Noether stratification of the diagonal entry. When χ 0, ≥ we set (v)t := F (v) h1(F) t (3) M { ∈ M | ≥ } and when χ 0 we use h0 instead. Every space (v+~i)t i in the t-th column admits a − ≤ M dominant rational morphism to the diagonal symplectic entry (v+~t) which is regular M away from the smaller stratum and realizes (v+~i)t i (v+~i)t i+1 (v+~t) (v+~t)1 − − M \M −→ M \M as a Grassmannian bundle. The fiber over E (v+~t) (v+~t)1 is G(t i,H0(E)). ∈ M \M − As a subvariety of (v), the codimension of each Grassmannian bundle is equal to the M 5 dimension of the Grassmannian fiber. Hence, the projectivized normal bundle of each Grassmannian bundle is isomorphic to its relative projectivized cotangent bundle. The latter is a bundle of homomorphisms and admits a canonical determinantal stratification. The stratified elementary transformation is performed by blowing up the matrix (2) column by column from right to left. The key to the success of the recursion is the fact that the intersection of the strict transform of the Brill-Noether stratification with a new exceptionaldivisor(whichisfiberedbyprojectivizedcotangentbundlesofgrassmannians) is precisely the determinantal stratification of the latter. The existence of two dual sequences of blow-downs on the top iterated blow-up can be seen already on the level of the projectivised cotangent bundle of a single grassmannian: Given a vector space H, the top iterated blow-up of PT G(t,H) is naturally isomorphic to that of PT G(t,H ). ∗ ∗ ∗ The top iterated blow-up B[1]PT G(t,H) is a particularly nice compactification of the ∗ open dense GL(H)-orbit in PT G(t,H). It admits an interpretation as a moduli space ∗ of complete collineations and its cohomology ring is well understood [BDP]. Complete collineations and complete quadrics play an important role in enumerative geometry [Lak1]. They received a modern treatment in [Vain, Lak2, KT, Th2]. The top iterated blow-up B[1] (v) is a coarse moduli space of complete stable S M sheaves. A complete sheaf (F,ρ ,ρ ,...,ρ ) consists of a sheaf F on S and a sequence of 2 3 k non-zero homomorphisms, up to a scalar factor, defined recursively by ρ : H0(F) H1(F) and 2 → ρ : ker(ρ ) coker(ρ), 2 i k 1, i+1 i i → ≤ ≤ − such that the last homomorphism ρ is surjective if χ(F) 0 and injective if χ(F) 0. k ≥ ≤ The sequence is empty if either H0(F) or H1(F) vanishes. The sequence ρ ,ρ ,...,ρ 2 3 k { } should be viewed as a truncation of a complete collineation ρ ,ρ ,...,ρ . The latter 1 2 k { } behaves well when the sheaf F varies in a flat family. A choice of a section γ of a sufficiently ample line bundle on S gives rise to a homomorphism ρ : V (F) V (F) 1 0 1 → between vector spaces which ranks depend only on the Mukai vector v(F) (see (70)). The kernel of ρ is H0(F) and cokernel is H1(F). The notion of families of complete 1 collineations is subtle but well understood [Lak2, KT]. It leads to the notion of families of complete sheaves once we fix a section γ as above. Thaddeus’ work on complete collineations [Th2] combined with Theorem 1 suggests that the stratified elementary transformations in Theorem 1 come from a variation of Geometric Invariant Theory quotients in the sense of [DH, Th1]. The moduli space of complete sheaves B[1] (v) is instrumental in studying the S M intersection theory on moduli spaces. Indeed, it plays a central role in the proof of Theorem 2. This is not surprising, considering the important role played by complete collineations in classical enumerative geometry. B[1] (v) is different in general from S M the closure of the graph of the birational isomorphism in (v) (σ(v)). The latter M ×M is the moduli space of coherent systems G0(χ(v), (σ(v))) discussed in section 5.4. The M two are equal only in the case of a Mukai elementary transformation, i.e., when µ(v) 1. ≤ 6 While both moduli spaces are useful in the proof of Theorem 1, it is the moduli space of complete sheaves which admits the full recursive structure which makes transparent the analogy with dual Springer resolutions of nilpotent orbits. A correspondence inducing isomorphism of cohomology rings: Birational hyperkahler varieties M , M are especially similar (under mild conditions ′ ′′ which are satisfied by our stratified elementary transformations): There exists a family D of hyperkahler varieties over the punctured disk with two extensions , to × ′ ′′ M → M M smooth families over the disk with special fibers M , M respectively[Huy]. Consequently, ′ ′′ the Hodge structure, the cohomology ring structure and all continuous invariants of M ′ and M coincide. The Hodge conjecture suggests the existence of a correspondence (an ′′ algebraic cycle in M M ) which induces the isomorphism of Hodge structures. This ′ ′′ × correspondence contains the closure of the graph of the birational isomorphism as a component. It is easy to see that there are other components. For example, we saw that Jg and S[g] are birational σ(Jg) = S[g] (see Figure 1). The g-th symmetric product C[g] of a smooth member (curve) C of (1) , is a subvariety in S[g]. The birational image of S |O | C[g] in Jg is the Jacobian Jg which has self intersection 0. However, the self intersection C of C[g] in S[g] is 2g−2 . Recently, progress has been made in understanding the ring (cid:18) g (cid:19) structure ofthe cohomologyoftheHilbert schemes [ES,Leh]. Inorder to translatethisto an understanding of the cup-product in the cohomology of birational components such as σ(S[n]), one needs to compute explicitly the isomorphism H (S[n],Z) ∼= H (σ(S[n]),Z). ∗ ∗ → Theorem 2 Let M(i)j W(i)j be a stratified elementary transformation be- { } ←→ { } tween two stratified dualizable collections associated with irreducible projective symplectic varieties M = M(0)0 and W = W(0)0. Then, the natural isomorphism H (M,Z) = ∗ ∼ H (W,Z) is induced by a correspondence which, as a cycle ∆ in M W, is a sum ∗ × µ ∆ = ∆ + ∆ (4) 0 t Xt=1 where ∆ is the closure of the graph of the birational transformation M W and ∆ is 0 t ↔ the closure of the fiber product Mt Wt of the two grassmannian bundles. M(t) × Note that the dimension of ∆ is equal to dim(M). The Theorem is proven in Section t 4.2. A few applications of Theorem 2 are discussed in Section 5.2. Auto-equivalences of the derived category: Let S be a K3 surface and consider the group G of Hodge isometries of the Mukai lattice H(S,Z) of S. These are integral isometries which send H2,0(S) to itself when the lattice is complexified. Any Hodge-isometry φ leaves the algebraic sublattice H (S,Z) e Alg invariant. If φ restricts to the identity automorphism of H (S,Z) then it is induced by Alg e an automorphism of S (the Torelli theorem). It follows that G fits in the exact sequence: e 0 Aut(S,Pic(S)) G Aut(H (S,Z)) 0. (5) Alg → → → → e 7 Let G be the group of (covariant and contravariant) auto-equivalences of the bounded Der derived category of S. We have a map η : Db(S) H(S,Z) → (Fi,∂i) e ( 1)iv(Fi) 7→ − Xi sending an object represented by a complex (F ,∂ ) to the Mukai vector of the associated i i class in K-theory, i.e., to the alternating sum of the Mukai vectors of its coherent sheaves. Notice that η is equivariant with respect to the cyclic subgroup T G generated by Der h i ⊂ the translation auto-equivalence T : Db(S) Db(S) once we send T to Id G. Orlov → − ∈ proved that there is a surjective homomorphism G / T2 G. (6) Der h i → Any auto-equivalence Φ of the derived category Db(S) induces a Hodge isometry φ of H(S,Z) and η is (Φ,φ)-equivariant ([Or], Theorem 2.2). Moreover, any Hodge isometry canbeliftedtoanauto-equivalenceofthederivedcategoryDb(S)([Or],proofofTheorem e 3.13). For example, 1. Tensorization by a line bundle γ on S is induced by the Fourier-Mukai functor L Rπ π ( ) : Db(S) Db(S) associated to the sheaf on S S where is · 2∗E ⊗ 1∗ · → E × E supported on the diagonal as the line bundle γ. 2. The Hodge-isometry τ (τ as in Theorem 1) is induced by the Fourier-Mukai − functor associated to the ideal sheaf of the diagonal. 3. The Hodge isometry σ τ is induced by the contravariant involutive functor − ◦ R om ( , ) : Db(S) Db(S)op. · S S H · O → Assume that the K3 surface S has a cyclic Picard group and let us introduce yet a third group G . Let bir ( ) := (v) : v H (S,Z) is a primitive Mukai vector, v,v 2 S S Alg M • ∪{M ∈ h i ≥ − } e be the disjoint union of moduli spaces of stable sheaves on S with primitive algebraic Mukai vectors of non-negative dimension. Note that these moduli spaces are all smooth and compact. We use the convention that, if rank(v) 0, then (v) is equal to S ≤ M ( v). The group G is defined to be the group of all possible lifts of elements of G S bir M − to birational automorphisms of ( ). A natural question arises: S M • Question: Does the homomorphism (6) factor through G ? bir TheFourier-Mukaifunctorliftinganisometrytoanauto-equivalencesends, ingeneral, a stable sheaf to the class of a complex supported at more than one degree. Nevertheless, there seems to exist, at least, a surjective homomorphism G G. Clearly, Pic and bir S → 8 Aut(S) lift to G . The element Id lifts, by definition. Theorem 1 suggests that the bir − Hodge isometries σ and τ lift to G . In the Theorem the lift is carried out only for bir the collection with c = (1), but our definitions of σ and τ on the Zariski open Brill- 1 S O Neother stratum seem to extend to birational isomorphisms for other values of c . If 1 indeed σ and τ lift, then the image of G G has at most a finite index (and is bir → surjective if g = 2). This follows from the exactness of (5) and the fact that Id,σ,τ {− } and Pic generate a finite index subgroup Γ of the group of isometries of the rank 3 S lattice H (S,Z). If g = 2, Γ is the whole group, but in general it is a proper subgroup. Alg For example, if g = 7 the isometry e r 2 12 3 r  d   1 5 1  d  s 7→ 3 12 2 s      is not in the subgroup Γ because any isometry in Γ takes (0,0,1) to a vector (r,d,s) such that (g 1) divides exactly one of the two integers r,s and is relatively prime to − { } the other. A much harder problem would be to resolve the birational isomorphisms in G . It is bir naturalto trytogeneralize our results tootherreflectionsoftheMukailattice. Thegroup of isometries of the rank 3 lattice H (S,Z) contains a finite index subgroup W G Alg ⊂ generated by reflections with respect to Mukai vectors with v,v = 2. In fact, setting e h i ± σ := (1) σ ( 1) and τ := (1) τ ( 1) we have the equality ′ ′ O ◦ ◦O − O ◦ ◦O − (2) = σ τ στ W. ′ ′ O ∈ Our results suggest a relationship between dual pairs of hyperkahler resolutions of singu- larities and reflections in G . While at the level c = (1) the reflections σ and τ corre- bir 1 O spond to Springer resolutions of a nilpotent orbit with a simple (well ordered) stratifica- tion, itseems thatforothervaluesofc morecomplicatedsingularities willarise. Itwould 1 be interesting, for example, to interpret the resolution (1,0,1 n) = S[n] SymnS S ∼ M − → of the symmetric product as a special case of a lift of some reflection (τ ?) to G . bir The rest of the paper is organized as follows. Sections 2, 3, and 4 are devoted to the general study of stratified elementary transformations. In section 2 we construct the stratified elementary transformations. Section 3 contains the background information on determinantal varieties needed for the proof of the construction. In section 4 we complete the proof of the construction. We also prove Theorem 2 identifying the correspondence inducing the cohomology ring isomorphism. Section 5 is devoted to our main example, the moduli spaces of sheaves on a K3 surface. The organization of section 5 is described at the beginning of that section. Acknowledgments: It is a pleasure to acknowledge fruitful conversations with D. Cox, R. Donagi, B. Fantechi, L. G¨ottsche, V. Ginzburg, D. Huybrechts, S. Kleiman, I. Mirkovic, T. Pantev and M. Thaddeus. 9 2 Stratified Elementary Transformations In section 2.1 we introduce the prototypical example of the symplectic birational iso- morphisms considered in this paper. In section 2.2 we introduce a global analogue for projective symplectic varieties. 2.1 Dual resolutions of the closure of a nilpotent orbit Let H be a vector space of dimension h, t h/2 an integer and consider the natural ≤ morphism π : T G(t,H) End(H) onto the closure t in End(H) of the nilpotent 1 ∗ → N orbit t of square-zero nilpotent elements of rank t. The natural isomorphism End(H) = ∼ N End(H ) provides another resolution ∗ T G(t,H) π1 t π2 T G(t,H ). (7) ∗ ∗ ∗ −→ N ←− There is a simultaneous deformation of (7) over C which smoothes t away from N the special fiber and deforms π to isomorphisms. The cotangent bundle of G(t,H) i is isomorphic, as a bundle of Lie subalgebras of nd(H ), to the nilpotent radical G(t,H) E of the corresponding parabolic subalgebra nd(H ). We have a unique G(t,H) G(t,H) P ⊂ E non-trivial extension 0 T E(H) 0. (8) → G∗(t,H) → → OG(t,H) → A non-zero section γ of determines a symplectic T torsor X embedded in OG(t,H) G∗(t,H) γ E(H). X does not have any section. Hence the lagrangian zero-section of T does γ G∗(t,H) not deform. The vector bundle E(H) admits an embedding as a bundle of subalgebras in End(H ). Each point in G(t,H) determines a decomposition of the Levi factor of the G(t,H) parabolic subalgebra (with ranks t and h t). E(H) is the subalgebra of which G(t,H) − P projects to the center in the first factor and to zero in the second factor. In other words, E(H) is the subalgebra of endomorphisms leaving the subbundle τ(H) invariant, whose image in nd(τ(H)) is a scalar multiple of the identity and whose image in nd(q(H)) E E is zero. The morphism E(H) End(H) restricts to an embedding of X , γ = 0, as a γ → 6 smooth closed orbit. The birational isomorphism (7) is a special case of the setup of Section 2.2. Consider the square (t+1) (t+1) upper triangular matrix whose rows are the determinantal × stratification of T G(k,H), 0 k t, indexed as follows. T G(k,H) is isomorphic ∗ ∗ ≤ ≤ to the homomorphism bundle om(q ,τ ) from the tautological quotient bundle to the k k H tautological sub-bundle. The generic point corresponds to a surjective homomorphism. T G(k,H)i denotes the locus with i-dimensional cokernel. ∗ 10