Symplectic Moduli Spaces

Transcription

1 Symplectic Moduli Spaces M. Lehn Fachbereich Mathematik, Johannes-Gutenberg-Universität Mainz, Germany Lectures given at the School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 LNS

2 Abstract Most examples of irreducible holomorphic symplectic manifolds arise as moduli spaces of sheaves. We will briefly introduce the notion of an irreducible holomorphic symplectic manifold and then discuss Hilbert schemes, generalised Kummer varieties, moduli of sheaves and O Grady s new examples of symplectic manifolds.

3 Contents 1 Irreducible holomorphic symplectic manifolds Hyperkähler manifolds Symplectic structures K3-surfaces Two dimensional tori Hilbert schemes Fujiki s example The Kummer variety revisited The Quot-scheme Hilbert schemes Punctual Hilbert schemes Beauville s theorem Generalised Kummer varieties Moduli spaces of semistable sheaves Semistable sheaves Moduli spaces Local description of the moduli space Semistable sheaves on K3-surfaces O Grady s first example The global picture The local picture Resolution of the singularities Epilogue References 183

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5 Symplectic Moduli Spaces Irreducible holomorphic symplectic manifolds 1.1 Hyperkähler manifolds A simply connected Riemannian manifold X is called a hyperkähler manifold, if its holonomy group is the symplectic group Sp(m), dim R (M) = 4m. In this case there exist three complex structures I, J and K satisfying the relations I 2 = J 2 = K 2 = IJK = id such that the metric of X is Kähler with respect to all three complex structures. Hyperkähler manifolds arise in the classification of Ricci-flat Kähler manifolds due to the following decomposition theorem Theorem 1.1 Let X be a compact Ricci-flat Kähler manifold. Then there is a finite cover X X such that X is isomorphic to a product A Y 1... Y l X 1... X m where A is a flat Kähler torus, each Y i is a compact simply-connected manifold with holonomy SU(n i ), a so-called Calabi-Yau manifold, and each X j is a compact simply-connected Hyperkähler manifold. For detailed information on hyperkähler manifolds, the decomposition theorem and the relation to the Calabi conjecture I refer to the lecture notes of Joyce in [12]. In this conference volume the interested reader also finds the lecture notes of Huybrechts on moduli spaces of hyperkähler manifolds [14]. It turns out that there are plenty of examples of Calabi-Yau manifolds, but only very few known examples of compact hyperkähler manifolds. The purpose of this lecture course is to give an elementary introduction to the known examples in the language of algebraic geometry. This is possible as the differential-geometric notion of a hyperkähler manifold can be translated into the algebraic-geometric or complex-geometric notion of an irreducible holomorphic symplectic manifold. In the following we will only use that notion. We will discuss the topology and geometry of these examples and thus hope to provide some background information for the lecture course of Huybrechts in this summer school. 1.2 Symplectic structures Let X be a complex manifold and let σ Γ(X, Ω 2 X ) be a global holomorphic 2-form. σ is non-degenerate, if the induced skew-symmetric pairing T X

6 144 M. Lehn T X O X is non-degenerate at every point x. Equivalently, the adjoint homomorphism σ : T X Ω X is required to be an isomorphism. Because of the skew-symmetry, a necessary condition for σ to be non-degenerate is that X be even dimensional, say dim(x) = 2n. Forming σ n = σ... σ Γ(X, Ω 2n X ), we can also express the non-degeneracy of σ by saying that σn should be a nowhere vanishing section of the canonical sheaf K X = Ω 2n X. In particular, another necessary condition for the existence of a symplectic structure on X is that K X be trivial. Definition 1.2 A holomorphic 2-form σ is said to be a symplectic structure on X, if dσ = 0 and if σ is non-degenerate. It is very easy to construct non-compact examples of manifolds with symplectic structures: Let Y be an arbitrary complex manifold. On the cotangent bundle X := T Y Y, there is a tautological 1-form θ Γ(X, Ω 1 X ). Then σ := dθ is a symplectic structure on X. It is much harder to find compact examples. Definition 1.3 An irreducible holomorphic symplectic manifold is a simply-connected complex manifold X of Kähler type such that H 0 (X, Ω 2 X ) is generated by a symplectic structure σ. Note that we require that X could be endowed with a Kähler metric but that the Kähler metric is not part of the structure. 1.3 K3-surfaces Symplectic manifolds are necessarily even-dimensional. The lowest possible dimension is two. A glance at Kodaira s classification of compact Kähler surfaces shows that the only surfaces with trivial canonical bundle are K3- surfaces and 2-dimensional tori. Definition 1.4 A smooth compact complex surface is a K3-surface if H 1 (O X ) = 0 and if the canonical bundle is trivial. Example 1.5 Let f C[x 0, x 1, x 2, x 3 ] be a general homogeneous polynomial of degree 4, and let X P 3 be the zero-set of f. By Bertini s theorem X is a smooth irreducible surface. The Fubini-Study metric on P 3 restricts

7 Symplectic Moduli Spaces 145 to a Kähler metric on X. By Lefschetz theorem on hyperplane sections, π 1 (X) = 0. From the exact sequence one gets an exact sequence 0 O P 3( 4) O P 3 O X 0 H 1 (P 3, O P 3) H 1 (X, O X ) H 2 (P 3, O P 3( 4)) and concludes that H 1 (X, O X ) = 0. Finally, the adjunction formula implies K X = ( KP 4 O P 3(4) ) X = OX. Example 1.6 Let C P 2 be a smooth curve of degree 6. Let π : X P 2 be the 2 : 1-cover ramified along C. Explitly X can be constructed as follows: let A = O X O X ( 3) be the sheaf of algebras where the multiplication O X ( 3) O X ( 3) O X is given by the equation of C. Now let X be the relative affine spectrum SpecA. A local calculation shows that X is non-singular, because C is a smooth curve. Since π is finite, X is projective and therefore a Kähler surface. The canonical sheaf of X is K X = π (K P 2 O P 2(3)) = O X. Moreover, H 1 (X, O X ) = H 1 (P 2, π (O X )) = H 1 (P 2, O P 2 O P 2( 3)) = 0. The topology of a K3-surface X is completely determined by its definition: For any compact smooth complex surface X, the Hodge numbers h p,q = dim H q (X, Ω p X ) satisfy the conditions 2h1,0 b 1 = h 1,0 + h 0,1 [1, p. 116]. Therefore, the vanishing assumption h 0,1 = 0 for a K3-surface implies that h 1,0 = 0 = b 1 (X). Next, Serre duality implies that H 2 (O X ) = H 0 (K X ) = C. Hence the holomorphic Euler characteristic is given by χ(o X ) = h 0,0 h 0,1 + h 0,2 = = 2. Since c 1 (X) = 0, we deduce from Noether s formula that the topological Euler-Poincaré characteristic is e(x) = c 2 (X) = c 2 (X) + c 1 (X) 2 = 12 χ(o X ) = 24. On the other hand we know that b 1 (X) = 0, and conclude by Poincaré duality that b 3 (X) = 0. This fixes the second Betti number: 24 = e(x) = 1 + b 2 (X) + 1, and thus b 2 (X) = 22. Furthermore, H 1 (X; Z) = 0. For assume on the contrary that ξ H 1 (X; Z) were a nontrivial element. Since b 1 (X) = 0, ξ must be a torsion element and gives rise to a finite étale cover f : X X of degree, say, d. Then K X = g K X = OX and, because of Serre duality, one gets h 2 (O X ) = h 0 (K X ) = 1. Now on the one hand, we have

8 146 M. Lehn χ(o X ) = 2 h 1 (O X ). On the other hand, the Hirzebruch-Riemann-Roch theorem implies χ(o X ) = td(t X ) = f td(t X ) = d td(t X ) = dχ(o X ) = 2d. X X X This is impossible unless d = 1 and ξ = 0. Knowing that H 1 (X; Z) = 0, it follows that H 2 (X; Z) is torsion free and hence a unimodular lattice. The lattice is even since w 2 (X) c 1 (X) 0mod Z/2. Hirzebruch s signature theorem implies that b + b = 1 3 (c2 1 2c 2 ) = 16. It follows from the classification of even unimodular lattices that H 2 (X; Z) ( 0 1 = ) 2( E 8 ). It requires more work to show that an arbitrary K3-surface can be deformed into a smooth quartic hypersurface as in example 1.5. In particular, all K3-surfaces are diffeomorphic and hence simply connected, because smooth quartics are simply connected. Finally, a non-trivial theorem of Siu states that every K3-surface is Kähler. For detailed information on K3-surfaces I refer to the seminar notes [8] and the text book [1]. Most of the symplectic manifolds that we will encounter in these lectures are based on constructions on a K3 surface. All others are based on 2- dimensional tori. 1.4 Two dimensional tori The second surface in Kodaira s list of compact Kähler surfaces with trivial canonical divisor are 2-dimensional tori. A torus is a quotient A = C 2 /Γ for a some lattice Γ C 2. If Γ satisfies the Riemann conditions then A is projective. If this is the case A is called an abelian surface. A 2-dimensional torus is equipped with a unique symplectic structure (up to constant factors), namely, if z 1 and z 2 are linear coordinates on C 2, then dz 1 dz 2 is a translation invariant symplectic structure on C 2 and therefore descends to a symplectic structure on A.

9 Symplectic Moduli Spaces 147 However, A fails to be irreducible holomorphic symplectic for the trivial reason that it is not simply connected. In fact, π 1 (A) = Γ = Z 4. Nevertheless, complex tori will be very useful for the construction of higher dimensional symplectic manifolds. As a warm-up, we recall the construction of non-singular Kummer varieties: The group Z/2 acts on A by means of the involution ι : A A, x x. This action is free except for the sixteen 2-torsion points on A. If we pass to the quotient Y = A/ι, each of these fixed points contributes an A 1 - singularity to Y. Locally near 0 Y, we have coordinates a = z 2 1, b = z 1z 2, c = z 2 2, subject to the relation ac b 2 = 0. Let f : K(A) Y be the blow-up of the singularities. The exceptional divisors are ( 2)-curves. What happens to the symplectic structure of A? As σ = dz 1 dz 2 is Z/2-invariant it descends to a symplectic structure σ on Y reg. Near 0, we can write it as σ = da db 2a = db dc. 2c In local coordinates a and β = b/a (with c = β 2 a) on K(A) the pull-back of σ is given by 1 2 da dβ. This shows that f σ extends over the exceptional divisors without zeroes and hence is a symplectic structure on K(A). One can check that π 1 (X) = 0, so K(A) is again a K3-surface. 2 Hilbert schemes The first higher dimensional example of an irreducible symplectic manifold was given by Fujiki, namely the blow-up of the diagonal in S 2 (X) = X 2 /S 2 for a K3-surface X. This example was generalised by A. Beauville. He showed that all Hilbert schemes Hilb n (X) of generalised n-tuples of points on a K3-surface X are irreducible holomorphic symplectic manifolds. Fujiki s example is the second instance of this series. Beauville also constructed a second series of so-called generalised Kummer varieties K n (A) by modifying the Hilbert schemes associated to a 2-dimensional torus A. In this section we will review Beauville s constructions and discuss the geometric and topological properties of Hilbert schemes and generalised Kummer varieties.

10 148 M. Lehn 2.1 Fujiki s example We begin with a special case of the Hilbert scheme and the associated generalised Kummer variety, namely Hilb 2 (X) and K 1 (A). The reason is that for the case of pairs of points all calculations can be done very explicitly, without introducing general notions, and that all the same the calculations show almost all phenomena that one meets in the general case. Also historically, Hilb 2 (K3) was the first higher dimensional irreducible holomorphic symplectic manifold discovered. Let X be a connected smooth projective surface. The group Z/2 acts on the product X X by exchanging the factors: ι : X X X X, (x 1, x 2 ) (x 2, x 1 ). This action is free except along the diagonal = {(x, x) x X}. Let ρ : Z X X be the blow-up of the diagonal, and let E := ρ 1 denote the exceptional divisor. The Z/2 action extends to Z. Let Y := Z/(Z/2) and S 2 (X) := X X/(Z/2) denote the quotient varietes. Thus we have the following diagram ρ Z X X p p Y ρ S 2 (X). The morphism ρ is the blow-up of S 2 (X) along := p( ) with exceptional divisor E = p(e ) = ρ 1 ( ). Note that Y is smooth even though E is the set of fixed points for the Z/2-action on Z. However, in appropriate coordinates near a point in E, the action looks like (z 1, z 2, z 3, z 4 ) (z 1, z 2, z 3, z 4 ). The smoothness of Y follows easily. Lemma 2.1 π 1 (Y ) = π 1 (X)/[π 1 (X), π 1 (X)] = π 1 (X) ab. Proof. Let α : [0, 1] X be a path with distinct end points x 0 := α(0) and x 1 = α(1). Then α 1 (t) = (x 0, α(t)) and α 2 (t) = (α(t), x 0 ) are two paths in X X that connect (x 0, x 0 ) to (x 0, x 1 ) and (x 1, x 0 ), respectively. The path β := α 1 1 α 2 then connects (x 0, x 1 ) and (x 1, x 0 ). Note that β = (ι β 1 ). Since has real codimension 2 there is a path γ that is homotopic (relative to the end points) in X X to β and does not intersect

11 Symplectic Moduli Spaces 149. Moreover, (ι γ) 1 (ι β) 1 = β γ by a homotopy relativ end points in X X. But again, as has real codimension 3 in X X, we can make this homotopy disjoint from. It follows that γ (ι γ) 1 in X X \ E. Since Z \ E X X \ is an isomorphism, we can think of γ as a path in Z \ E connecting z 0 := ρ 1 (x 0, x 1 ) and ι(z 0 ). Then γ := p γ is a loop in Y \ E with base point y 0 := p (z 0 ) and satisfies ( γ) 1 = p γ 1 = p ι γ 1 p γ = γ. This means that the class τ := [ γ] π 1 (Y \ E, y 0 ) is an element of order 2. Since has real codimension 3 in X X there are isomorphisms π 1 (Z \ E =, z 0 ) π 1 (X X \ =, (x 0, x 1 )) π 1 (X X, (x 0, x 1 )) α 1 = π1 (X X, (x 0, x 0 )) π 1 (X, x 0 ) π 1 (X, x 0 ). On the other hand, p : Z \ E Y \ X is a regular topological covering with automorphism group Z/2. Therefore, there is an exact sequence 1 π 1 (Z \ E, z 0 ) π 1 (Y \ E, y 0 ) Z/2 1. Now by the constrution above, γ lifts γ and has distinct end points. Hence the class τ of γ surjects onto the generator of Z/2. Moreover, τ 2 = 1. This means that the sequence splits, that there is an isomorphism π 1 (Y \ E, y 0 ) = π 1 (Z \ E, z 0 ) Z/2, and that τ acts on the normal subgroup by [v] [γ (ι v) γ 1 ]. Now, one can check that under the isomorphism α 1, this action turns into flipping the factors of π 1 (X, x 0 ) 2. Thus we can summarise the discussion up to now by stating: π 1 (Y \ E, y 0 ) = π 1 (X, x 0 ) 2 Z/2 with τ([v], [w])τ = ([w], [v]). Let U be tubular neighbourhood of E and let q : U E denote the retraction to E. Then q is a homotopy equivalence and q 0 : U \ E E is a fibration with fibres homotopy equivalent to S 1. We can choose the set-up above in such a way that y 0 U \ E and that γ is a generator of the fundamental group of F = q0 1 (q(y 0)). By the Seifert-van Kampen theorem we obtain π 1 (Y, y 0 ) as the push-out π 1 (U \ E, y 0 ) π 1 (Y \ E, y 0 ) π 1 (U, y 0 ) = π 1 (E, q(y 0 )) π 1 (Y, y 0 )

12 150 M. Lehn On the other hand we have an exact sequence Z[ γ] = π 1 (F, y 0 ) π 1 (U \ E, y 0 ) π 1 (E, q(y 0 )) 1. The image of [ γ] in π 1 (Y \ E, y 0 ) is τ. We conclude that π 1 (Y, y 0 ) = π 1 (Y \ E, y 0 )/ τ = π 1 (X, x 0 ) 2 τ / τ. The rest is purely algebraic: for any two elements g and h in π 1 (X, x 0 ) we have τ(g, 1)τ = (1, g), and (g, 1) and (1, h) commute. Thus introducing the relation τ = 1 leads to an identification (g, 1) (1, g) and makes the resulting class commute with (h, 1). This shows that the homomorphism π 1 (X, x 0 ) 2 (Z/2)/ τ π 1 (X, x 0 ) ab, (g, h, t) [gh], is an isomorphism. Lemma 2.2 H 0 (Y, Ω 2 Y ) = H 0 (Ω 2 X ) Λ2 H 0 (Ω 1 X ). Moreover, if σ is a symplectic structure on X, then the form σ Y induced on Y by this isomorphism is again a symplectic structure. Proof. The quotient map p : Z Y induces an injective linear map p : Γ(Y, Ω 2 Y ) Γ(Z, Ω2 Z )Z/2 into the invariant part of the space of holomorphic forms. This map is surjective: this is clearly a local question. Let z E Z be an arbitrary point. We can choose local coordinates e 1, e 2, e 3, z such that the e i are invariant, z z, and E = {z = 0}. A Z/2-invariant holomorphic 2-form ψ can be written as ψ = i>j ψ ij de i de j + i ψ i zdz de i, where ψ ij and ψ i are invariant holomorphic functions. As such, they are functions in e 1, e 2, e 3 and e 4 := z 2. Now e i, i = 1,..., 4, are local coordinates near p (z) Y. As zdz = 1 2 p (de 4 ), we see that there is a holomorphic 2- form ψ such that ψ = p ψ. This shows: Γ(Y, Ω 2 Y ) Γ(Z, Ω 2 Z )Z/2. On the other hand, the blow-up map ρ : Z X X induces a homomorphism ρ : Γ(X X, Ω 2 X ) Γ(Z, Ω2 Z ). This is an isomorphism. To see surjectivity note that for any ϕ Γ(Z, Ω 2 Z ) the restriction ϕ Z\E is a section of Ω X X X X\. But since has complex codimension 2, this section extends to all of X X. This shows: Γ(Z, Ω 2 Z) = Γ(X X, Ω 2 ) = pr 1Γ(X, Ω 2 X) pr 2Γ(X, Ω 2 X) (pr 1Γ(X, Ω 1 X) pr 2Γ(X, Ω 1 )).

13 Symplectic Moduli Spaces 151 Z/2 acts on this space as follows: ι (pr i ϕ) = pr 3 i(ϕ), ι (pr 1α pr 2β) = pr 1β pr 2α. Hence the invariant part is isomorphic to Γ(X, Ω 2 X ) Λ2 Γ(X, Ω 1 X ). Let σ Γ(X, Ω 2 X ) be a symplectic structure. Then σ X X = pr 1 (σ) + pr 2 (σ) is a symplectic structure on X X. Suppose that z, w are local coordinates at x X. Then we can write σ = f(z, w)dz dw with some holomorphic function f such that f(0, 0) 0. Let z i := z pr i and w i := w pr i be the associated coordinates near (x, x) X X. Then σ 2 X X = 2f(z 1, w 1 )f(z 2, w 2 )dz 1 dw 1 dz 2 dw 2. Note that u = (z 1 + z 2 )/2 and v = (w 1 + w 2 )/2 are invariant coordinates, whereas s = (z 1 z 2 )/2 and t = (w 1 w 2 )/2 change signs under the Z/2 action. In a local chart of Z we may write t = s t for an even coordinate t. Then dz 1 = du+ds, dz 2 = du ds, dw 1 = dv+ds t+s d t, dw 2 = dv ds t s d t. Expressing σ Z := ρ σ X X in these coordinates we see that σ 2 Z = f(u + s, v + s t)f(u s, v s t)4sds du d t dv. The coefficient function is an even function with respect to s and can therefore be expressed as a function in u, v, t and s := s 2. Thus σ 2 Z = 2g(u, v, t, s )ds du d t dv. But u, v, t, s are coordinates on Y. This shows that σ Z descends to 2-form on Y that is non-degenerate. If we specialise to K3-surfaces we get our first higher dimensional irreducible holomorphic symplectic manifold: Theorem 2.3 Let X be a K3-surface. Then π 1 (Y ) = 0 and H 2 (Y, Ω 2 Y ) = Cσ Y for a symplectic structure σ Y. In particular, Y is an irreducible holomorphic symplectic manifold. For completeness, let us compute the cohomology of Y : Lemma 2.4 H (Y, Q) = S 2 H (X; Q) H (X; Q) [E].

14 152 M. Lehn Here the symmetric product is to be taken in the graded sense, as will become clear in the proof. Proof. According to the Künneth formula, H (X X; Q) = H (X; Q) H (X; Q). Blowing up the diagonal gives H (Z; Q) = H (X; Q) H (X; Q) H ( ; Q) [E], where [E] is the cohomology class Poincaré dual to the fundamental class of the divisor E. The group Z/2 acts trivially on the second factor and interchanges the first two factors. We obtain the rational cohomology of Y as the invariant part of H (Z; Q). This proves the lemma. However, we have to be careful about the signs here: For example, H 2 (Z; Q) = 1 H 2 (X; Q) H 2 (X; Q) 1 H 1 (X; Q) H 1 (X; Q) Q[E]. The involution exchanges the first two summands, exchanges the two factors of the third summand, introducing a ( 1) sign at the same time, and leaves the last summand fixed. It follows: H 2 (Y ; Q) = H 2 (X; Q) Λ 2 H 1 (X; Q) C[E]. More generally, if we split the cohomology of X into its even and odd part, H (X; Q) = H ev H odd, then S 2 (H (X; Q)) := S 2 (H ev (X; Q)) (H ev (X; Q) H odd (X; Q)) Λ 2 (H odd (X; Q)), where S and Λ on the right hand side have there ordinary meaning and S is taken in the Z/2-graded sense on the left hand side. 2.2 The Kummer variety revisited Assume now that A is a 2-dimensional torus, A = C 2 /Γ. We keep the notation from the previous section: Z is the blow-up of A A along the diagonal, and Y = Z/(Z/2). In this situation we have π 1 (A) = Γ = Z 4 and H i (A; Q) = Λ i Hom Z (Γ, Q). Moreover, Γ(A, Ω i A ) = Λi C dz 1, dz 2, where z 1, z 2 are coordinates on the cover C 2 of A. It follows from the discussion above, that π 1 (Y ) = π 1 (A) ab =

15 Symplectic Moduli Spaces 153 π 1 (A), and that dim Γ(Y, Ω 2 Y ) = 2. More precisely, Γ(Y, Ω2 Y ) is generated by two 2-forms σ and σ that are characterised by and p σ = ρ (pr 1 (dz 1 dz 2 ) + pr 2 (dz 1 dz 2 )) p σ = ρ (pr 1dz 1 pr 2dz 2 pr 1dz 2 pr 2dz 1 ). Note that σ σ = 0 and σ 2 = σ 2 = 2pr 1 (dz 1 dz 2 ) pr 2 (dz 1 dz 2 ). Let + denote the group law on A. As + is commutative, the map Z A A + A is Z/2-invariant and factors through p : Z Y and Σ : Y A. Let K, K Z and F denote the fibres over 0 A of the morphisms Y A, Z A and A A + A, respectively. Clearly, F = {(x, x) x A}, and under the obvious isomorphism F = A the involution ι on F corresponds to x x on A. Next, F meets the diagonal A A transversely in the sixteen 2-torsion points, so that K Z is isomorphic to the blow-up of A in the 2- torsion points. Passing to the quotient for Z/2-action it follows that K is isomorphic to the Kummer variety introduced in section 1.4. Let us pretend for a moment that we don t yet know that K is a K3- surface and let us compute π 1 (K) and Γ(K, Ω 2 K ) directly. This will be useful when we later go on to generalised Kummer varieties. Lemma 2.5 π 1 (K) = 0. Proof. Let t a : A A denote the translation map x x + a. The map t a t a : A A A A preserves the diagonal and, by the universal property of the blow-up, induces maps t Z a : Z Z and t [2] a : Y Y. The following diagram is cartesian: A K m Y Σ pr 1 A m 2 A, where m(a, y) := t [2] a (y) and m 2 is multiplication by 2: m 2 (a) = a + a. It follows that Σ is a fibre bundle with fibre K that is locally trivial in the étale topology. The first conclusion that we draw from this is that K is smooth. Next, every topological fibration gives rise to a long exact sequence of homotopy groups (sets): π 2 (A) π 1 (K) π 1 (Y ) π 1 (A) π 0 (K)

16 154 M. Lehn Since K is connected and π 2 (A) = 0, the following sequence is exact: 1 π 1 (K) π 1 (Y ) π 1 (A) 1. But we have already seen that π 1 (Y ) = Γ = π 1 (A). π 1 (K) = 0. Hence, necessarily, Lemma 2.6 Γ(K, Ω 2 K ) is one-dimensional and generated by the restriction of σ to K. This form is non-degenerate. Proof. We keep the notation of the proof of Lemma 2.2: In a neighbourhood of a point in the fibre ρ 1 ( 0) Y, we have coordinates u, v, s and t, and the forms σ and σ are given by the formulae σ = 2du dv + ds d t and σ = 2du dv ds d t In these coordinates, K is given by the equations u = 0 and v = 0. Thus s and t are coordinates on K, and we see that the form σ K = σ K = ds d t is non-degenerate. 2.3 The Quot-scheme Quotient schemes were introduced by Grothendieck as a technical tool for many constructions in algebraic geometry. They generalise the notion of a Grassmann variety. A Grassmann variety Grass(W, d) parametrises all quotient spaces of a fixed dimension d of a given vector space W, and a quotient scheme Quot X,H (G, P ) parametrises all quotient sheaves with a fixed Hilbert polynomial P of a given coherent sheaf G. We will need Quot schemes twice: Hilbert schemes are special cases of Quot schemes, and moduli spaces of semistable sheaves are constructed as quotients of Quot schemes by an action of a reductive group. Let X be a projective scheme with an ample divisor H. If F is a coherent sheaf, the function P (F ) : n χ(f O X (nh)) is a polynomial, the Hilbert polynomial of F. Fixing such a polynomial P essentially amounts to fixing the topological invariants of F, i.e. rank and

17 Symplectic Moduli Spaces 155 Chern classes. There are too many coherent sheaves F with P (F ) = P to be parameterised by a noetherian scheme. However, such a parameterisation is possible if we restrict ourselves to sheaves that are quotients of a given sheaf G. A family of quotient sheaves of G, parameterised by a scheme S, is a coherent sheaf F on S X together with a surjective homomorphism q : O S G F such that F is S-flat and for each s S the restriction k(s) F of F to the fibre {s} X has Hilbert polynomial P. Two such pairs (F, q) are equivalent, if they have the same kernel. The reason why one works with quotient sheaves rather than subsheaves is that the tensor product is right exact but not left exact. Hence if f : S S is a base change map, then the pull-back (f id X ) q of a surjective map q is again surjective, whereas the pull-back of an injective homomorphism is, in general, no longer injective. Formally, we get a functor Quot X,H (G, P ) := (Schemes) o (Sets) S {O S G F F is S-flat, P (F s ) = P for all s S}. Theorem 2.7 (Grothendieck [13]) The functor Quot X,H (G, P ) is represented by a projective scheme Quot X,H (G, P ). Let Q := Quot X,H (G, P ). That Q represents the functor means that there is family q : O Q G F that is universal in the following sense: For any family q : O S G F there is a unique morphism ψ : S Q such that q = (ψ id X ) q. In this way families of quotient sheaves parameterised by S correspond bijectively to morphisms S Quot X,H (G, P ). Taking S = Spec(C), it follows that closed points in Quot X,H (G, P ) correspond to isomorphism classes of quotients q : G F with P (F ) = P. Next, taking S = SpecC[ε] with C[ε] = C[t]/(t 2 ), we get an intrinsic description of the Zariski tangent space of of the quot scheme at a point [q : G F ]. Corollary 2.8 [q] be a closed point in Quot X,H (G, P ) represented by a surjective homomorphism q : G F with kernel K. Then there is a natural isomorphism T [q] Quot X,H (G, P ) = Hom OX (K, F ).

18 156 M. Lehn Proof. A tangent vector to Q at [q] is an C[ε] valued point in Q, i.e. an epimorphism q : G C[ε] F that restricts to q at the special point of Spec C[ε]. Thus tangent vectors correspond to diagrams K G q F 0 0 K G C[ε] q F 0. 0 εk εg εf The flatness of F over C[ε] is equivalent to the requirement that the first and the third horizontal sequences are isomorphic. To give q is the same as to fix K. As K always contains εk, it is determined by giving a homomorphism K εf. It follows that there is a natural isomorphism T [q] Quot X,H (G, P ) = Hom OX (K, F ). The group Aut(G) naturally acts on Quot X,H (G, P ) from the right: if g Aut(G) then [q] g := [q g]. (1) Now, g belongs to the isotropy group of [q], if and only if q and q g represent equivalent surjective maps, i.e. if there is an automorphism ϕ : F F such that the diagram G g G q q commutes. As q is surjective, ϕ is uniquely determined by g. In this way we obtain an injective group homomorphism F ϕ F Aut(G) [q] Aut(F ) (2) Quotient schemes will reappear in the context of moduli spaces of sheaves. For the moment we specialise to Hilbert schemes of points.

19 Symplectic Moduli Spaces Hilbert schemes Let X be an irreducible projective surface and let n N 0 be a natural number that we consider as a constant polynomial. Definition 2.9 Hilb n (X) := Quot X,H (O X, n) is called the n-th Hilbert scheme of X. Remark 2.10 If X is a surface of Kähler type, the complex analytic analogues of Quot-schemes were constructed by Douady [5]. The resulting moduli spaces are usually called Douady spaces. It follows from results of Varouchas [23] that these Douady spaces are again Kähler. The name Hilbert schemes is due to Grothendieck. As we will work in the algebraic category throughout the lectures, we will stick to this terminology. All topological results are valid in both categories. Closed points in Hilb n (X) correspond to exact sequences 0 I ξ O X O ξ 0, where I ξ is the ideal sheaf of a zero-dimensional closed subscheme ξ X of length l(ξ) := dim C H 0 (O ξ ) = n. Since Hilb n (X) represents the functor of families of such subschemes there is a universal subscheme Ξ n Hilb n (X) X. Any set of pairwise distinct points x 1,..., x n X defines a point in Hilb n (X). More precisely, let U X n denote the open subset of all n- tuples of pairwise disjoint points. The symmetric group S n acts freely on U. The configuration space C n := U/S n is a smooth variety that embeds as an open subscheme into Hilb n (X). Thus we can think of the Hilbert scheme as a particular way of compactifying the configuration space of unordered n-tuples of distinct points on X. We will see shortly what happens when some of the points collide. Corollary 2.8 specialises to Corollary 2.11 Let [ξ] Hilb n (X) be a point corresponding to a closed subscheme ξ X with ideal sheaf I ξ and structure sheaf O ξ = O X /I ξ. Then T [ξ] Hilb n (X) = Hom Oξ (I ξ /I 2 ξ, O ξ). The following theorem is at the base of all the miraculous properties of the Hilbert schemes. One should point out, that it fails if dim(x) 3 and n 4:

20 158 M. Lehn Theorem 2.12 (Fogarty [7]) If X is an irreducible smooth surface, then Hilb n (X) is a smooth irreducible variety of dimension 2n. Proof. Since Hilb n (X) contains the configuration space as a smooth open subscheme of dimension 2n, it suffices to prove the following assertions: 1. Hilb n (X) is connected. 2. dim C T [ξ] Hilb n (X) = 2n for all [ξ] Hilb n (X). Ad 1. In fact, this assertion is true for an arbitrary connected variety X. It can be proved easily by induction on n, the case Hilb 1 (X) = X red being obvious. Let ξ X be a subscheme of length n, let x X be an arbitrary point and λ : I ξ k(x) a surjective homomorphism. Then ker(λ) is the ideal sheaf of a subscheme ξ of length n + 1. Moreover, any subscheme of length n + 1 arises in this way for some triple (ξ, x, λ). Let I Ξ be the ideal sheaf of the universal subscheme Ξ Hilb n (X) X. Recall that a point in the fibre of P(I Ξ ) over (ξ, x) Hilb n (X) X is precisely a surjective homomorphism λ : I ξ k(x). Mapping λ to the subscheme given by ker(λ) defines a surjective morphism ψ : P(I Ξ ) Hilb n+1 (X). By induction, Hilb n (X) is connected. The fibres of the projection P(I Ξ ) Hilb n (X) X are projective spaces P(I Ξ k(x)) (of varying dimension) and hence connected. It follows that P(I Ξ ) and its image Hilb n+1 (X) are connected. Ad 2. Using standard exact sequences we get T [ξ] Hilb n (X) = Hom(I ξ, O ξ ) = Ext 1 (O ξ, O ξ ). We need to show that dim Ext 1 (O ξ, O ξ ) = 2n. Since Hom(O ξ, O ξ ) = H 0 (O ξ ) = C n and, by Serre duality, Ext 2 (O ξ, O ξ ) = H 0 (O ξ K X ) = C n, it suffices to show that 2 χ(o ξ, O ξ ) = dim Ext i (O ξ, O ξ ) = 0. i=0 This follows from Hirzebruch-Riemann-Roch. For small n, one can describe Hilb n (X) more explicitly. The cases n = 0, 1 are trivial: Hilb 0 (X) consists of a single point, the empty subscheme in X, and subschemes of length 1 are closed points, so Hilb 1 (X) = X.

21 Symplectic Moduli Spaces 159 Points in Hilb 2 (X) either correspond to an unordered pair of distinct points in X, or to a subscheme ξ of length 2 supported at a point p X. If m is the ideal sheaf of p and I the ideal sheaf of ξ, we must have m I m 2. Thus I is determined by a one-dimensional linear subspace I/m 2 in the cotangent space T p X = m/m 2. These lines form a P 1. Intuitively, we may think of ξ as the limit of a sequence of pairs of points (p t, p t ) that approach p along a fixed tangent line through p. Recall the construction of section 2.1: Let ρ : Z X X be the blowup along the diagonal and let p : Z Y be the quotient map for the Z/2-action on Z. Then the map p (pr 1 )ρ is a closed immersion of Z into Y X. The projection p : Z Y is flat and and finite of degree 2. Associated to this family there is a classifying morphism ψ : Y Hilb 2 (X). This is a bijection of smooth varieties and hence an isomorphism. For n 3, similar explicit constructions are not so readily at hand. 2.5 Punctual Hilbert schemes We have two compactifications of the configuration space C n : The Hilbert scheme Hilb n (X) and the symmetric product S n (X). They are related as follows. There is a morphism ρ n : Hilb n (X) S n (X) = X n /S n, ξ x ξ l(o ξ,x ) x that maps ξ to its weighted support, in other words: ρ remembers the underlying subspace of ξ and the multiplicities but forgets the subscheme structures. This morphism is called the Hilbert-Chow morphism. ρ n is an isomorphism over the open subscheme C n S n (X). We need to get some understanding of the fibres of ρ n. The worst fibre is H n = ρ 1 n (n p) for some point p X. It is clear that up to isomorphism H n does not depend on X or p. More general fibres of ρ n can then be expressed in terms of these H ν : if u := n i x i S n (X) with distinct points x 1,..., x s and multiplicities n 1,..., n s, then ρ 1 n (u) = H n1... H ns. For small n, H n can be described explicitly. Let m denote the maximal ideal in O X,p for p X. Elements in H n correspond to ideals I m such that dim C O X /I = n. Clearly, H 1 consists of a single point {m}. And we have seen above that H 2 = P 1 = P(T p X) is the space of tangent directions at p. Let n = 3. There are two different types of points in H 3 distinguished by dim T p ξ {1, 2}: There is a unique point ξ 0, given by the ideal sheaf

22 160 M. Lehn m 2, with dim T p ξ 0 = 2. All others correspond to ideals of the form I = (y + αx + βx 2, x 3 ) for some regular parameters x, y m. One can show that the set of points of this type forms a line bundle over the space P(m/m 2 ). Adding the point ξ 0 compactifies this line bundle. In fact, H 3 is isomorphic to a cone in P 4 of a twisted cubic line in P 3 : explicitly, let a, b, c, d, w be homogeneous coordinates in P 4, let S be the closed subscheme cut out by the polynomials ac b 2, ad bc, bd c 2, and let Σ S C 2 be the closed subscheme defined by the ideal. I = (ax + by + uy 2, bx + cy uxy, cx + dy + ux 2 ) + (x, y) 3. Then Σ is an S-flat family of subschemes of length 3 in X. The point (0 : 0 : 0 : 0 : 1) S parameterises the ideal (x, y) 2, and the point (1 : s : s 2 : s 3 : t) parameterises the ideal (x + sy + ty 2, y 3 ). Now S = H 3. For n 4 the picture essentially remains the same though it gets more complicated in the details: there is an open subset H n H n that parameterises subschemes ξ with dim T p ξ = 1. Such ξ can be characterised by the fact that ξ is contained in a smooth curve through p. They are called curvilinear. In appropriate coordinates the ideal sheaf of ξ can be written as (y + a 1 x +..., a n 1 x n 1, x n ). It is not difficult to see that H n is isomorphic to an affine bundle over P(T p X) with fibre A n 2. Subschemes ξ that are fat in the sense that dim T p ξ = 2 are difficult to deschribe. The following theorem is therefore of great importance. One of its consequences is that we have at least a dimension bound for the locus of non-curvilinear points. This suffices for many purposes. Theorem 2.13 (Briançon [3]) The subscheme H n of curvilinear subschemes is open and dense in H n. In particular, H n is irreducible of dimension n 1. Briançons s theorem allows one to derive dimension estimates for the various strata of the Hilbert scheme. Let Z n1,...,n s denote the locally closed subset of Hilb n (X) that consists of all ξ such that ρ(ξ) = i n ix i with pairwise distinct points x i. Corollary 2.14 dim(z n1,...,n s ) = n + s. Proof. The image ρ(z n1,...,n s ) is a smooth locally closed subscheme of S n (X) of dimension 2s. Moreover, for any point i n ix i in the image, the fibre

23 Symplectic Moduli Spaces 161 ρ 1 ( i n ix i ) = i H n i has dimension i (n i 1) = n s, by Briançon s theorem. It follows that dim(z n1,...,n s ) = n + s. It follows that the locus of all ξ Hilb n (X) where at least two points coincide is a divisor, and that the locus of worse collisions than that is of codimension Beauville s theorem We are now ready to state and prove Beauville s theorem. Theorem 2.15 (Beauville) Let X be a smooth projective surface and let n π 1 (Hilb n (X)) = π 1 (X) ab. 2. There is an isomorphism H 2 (Hilb(X); C) = H 2 (X; C) Λ 2 H 1 (X; C) C[E]. 3. If X is a K3-surface, then Hilb n (X) is an irreducible holomorphic symplectic manifold of dimension 2n 4 and second Betti number b 2 (X) = 23. Recall that we already discussed the case n = 2 in section 2.1 in detail. The point is that by means of dimension arguments, we can essentially reduce everything to this special case. Proof. Let X n be the open subset of X n consisting of all n-tuples (x 1,..., x n ) where at most two of the x i are equal. Then the complement of X n has codimension 4 in X n. Moreover, let ij = {(x 1,..., x n ) X n x i = x j } for i j, and let = i>j ij. Let ρ : B X n be the blowing-up of X n along, and let E ij := ρ 1 ( ij ) be the exceptional divisors, E = i>j E ij. The symmetric group S n acts on B so that ρ becomes equivariant. Next, let p : X n S n (X) be the quotient map for the S n action, let S n (X) := p(x n) and := p( ). Note that = p( ij ) for any pair i > j. Now observe that locally near a point in, S n (X) is, up to an étale cover, isomorphic to S 2 (X) X n 2. Thus, up to an étale cover, the Hilbert-Chow morphism ρ n : Hilb n (X) S n (X) is isomorphic to ρ 2 id X n 2 : Hilb 2 (X) X n 2 S 2 (X) X n 2.

24 162 M. Lehn Over the open subset S n (X) the Hilbert-Chow morphism is the blow-up along. (This is infact true for all of S n (X), due to a theorem of Haiman, but much more difficult to prove.) It follows that the quotient of B by S n is isomorphic to Hilb n (X) := ρ 1 n (S n (X) ). We get a commutative diagram B p Hilb n (X) ρ (X n ) p ρ n S n (X), Let E = ρ 1 n ( ) = p (E ). Note that the complements of Hilb n (X) and (X n ) in Hilb n (X) and X n have complex codimension 2 and 4, respectively. It follows that π 1 (Hilb n (X) ) = π 1 (Hilb n (X)) and π 1 (X n ) = π 1 (X n ) = π 1 (X) n. Arguing precisely along the same lines as in section 2.1, it follows that π 1 (Hilb n (X) \ E) = π 1 (X) n S n, and that gluing in a tubular neighbourhood of E introduces a transposition as additional relation. It follows that π 1 (Hilb n (X)) = π 1 (Hilb n (X) ) = π 1 (X) n S n / S n = π 1 (X) ab. The last isomorphism is again a consequence of the following algebraic lemma. Lemma 2.16 Let G be any group and n 2. Then G n S n / S n = G/[G, G] = G ab. Proof. Exercise. This proves part 1. of the theorem. In particular, if X is a K3-surface then π 1 (Hilb n (X)) = 0. For the same codimension reasons the inclusions X n Xn and Hilb n (X) Hilb n (X) induce isomorphisms for all cohomology groups H i, i 2. Blowingup X n adds a direct summand Q to H 2 (X n ; Q) for each exceptional divisor E ij. One finds H 2 (B; Q) = i pr i H 2 (X; Q) i>j ( pr i H 1 (X; Q) pr j H 1 (X; Q) Q[E ij ] ).

25 Symplectic Moduli Spaces 163 Then H 2 (Hilb n (X); Q) = H 2 (Hilb n (X) ; Q) is the S n -invariant part of this vector space. Note that S n permutes all summands in the expected way but introduces an additional sign whenever the two factors of a summand of the form H 1 H 1 are exchanged. This yields H 2 (Hilb n (X); Q) = H 2 (X; Q) Λ 2 H 1 (X; Q) Q[E]. In particular, if X is a K3-surface, then H 2 (Hilb n (X); Q) = H 2 (X, Q) Q[E]. Next, the restriction homomorphism Γ(Hilb n (X), Ω 2 ) Γ(Hilb n (X), Ω 2 ) is an isomorphism by Hartogs s theorem, since the complement of Hilb n (X) has complex codimension 2. Moreover, if σ is a 2-form on Hilb n (X) such that σ Hilb n (X) is non-degenerate then σ is also non-degenerate: namely, the degeneracy locus of σ is the locus where the adjoint map T Hilb n (X) Ω 1 Hilb n (X) fails to have maximal rank. This locus is therefore either empty or a divisor cut out by the determinant of this bundle map. For dimension reasons no divisor can be contained in the complement of Hilb n (X). This proves the claim. Thus symplectic structures on Hilb n (X) and on Hilb n (X) are the same. But now the same local calculations an in section 2.1 show that there is a natural isomorphism Γ(Hilb n (X), Ω 2 ) = Γ(Hilb n (X), Ω 2 ) = Γ(X n, Ω2 ) Sn = Γ(X, Ω 2 X ) Λ2 Γ(X, Ω 1 X ). In particular, if X is a K3-surface, then Γ(X, Ω 1 X ) = 0 and hence Γ(Hilbn (X), Ω 2 Hilb n (X) ) = Γ(X, Ω2 X ). By the arguments just given, the symplectic structure on X induces a symplectic structure on Hilb n (X). There are general results about the structure of H (Y ) for irreducible symplectic manifolds Y due to Bogomolov, Beauville, and others (see the lectures of Huybrechts in [12]). For example, there is an integral quadratic form q : H 2 (Y ; Z) Z of signature (3, b 2 3) that generalises the intersection pairing of H 2 (K3; Z). By a theorem of Verbitsky, the subring in H (X; C), that is generated by H 2 (Y ; C), is isomorphic to S H 2 (Y ; C)/I, where I is the ideal generated by α n+1, for all α H 2 (Y ) with q(α) = 0. On the other hand, much more is known for Hilbert schemes of K3- surfaces. By results of Göttsche [9], Nakajima [20], and Grojnowski [11], there is a natural isomorphism of bigraded vector spaces 4n n=0 i=0 H i (Hilb(X; Q))t i q n = S ( m=1 j=0 4 H j (X; Q)t 2m+j 2 q m) (3)

26 164 M. Lehn for an arbitrary smooth projective surface X, where formal variables t and q have been thrown in in order to make the relevant bigrading transparent. The symmetric product has to be taken in the super sense. This means that it is the symmetric product in the ordinary sense on the even part of the cohomology (with respect to the cohomological grading) and the alternating product on the odd part of the cohomology. Moreover, Nakajima and Grojnowski show that a certain infinite dimensional Heisenberg algebra acts naturally on both sides of equation (3) and that (3) is an isomorphism of representations of this Lie algebra. The ring structure on H (Hilb n (X); Q) for a K3-surface X was computed by Lehn and Sorger [17]. We refer to the lecture notes by Ellingsrud and Göttsche for a discussion of these results [6]. 2.7 Generalised Kummer varieties We come to Beauville s second series. Let A = C 2 /Γ be an abelian surface. The group structure + on A provides us with a summation morphism A... A A, and since + is commutative, this morphism factors through S n (A) A. Let Σ n : Hilb n (A) S n (A) A be the composition with the Hilbert-Chow morphism. Definition 2.17 Let n 1. Then K n (A) := Σ 1 n+1 (0) is called the n-th generalised Kummer variety. For any a A we have a translation morphism t a : A A, x x + a, and an induced morphism t [n] a : Hilb n (A) Hilb n (A). Let m : A K n Hilb n+1 (A) be the morphism m(a, x) := t [n+1] a (x). The following diagram is cartesian: m A K n (A) Hilb n+1 (A) pr 1 A t (n+1)a Σn+1 This shows that Σ n+1 is a fibration with fibre K n (A) that is locally trivial in the étale topology. In particular, K n (A) is a compact complex manifold of dimension 2n. We have seen that K 1 (A) is the classical Kummer variety. This explains the name for the higher dimensional K n (A). Theorem 2.18 (Beauville) Let n 2. Then K n (A) is an irreducible holomorphic symplectic manifold of dimension 2n with b 2 = 7. Furthermore, H 2 (K n (A); C) = H 2 (A; C) C[E ], A

27 Symplectic Moduli Spaces 165 where E = E K n (A), E Hilb n+1 (A) denoting the exceptional divisor. Proof. By the same arguments as in section 2.2 there is a short exact sequence 0 π 1 (K n (A)) π 1 (Hilb n+1 (A)) π 1 (A) 0, from which one deduces that π 1 (K n (A)) since π 1 (Hilb n+1 (A)) = π 1 (A) = Z 4. Recall that the reasoning of the previous section show that there is an isomorphism H 2 (Hilb n+1 (A); C) = H 2 (A; C) Λ 2 H 1 (A; C) C[E]. For an abelian surface H 2 (A; C) and Λ 2 H 1 (A; C) are naturally isomorphic. There is a spectral sequence for the cohomology with twisted coefficients E pq 2 = Hp (A; {H q (K n (A); C)}) H p+q (Hilb n+1 (A); C). We know already that H 1 (K n (A); C) = 0, because K n (A) is simply-connected. The spectral sequence therefore yields an exact sequence 0 H 2 (A, C) Σ n+1 H 2 (Hilb n+1 (A); C) H 2 (K n (A); C) Γ. One can show using Mayer-Vietoris type arguments that the natural map H 2 (Hilb n+1 (A); C) H 2 (K n (A); C) is surjective. It follows that H 2 (K n (A); C) is obtained from H 2 (Hilb n+1 (A)) by cancelling one of the two components H 2 (A; C). Finally, a local calculation shows that the restriction of the symplectic structure to K n (A) remains non-degenerate. The complete cohomology groups of K n (A) have been computed by Göttsche and Soergel [10]. The ring structure on H (K n (A); C) has been described by Britze [4]. 3 Moduli spaces of semistable sheaves There is a different approach to Hilbert schemes that also gives a more conceptual explanation why Hilbert schemes of K3 surfaces and generalised Kummer varieties have a natural symplectic structure. To this end we need to introduce the concept of moduli spaces of sheaves.

28 166 M. Lehn In general, the classification of coherent sheaves on a projective manifold splits into a discrete and a continuous part: there is a discrete set of possible topological invariants of a sheaf F like its rank r(f ), i.e. the dimension of the fibre F η at the generic point η X, or its Chern classes c i (F ) H 2i (X, Z). Once the topological data are fixed, sheaves of the same data appear in continuous families. More precisely, a family of sheaves on X parametrised by a scheme S is an S-flat coherent sheaf F Coh(S X). For each closed point s S one gets a sheaf F s := k(s) F on X, and we think of S as parameterising the set {F s } s S. As we mentioned before, without further restrictions there are too many sheaves to be parameterised by a noetherian scheme. The relevant restriction that is usually imposed is that the sheaves be semistable. For a detailed discussion of semistable sheaves and their moduli theory I refer to [15] and the references given therein. 3.1 Semistable sheaves Let X be a projective manifold and let H be an ample divisor. The Hirzebruch- Riemann-Roch theorem allows one to express the holomorphic Euler characteristic of coherent sheaves F and G in terms of their ranks and Chern classes, namely χ(f ) := χ(f, G) := dim(x) i=0 dim(x) i=0 ( 1) i dim H i (X, F ) = X ( 1) i dim Ext i (F, G) = X ch(f )td(x) ch(f ) ch(g)td(x), where = ( 1) i : H 2i (X; Q) H 2i (X; Q). In particular, the Hilbert polynomial P (F, n) = e nh ch(f )td(x) X is fixed by r(f ) and c i (F ). For a surface X and a coherent sheaf F of rank r and with Chern classes c 1, c 2, one gets P (F, n) = r 2 H2 n 2 + (Hc 1 r 2 K XH)n + (rχ(o X ) 1 2 c 1K X c2 1 c 2 ) χ(f, F ) = r 2 χ(o X ) (2rc 2 (r 1)c 2 1 ). Definition 3.1 Let X be projective manifold, H an ample divisor. A coherent sheaf of rank r(f ) > 0 is called stable (respectively semistable) if

29 Symplectic Moduli Spaces 167 F is torsion free and if for all subsheaves F, 0 F F, one has (resprectively ). P (F ) r(f ) < P (F ) r(f ) Here and in the following we write f < g (f g) for two polynomials f and g if f(n) < g(n) (respectively ) for all n 0. The quotient p(f ) = P (F )/r(f ) is called the reduced Hilbert polynomial. Any torsion free coherent sheaf F has a unique filtration, the so-called Harder-Narasimhan filtration, 0 = F 0 F 1... F l = F such that all factors F i /F i 1 are semistable with a decreasing sequence of reduced Hilbert polynomials: p(f 1 /F 0 ) > p(f 2 /F 1 ) >... > p(f l /F l 1 ) In this sense semistable sheaves can be thought of as building blocks for arbitrary torsion free sheaves. We have defined (semi)stability in terms of subsheaves. It is easy to see that one can characterise (semi)stables sheaves equivalently by the following property: A torsion free sheaf F is stable (respectively semistable) if for all surjections π : F F to a torsion free sheaf F with F 0, ker(π) 0, one has P (F ) r(f ) < P (F ) r(f ) (respectively ). Let F and F be semistable sheaves with P (F )/r(f ) = P (F )/r(f ) and let ϕ : F F be a nontrivial Homomorphism. Being both a quotient sheaf of F and a subsheaf of F, the image sheaf im(ϕ) must satisfy the following inequalities: P (F ) r(f ) P (im(ϕ)) r(im(ϕ)) P (F ) r(f ). Since the outer terms are equal, we have equality everywhere. In particular, if F is stable, it follows that F = im(ϕ), i.e. ϕ must be injective, and if F is stable, then im(ϕ) = F, i.e. ϕ must be surjective, and if both F and F are stable, ϕ must be an isomorphism. We conclude: Lemma 3.2 If F and F are stable sheaves with P (F ) = P (F ) then { Hom(F, F C, if F = F ) =, and 0 else.

30 168 M. Lehn Proof. The arguments given above show that if F = F then there are no non-trivial homomorphisms from F to F. Assuming to the contrary that F = F we must argue that End(F ) = C. But again the discussion showed that any non-trivial endomorphism of F is an isomorphism. This means that End(F ) is a finite dimensional skew field over C. But the only finite dimensional skew field over C is C itself. A semistable sheaf F is said to be strictly semistable, if it is not stable. By definition, a strictly semistable sheaf F admits an exact sequence 0 F F F 0, with semistable sheaves F and F that satisfy P (F )/r(f ) = P (F )/r(f ) = P (F )/r(f ). If F or F are not stable, we can further decompose these sheaves, and iterating the process we end up with a filtration, a so-called Jordan-Hölder filtration, 0 = F 0 F 1... F s = F with stable factors gr i (F ) = F i /F i 1, all of the same reduced Hilbert polynomial. This filtration is not unique, but the associated sheaf gr(f ) = i gr i (F ) is. Definition 3.3 Two semistable sheaves F and F are S-equivalent if gr(f ) = gr(f ). The importance of S-equivalence is based on the following phenomenon: Lemma 3.4 Let F be a semistable sheaf and let gr(f ) be the sheaf associated to a Jordan-Hölder filtration of F. Then there exists a family F parameterised by A 1, such that F 0 = gr(f ) and F A 1 \{0} = O A 1 \{0} F. Proof. For simplicity, we prove the lemma only for the case when the Jordan- Hölder filtration has length 2, i.e. when F fits into a short exact sequence 0 F F F 0 with stable sheaves F, F with the same reduced Hilbert polynomial. The general case follows from an easy modification of the argument (exercise). We write A = SpecC[t]. Consider the subsheaf F = F C[t] + F tc[t] in the trivial SpecC[t]-family F C[t]. Then F is flat over C[t] and