TAUTOLOGICAL SHEAVES : STABILITY, M TitleSPACES AND RESTRICTIONS TO GENERALI KUMMER VARIETIES. Citation Osaka Journal of Mathematics.

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1 TAUTOLOGICAL SHEAVES : STABILITY, M TiteSPACES AND RESTRICTIONS TO GENERALI KUMMER VARIETIES Author(s) Wande, Mate Citation Osaka Journa of Mathematics. 53(4) Issue Date Text Version pubisher URL DOI Rights Osaka University

2 Wande, M. Osaka J. Math. 53 (2016), TAUTOLOGICAL SHEAVES: STABILITY, MODULI SPACES AND RESTRICTIONS TO GENERALISED KUMMER VARIETIES MALTE WANDEL (Received January 16, 2014, revised September 9, 2015) Abstract Resuts on stabiity of tautoogica sheaves on Hibert schemes of points are extended to higher dimensions and to the restriction of tautoogica sheaves to generaised Kummer varieties. This provides a big cass of new exampes of stabe sheaves on higher dimensiona irreducibe sympectic manifods. Some aspects of deformations of tautoogica sheaves are studied. Contents 0. Introduction Summary of the resuts Structure of the paper Notations and Conventions Preiminaries Geometric considerations Tautoogica sheaves Poarisations and sopes Higher n Restriction to generaised Kummer varieties Restriction to the Kummer surface Generaised Kummer varieties of dimension four Deformations and modui spaces of tautoogica sheaves Deformations of tautoogica sheaves The additiona deformations and singuar modui spaces Deformations of the manifod X [n] Acknowedgements References Introduction In the theory of compact Ricci fat manifods one kind of basic buiding bocks for these manifods are irreducibe hoomorphic sympectic (short: irreducibe sympectic) manifods. These are compact käher manifods that are simpy connected and admit a up to scaar unique everywhere non-degenerate hoomorphic two-form. One of the fundamenta aspects in the theory of irreducibe sympectic manifods is the fact that ony few exampes have been constructed. Up to now, we are ony aware of the 2010 Mathematics Subject Cassification. 14D20, 14F05, 14J28, 14J60, 14K05.

3 890 M. WANDEL existence of two infinite series of deformation casses due to Beauvie ([1]) and two more sporadic exampes found by O Grady ([15, 16]). Modui spaces of sheaves on sympectic surfaces pay an important roe in the construction of irreducibe sympectic manifods. The fundamenta resut of Mukai ([14]) states that the modui space of stabe sheaves on a K 3 or abeian surface is a smooth sympectic variety. In fact, under mid technica conditions these spaces turn out to be irreducibe sympectic manifods. A natura question that arises is the foowing: Can we iterate this process? That is, can we construct new exampes of irreducibe sympectic manifods using modui spaces of sheaves on known higher dimensiona irreducibe sympectic manifods such as Hibert schemes of points on K 3 surfaces or generaised Kummer varieties? Certainy it is difficut to answer this question in this generaity. On the other hand amost no exampes of stabe sheaves on higher dimensiona irreducibe sympectic manifods had been encountered. In [18] the first exampe of a rank two stabe vector bunde on the Hibert scheme of two points on a K 3 surface was found. In [19] this resut was drasticay generaised continuing aong the foowing concept: Start with a stabe sheaf on a K 3 surface, transfer this sheaf to the Hibert schemes of points using the universa property of the atter and obtain what is caed a tautoogica sheaf and, finay, prove its stabiity. This artice is to be understood as a seque to [19]. We further extend its resuts and transfer them to the case of generaised Kummer varieties. The atter are cosed subvarieties of the Hibert schemes of points on abeian surfaces. We study the restriction of tautoogica sheaves to Kummer varieties and obtain non-trivia exampes of stabe sheaves on these manifods. Summary of the resuts. Let X be a reguar (i.e. h 1 (X, O X ) 0) smooth projective surface. We study the stabiity of tautoogica sheaves with respect to an ampe cass H N on the Hibert scheme which is naturay associated with an ampe cass H on the underying surface and depending on an integer N 0. Proposition (Proposition 2.4). Let F be a torsion-free H -stabe sheaf on X. Assume that its refexive hu F O X. Then the tautoogica sheaf F [n] on X [n] does not contain HN -destabiising subsheaves of rank one for a N 0. From this we can deduce: Theorem (Theorem 2.5). Let F be a torsion-free rank one sheaf on X satisfying detf O X. Then for a sufficienty arge N the associated rank three sheaf F [3] on X [3] is HN -stabe. If X is abeian, then inside the Hibert scheme X [n] there is the generaised Kummer variety K n (X). Let us denote the embedding by j. On X we have the natura invoution

4 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 891 from the group structure. A sheaf H is caed symmetric if H H. We have the foowing resuts: Theorem (Theorems 3.7, 3.9 and 3.12). Let F be a H -stabe sheaf on X such that detf O X. There is a poarisation on K 3 (X) such that if F is of rank one, j F [3] is -stabe of rank three. Furthermore, there is a poarisation on K 2 (X) (the Kummer surface associated with X) such that if detf is not symmetric and F is of rank one (rank two), the restriction j F [2] is -stabe of rank two (rank four). Furthermore, we have the foowing reation between modui spaces of sheaves on K 3 surfaces and modui spaces of tautoogica sheaves: Proposition (Proposition 4.4). Let F be a stabe sheaf on a K 3 surface X of Mukai vector Ú such that F [2] is stabe (of Mukai vector Ú [2] ). We have an embedding of modui spaces M s (Ú) M s (Ú [2] ). Structure of the paper. The paper is organised as foows: We begin in Section 1.1 by coecting known resuts on the geometry of Hibert schemes of points on a surface and prove a few technica emmata which wi be needed ater. Next, in Section 1.2 we introduce the main objects of this artice, the tautoogica sheaves. In Section 1.3 we introduce poarisations on the Hibert schemes and compute the sopes of tautoogica sheaves with respect to these poarisations. In Section 2 we anayse destabiising subsheaves of tautoogica sheaves on Hibert schemes of three or more points. The case of generaised Kummer varieties is treated in Section 3. We prove the stabiity of the restriction of certain tautoogica sheaves to the Kummer surface (Section 3.1) and the generaised Kummer variety of dimension four (Section 3.2). We concude the paper by studying deformations of tautoogica sheaves in Section 4. We show that the modui spaces of tautoogica sheaves can be singuar in Section 4.2 and investigate in which way we may deform tautoogica sheaves together with the underying manifod in Section 4.3. Notations and conventions. The base fied of a varieties and schemes is the fied of compex numbers. A functors such as pushforward, puback, oca and goba homomorphisms and tensor product are not derived uness mentioned otherwise. By M s (Ú) we denote the modui space of -stabe sheaves with numerica invariants Ú. (We assume a poarisation has been fixed.) 1. Preiminaries Throughout this chapter we consider a smooth projective surface X together with a poarisation H ¾ NS(X).

5 p 892 M. WANDEL 1.1. Geometric considerations. For n 2 the geometry of the Hibert scheme points on a surface is very we accessibe: In fact, X [2] is the bowup of the symmetric square S 2 X aong the diagona. If n 2, the situation is a itte bit more compicated but by [8, Proposition 2.6] the Hibert scheme is sti the bowup aong the big diagona. This is important, especiay if we want to determine the Picard group of the Hibert scheme. Let us summarise the most important resuts. Foowing [4, Section 1], we consider the foowing diagram: X [n 1,n] n X n 1 X X [n 1] X [n] X n X [n 1] X S n X. Û p q q Here we denote by n Ï {(x, ) x ¾ } X X [n] the universa subscheme and by X [n 1,n] Ï {( ¼, ) ¼ } X [n 1] X [n] the so-caed nested Hibert scheme. We have the fat degree n covering p Ï n X [n] which is, in fact, the restriction of the second projection X X [n] X [n]. Furthermore, X [n 1,n] is isomorphic to [n 1] the bowup of X X aong the universa subscheme n 1. Denote this bowup morphism by and the projections from X X [n 1] to X [n 1] and X by p and q, respectivey. By [5, Proposition 2.1] the second projection Ï X [n 1,n] X [n] factors through n and from [9, Proposition 3.5.3] it foows that Û is an isomorphism outside codimension four subschemes. Thus the morphism is fat outside codimension four. Finay, we have q Æ q Æ Û and Û p. We have and embeddings Pic 0 X [n] Pic 0 X and ( ) X [n] Ï NS X NS X [n], X [n] Ï ( n ) S n ( ) X [n] Ï Pic X Pic X [n]. Furthermore, there is a cass Æ ¾ NS X [n], such that 2Æ is the cass of the divisor consisting of a non-reduced subschemes X.

6 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 893 Now, using Section 2 of [4] and the proof of Theorem 4.2 of [13], we can deduce the foowing formuas: Lemma 1.1. Let D be the exceptiona divisor of and et be a cass in NS X. We have and Æ [D] p Æ X [n] (p X [n 1] q ). If X is reguar (i.e. h 1 (X, O X ) 0), we have NS X [n] NS X Æ by a resut of Fogarty (cf. [6]). Finay, et us briefy introduce the generaised Kummer varieties. If one mimics the construction of Hibert schemes to the case of abeian surfaces, one again obtains Ricci fat manifods. But they are not simpy connected and contain an additiona factor in the Beauvie Bogomoov decomposition. To get rid of this factor we consider (for an abeian surface A) the fibre K n (A) Ï m 1 (0) and ca it generaised Kummer variety. It is a (2n 2)-dimensiona irreducibe sympectic manifod (cf. [1]). In the case n 2 this just gives the Kummer surface Km A. Let us denote the incusion K n (A) A [n] by j. It is a we known fact (again cf. [6]) that we have: NS(K n (A)) j NS(A) j Æ, where we embedded, as before, NS(A) into NS(A [n] ) Tautoogica sheaves. Let us give the definition of tautoogica sheaves, the objects of main interest in this artice. Fix a sheaf F on X and reca that there is the universa subscheme n X X [n]. Furthermore we have the two projections pï X X [n] X [n] and q Ï X X [n] X. DEFINITION 1.2. The tautoogica sheaf associated with F is defined as F [n] Ï p (q F ÅO n ). REMARK 1.3. Very important for the study of tautoogica sheaves is the foowing observation: The universa subscheme n and the nested Hibert scheme [n 1,n] X

7 894 M. WANDEL are isomorphic outside codimension four subschemes (cf. Section 1.1). Let U denote the open subset where they are actuay isomorphic. The restrictions of q F and q F to U are naturay isomorphic. Thus the restriction of F [n] to the image p(u) in X [n] is isomorphic to F [n] Ï q F (restricted to (U) p(u)). Hence we can use F [n] instead of F [n] as ong as we want to study properties that are not sensibe with respect to modifications in codimension four. In the case n 2 we, in fact, have F [2] F [2]. The restriction of p to n is a fat covering of degree n. Hence the tautoogica sheaf F [n] is ocay free whenever F is. For the tautoogica sheaf associated with the dua sheaf F we have the foowing formuas which wi be important ater. Lemma 1.4. (1) (F ) [n] p Hom(q F, O ) and (2) (F [n] ) p Hom(q F, p ). Again, from [13] we deduce: Lemma 1.5. We have the foowing formua for the first Chern cass of F [n] : c 1 (F [n] ) c 1 (F) X [n] rk(f)æ. Next, we want to summarise the resuts of Scaa and Krug about goba sections and extensions of tautoogica sheaves. These formuas turn out to be a powerfu too to anayse stabiity and deformations of these sheaves. Theorem 1.6. For every sheaf F and every ine bunde L on X we have H (X [n], F [n] ÅL X [n]) H (X, F ÅL) Å S n 1 H (X, L). Proof. [17, Coroary 4.5], [12, Theorem 6.17]. We continue by stating Krug s formua for the extension groups of tautoogica sheaves: Theorem 1.7. Let F and E be sheaves and L and M be ine bundes on X. We have Ext X [n](e [n] ÅL X [n], F [n] ÅM X [n]) (3) Ext X (E ÅL, F ÅM) Å Sn 1 Ext X (L, M) Ext X (E ÅL, M) Å Ext X (L, F ÅM) Å Sn 2 Ext X (L, M).

8 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 895 Proof. [12, Theorem 6.17]. Krug aso gave a description how to compute Yoneda products on these extension groups (cf. [12, Section 7]). The genera formuas are extremey ong. We wi give a more detaied account on them as needed. Let us finish this section by deriving a specia case of formua (3). Coroary 1.8. Let X be a K 3 surface and et F be a sheaf on X satisfying h 2 (F) 0. Then we have (4) Hom X [n](f [2], F [2] ) Hom X (F, F), Ext 1 X [n](f [2], F [2] ) Ext 1 X (F, F) H0 (X, F) Å H 1 (X, F). REMARK 1.9. From these equations we can deduce that tautoogica sheaves F [2] associated with stabe sheaves F O X are aways simpe: By Serre duaity a stabe sheaf F O X on a K 3 surface satisfies either h 2 (F) 0 or h 0 (F) 0 and by twisting with a suitabe ine bunde we may assume that h 2 (F) 0. This is a first indication that tautoogica sheaves might be stabe Poarisations and sopes. In this section we sha tak about poarisations on the Hibert scheme of points on a surface. In genera the ampe cone of these varieties is not competey known. Nevertheess, if we fix a poarisation H on our surface X, we wi define poarisations H N on X [n], depending on H and an integer N. Furthermore, we sha derive and discuss the sopes of tautoogica sheaves with respect to these poarisations. This wi be important when we want to study the stabiity of these sheaves in Sections 2 and 3. Fix a smooth projective surface X and an ampe cass H ¾ NS X. For any integer N we consider the cass H N Ï N H X [n] Æ ¾ NS X [n]. Using induction one easiy shows that H N is ampe for arge N. Thus we have a natura candidate for a poarisation of the Hibert scheme and, as it turns out, in many cases tautoogica sheaves are stabe with respect to these poarisations (cf. [19]) for n 2. Next, we want to compute sopes of tautoogica sheaves aso in the case n 2. Hence we need to compute intersection numbers. We have the foowing genera resut: Lemma Let be a cass in NS X. We have (5) X [n] H 2n 1 (2n 1)! X [n] (n 1)! 2 ( H)(H 2 ) n 1 n 1

9 896 M. WANDEL and (6) Æ H 2n 1 X [n] 0, where on the right hand side of (5) we consider the intersection in NS X. We wi abbreviate the factor (2n 1)!((n 1)! 2 n 1 ) by c n. Proof of Lemma Note that (6) hods triviay since H X [n] is a puback aong the Hibert Chow morphism. Let us prove (5). We pu back X [n] and H X [n] aong the n!-fod covering X n S n X and obtain the casses n and H n, respectivey. We have X [n] H 2n 1 X [n] 1 n! ( n )(H n ) 2n 1 1 n! (2n 1)! (n 1)! 2 n 1 ( H)(H 2 ) n 1. 2n 1 1, 2,, 2 n( H)(H 2 ) n 1 Coroary Let L be a ine bunde on X with first Chern cass and F a sheaf of rank r and first Chern cass f. We have the foowing expansions for the sopes of F [n] and L with respect to H N : and HN (L X [n]) N 2n 1 c n ( H)(H 2 ) n 1 O(N 2n 2 ) HN (F [n] ) N 2n 1 c n 1 nr ( f H)(H 2 ) n 1 O(N 2n 2 ). 2. Higher n In this chapter we try to generaise the resuts on destabiising ine subbundes in [19, Section 3] to higher n. From this generaisation we wi be abe to prove the stabiity of rank three tautoogica sheaves on X [3]. In this chapter we fix a poarised reguar surface (X, H). Let F be a torsion-free H -stabe sheaf on X. Denote its rank by r and its first Chern cass by f. We want to show that the associated tautoogica sheaf F [n] on X [n] has no destabiising subsheaves of rank one. We wi first assume that F is refexive, i.e. ocay free. Thus we may assume that a destabiising rank one subsheaf of F [n] is aso refexive, that is, a ine bunde. Proposition 2.1. For sufficienty arge N, there are no HN -destabiising ine subbundes in F [n] of the form L X [n], (L ¾ Pic X), except the case r 1 and L F O X. Proof. Denote the first Chern cass of L by. Using Scaa s cacuations of cohomoogy groups of tautoogica sheaves with twists as stated in Theorem 1.6 we can

10 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 897 immediatey deduce the foowing formua for homomorphisms from ine bundes of the form L X [n] to tautoogica sheaves F [n] : Hom X [n](l X [n], F [n] ) Hom X (L, F) Å S n 1 Hom X (L, O X ). Let us first assume r 1. Since F is H -stabe, we have necessary conditions for the existence of a ine subbunde of F [n] : (7) H f H r and H 0. The first inequaity is due to the stabiity of F and the second comes from the fact that if a ine bunde has a section, its first Chern cass has non-negative intersection with any ampe cass H. If L X [n] F [n] is destabiising, by Coroary 1.11 we must have H f H. nr But this is certainy a contradiction to (7). If r 1, we can proceed as above but additionay have to consider the specia case L F, i.e. H f H. The destabiising condition together with H 0 immediatey yieds H 0. But now Hom X (L,O X ) can ony be nontrivia if L O X. We wi need the anaogue of Proposition 3.1 in [19] which aows to reduce the genera case to Proposition 2.1 above. We therefore first ook at the foowing more genera set-up which wi be usefu ater, too. The proof is a straightforward induction. Lemma 2.2. Let Ï Y Z be a bow-up morphism of a smooth variety in a smooth codimension two center. Then for a a ¾. R 0 ( Åa ) O Z Appying this to our situation, we easiy find: Lemma 2.3. Let L X [n] ÅO(aÆ) be a ine bunde on X [n] (a ¾ arbitrary), then for any ocay free sheaf F on X we have Hom X [n](l X [n] ÅO(aÆ), F [n] ) Hom X [n](l X [n], F [n] ). We can thus deduce the first main resut of this section.

11 898 M. WANDEL Proposition 2.4. Let F be a torsion-free H -stabe sheaf on X. Assume that its refexive hu F O X. Then F [n] does not contain HN -destabiising subsheaves of rank one for a N 0. Proof. We can easiy reduce to the case where F is ocay free and then appy Proposition 2.1 and Lemma 2.3 above. Since the tautoogica sheaf on X [3] associated with a rank one sheaf has rank three, the above proposition is enough to show that these sheaves are stabe (except, of course). O [3] X Theorem 2.5. Let F be a torsion-free rank one sheaf on X satisfying detf O X. Then for a sufficienty arge N the associated rank three sheaf F [3] on X [3] is HN -stabe. Proof. As usua we can reduce to the case that F is ocay free. We have seen that F [3] cannot contain destabiising subsheaves of rank one. But any destabiising subsheaf of rank two yieds a rank one destabiising subsheaf of the dua sheaf. It is now enough to prove that for any ine bunde L on X [3] we have Hom(L, (F [3] ) ) Hom(L, (F ) [3] ). To show this formua we use equations (1) and (2), then adjunction p p and finay we use Lemma 2.2, keeping in mind that Û is an automorphism outside codimension four and Û p. 3. Restriction to generaised Kummer varieties In this section we study the stabiity the restrictions of tautoogica sheaves to the associated generaised Kummer varieties Restriction to the Kummer surface. In this section we sha prove the stabiity of the restriction of certain tautoogica sheaves from the Hibert scheme of two points on an abeian surface to the associated Kummer surface. Throughout this section we fix a poarised abeian surface (A, H). Let bï É A A denote the simutaneous bowup of a fixed points of the invoution on A and denote by E 1,, E 16 the exceptiona divisors. On É A we sti have an invoution which fixes the E pointwise. We consider the quotient Ï É A Km A which is a degree two covering onto the associated Kummer surface. By [2, VIII Proposition 5.1] we have a monomorphism «! b Ï H 2 (A, ) H 2 (Km A, )

12 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 899 satisfying «(x)«(y) 2xy for a x, y ¾ H 2 (A, ). We set È N (E ). It is we known that E 2 È 1 and N 2 2. Furthermore, the cass N is 2-divisibe and we have (12) N È E and N 2E. Finay, we have (8) NS É A b NS A Å16 1 E and Pic 0 É A b Pic 0 A. We define the cass H N Ï N «(H) 1 2 N on Km A, which is ampe for sufficienty arge N. (This is the restriction to Km A of the cass H N defined on A [2] in Section 1.1.) DEFINITION 3.1. Let F be a sheaf on A. We set F Km Ï b F. One easiy shows: Lemma 3.2. The sheaf F Km is the restriction of the tautoogica sheaf F [2] aong the incusion j Ï Km A A [2] : j F [2] ³ F Km. Now we want to prove the stabiity of F Km in the case that F is of rank one or two. As in the previous cases we begin with the anaysis of ine subbundes in the puback b F: Proposition 3.3. Let F be a H -stabe sheaf on A of rank r and first È Chern cass f ¾ NS A. Then b F does not contain any ine bunde L ¼ b G ÅO a E with G ¾ Pic(A), c 1 (G) g satisfying but in the case r 1, G ³ F. H g 1 r H f Proof. As usua we may assume that F is ocay free. We want to show that Hom ÉA b G ÅO a E, b F

13 900 M. WANDEL vanishes. Using adjunction (b b ) and a simiar induction argument as in the proof of [19, Proposition 3.1], we see that it is enough to prove that Hom A (G, F) 0. This easiy foows from the stabiity of F if F G. Next, we wi show that Proposition 3.3 impies that there are no destabiising ine subbundes in F Km. We ony need to cacuate sopes. Lemma 3.4. Let F be a sheaf on A of rank r and first Chern cass f. We have c 1 ( b F) «( f ) r 2 N. Proof. È We have c 1 ( ÉA ) E. Thus the Grothendieck Riemann Roch theorem reads ch( b F) (ch(b F) td ) (r, b f, ) r, b f r E,. 2 1, 1 2 E, Let L be a ine bunde on Km A. By equation (8) there is a ine bunde G on A and integers a such that L ³ b G ÅO a E. Set g Ï c 1 (G). Note that since L comes from Km A, the ine bunde G has to be symmetric, i.e. G ³ G. Coroary 3.5. Let L be a ine bunde on Km A as above. We have HN (F Km ) 1 r N H. f 4 and HN (L) N H g 1 2 Proof. We puback a casses to A: É È Note that (12) N E and «( f ) 2b f for a f ¾ NS A. Thus we have «( f ) «(H) 2 f H. Furthermore, we a. È

14 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 901 have È E 2 16 ( 1) 16 and È E È a E È a. Finay, we have to divide everything by two because we pued back aong a degree two covering. Coroary 3.6. Let F be a non-symmetric (i.e. F F) H -stabe sheaf on A. Then F Km does not contain H Km-destabiising ine subbundes for a N 0. N Proof. Let L be a destabiising ine subbunde of F Km. Again, we can write L ³ b G Å O È a E for a symmetric ine bunde G ¾ Pic A. The destabiising condition yieds H g 1 r H. f. As usua we use adjunction to obtain a homomorphism L b F. This gives a contradiction to Proposition 3.3 but in the case r 1, G ³ F. But this cannot be since F was chosen not to be symmetric. We immediatey deduce: Theorem 3.7. Let F be a non-symmetric rank one torsion-free sheaf on A. Then for a N sufficienty arge, F Km b F is a rank two H Km-stabe sheaf. N EXAMPLE 3.8. We appy the theorem to the case c 1 (F) 0. Denote by A Ç the dua abeian variety and by A[2] Ç its two-torsion points. The assignment Pic 0 A F F Km gives a map where M Ï M H Km N ÇA Ò Ç A[2] M, Km (Ú) is the modui space of H -stabe sheaves with Ú 2, 1 2 N N, 2 È Note that Ú 2 0 and the first Chern cass (12) N is primitive. Hence M is smooth of dimension two. Since F Km ³ ( F) [Km], this map is two-to-one. Furthermore, et us consider the case that F is symmetric, i.e. F ¾ A[2]. Ç We concentrate on the case F O A. We have extensions 0 O 1 N E O 0. 2 The sheaf O Km X is isomorphic to the trivia extension O (12) È N O (cf. [2, Lemma 17.2]), which is not stabe. On the other hand one can show that every nontrivia extension is H -stabe. The vector space of extensions E is two-dimensiona and.

15 902 M. WANDEL thus we have a È 1 M parametrising the E. Atogether we see that M is isomorphic to the Kummer surface Km Ç A of the dua abeian surface Ç A. If F has nontrivia first Chern cass f ¾ NS A, we may choose a symmetric ine bunde L satisfying c 1 (L) f. Then F ÅL is in Pic 0 (A) and (F ÅL) [Km] ³ F Km ÅO(«( f )). Thus the modui space containing F Km is isomorphic to Km Ç A, too. Note that by [7, Theorem 1.5] the Kummer surfaces Km A and M Km Ç A are isomorphic. We finish the section by proving the anaogue of Theorem 4.4 of [19]. Theorem 3.9. Let F be a H -stabe rank two sheaf on A such that detf is not symmetric. Then F Km is a HN -stabe rank four sheaf on Km A. Proof. We exacty imitate the proof of [19, Theorem 4.4]. Assume that F is ocay free and et f Ï c 1 (F). È Let E be a refexive semistabe rank two subsheaf of F Km and write c 1 (E) «(e) a N. The destabiising condition thus impies 2H e H f. We have a homomorphism Ï E b F. Again, the ony difficut case is when ker 0: If the first Chern cass of the Q Ï coker is trivia, we see that the homoogica dimension of Q is 2. Since b F is ocay free, this woud contradict the fact that E is refexive. Thus Q 0 and has to be an isomorphism. But since E is symmetric and F is not, we are done. If there is an effective divisor with first Chern cass c 1 (Q), the ine bunde a N b detf Å dete must have a section. Hence either a 0 and detf ³ O A (which we excuded) or a 0 and which contradicts the stabiity condition. H f 2H e 3.2. Generaised Kummer varieties of dimension four. Let (A, H) be a poarised abeian surface. In this section we prove some resuts concerning the stabiity of the restriction of tautoogica sheaves from the Hibert scheme of three points on A to the four dimensiona generaised Kummer variety j Ï K 3 (A) A [3].

16 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 903 We have an isomorphism NS(K 3 (A) j NS(A) j Æ and we define a poarisation on K 3 (A) H N Ï j H N N j H A [3] j Æ. Lemma We have (H N ) 3 j Æ 0 O(N 4 ). Proof. By definition of H N we have (H N ) 3 j Æ j ((H N ) 3 Æ). Now the emma foows from equation (6) in Lemma Proposition Let F be a H -stabe sheaf on A of rank r and first Chern cass f. If F O A, then for N sufficienty arge F K 3 Ï j F [3] does not contain any HN -destabiising subsheaves of rank one. Proof. We may assume that F is ocay free. Since a ine bundes on K 3 (A) come from A [3], we may assume that a destabiising ine bunde is of the form j M ¼ with c 1 (M ¼ ) ¾ NS(A) Æ. By a simiar reasoning as in [19, Proposition 3.1] we can reduce to the case where we have no contribution of the Æ-summand neither. Thus there is, in fact, a ine bunde M on A such that M ¼ M A [3]. Let m Ï c 1 (M). We denote the projections A 3 A by i. Furthermore et us denote the incusion of the zero fibre s 1 (0) A 3 of the group aw by i. Using adjunctions, fatness and faithfuness of the puback aong finite coverings, it is straightforward to prove Hom( j M, j F [3] ) Hom(i M 3, i 1 F) H 0 (i (M 3 Å 1 F)). In order to proceed, we choose an isomorphism s 1 (0) A 2 by sending (x, y, z) to (x, z) and denote the projections A 2 A by Ç i. In this picture we have the identifications: 1 Æ i Ç 1, 2 Æ i Ç Æ s and 3 Æ i Ç Ç 2. (Reca that denotes 1 on A.) Thus pushing forward aong 1 ( 2 in the second ine), we have (9) (10) H 0 (i (M 3 Å 1 F)) H0 (F ÅM Å Ç 1 ( Ç 2 M Å s M )) H 0 (M Å Ç 2 ( Ç 1 (F ÅM ) Å s M )). By Lemma 3.10 the destabiising condition of j M in j F [3] impies m H f H. 3r

17 904 M. WANDEL If m H 0, we see that the right hand side of (9) vanishes but in the case M O A F. If m H 0, the destabiising condition impies 2m H f H. r In this case the right hand side of (10) has to vanish. As usua, from Proposition 3.11 we can deduce the stabiity of rank three restricted tautoogica sheaves associated with rank one sheaves: Theorem Let F be a torsion-free rank one sheaf on A. Assume detf O A. Then for a sufficienty arge N the sheaf j F [3] is K N -stabe. 4. Deformations and modui spaces of tautoogica sheaves This chapter coects a few resuts on different aspects of the behaviour of tautoogica sheaves under deformations Deformations of tautoogica sheaves. In this section we wi make the foowing genera assumption: ASSUMPTION. X is a K 3 surface and F a stabe sheaf on X with invariants Ú such that for every (-stabe sheaf G ¾ M s (Ú) the associated tautoogica sheaf G [n] is aso stabe. Note that in the cases where the stabiity of tautoogica sheaves has been expicity proven the tautoogica sheaf associated with a sheaf F is stabe if and ony if it is true for every other (-stabe) G in the same modui space. (We are ony considering sheaves on K 3 surfaces.) Denote by Ú [n] ¾ H (X [n], É) the Mukai vector of F [n]. The assignment yieds a morphism F F [n] [ ] [n] Ï M s (Ú) M s (Ú [n] ). We sha mainy discuss the case n 2. Let us prove the foowing emma which shows that [ ] [2] is injective on cosed points. Lemma 4.1. For every sheaf F on X we have F Tor 1 O X X (O ½, F [2] ).

18 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 905 Thus we can reconstruct the origina sheaf F from the tautoogica sheaf F [2]. Proof. Reca that we have an exact sequence on X X: (11) 0 F [2] F 2 ½ F 0. We tensor this sequence with the structure sheaf of the diagona ½ X X. Of course we have 1 F ½ ½ 1 F F and the higher Tors TorO i X X (O ½, 1 F) vanish. Therefore we have an isomorphism By Proposition 11.8 in [11] we find Tor 1 O X X (O ½, F [2] ) Tor 2 O X X (O ½, ½ F). Tor i O X X (O ½, ½ F) F i 0, 2 and F Å Đ X i 1. REMARK 4.2. If we tensor (11) with O ½ as above, the first terms of the resuting ong exact Tor-sequence yied a short exact sequence 0 F Å Đ X F [2] ½ F 0. It is not cear if this exact sequence is spit or if it is equivaent to the natura extension corresponding to the Atiyah cass of F. Let us consider a stabe sheaf F on a K 3 surface X. The stabiity impies that either h 0 (X, F) or h 2 (X, F) h 0 (X, F ) vanishes. Let us assume the former is the case. (The case h 2 (X, F) 0 can be treated in exacty the same way.) Coroary 1.8 shows that we have a natura monomorphism [ ] [2] Ï Ext 1 (F, F) Ext 1 (F [2], F [2] ), which maps an infinitesima deformation of F to its induced deformation of F [2]. DEFINITION 4.3. We ca an infinitesima deformation of F [2], the cass of which ies in the image of [ ] [2] above, a surface deformation. Deformations ying in the other summand of equation (4) in Coroary 1.8 are referred to as additiona deformations. We concude:

19 906 M. WANDEL Proposition 4.4. We have an embedding of modui spaces M s (Ú) M s (Ú [2] ). The additiona deformations are isomorphic to H 0 (X, F) Å H 1 (X, F). Coroary 4.5. Let F be such that h 1 (X, F) 0. Then we have a oca isomorphism of the corresponding modui spaces. Coroary 4.6. Let F be such that M s (Ú) is compact and h 1 (X, G) 0 for a G ¾ M s (Ú). Then we have an isomorphism of M s (Ú) with a connected component of M s (Ú [2] ) The additiona deformations and singuar modui spaces. In the ast section we have seen that the surface deformations of tautoogica sheaves are unobstructed. This is not true for a deformations. Indeed, in this section we wi give an expicit construction of an exampe of a sheaf F on an eipticay fibred K 3 surface such that F [2] is stabe and the corresponding point in the modui space is singuar. To prove this statement et us reca the most basic properties of the Kuranishi map: The genera idea of the deformation theory of a stabe sheaf F is that infinitesima deformations are parametrised by Ext 1 (F,F) and the obstructions ie in Ext 2 (F,F). This is formaised by the so-caed Kuranishi map. More precisey it can be shown that there is a map Ï Ext 1 (F, F) Ext 2 (F,F) such that the competion of the oca ring of the point of the modui space corresponding to F is isomorphic to the oca ring of 1 (0) in 0. In genera there is no direct geometric description of the Kuranishi map but it is known that the constant and inear terms of the power series expansion of vanish and that its quadratic part is given by 2 Ï Ext 1 (F, F) Ext 2 (F, F), e (12)(e Æ e). For a K 3 surface this quadratic term aways vanishes since it is exacty the Serre duaity pairing which is known to be aternating. But if we consider a tautoogica sheaf F [2] the quadratic part of the Kuranishi map may be non-trivia. This woud correspond to the existence of a quadratic part in the equation of the tangent cone of the point in the modui space corresponding to F [2]. Consequenty, the tangent cone woud be stricty smaer than the tangent space and we woud end up with a singuarity. EXAMPLE 4.7. Let X be an eipticay fibred K 3 surface with fibre cass E and section C. Consider the ine bunde G Ï O(k F), k 2. We have h 0 (G) k 1 and h 1 (G) k 1. Certainy G is stabe and the modui space is a reduced point. The rank two tautoogica sheaf G [2] is aso stabe and the tangent space of its modui space at the point corresponding to G [2] is isomorphic to H 0 (X, G) Å H 1 (X, G), which has dimension k 2 1. The quadratic term of the Kuranishi map vanishes identicay but it is not cear if we can deform G [2] aong any of these infinitesima directions.

20 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 907 EXAMPLE 4.8. We continue with the same eiptic K 3 as above. From [3] we earn that the inear system of the ine bunde L with first Chern cass C k E has C as a base component for k 2. We have h 0 (G) k 1 and h 1 (G) 0. Now et p be a point on the curve C and denote by I p the corresponding idea sheaf. We set F Ï L Å I p. Certainy F is a torsion-free rank one sheaf with nonvanishing first Chern cass. Hence F [2] is stabe by Theorem [19, Theorem 4.2]. Theorem 4.9. The point in the modui space corresponding to F [2] is singuar. By the above considerations we have to prove the foowing emma: Lemma For the exampe F [2] (LÅI p ) [2] the quadratic part of the Kuranishi map does not vanish. Proof. We have to anayse the Yoneda square Ext 1 (F [2], F [2] ) Ext 2 (F [2], F [2] ), x x Æ x. Therefore et us use Krug s formua (4) in Coroary 1.8 to write down the extension groups expicity. Note that h 2 (F) 0. Ext 1 (F [2], F [2] ) Ext 1 (F, F) H 1 (F) Å H 0 (F), Ext 2 (F [2], F [2] ) Ext 2 (F, F) H 0 (F) Å H 0 (F) H 1 (F) Å H 1 (F). According to this decomposition we can decompose the Yoneda square as we foowing the detaied formuas in [12, Section 7]: Ext 1 (F, F) H 1 (F) ÅH 0 (F) Ext 2 (F, F) H 0 (F) ÅH 0 (F) H 1 (F) ÅH 1 (F), e a Å b e Æ e 0 Hence we need to show that the map Ext 1 (F, F) H 0 (F) H 1 (F) (a Æ e) Å b a Å (e Æ b). is not the zero map. The geometric interpretation of this map is the foowing: Let e ¾ Ext 1 (F,F) be an infinitesima deformation of F and ³ ¾ H 0 (F) be a goba section. Then ³ Æ e is zero if and ony if we can deform the section ³ aong e. It is time to return to the geometry of our exampe. Since p is on the base curve C, we have H 0 (F) H 0 (L). The deformations of F are those of I p, which correspond to deforming the point p in X. Now if we deform p into a direction norma to C, the space of goba sections wi shrink since the point wi fai to be a base point of L and thus we can find a section ³ ¾ H 0 (F) that does not deform with e.

21 908 M. WANDEL The Zariski tangent space is (k 3)-dimensiona and we can expicity derive the quadratic equation of the tangent cone. It is equivaent to the intersection of a pane (corresponding to the surface deformations) and a hyperpane (the additiona deformations and the curve C) in a ine (the curve C) Deformations of the manifod X [n]. A question which has not been touched so far, is the foowing: The manifod X [n] has an unobstructed deformation theory. Does the tautoogica sheaf F [n] deform with X [n]? The technique to answer this question is presented in [10]. We can summarise as foows: Theorem 4.11 (Huybrechts Thomas). Let Y be a projective manifod and E a sheaf on Y. Let ¾ H 1 (Y, T Y ) Ext 1 (Đ Y, O Y ) be the Kodaira Spencer cass of an infinitesima deformation of Y and denote by At(E) ¾ Ext 1 (E,E Å Đ Y ) the Atiyah cass of E. The sheaf E can be deformed aong if and ony if 0 ob(, E) Ï ( Å id E ) Æ At(E) ¾ Ext 2 (E, E). For every sheaf E on Y there are natura trace maps and trï Ext 1 (E, E Å Đ Y ) H 1 (Y, Đ Y ) trï Ext 2 (E, E) H 2 (Y, O Y ), which up to a sign commute with the Yoneda product. known that tr(at(e)) c 1 (E) and tr(ob(, E)) ob(, dete). Furthermore, it is we Appying this theorem to our situation, we get the foowing picture: The tangent space of the Kuranishi space at the point corresponding to X [n] is isomorphic to H 1 (X [n], T X [n]) H 1 (X [n], Đ X [n]) H 1 (X, Đ X ) Æ n. We write a cass in H 1 (X [n], Đ X [n]) as (, a) with ¾ H 1 (X, Đ X ) the cass of an infinitesima deformation of the surface X and a ¾. Unfortunatey there is no decomposition of the Atiyah cass At(F [n] ) at hand. But we can at east study its trace tr(at(f [n] )) c 1 (F [n] ) c 1 (F) X [n] r Æ n, where we set r Ï rkf. We have: tr(ob((, a), F [n] )) ob(, detf) raæ 2 n ¾ H 2 (X [n], O X [n]).

22 TAUTOLOGICAL SHEAVES: STABILITY AND MODULI SPACES 909 But we have ob(,detf) c 1 (F) and Æ 2 n 2(1 n), where we consider the Beauvie Bogomoov pairing. Thus we see: If F deforms aong, then surey the tautoogica sheaf F [n] deforms aong (, 0). If the determinant ine bunde detf does not deform aong, then F [n] does not deform aong (, 0). If c 1 (F) 2(1 n)ra, the tautoogica sheaf F [n] does not deform aong (, a). Thus there is an interesting hyperpane inside the space of infinitesima deformations of X [n] consisting of a pairs (, a) such that c 1 (F) 2(1 n)ra: It is an open question if the tautoogica sheaf deforms aong these directions. ACKNOWLEDGEMENTS. I want to thank Kaus Huek and David Poog for their support, their hep and for many interesting discussions. Thanks to Manfred Lehn and Jesse Kass for their good advice. This artice has been written at Leibniz Universität Hannover, Germany. The author was supported by the DFG Research Training Group GRK 1463 Anaysis, Geometry and String Theory. References [1] A. Beauvie: Variétés Käheriennes dont a première casse de Chern est nue, J. Differentia Geom. 18 (1983), [2] W.P. Barth, K, Huek, C.A.M. Peters and A. Van de Ven: Compact Compex Surfaces, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, A Series of Modern Surveys in Mathematics 4, Springer, Berin, [3] R. Donagi and D.R. Morrison: Linear systems on K 3-sections, J. Differentia Geom. 29 (1989), [4] G. Eingsrud, L. Göttsche and M. Lehn: On the cobordism cass of the Hibert scheme of a surface, J. Agebraic Geom. 10 (2001), [5] G. Eingsrud and S.A. Strømme: An intersection number for the punctua Hibert scheme of a surface, Trans. Amer. Math. Soc. 350 (1998), [6] J. Fogarty: Agebraic famiies on an agebraic surface, II, The Picard scheme of the punctua Hibert scheme, Amer. J. Math. 95 (1973), [7] V. Gritsenko and K. Huek: Minima Siege moduar threefods, Math. Proc. Cambridge Phios. Soc. 123 (1998), [8] M. Haiman: t,q-cataan numbers and the Hibert scheme, Discrete Math. 193 (1998), [9] M. Haiman: Hibert schemes, poygraphs and the Macdonad positivity conjecture, J. Amer. Math. Soc. 14 (2001), [10] D. Huybrechts and R.P. Thomas: Deformation-obstruction theory for compexes via Atiyah and Kodaira Spencer casses, Math. Ann. 346 (2010), [11] D. Huybrechts: Fourier Mukai Transforms in Agebraic Geometry, Oxford Mathematica Monographs, Oxford Univ. Press, Oxford, [12] A. Krug: Extension Groups of Tautoogica Sheaves on Hibert Schemes, ag-geom/ , (2011). [13] M. Lehn: Chern casses of tautoogica sheaves on Hibert schemes of points on surfaces, Invent. Math. 136 (1999), [14] S. Mukai: Sympectic structure of the modui space of sheaves on an abeian or K 3 surface, Invent. Math. 77 (1984),

23 910 M. WANDEL [15] K.G. O Grady: Desinguarized modui spaces of sheaves on a K 3, J. Reine Angew. Math. 512 (1999), [16] K.G. O Grady: A new six-dimensiona irreducibe sympectic variety, J. Agebraic Geom. 12 (2003), [17] L. Scaa: Some remarks on tautoogica sheaves on Hibert schemes of points on a surface, Geom. Dedicata 139 (2009), [18] U. Schickewei: Stabiity of tautoogica vector bundes on Hibert squares of surfaces, Rend. Semin. Mat. Univ. Padova 124 (2010), [19] M. Wande: Stabiity of tautoogica bundes on the Hibert scheme of two points on a surface, Nagoya Math. J. 214 (2014), Research Institute for Mathematica Sciences Kyoto University Kitashirakawa-Oiwakecho, Sakyo-ku Kyoto, Japan e-mai: wande@kurims.kyoto-u.ac.jp