Representations of the Heisenberg algebra and moduli spaces of framed sheaves
aa r X i v : . [ m a t h . AG ] A p r REPRESENTATIONS OF THE HEISENBERG ALGEBRA ANDMODULI SPACES OF FRAMED SHEAVES
FRANCESCO SALA AND PIETRO TORTELLA
Abstract.
Let M X,D ( r, c, n ) be the moduli spaces of torsion free sheaves on a com-plex smooth connected projective surface X , framed along a smooth connected genuszero curve D. This paper gives a ‘geometrical’ construction of an action of a Heisen-berg algebra on the homology of M X,D ( r, c, n ) (more precisely, on the direct sum L n H ∗ ( M X,D ( r, c, n ))) using correspondences. This result generalizes Nakajima’s con-struction for the Hilbert schemes of points. Introduction
Let X be a complex smooth connected quasi-projective surface and X [ n ] the Hilbertscheme of length n zero-dimensional subschemes of X . In [17] Nakajima builds a rep-resentation of the Heisenberg-Clifford super-algebra on the direct sum of the homologygroups L n ≥ H ∗ (cid:0) X [ n ] (cid:1) , showing that the generating function of the Poincar´e polynomi-als, found by G¨ottsche and Soergel in [11], coincides with the character formula of therepresentation. Nakajima’s result gives a supporting evidence to the ‘S-duality conjec-ture’ proposed by Vafa and Witten in [21].The Hilbert scheme X [ n ] can be viewed as the moduli space of rank one sheaves over X with trivial determinant and second Chern class equals to n . Let X be a smoothprojective surface; in [3] Baranovsky generalizes Nakajima’s result to the moduli spaces M X ( r, L, n ) of Gieseker-stable torsion free sheaves on X of rank r , determinant L andsecond Chern class n. Let D be a big and nef curve in X . In [5] Bruzzo and Markushevich built a finemoduli space M X,D ( r, c, n ) for torsion free sheves on X with invariants ( r, c, n ), framedalong the divisor D. The Hilbert scheme of points ( X \ D ) [ n ] can be viewed as the rankone case of this moduli space.Moreover moduli spaces of framed sheaves are studied because they provide a desingu-larization of the moduli spaces of ideal instantons, so their equivariant cohomology undersuitable toric actions is relevant to the computation of partition functions in topologicalquantum field theory (e.g. see [19], [4], [18] and [7]).The aim of this work is to generalize Nakajima’s construction to these moduli spaces:we explicitly construct Nakajima’s operators on the direct sum of the homology groups L n H ∗ ( M X,D ( r, c, n )) and show that they satisfy the Heisenberg commutation relationsand hence we get a representation of the Heisenberg algebra generated by Nakajima’soperators on this space.Using this construction one can expect to generalize to higher rank case Carlsson andOkunkov result ([8]): they use Nakajima’s operators for Hilbert schemes X [ n ] to define aclass of vertex operators, whose trace gives the Nekrasov’s partition function in the casewe have a toric action on X. This could be relevant to provide a mathematical motivationfor the conjecture about a correspondence between the Nekrasov’s partition function forsupersymmetric Yang-Mills topological field theories and the conformal blocks of a 2-dimensional conformal field theory (e.g. see [2]).The article is structured in the following way: in section 2 we introduce framed sheavesover a smooth connected projective surface and we outline how one can construct modulispaces of these; in section 3 we construct the
Nakajima operators , wich will lead us tothe representation we are seeking; in section 4 we state and prove a generalization ofNakajima’s theorem for the case of moduli spaces of framed sheaves, postponing to thelast two sections a proof of the more technical propositions.
Technical remark.
In this paper we shall assume that the framing divisor D is asmooth connected genus zero curve and this forces X to be a rational surface; we assumethis hypothesis because we need the moduli spaces of framed sheaves to be smoothvarieties and for the technical lemma 25. A priori, one can replace this assumptionby ‘ let X and D be such that the moduli spaces M X,D ( r, c, n ) are smooth ’ and one canconstruct the operators in the same way, but we do not know how to prove the Heisenbergcommutator relations in this general case. Anyway the main explicit examples of framedsheaves and their moduli (e.g. see chapter 2 in [17], [1], [20]) satisfy our hypothesis. Conventions and notation.
All schemes we are dealing with are of finite type over C , by ‘variety’ we mean a reduced separated scheme; a ‘sheaf’ is always coherent. Acknowledgements.
We would like to thank Ugo Bruzzo for suggesting us this prob-lem and for constant support. Thanks to Vladimir Baranovsky for useful remarks abouthis article and suggestions, Claudio Rava and Emanuele Macr`ı for useful conversations.2.
Framed Sheaves and their Moduli
In this section we introduce the notion of framed sheaf and we give a construction ofmoduli spaces of these objects.
EISENBERG ALGEBRA AND FRAMED SHEAVES 3
Generalities.
Let X be a smooth connected projective surface. Fix a smooth connectedcurve D in X . Definition 1. A framed sheaf of rank r on X with framing divisor D is a pair E =( E, φ E ) where E is a torsion free sheaf of rank r on X , φ E is a morphism φ E : E → O ⊕ rD such that φ E | D is an isomorphism. If ( E, φ E ) and ( F, φ F ) are two framed sheaves of thesame rank r , we define a morphism of framed sheaves as a morphism α : E → F suchthat for some λ ∈ C we have a commutative diagram(1) E α / / φ E (cid:15) (cid:15) F φ F (cid:15) (cid:15) O ⊕ rD · λ / / O ⊕ rD Clearly the set of morphisms of framed sheaves is a linear subspace of the space ofmorphisms between the corresponding underlying sheaves.Now let (
E, φ E ) be a framed sheaf of rank r on X. Because of the framing, the stalksat the points on the divisor D are free of rank r , so E is locally free in a neighborhood of D , hence in this neighborhood the sheaves E and E ∨∨ are isomorphic. Thus we have anatural framing for E ∨∨ . Moreover the support of A is disjoint from D . In the followingwe denote by E ∨∨ the framed sheaf ( E ∨∨ , φ E ). Note that the inclusion of E in E ∨∨ induces a morphism between the corresponding framed sheaves. Construction of the moduli space.
We now come to the construction of the modulispace of framed sheaves on a surface.
Definition 2.
Let S be a scheme. A family of rank r framed sheaves parametrized by S is a pair ( G, φ G ) where G is a torsion free sheaf of rank r on X × S flat over S , φ G isa morphism φ G : G → p ∗ X O ⊕ rD such that φ G | D : G | D × S → p ∗ X O ⊕ rD is an isomorphism.Remark that a family of rank r framed sheaves over X parametrized by S with framingdivisor D is in particular a rank r framed sheaf over the product X × S with framingdivisor D × S. We say that two families G and G ′ of rank r framed sheaves over X parametrized by S are isomorphic if they are isomorphic as rank r framed sheaves over X × S .Now fix invariants r, n ∈ Z with r > c ∈ NS( X ); we can define the functor M X,D ( r, c, n ) : ( Schemes ) ◦ −→ ( Sets )that associates to every scheme S the set of isomorphism classes of families ( G, φ G ) ofrank r framed sheaves parametrized by S such that the fibres G s have rank r and Chernclasses c and n , while to every morphism of schemes f : T → S associates the pull-back f ∗ that sends families parametrized by S in families parametrized by T . FRANCESCO SALA AND PIETRO TORTELLA
Remark . A necessary condition for this functor to be nonempty is Z D c = 0, indeed ingeneral Z D c ( E ) = c ( E | D ) and if E is trivial along D , then c ( E | D ) = c ( O ⊕ rD ) = 0.In [5] Bruzzo and Markushevich give (in a more general setting) a proof of the repre-sentability of the functor M X,D ( r, c, n ); using their result we have the following: Theorem 4.
Let X be a smooth connected projective surface and D a smooth connectedcurve in X which is a big and nef divisor. Then there exists a quasi-projective separatedscheme M X,D ( r, c, n ) that represents the functor M X,D ( r, c, n ) . Note that M X,D ( r, c, n ) is a fine moduli space for framed sheaves on X with framingdivisor D and topological invariants r, c and n . The adjective ‘fine’ means the existenceof a universal framed sheaf ¯ E = ( ¯ E, φ ¯ E ), i.e. a family of framed sheaves parametrized by M X,D ( r, c, n ) with the following universal property: for any family ( G, φ G ) of framedsheaves parametrized by S there exists a unique morphism g : S → M X,D ( r, c, n ) andan isomorphism α : G → (id × g ) ∗ ( ¯ E ) such that φ G | D × S = ((id × g ) ∗ φ ¯ E ◦ α ) | D × S . From now on we suppose that D is a smooth connected curve in X that satisfies thefollowing conditions:(a) D is a big and nef divisor,(b) D ∼ = CP . Remark . Since D is isomorphic to the complex projective line and it has positive selfintersection, X is a rational connected surface, hence it is a rational surface. Moreoverthe arithmetic genera of D and X are zero and H i ( X, C ) = 0 for i = 1 , . From these conditions follows the lemma:
Lemma 6.
Let ( E, φ E ) and ( F, φ F ) be framed sheaves on X . Then Hom(
E, F ) ∼ = End( C r ) . Proof.
Let f be a non trivial morphism in Hom( E, F ); since E | D and F | D are trivial, f | D is a constant matrix, i.e. is an element of End( C r ) . Consider the complete linear system | D | and let Y be its base locus: we can define amorphism π : X \ Y → | D | that associates to each point x of X \ Y the hyperplane in | D | consisting of those sections vanishing at x. Let V be the open set of divisor C ∈ | D | such that C is a smooth curve of genuszero. Let U be the set consisting on curves C ∈ V such that E | C and F | C are the trivialbundle. By lemma 2.3.1 in [14] and semicontinuity theorem U is open. We have that f | C is a constant matrix for all C ∈ U. So we get that f is a constant matrix in π − ( U ), EISENBERG ALGEBRA AND FRAMED SHEAVES 5 that is a dense open subset of X . Thus f is constant in the whole X and the thesis isproved. (cid:3) Now we want to study the local properties of the moduli space, in particular we wanta description of its tangent space. We begin with a vanishing lemma:
Lemma 7.
Let ( E, φ E ) be a framed sheaf on X with framing divisor D . Then (a) Hom( E, E ( − D )) = 0 , (b) Ext ( E, E ( − D )) = 0 . Proof. (a)
Let f be a non trivial morphism in Hom( E, E ( − D )); since E | D is trivial wehave thatHom( E | D , E ( − D ) | D ) ∼ = H ( D, H om ( O ⊕ rD , O ⊕ rD ⊗ O X ( − D ))) ∼ = H ( D, O D ( − a )) ⊕ r where a = D.D . Since D is a big and nef divisor, we get that a is positive, henceH ( D, O D ( − a )) = 0 and therefore f is zero along D .As before, consider the morphism π : X \ Y → | D | . Let T be the set of divisor C ∈ | D | such that E | C is a semistable sheaf on C . Note that T = ∅ because D belongs to T and T is open in | D | because semistability is an open property. Since p ( E | C , m ) > p ( E ( − D ) | C , m ) for all C ∈ | D | , by proposition 1.2.7 in [13] we get thatHom( E | C , E ( − D ) | C ) = 0 for all C ∈ T and therefore f | C = 0. So we can use the sameargument of lemma 6 and therefore f is zero in the whole X. (b) Using Serre’s duality we haveExt ( E, E ( − D )) ∼ = Hom( E, E ⊗ O X ( D ) ⊗ ω X ) ∨ . We have ω D ∼ = ( ω X ⊗ O X ( D )) | D and deg ω D < f in Hom( E, E ⊗ O X ( D ) ⊗ ω X ) is zero when restricted to D ; in the same way, for all C ∈ T we get deg O X ( K X + D ) | C < (cid:3) Thanks to this lemma and to theorem 4.3 in [5], we get
Theorem 8.
Let E = ( E, φ E ) be a framed sheaf with invariants r, c and n . Then theZariski tangent space at the corresponding point [ E ] of M X,D ( r, c, n ) is T [ E ] M X,D ( r, c, n ) ∼ = Ext ( E, E ( − D )) . Moreover M X,D ( r, c, n ) is a smooth quasi-projective variety of dimension dim C M X,D ( r, c, n ) = 2 rn − ( r − Z X c . From now on we put b := − ( r − Z X c . FRANCESCO SALA AND PIETRO TORTELLA
Remark . Note that there does not exist a general theorem about the nonemptiness ofthese moduli spaces. There are few known cases of nonempty moduli spaces of framedsheaves. The simplest case is given by r = 1, c = 0 and n >
0: the moduli space M X,D (1 , , n ) is isomorphic to the Hilbert scheme of points X [ n ]0 , where X = X \ D .This space is a smooth quasi-projective variety of dimension 2 n . Another case is givenby X = P and D a line in P (see for example [17], chapter II); in this case oneshows that M P ,D ( r, c, n ) is nonempty if and only if c = 0 and n >
0. A recentlyknown case is one on the Hirzebruch surfaces: let X = F p be the p -th Hirzebruchsurface and D be the ‘line at infinity’, let c = aE , where a ∈ Z , 0 ≤ a < r − E the only curve in F p with negative self-intersection, in [20] Rava proves thatthe moduli space M F p ,D ( r, c, n ) is nonempty if and only if the number n + r − r pa is an integer and n + pa ( a − ≥
0. Moreover he proves that when equality holds, M F ,D ( r, c, n ) = Grass ( a, r ) and M F p ,D ( r, c, n ) = T ( Grass ( a, r )) ⊕ ( p − for p >
1, where
Grass ( a, r ) is the Grassmannian of a -planes inside C r . In the following we assume that
X, D, r, c, n are such that the moduli space M X,D ( r, c, n )is nonempty. Tangent bundle of M X,D ( r, c, n ) . First of all we want to define the
Kodaira-Spencermap for framed sheaves in terms of cocycles. Let S be a scheme.Let p S and p X be the projections from Y = X × S to S and X respectively. Considera family ( G, φ G ) of framed sheaves parametrized by S . Let G • be a finite locally freeresolution G • → G .Following chapter 10 in [13], we can define the Atiyah class A ( G • ) in terms of con-nections: let p , p : Y × Y → Y be the projections to the two factors. Let I be theideal sheaf of the diagonal ∆ ⊂ Y × Y and let O denote the structure sheaf of the firstinfinitesimal neighbourhood of ∆.Choose an open affine covering U = { U i | i ∈ I } such that the restriction of the sequence0 −→ G q ⊗ Ω Y −→ ( p ) ∗ ( p ∗ G q ⊗ O ) −→ G q −→ U i splits for all q and i . Thus there are local connections ∇ qi : G q | U i → G q ⊗ Ω Y | U i .Note that the difference of two local connections is an O Y -linear map. Define cochains α ′ ∈ C ( H om ( G • , G • ⊗ Ω Y ) , U ) and α ′′ ∈ C ( H om ( G • , G • ⊗ Ω Y ) , U ) as follows: α ′ qi i = ∇ qi | U i i − ∇ qi | U i i ,α ′′ qi = d G • ◦ ∇ qi − ∇ q +1 i ◦ d G • , where d G • is the differential of the complex G • . The element α = α ′ + α ′′ is a cocyclein the total complex associated to the double complex C • ( H om • ( G • , G • ⊗ Ω Y ) , U ). Thecohomology class of α in E xt ( G • , G • ⊗ Ω Y ) is the Atiyah class A ( G • ) of the sheaf G . EISENBERG ALGEBRA AND FRAMED SHEAVES 7
Now, we want to care about the morphism φ G . The connecting morphism ε : G → G induces a cohomology class [ ε ⊗ id Ω Y ] in E xt ( G • ⊗ Ω Y , G ⊗ Ω Y ) and therefore [ α ] := A ( G • ) ⊗ [ ε ⊗ id Ω Y ] is a cohomology class in E xt ( G • , G ⊗ Ω Y ), where ⊗ is the Yonedaproduct for Ext-groups of complexes of sheaves.Since H om ( G • , G ⊗ Ω Y ) = H om ( G , G ⊗ Ω Y ) and H om n ( G • , G ⊗ Ω Y ) = 0 for n > α is an element in C ( H om ( G , G ⊗ Ω Y )) ⊂ C ( H om ( G • , G ⊗ Ω Y φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ Ω Y )) . We define morphisms ∇ i : G | U i → ( G ⊗ Ω Y ) | U i as compositions G | U i ∇ i −→ ( G ⊗ Ω Y ) | U i ε ⊗ id −→ ( G ⊗ Ω Y ) | U i and morphisms e φ i : G | U i → ( p ∗ X O ⊕ rD ⊗ Ω Y ) | U i as compositions G | U i ∇ i −→ ( G ⊗ Ω Y ) | U i φ G ⊗ id −→ ( p ∗ X O ⊕ rD ⊗ Ω Y ) | U i hence we obtain a cochain e φ ∈ C ( H om ( G • , G ⊗ Ω Y φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ Ω Y )).Since f φ i | U i i − f φ i | U i i = (( φ G ◦ ε ) ⊗ id Ω Y ) | U i i ◦ α ′ i i , we have that A ( G, φ G ) := α + e φ defines an element of E xt ( G • , G ⊗ Ω Y φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ Ω Y ) = E xt ( G, G ⊗ Ω Y φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ Ω Y ) . If we consider the induced section of A ( G, φ G ) under the global-to-local map E xt ( G, G ⊗ Ω Y φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ Ω Y ) → H ( S, E xt p S ( G, G ⊗ Ω Y φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ Ω Y ))where E xt p S is the first right derived functor of ( p S ) ∗ ◦ H om , then the direct sum de-composition Ω Y = p ∗ S Ω S ⊕ p ∗ X Ω X leads to an analogous decomposition A ( G, φ G ) = A ( G, φ G ) ′ + A ( G, φ G ) ′′ . The Kodaira-Spencer map associated to (
G, φ G ) is KS : Ω ∨ S A ( G,φ G ) ′ −→ Ω ∨ S ⊗E xt p S ( G, G ⊗ p ∗ S Ω S φ G ⊗ id −→ p ∗ X O ⊕ rD ⊗ p ∗ S Ω S ) → E xt p S ( G, G φ G → p ∗ X O ⊕ rD ) Remark . It is easy to prove that the complex G φ G → p ∗ X O ⊕ rD is quasi-isomorphic to G ⊗ p ∗ X O X ( − D ), hence E xt p S ( G, G φ G → p ∗ X O ⊕ rD ) ∼ = E xt p S ( G, G ⊗ p ∗ X O X ( − D )).Now we are able to give the following characterization of the tangent bundle of M X,D ( r, c, n ). Theorem 11.
Let ( ¯
E, φ ¯ E ) be the universal family of M X,D ( r, c, n ) and let p be theprojection from M X,D ( r, c, n ) × X to M X,D ( r, c, n ) . The Kodaira-Spencer map for M X,D ( r, c, n ) induces a canonical isomorphism T M X,D ( r,c,n ) ∼ = E xt p ( ¯ E, ¯ E ⊗ p ∗ X O ( − D )) . FRANCESCO SALA AND PIETRO TORTELLA
Proof.
To prove this theorem, we can use a similar argument of theorem 10.2.1 in [13]:by lemma 7 we get that the sheaf E xt p ( ¯ E, ¯ E ⊗ p ∗ X O ( − D )) commutes with base changeand E xt p ( ¯ E, ¯ E ⊗ p ∗ X O ( − D )) is locally free.Moreover, by example 10.1.9 in [13] we get that the Kodaira-Spencer map is an iso-morphism on the fibres. (cid:3) Donaldson-Uhlenbeck partial compactification.
Let M regX,D ( r, c, n ) be the opensubset in M X,D ( r, c, n ) consisting of framed sheaves ( F, φ F ) with F locally free.Now remember that each framed sheaf E = ( E, φ E ) has a natural inclusion in E ∨∨ =( E ∨∨ , φ E ) and the quotient E ∨∨ / E is a zero-dimensional sheaf whose support is disjointfrom D . In this way to any framed sheaf E with invariants r, c, n we can associate thepair ( E ∨∨ , P i a i [ x i ]) where x i are the points of the support of the quotient and a i is thelength of the quotient at x i ; remark that if c ( E ∨∨ ) = s then length( E ∨∨ / E ) = n − s ,hence P i a i x i is an element of the ( n − s )-th symmetric product S n − s X and E ∨∨ is anelement of M regX,D ( r, c, s ).This association gives a map(2) π r : M X,D ( r, c, n ) −→ \ M X,D ( r, c, n ) := n a s =0 M regX,D ( r, c, s ) × S n − s X ;in [6] the right hand side space is given a structure of scheme (similar to the Donaldson-Uhlenbeck partial compactification of instantons) such that the map π r is a projectivemorphism.Now we are going to study the fibers of this map, that will be useful later on: let p =(( G, φ G ) , P j m j [ x j ]) be a point in M regX,D ( r, c, s ) × S n − s X , the fiber π − r ( p ) parametrizesframed sheaves ( E, φ E ) such that E ∨∨ ∼ = G , the framing φ E is just the composition ofthe injection in the double dual with φ G (and hence is fixed if we fix G and E ) andthe quotient A = G / E is supported at x j , with length over x j equal to m j . We candecompose A = L A j as sum of skyscraper sheaves supported at x j of length m j ; eachof this A j may be seen as an Artin quotient of the trivial sheaf O ⊕ rX supported only at x j and of length m j , hence the possible choice of A j are parametrized by the scheme Quot x j ( r, m j ); these schemes have been studied in the appendix of [3], we recall thefollowing result: Proposition 12.
Let X be a smooth projective surface, x ∈ X a point and r, d a pairof positive integers. Then the scheme Quot x ( r, d ) parametrizing Artin quotients of thetrivial sheaf O ⊕ rX supported at x of length d is irreducible of dimension rd − . We get as a corollary the following:
EISENBERG ALGEBRA AND FRAMED SHEAVES 9
Corollary 13.
Let p = ( G , P lj =1 m j [ x j ]) be a point of M regX,D ( r, c, s ) × S n − s X . Then π − r ( p ) is isomorphic to the scheme Q j Quot x j ( r, m j ) as an irreducible closed subschemeof M X,D ( r, c, n ) of dimension P j ( rm j −
1) = r ( n − s ) − l . Nakajima operators on moduli spaces of framed sheaves
From now on we fix the rank r and the first Chern class c and we consider the modulispaces M X,D ( r, c, n ) all together by varying the second Chern class n . To emphasizethis fact in the following we denote by M r,cX,D ( n ) the moduli space M X,D ( r, c, n ) as ifwe consider n a variable and r, c constants.Let H • ( M r,cX,D ( n )) be the homology group of M r,cX,D ( n ) with complex coefficients.Denote by H ( X ) the direct sum L n H • ( M r,cX,D ( n )) of all the homology groups of all themoduli spaces M r,cX,D ( n ) . In this section for any i ∈ Z and α ∈ H • ( X ) we define operators P α [ i ] : H ( X ) → H ( X )that we shall call Nakajima operators and in the following section we prove that they sat-isfy the Heisenberg commutator relations: in this way we define an action of a Heisenbergalgebra associated to H • ( X ) on the space H ( X ) in a geometric way.First for any i, n ∈ Z and α ∈ H • ( X ) we are going to construct operators P nα [ i ] : H • ( M r,cX,D ( n )) −→ H • ( M r,cX,D ( n − i ))as correspondence for a suitable subvariety of M r,cX,D ( n − i ) × M r,cX,D ( n ). Moreover, fortechnical reasons due to the non-projectivity of the moduli spaces M r,cX,D ( n ) we have touse the Borel-Moore homology groups of our moduli spaces (for technical details aboutBorel-Moore homology we refer to Appendix B in [9]).Now we recall briefly the construction of correspondences. Let M , M be two orientedmanifolds of dimension d , d respectively and Z ⊆ M × M an oriented submanifoldsuch that the projection p : Z → M is a proper map. We denote by H lfj ( M a ) theBorel-Moore homology group of M a for a = 1 , . We can define operators e γ Z : H lfj ( M ) −→ H lfj +dim( Z ) − d ( M ) η p ∗ ( p ∗ η ∩ [ Z ])and for any α ∈ H lf • ( Z ) e γ α : H lfj ( M ) −→ H lfj +deg( α ) − d ( M ) η p ∗ ( p ∗ η ∩ α )where the ∩ -product is the one on Borel-Moore homology. Moreover, using the fact that the singular homology group of M a is the direct limitof the Borel-Moore homology group of M a with respect to the compact subsets of M a (for a = 1 , correspondence ) γ Z : H j ( M ) −→ H j +dim( Z ) − d ( M )and for any α ∈ H lf • ( Z ) γ α : H j ( M ) −→ H j +deg( α ) − d ( M ) . Now we want to study the composition of two such homomorphism: if M , M , M arethree oriented manifolds and Z ⊆ M × M , W ⊆ M × M two oriented submanifoldssuch that the projections Z → M , W → M are proper maps, we have the compositionof the correspondences γ Z ◦ γ W : H • ( M ) → H • ( M ): these turn out to be equal to thecorrespondence defined by the subvariety A ⊆ M × M , A = p ( p − ( Z ) ∩ p − ( W )) , where p ab are the projections M × M × M → M a × M b .If we have α ∈ H • ( Z ) and β ∈ H • ( W ), the composition of the correspondences γ β and γ α is given by the correspondence induced by the class ǫ ∈ H • ( A ) defined by ǫ = p ∗ ( p ∗ α ∩ p ∗ β ) . Now consider the moduli spaces M r,cX,D ( n ); let n, i ∈ Z , i > P n [ i ] ⊆ M r,cX,D ( n ) × M r,cX,D ( n + i ) × X defined by P n [ i ] = ( E , E , x ) ∈ M r,cX,D ( n ) × M r,cX,D ( n + i ) × X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) • E ∨∨ ∼ = E ∨∨ , • E ⊆ E , • supp ( E / E ) = { x } . We want to define corrispondences using the fundalmental class of P n [ i ]. Throughdirect computations one can prove that the dimension of the closed subvariety P n [ i ] is2 rn + b + r + 1.Note that if π : M r,cX,D ( n ) × M r,cX,D ( n + i ) × X → M r,cX,D ( n ) × M r,cX,D ( n + i ) is theprojection, then the restriction π | P n [ i ] is a proper injective map and therefore π ( P n [ i ])is a closed subvariety.Thus for any α ∈ H • ( X ) we have a Borel-Moore homology class P nα [ i ] ∈ H lf • ( π ( P n [ i ]))defined by P nα [ i ] = π ∗ ( τ ∗ α ∩ [ P n [ i ]])where τ : M r,cX,D ( n ) × M r,cX,D ( n + i ) × X → X is the projection on the surface. (Withthe previous notation: P nα [ i ] = e γ P n [ i ] ( α )). EISENBERG ALGEBRA AND FRAMED SHEAVES 11
Since the restriction to π ( P n [ i ]) of the projection from M r,cX,D ( n ) × M r,cX,D ( n + i ) to M r,cX,D ( n + i ) is a proper map, the class P nα [ i ] induces a correspondence operator P nα [ i ] : H j ( M r,cX,D ( n + i )) −→ H j +deg( α ) − ri − ( M r,cX,D ( n ))(With the previous notation this would be γ P nα [ i ] ).In a similar way we can define for i ∈ Z , i > P n [ − i ] ⊆ M r,cX,D ( n ) × M r,cX,D ( n − i )as before exchanging the roles of E and E ; these subvarieties have dimension 2 rn + b − ri + 1 . We can repeat the previous construction and for any β ∈ H • ( X ) we get theoperator P nβ [ − i ] : H j ( M r,cX,D ( n − i )) −→ H j +deg( β )+2 ri − ( M r,cX,D ( n )) . For i = 0 we set P nα [0] equal to the identity map for any n ∈ Z and α ∈ H • ( X ).Now we can extend these operators to H ( X ): for any i ∈ Z and α ∈ H • ( X ) defineoperators P α [ i ] : H • ( X ) −→ H • ( X )acting on every summand H • ( M r,cX,D ( n )) with P nα [ i ].4. Main Theorem
Now we want to study the commutation relations between the operators P α [ i ] weconstructed. In particular we want to prove the following result: Theorem 14.
For any i, j ∈ Z , α, β ∈ H • ( X ) the following holds: [ P α [ i ] , P β [ j ]] = ( − ri − riδ i + j, < α, β > idThis provides a representation of the Heisenberg algebra generated by P α [ i ] / ( − ri − r and P β [ j ] on H ( X ) . The first step to get this result is the following:
Proposition 15.
There exist constants c r,i,n such that for any v ∈ H • ( M r,cX,D ( n )) wehave: [ P α [ i ] , P β [ j ]] v = c r,i,n δ i + j, < α, β > v A proof of this proposition may be given directly by investigating the cycles P n − jα [ i ] P nβ [ j ]and P n − iβ [ j ] P nα [ i ] in M r,cX,D ( n − i − j ) ×M r,cX,D ( n ), and observing that they act in the sameway whenever i = j , while when i = j they differ by a constant times the intersectionproduct of the cycles; in [17] one can find the details of this proof for the case when r = 1, i.e. the Hilbert scheme of points, and this directly generalize to our case. Lemma 16.
The constants c r,i,n of the previous proposition are independent of n . (Andso, from now on we shall denote them simply by c r,i )Proof. Let i, k ∈ Z be such that k = ± i , α, β, γ ∈ H • ( X ) and v ∈ H • ( M r,cX,D ( n )). Since P γ [ k ] v ∈ H • ( M r,cX,D ( n − k )), we get[ P γ [ k ] , [ P α [ i ] , P β [ − i ]]] v = P γ [ k ][ P α [ i ] , P β [ − i ]] v − [ P α [ i ] , P β [ − i ]] P γ [ k ] v == < α, β > ( c r,i,n − c r,i,n − k ) v But k = ± i , hence [ P γ [ k ] , [ P α [ i ] , P β [ j ]]] = 0, and we can choose α, β, γ, k, v so that < α, β > = 0 and P γ [ k ] v = 0 . So c r,i,n = c r,i,n + k ; for i = 0, we can take k = 1 and have c r,i,n = c r,i,n +1 that impliesthe thesis; for i = 1 taking k = 2 , c r, ,n = c r, ,n +2 = c r, ,n +3 , and againwe have the thesis. (cid:3) Calculation of constants
In this section we compute explicitly the constants c r,i : following Grojnowski andNakajima, we find an isomorphism of the algebra of symmetric functions with somesubspaces of H • ( X ), such that the Newton polynomials correspond to the Nakajima op-erators P [ C ] [ i ] for some smooth connected curves C ⊆ X , C = D. In the following sectionwe recall some facts about the algebra of symmetric functions (for more informationssee chapter one in [16]).
Notations:
We use two different conventions for µ partition of a natural number k :if we write µ = ( µ , . . . , µ t ) we mean that P a µ a = k and assume that µ a ≥ µ a +1 > length | µ | of the partition µ the natural number t. If we write µ =(1 m , . . . , k m k ) we mean that m l = { a | µ a = l } . Symmetric Functions.
We call algebra of symmetric polynomial in N variables thesubspace Λ N of C [ x , . . . , x N ] stable by the action of the N -th group of permutations σ N . We have that Λ N is a graded ring:Λ N = M n Λ nN where Λ nN is the ring of homogeneous symmetric polynomials in N variables of degree n (together with the zero polynomial).For any M > N we have morphisms ρ MN : Λ M → Λ N mapping the variables x N +1 , . . . , x M to zero. Moreover the morphisms ρ MN preserve the grading, hence wecan define ρ nMN : Λ nM → Λ nN ; this allows us to defineΛ n := lim ← N Λ nN EISENBERG ALGEBRA AND FRAMED SHEAVES 13 and the algebra of symmetric functions in infinitely many variables asΛ := M n Λ n . Now we want to define a basis for Λ, to do this we start by defining a basis in Λ N .Let µ = ( µ , . . . , µ t ) be a partition with t ≤ N , we define the polynomial m µ ( x , . . . , x N ) = X τ ∈ σ N ( µ ) x τ · · · x τ N N where we set µ j = 0 for j = t + 1 , . . . , N . The polynomial m µ is symmetric, moreoverthe set of all m µ for all the partitions µ with | µ | ≤ N is a basis of Λ N . If we considerthe set of all m µ for all the partitions µ with | µ | ≤ N and P i µ i = n , then this set is abasis of Λ nN .Since for M > N ≥ t we have ρ nMN ( m µ ( x , . . . , x M )) = m µ ( x , . . . , x N ), by usingthe definition of inverse limit we can define the monomial symmetric functions m µ . Byvarying µ partition of n , these polynomials form a basis for Λ n . Now we want do define particular families of symmetric functions. Let n ∈ N , n ≥ elementary symmetric function e n as e n = m (1 n ) = X i <...
For any i ∈ N we have p i m µ = X ν a µν m ν where the summation is over partitions ν of | µ | + i which are obtained as follows: add i to a term in µ ,say µ j (possibly zero),and then arrange it in descending order. Thecoefficient a µν is { l | ∃ k s.t. ν l = µ k + i } . Computation of c r,i . We need to introduce some subvarieties of our moduli spaces:let C be a smooth connected curve in X , C = D , s, k ∈ N ; set L kC ( s ) ◦ = F = ( F, φ F ) ∈ M r,cX,D ( s + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) • F ⊂ H for some H ∈ M r,cX,D ( s ) , • H / F is an Artin sheaf of length k, • supp( H / F ) ⊆ C. and L kC ( s ) its closure in M r,cX,D ( s + k ); for µ partition of k , set L µC ( s ) ◦ = F = ( F, φ F ) ∈ M r,cX,D ( s + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) • F ⊂ H for some H ∈ M r,cX,D ( s ) , • supp( H / F ) = { x , . . . , x t } ⊆ C, • length x a ( H / F ) = µ a for a = 1 , , . . . , t. and L µC ( s ) its closure in M r,cX,D ( s + k ).Now let G = ( G, φ G ) ∈ M regX,D ( r, c, s ) and set L kC ( G ) ◦ = (cid:26) F = ( F, φ F ) ∈ M r,cX,D ( s + k ) (cid:12)(cid:12)(cid:12)(cid:12) • F ∨∨ = G , • supp( G / F ) ⊆ C. (cid:27) and L kC ( G ) its closure in M r,cX,D ( s + k ); for µ partition of k set L µC ( G ) ◦ = F = ( F, φ F ) ∈ M r,cX,D ( s + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) • F ∨∨ = G , • supp( G / F ) = { x , . . . , x t } ⊆ C, • length x a ( G / F ) = µ a for a = 1 , , . . . , t. and L µC ( G ) its closure.The first result we have is similar to that of proposition 4.5 in [3]: EISENBERG ALGEBRA AND FRAMED SHEAVES 15
Proposition 18.
The irreducible components of L kC ( s ) are the closed subvarieties L µC ( s ) for µ a partition of k , each of which has dimension rs + rk .For any G ∈ M regX,D ( r, c, s ) , the irreducible components of L kC ( G ) are the closed subva-rieties L µC ( G ) , for µ a partition of k , each of which has dimension rk . Now we remark the following: let α = [ C ] ∈ H ( X ) and consider the correspondingoperator P [ C ] [ − i ] : H j ( M r,cX,D ( n )) → H j +2 ri ( M r,cX,D ( n + i ))we have that the subspace of H ( X ) of middle degrees , i.e. L n H rn ( M r,cX,D ( n )), is pre-served by the action of P [ C ] [ − i ].Moreover, since P [ C ] [ − i ] ‘changes the sheaves only at a point of C adding i to the lengthof a zero-dimensional quotient’, if µ is a partition of k we have that P [ C ] [ − i ][ L µC ( G )] ∈ H r ( k + i ) ( L k + iC ( G )); the next proposition calculates the coefficients of this element withrespect to the basis of H r ( k + i ) ( L k + iC ( G )) consisting of the classes of the irreducible com-ponents of L k + iC ( G ) . Proposition 19.
For any s, k ∈ N , µ a partition of k , C ⊆ X smooth connected curve, C = D , the following holds: P [ C ] [ − i ][ L µC ( s )] = X ν partition of | µ | + i a µν [ L νC ( s )] where the constants a µν are the same as in lemma 17.For any G ∈ M
X,D ( r, c, s ) we get P [ C ] [ − i ][ L µC ( G )] = X ν partition of | µ | + i [ L νC ( G )] with the same constants a µν . The proof of this proposition is given in section 6.Let [ vac ] be the class of the point G and [ V AC ] the fundamental class of M r,cX,D ( s ).Let L ( G ) (resp. L ( s )) be the subspaces of H ( X ) generated by the classes [ L µC ( G )] and[ vac ] (resp. [ L µC ( s )] and [ V AC ]). We define a C -linear isomorphism from L ( G ) (resp. L ( s )) onto the algebra Λ of symmetric functions in infinitely many variables, sending[ vac ] (resp. [ V AC ]) to 1 ∈ Λ and [ L µC ( G )] (resp. [ L µC ( s )]) to the monomial symmetricfunction m µ .By proposition 19, the Nakajima operator P [ C ] [ − i ] corresponds under this isomor-phism to multiplication by the i-th Newton function p i ∈ Λ. By this isomorphism, the elementary symmetric function e n corresponds to [ L (1 n ) C ( G )](resp. [ L (1 n ) C ( s )]) and the equation 3 gives the following equalities: ∞ X n =0 z n [ L (1 n ) C ( s )] = exp ∞ X i =1 z i P [ C ] [ − i ]( − i − i ! [ V AC ] , (4) ∞ X n =0 z n [ L (1 n ) C ( G )] = exp ∞ X i =1 z i P [ C ] [ − i ]( − i − i ! [ vac ] . (5)Now we need the following proposition, whose proof will be given in section 7: Proposition 20.
Let L be a very ample divisor in X . Let C and C ′ be two smoothconnected curves different from D , such that C ∩ C ′ ∩ D = ∅ . Assume that C and C ′ meets trasversally and are in | L | . Then the following holds: ∞ X n =0 z n < [ L (1 n ) C ( G )] , [ L (1 n ) C ′ ( s )] > = (1 − ( − r z ) r< [ C ] , [ C ′ ] > . We can finally conclude the proof of theorem 14, via the same computations of theproof of theorem 4.1 in [3]:
Proposition 21.
The constants c r,i of proposition 15 satisfy c r,i = ( − ri − ri. Proof.
For any C smooth connected curve in X , C = D , set C + ( z ) = ∞ X i =1 P [ C ] [ i ] z i ( − i − i , C − ( z ) = ∞ X i =1 P [ C ] [ − i ] z i ( − i − i . Since P [ C ] [ i ] is the adjoint to P [ C ] [ − i ] with respect to the intersection product on H ( X ), C + ( z ) is the adjoint to C − ( z ) (with respect to the intersection product extendedby linearity to power series).Now, by proposition 20 and formulas 4 and 5 we have the equalities(1 − ( − r z ) < [ C ] , [ C ′ ] > = ∞ X n =0 z n < [ L (1 n ) C ( G )] , [ L (1 n ) C ′ ( s )] > == < ∞ X n =0 z n [ L (1 n ) C ( G )] , ∞ X n =0 z n [ L (1 n ) C ′ ( s )] > == < exp( C − ( z ))[ vac ] , exp( C ′− ( z ))[ V AC ] > == < exp( C ′ + ( z )) exp( C − ( z ))[ vac ] , [ V AC ] > == < exp( C − ( z )) (cid:2) exp( − ad C − ( z ))) (cid:0) exp( C ′ + ( z )) (cid:1)(cid:3) [ vac ] , [ V AC ] > EISENBERG ALGEBRA AND FRAMED SHEAVES 17 where the last equality follows from the general equality between operators exp( − A ) B exp( A ) =exp( − ad A )( B ) (we put A = C − ( z ) and B = exp( C ′ + ( z )).An explicit computation shows that[ C − ( z ) , exp( C ′ + ( z ))] = − Φ( z ) exp( C ′ + ( z ))with Φ( z ) = P ∞ n =0 c r,n n < [ C ] , [ C ′ ] > z n ; from this and the fact that exp( − ad A )( B ) =1 − ad A ( B ) + . . . one gets thatexp( − ad C − ( z ))) (exp( Cd + ( z ))) = exp(Φ( z )) exp( Cd + ( z )) , and so(1 − ( − r z ) < [ C ] , [ C ′ ] > = exp(Φ( z )) < exp( C − ( z )) exp( C ′ + ( z ))[ vac ] , [ V AC ] > == exp(Φ( z )) < exp( C − ( z ))[ vac ] , [ V AC ] > == exp(Φ( z )) . where the second of this equalities follows from the fact that since i > P [ C ′ ] [ i ][ vac ] = 0,while the last follows from the fact that every P [ C ] [ − i ] maps [ vac ] in the orthogonalcomplement of [ V AC ].So finally we get r < [ C ] , [ C ′ ] > log(1 − ( − r z ) = ∞ X n =1 c r,n n < [ C ] , [ C ′ ] > z n from which, developing in power series the left hand side, we get c r,n = ( − rn − rn , thatis the thesis. (cid:3) Proof of proposition 19
Now we want to give a proof of proposition 19; we explain how to do with L µC ( G ), thecase of L µC ( s ) is analogous.First of all we need the following lemma: Lemma 22.
Let λ be a partition of k. For F generic in L λC ( G ) (i.e. in a open densesubset), around each x j point of the support of G / F exist coordinates ( η j , ζ j ) such that (cid:18) GF (cid:19) x j ∼ = C [ ζ j ]( ζ j ) λ j . Proof.
It follows from the first paragraph of section 5 in [3]. (cid:3)
As we have already seen, P [ C ] [ − i ]([ L µC ( G )]) is an element of maximal degree in H • ( L k + iC ( G )).Since L νC ( G ) are the irreducible components of L k + iC ( G ) for ν partition of k + i , P [ C ] [ − i ]([ L µC ( G )])decomposes as a sum P ν b µν [ L νC ( G )] for some coefficients b µν . Put n = k + s. To compute the coefficients b µν , consider the intersection P n [ C ] [ − i ] ∩ p − n ( L µC ( G )) ∩ p − n + i ( L νC ( G ))where the projection morphisms are: M r,cX,D ( n + i ) × M r,cX,D ( n ) p n + i u u kkkkkkkkkkkkkkk p n ) ) RRRRRRRRRRRRRR M r,cX,D ( n + i ) M r,cX,D ( n )We prove the proposition: Proposition 23.
Let F ∈ L µC ( G ) , F ∈ L νC ( G ) be such that ( F , F ) ∈ P n [ C ] [ − i ] isa smooth point and such that they are generic in the sense of lemma 22. Then theintersection of the tangent spaces W = T ( F , F ) p − n ( L µC ( G )) ∩ T ( F , F ) P n [ C ] [ − i ] is isomorphic via p n + i ∗ to the tangent space T F L νC ( G ) . This result implies that the coefficients b µν may be computed by counting the numberof points in a generic fiber of the morphism p n + i | P [ C ] [ − i ] ∩ p − n ( L µC ( G )) ∩ p − n + i ( L νC ( G )) . Fix a generic (in the sense of lemma 22) E ∈ L νC ( G ) such that the quotient sheaf A = G / E is of the form stated in lemma 22. For a generic framed sheaf E ∈ L µC ( G )such that ( E , E ) ∈ P [ C ] [ − i ] the quotient A = G / E satisfy the equality ( A ) x i = C [ ζ i ]( ζ i ) µi ,with µ i = ν i for all i except for exactly one i , for which µ i = ν i − i. So the number ofgeneric E ∈ L µC ( G ) such that ( E , E ) ∈ P [ C ] [ − i ] is exactly a µν , thus b µν = a µν . To prove proposition 23 we need to give a description of the tangent spaces to L µC ( G )and P [ C ] [ − i ]. Tangent to L λC ( G ) . Let λ be a partition of k. Fix F = ( F, φ F ) ∈ L λC ( G ) . Remarkthat since F ∨∨ = G , the framing φ F is just the composition of the inclusion F ֒ → G with the framing φ G of G . So the deformations of F in L λC ( G ) (that in general wouldbe deformations of F and φ F ) are uniquely determined by deformations of the sheaf F with respect to the functor F ( L λC ( G )), where F is the ‘framing-forgetting’ morphismof functors M r,cX,D ( n ) → S , where S is the controvariant functor from the category ofschemes to the category of sets that associates to every scheme S the set consisting ofisomorphism classes of flat families of torsion free sheaves over X parametrized by S. The deformations of the sheaf F have been studied in [3], here we recall that study. EISENBERG ALGEBRA AND FRAMED SHEAVES 19
First note that the functor F ( L λC ( G )) is a subfunctor of the Quot-scheme functor Quot X ( G, | λ | ) . It is well known that deformations at a point F in Quot X ( G, | λ | ) areparametrized by Hom( F, A ), where A = G / F , that can be seen as a subspace of Ext ( F, F )via the connection morphism that comes applying the left-exact functor Hom( F, • ) tothe short exact sequence 0 → F → G → A → . Now, for F = ( F, φ F ) ∈ L λC ( G ) we have the decomposition A = L j A j with A j a skyscraper sheaf, supported at x j ∈ C of length λ j : this gives a decompositionHom( F, A ) = L j Hom(
F, A j ). If we assume that F is generic, around each x j we havea neighborhood in which we have an inclusion G ( − λ j C ) ֒ → F such that the diagram(6) G ( − λ j C ) (cid:31) (cid:127) / / F (cid:127) _ (cid:15) (cid:15) G ( − λ j C ) (cid:31) (cid:127) / / G commutes; it turns out that deformations of F with respect to F ( L λC ( G )) are the onethat preserves these diagrams. Now with deformation theory and diagram chasing onecan prove Proposition 24.
The set of deformations of a sheaf F with respect to the functor F ( L λC ( G )) that preserve the diagram 6 are parametrized by t M j =1 Hom FG ( − λ j C ) , (cid:18) GF (cid:19) x j ! . So the tangent space of L λC ( G ) at a generic framed sheaf F is T F ( L λC ( G )) = t M j =1 Hom FG ( − λ j C ) , (cid:18) GF (cid:19) x j ! . Remark that this is a subspace of Ext ( F, F ( − D )) via t M j =1 Hom FG ( − λ j C ) , (cid:18) GF (cid:19) x j ! ֒ → Hom(
E, A ) = Hom(
E, A ( − D )) ֒ → Ext ( F, F ( − D )) . Tangent to P n [ C ] [ − i ] . We recall that P n [ C ] [ − i ] is the subset of M r,cX,D ( n + i ) × M r,cX,D ( n )consisting of pairs ( F , F ) such that(a) F ∨∨ ∼ = F ∨∨ ;(b) F ⊆ F ;(c) supp( F / F ) = { x } with x ∈ C and length( F / F ) = i. Condition (a) implies that the framing φ F of F is uniquely determined by the inclusionmorphism and by the framing on F . So the deformations of ( F , F ) are one to one withtriples ( F ′ , F ′ , φ ′ ) where F ′ a is a deformation of F a for a = 1 , φ ′ is a deformationof the framing φ F . Moreover, note both pairs ( F ′ , φ ′ ) and ( F ′ , φ ′ ) defines deformationsof F and F , hence define elements of Ext ( F , F ( − D )) and Ext ( F , F ( − D )) respec-tively.The pair ( F ′ , F ′ ) is a deformation of ( F , F ) with respect to the functor F ( P n [ C ] [ − i ]) . This kind of deformations are studied by Baranovsky in [3], here we recall his study.Conditions (b) and (c) are equivalent to having in a neighborhood of x the commu-tative diagram F ( − iC ) (cid:31) (cid:127) / / F (cid:127) _ (cid:15) (cid:15) F ( − iC ) (cid:31) (cid:127) / / F where the lower arrow is the multiplication by i times a generator of the curve C. Soa deformation of ( F , F ) with respect to F ( P n [ C ] [ − i ]) is a pair of sheaves ( F ′ , F ′ ) over X × Spec ( C [ ε ] / ε ), flat over Spec ( C [ ε ] / ε ), with fibers over the closed point isomorphicrespectively to F and F satisfying the following conditions:(i) there is an inclusion F ′ ֒ → F ′ ;(ii) in a neighborhood of x we have an inclusion F ′ ( − iC ) ֒ → F ′ ;(iii) in this same neighborhood we have a commutative diagram(7) F ′ ( − iC ) (cid:31) (cid:127) / / F ′ (cid:127) _ (cid:15) (cid:15) F ′ ( − iC ) (cid:31) (cid:127) / / F ′ where the lower arrow is the multiplication by i times a generator of the curve C. Recall that for a = 1 ,
2, the sheaves F ′ a fit in an exact sequence0 → F a → F ′ a → F a → v a ∈ Ext ( F a , F a ) . By diagram chasing, we get thatcondition (i) is equivalent to the fact that the images of v and v in Ext ( F , F )coincide.Condition (ii) may be treated in the same way: first of all notice that v ∈ Ext ( F , F )defines an element ˜ v ∈ Ext ( F ( − iC ) , F ( − iC )). One can prove that condition (ii) isequivalent to the fact that the images of v and ˜ v in Ext ( F ( − iC ) , F ) coincide. EISENBERG ALGEBRA AND FRAMED SHEAVES 21
Finally we come to condition (iii). Assume that ( F ′ , F ′ ) satisfies conditions (i) and(ii) and label the morphisms:˜ ρ : F ′ ( − iC ) → F ′ , σ : F ′ ( − iC ) → F ′ and ρ the composition of ˜ ρ with the inclusion F ′ → F ′ . This allows us to define amorphism τ : F ( − iC ) → F that keeps track of the commutativity of the diagram 7,i.e. such that τ vanish if and only if condition (iii) holds.These three facts completely describe the deformations of pairs of sheaves ( F , F ) . So the tangent of P n [ C ] [ − i ] at point ( F , F ) is parametrized by pairs( v , v ) ∈ Ext ( F , F ) ⊕ Ext ( F , F )such that the corresponding extensions satisfy conditions (i), (ii) and (iii) and both v a ∈ Ext ( F a , F a ( − D )) for a = 1 , . Proof of proposition 23.
Let w ∈ W. By previous descriptions w = ( v , v ) ∈ Ext ( F , F ( − D )) ⊕ Ext ( F , F ( − D )) is such that the corresponding extensions sat-isfy conditions (i), (ii) and (iii) and moreover v ∈ L tj =1 Hom (cid:16) F G ( − µ j C ) , ( A ) x j (cid:17) , where A = G / F . We want to prove that v ∈ T F L νC ( G ) and that v = 0 only if v = 0 . Since F ∨∨ ∼ = F ∨∨ ∼ = G , we have that v is an element of Hom( F , A ), where A is thequotient sheaf G / F . We need to prove that actually v ∈ L tj =1 Hom (cid:16) F G ( − ν j C ) , ( A ) x j (cid:17) . Using the inclusion morphism from F a to its double dual G , for a = 1 ,
2, and theinclusion morphism F → F , we get the commutative diagram(8) Hom( F , A ) / / (cid:15) (cid:15) Ext ( F , F ) (cid:15) (cid:15) Hom( F , A ) / / Ext ( F , F )Hom( F , A ) O O / / Ext ( F , F ) O O Lemma 25.
The horizontal arrows of the previous diagram are injective morphisms.Proof.
Consider the first row: it comes from the exact sequence0 → Hom( F , F ) → Hom( F , G ) → Hom( F , A ) → Ext ( F , F )It suffices to show that the first arrow is an isomorphism, but it is trivial becauseby lemma 6 we have Hom( F , F ) ∼ = End( C r ) ∼ = Hom( F , G ) . By applying the sameargument to the other arrows, we get the thesis. (cid:3)
By condition (ii) the images of v and v in Ext ( F , F ) coincide. Since the diagram 8commutes and the second horizontal arrow is injective, the images of v and v actuallycoincide in Hom( F , A ) . Recall that the sheaves A and A are supported at points anddiffer only at x , so we can decompose A and A in the following way: A = M j A x j ⊕ B A = M j A x j ⊕ B where B , B are sheaves supported in x , and length( B ) − length( B ) = i (remark that B may be zero). So we can decompose the morphisms v , v in v = M j v ,j ⊕ u v = M j v ,j ⊕ u where v a,j : ( F a ) x j → A x j and u a : ( F a ) x → B a for a = 1 , . Since the images of v , v inHom( F , A ) coincide and F differs from F only at x , we have that v ,j = v ,j . Moreoverat each point x j the morphism v ,j factors through F / G ( − µ j C ) , so v ,j factors through F / G ( − ν j C ) ; so it remains to show only that u factors through F / G ( − ν j C ) knowing that u factors through F / G ( − µ j C ) , and ν j = µ j + i. Recall that in a neighborhood of x wehave an inclusion morphism F ( − iC ) ֒ → F : this induces a diagram similar to 8:Hom( F ( − iC ) , A ( − iC )) / / (cid:15) (cid:15) Ext ( F ( − iC ) , F ( − iC )) (cid:15) (cid:15) Hom( F ( − iC ) , A ) / / Ext ( F ( − iC ) , F )Hom( F , A ) O O / / Ext ( F , F ) O O with injective horizontal arrows. Condition (ii) assures that the images of ˜ v , v inHom( F ( − iC ) , A ) coincide, hence it follows that v factors through F / G ( − ν j C ) with ν j = µ j + i. Now assume that v = 0, we want to show that this implies that v = 0. From thecommutativity of diagram 7 we get the commutativity of the diagram F ( − iC ) / / ˜ v (cid:15) (cid:15) F v (cid:15) (cid:15) A ( − iC ) / / A The upper horizontal arrow in this diagram is injective by hypothesis and this impliesthat the lower one is injective too. Now from the fact that v = 0 follows that ˜ v = 0,hence v = 0 and this concludes the proof of proposition 19. EISENBERG ALGEBRA AND FRAMED SHEAVES 23 Proof of proposition 20
In this section we provide a proof of proposition 20; to achieve this, we compute theintersection number of the fundamental classes of the cycles L (1 n ) C ′ ( s ) and L (1 n ) C ( G ) in M r,cX,D ( n + s ).First note that any framed sheaf F in L (1 n ) C ( G ) corresponds to the kernel of a compo-sition G → G | C → A → A is an Artin sheaf of length n supported on C . Thus L (1 n ) C ( G ) is isomorphicto Quot nC ( G | C ), that is the Quot scheme that parametrizes the length n Artin quotientsheaves of G | C . Let χ : G | C → A be an element in Quot nC ( G | C ) . Denote by F the kernel of χ (notethat it is a locally free sheaf). By Serre’s duality we haveExt ( F , A ) = Hom( A ⊗ F ∨ , ω C ) = H ( C, A ⊗ F ∨ )and H ( C, A ⊗ F ∨ ) = 0 because A is a zero-dimensional sheaf, hence by proposition 2.2.8of [13] we get that Quot nC ( G | C ) is smooth at the point A . Thus L (1 n ) C ( G ) is smooth.Moreover there exists an open subset L (1 n ) C ′ ( s ) • in L (1 n ) C ′ ( s ) isomorphic to a fiber bundleover M regX,D ( r, c, s ) with fibers L (1 n ) C ′ ( H ) for H ∈ M regX,D ( r, c, s ) . Finally we observe that L (1 n ) C ( G ) does not intersect L (1 n ) C ′ ( s ) \ L (1 n ) C ′ ( s ) • because for any F ∈ L (1 n ) C ( G ) the length of the Artin sheaf F ∨∨ / F is equal to n while, on the other hand,if F ∈ L (1 n ) C ′ ( s ) \ L (1 n ) C ′ ( s ) • , then length ( F ∨∨ / F ) ≥ n + 1 . Now we compute the intersection in the case where X = CP and C, C ′ are twodistinct lines intersecting at x / ∈ D ; at the end of the section we explain how the generalcase follows from this. X = CP , C, C ′ distinct lines. Let F = ( F, φ F ) ∈ L (1 n ) C ( G ) ∩ L (1 n ) C ′ ( s ); then F ∨∨ ∼ = G ,the quotient G / F is of length n and supported on both C and C ′ , hence it is supportedonly at x ; moreover ( G / F ) x is a vector space of dimension n , and it is a quotient of G x , wich is an r -dimensional vector space. So the intersection L (1 n ) C ( G ) ∩ L (1 n ) C ′ ( s ) isparametrized by the n -dimensional quotients of G x , i.e. by the Grassmann variety Gr ( G x , n ) (in particular we observe that if n > r the intersection is empty). Now L (1 n ) C ( G ) and L (1 n ) C ′ ( s ) have complementary dimension in M r,cX,D ( s + n ), but theirset-theoretic intersection has positive dimension; however we are dealing with this situ-ation: Gr ( G x , n ) h & & NNNNNNNNNNNN g (cid:15) (cid:15) j / / L (1 n ) C ′ ( s ) f (cid:15) (cid:15) L (1 n ) C ( G ) i / / M r,cX,D ( n + s )where i is a regular imbedding of codimension 2 rs + rn + b and j is a regular imbeddingof codimension 2 rs + n + b by theorem 17.12.1 in [12]. Thus we can apply the excessintersection formula (see chapter 6 in [10]) to get(9) i ∗ [ L (1 n ) C ( G )] · f ∗ [ L (1 n ) C ′ ( s )] = h ∗ ( c n ( r − n ) ( V ) ∩ [ Gr ( G x , n )])where V = g ∗ N L (1 n ) C ( G ) ( M r,cX,D ( n + s )) N Gr ( G x ,n ) ( L (1 n ) C ′ ( s ))is a locally free sheaf of rank n ( r − n ). It is called excess bundle .The vector bundle V arises also from the following exact sequence0 / / T Gr ( G x ,n ) / / g ∗ T L (1 n ) C ( G ) ⊕ j ∗ T L (1 n ) C ′ ( s ) / / h ∗ T M r,cX,D ( n + s ) / / V / / c ( V ) = c ( h ∗ T M r,cX,D ( n + s ) ) c ( T Gr ( G x ,n ) ) c ( g ∗ T L (1 n ) C ( G ) ) c ( j ∗ T L (1 n ) C ′ ( s ) ) Remark . For any pair of flat family of sheaves F and F on M r,cX,D ( n + s ) × CP ,we denote by E xt ip ( F , F ) the i th right derived functor of p ∗ ◦ H om , where p is theprojection from M r,cX,D ( n + s ) × CP to M r,cX,D ( n + s ). We denote by q the projectionfrom M r,cX,D ( n + s ) × CP to the other factor. If for all x ∈ M r,cX,D ( n + s ) the global Extgroup E xt i ( p − ( x ); F , F ) on the fiber p − ( x ) is of constant dimension, then one canprove that E xt ip ( F , F ) is a vector bundle on M r,cX,D ( n + s ) which has the global Extgroup above as the fiber over point x (see [15]). Similar remarks apply to any closedsubspace of M r,cX,D ( n + s ).First we prove the following technical lemma: Lemma 27.
Let ( F, φ F ) be a framed sheaf on X. Then
Ext j ( F ∨∨ , F ( − D )) = 0 for j = 0 , . EISENBERG ALGEBRA AND FRAMED SHEAVES 25
Proof.
Consider the short exact sequence 0 → F → F ∨∨ → A → F ∨∨ , · ), we get0 → Hom( F ∨∨ , F ( − D )) → Hom( F ∨∨ , F ∨∨ ( − D )) → Hom( F ∨∨ , A ( − D )) → Ext ( F ∨∨ , F ( − D )) → Ext ( F ∨∨ , F ∨∨ ( − D )) → Ext ( F ∨∨ , A ( − D )) → Ext ( F ∨∨ , F ( − D )) → Ext ( F ∨∨ , F ∨∨ ( − D )) → Ext ( F ∨∨ , A ( − D )) → A is zero-dimensional, we have Ext i ( F ∨∨ , A ( − D )) = 0 for i = 1 ,
2, henceExt ( F ∨∨ , F ( − D )) ∼ = Ext ( F ∨∨ , F ∨∨ ( − D )) . By lemma 7 we get Ext j ( F ∨∨ , F ∨∨ ( − D )) =0 for j = 0 ,
2, hence we get the thesis. (cid:3)
Let ¯ E = ( ¯ E, φ ¯ E ) be the universal family on M r,cX,D ( n + s ) × CP . Recall that thetangent bundle to M r,cX,D ( n + s ) is E xt p ( ¯ E, ¯ E ⊗ q ∗ O CP ( − D )).We have the following result for the sheaf h ∗ E xt p ( ¯ E, ¯ E ⊗ q ∗ O CP ( − D )). Lemma 28.
Let Q be the universal quotient bundle on the Grassmann variety Gr ( G x , n ) .Then the full Chern class of h ∗ E xt p ( ¯ E, ¯ E ⊗ q ∗ O CP ( − D )) is equal to ( c ( Q ) c ( Q ∨ )) r , where c ( Q ) (resp. c ( Q ∨ ) ) is the full Chern class of Q (resp. of its dual).Proof. Let e E = ( h × id CP ) ∗ ¯ E ; then by universal property we get e E| { A }× CP ∼ = F where F is the kernel of the composition morphism G → G x → A for A ∈ Gr ( G x , n )and the framed sheaf F is F with the framing induced by inclusion to G .We denote by I x the ideal sheaf of x and by C x the quotient sheaf O X / I x . Recallthat the inclusion morphism from a torsion free sheaf on X in its double dual sheafinduces a short exact sequence of sheaves on X ; using the universal sheaf e E one can givea ‘universal version’ of this exact sequence and obtain an exact sequence of sheaves on Gr ( G x , n ) × CP :(11) 0 −→ e E −→ q ∗ G −→ B −→ B = p ∗ Q ⊗ q ∗ C x (we denote by p and q the projection morphisms from Gr ( G x , n ) × CP to the first factor and the second factor respectively).We want to compute the full Chern class of the sheaf h ∗ E xt p ( ¯ E, ¯ E ⊗ q ∗ O CP ( − D )) ∼ = E xt p ( e E, e E ⊗ q ∗ O CP ( − D ))Applying the functor H om p ( · , e E ⊗ q ∗ O CP ( − D )) to the exact sequence 11, we obtaina long exact sequence of sheaves on the Grassmann variety:0 −→ E xt p ( B, e E ⊗ q ∗ O CP ( − D )) −→ E xt p ( q ∗ G, e E ⊗ q ∗ O CP ( − D )) −→−→ E xt p ( e E, e E ⊗ q ∗ O CP ( − D )) −→ E xt p ( B, e E ⊗ q ∗ O CP ( − D )) −→ Indeed, the sheaf H om p ( e E, e E ⊗ q ∗ O CP ( − D )) is zero since its fiber over A ∈ Gr ( G x , n )is Hom( F, F ( − D )), that vanishes by lemma 7. The sheaf E xt p ( q ∗ G, e E ⊗ q ∗ O CP ( − D )) iszero because its fiber Ext ( G, F ( − D )) over any point A of the Grassmann variety, thatit is zero by lemma 27.Thus the full Chern class of E xt p ( e E, e E ⊗ q ∗ O CP ( − D )) is equal to: c ( E xt p ( B, e E ⊗ q ∗ O CP ( − D ))) c ( E xt p ( q ∗ G, e E ⊗ q ∗ O CP ( − D ))) c ( E xt p ( B, e E ⊗ q ∗ O CP ( − D )))Now we compute the three terms in the formula above.To compute c ( E xt p ( q ∗ G, e E ⊗ q ∗ O CP ( − D ))), first we tensor the exact sequence 11 bythe invertible sheaf q ∗ O CP ( − D ) and then apply the functor H om p ( q ∗ G, · ). We get0 −→ H om p ( q ∗ G, q ∗ G ⊗ q ∗ O CP ( − D )) −→ H om p ( q ∗ G, B ⊗ q ∗ O CP ( − D )) −→−→ E xt p ( q ∗ G, e E ⊗ q ∗ O CP ( − D )) −→ E xt p ( q ∗ G, q ∗ G ⊗ q ∗ O CP ( − D )) −→ A ∈ Gr ( G x , n ) of the sheaf H om p ( q ∗ G, e E ⊗ q ∗ O CP ( − D )) is equal toHom( G, F ( − D )), that vanishes by lemma 27. Thus the sheaf H om p ( q ∗ G, e E ⊗ q ∗ O CP ( − D ))is zero.The sheaf E xt p ( q ∗ G, B ⊗ q ∗ O CP ( − D ))) is zero, because its fiber over A ∈ Gr ( G x , n ) isH ( CP , G ∨ ⊗ A ( − D )) = 0. Moreover, the sheaf E xt p ( q ∗ G, q ∗ G ⊗ q ∗ O CP ( − D )) is a trivialvector bundle, because its fiber over all the points of Gr ( G x , n ) is Ext ( G, G ( − D )).Thus c ( E xt p ( q ∗ G, e E ⊗ q ∗ O CP ( − D ))) = c ( H om p ( q ∗ G, B ⊗ q ∗ O CP ( − D )))We have H om p ( q ∗ G, B ⊗ q ∗ O CP ( − D )) ∼ = H om p ( q ∗ G, q ∗ ( C x ⊗ O CP ( − D ))) ⊗ Q because Q is a locally free sheaf.Moreover the stalk of H om ( q ∗ G, q ∗ ( C x ⊗ O CP ( − D ))) is equal to Hom( G x , C x ) ∼ = C r over the points ( A, x ) ∈ Gr ( G x , n ) × CP and it is equal to zero over the points ( A, y )with y = x . Thus the sheaf H om p ( q ∗ G, q ∗ ( C x ⊗ O CP ( − D ))) is the constant sheaf C r on Gr ( G x , n ).Thus c ( H om p ( q ∗ G, B ⊗ q ∗ O CP ( − D ))) = c ( C r ⊗ Q ) = c ( Q ) r Following Baranovsky’s computations in the proof of lemma 6.1 in [3], we get c ( E xt p ( B, e E ⊗ q ∗ O CP ( − D ))) = c ( H om p ( B, B ⊗ q ∗ O CP ( − D ))) = c ( Q ∨ ⊗ Q ) EISENBERG ALGEBRA AND FRAMED SHEAVES 27 and c ( E xt p ( B, e E ⊗ q ∗ O CP ( − D ))) = c ( Q ∨ ⊗ Q ) c ( Q ∨ ) r Summing up the results of our computation, we get c ( E xt p ( e E, e E ⊗ q ∗ O CP ( − D ))) = c ( Q ∨ ⊗ Q ) c ( Q ∨ ) r c ( Q ) r c ( Q ∨ ⊗ Q ) = ( c ( Q ∨ ) c ( Q )) r (cid:3) By this lemma we get c ( h ∗ T M r,cX,D ( n + s ) ) = c ( h ∗ E xt p ( E, E ⊗ q ∗ O CP ( − D ))) = ( c ( Q ∨ ) c ( Q )) r A similar approach can be used with T L (1 n ) C ( G ) and T L (1 n ) C ′ ( s ) (as sheaves on Gr ( G x , n ) × C and on Gr ( G x , n ) × C ′ ). We obtain the following result: Lemma 29.
Let Q be the universal quotient bundle on the Grassmann variety Gr ( G x , n ) .Then c (cid:16) g ∗ T L (1 n ) C ( G ) (cid:17) = c (cid:18) j ∗ T L (1 n ) C ′ ( s ) (cid:19) = c ( Q ) r Proof.
Note that we can consider L (1 n ) C ( G ) (resp. L (1 n ) C ′ ( s ) • ) as a variety that parametrizesArtin quotient of G | C on C of length n (resp. Artin quotient of G ′ | C ′ on C ′ of length n , where G ′ ∈ M regX,D ( r, c, s )). From this point of view, we can use the same result ofBaranovsky (see lemma 6.2 in [3]) and get the thesis. (cid:3) Let S be the universal subbundle on the Grassmann variety Gr ( G x , n ). Recall that T Gr ( G x ,n ) ∼ = S ∨ ⊗ Q . Now using formula 10, we get c ( V ) = ( c ( Q ∨ ) c ( Q )) r c ( S ∨ ⊗ Q ) c ( Q ) r = c ( Q ∨ ) r c ( S ∨ ⊗ Q ) c ( Q ) r Since c ( Q ) r = c ( S ∨ ⊗ Q ) c ( Q ∨ ⊗ Q ) and c ( Q ∨ ) r = c ( S ⊗ Q ∨ ) c ( Q ⊗ Q ∨ ), we obtain c ( V ) = ( c ( Q ∨ ) c ( S ∨ ⊗ Q ) c ( Q ) r = c ( S ∨ ⊗ Q ) c ( S ⊗ Q ∨ ) c ( Q ∨ ⊗ Q ) c ( S ∨ ⊗ Q ) c ( Q ∨ ⊗ Q ) = c ( S ⊗ Q ∨ )The vector bundle S ⊗ Q ∨ is the cotangent bundle on the Grassmann variety, hencewe have ( c n ( r − n ) ( V ) ∩ [ Gr ( G x , n )]) = ( − n ( r − n ) Z Gr ( G x ,n ) c n ( r − n ) ( T Gr ( G x ,n ) )and therefore i ∗ [ L (1 n ) C ( G )] · f ∗ [ L (1 n ) C ′ ( s )] = ( − n ( r − n ) χ ( Gr ( G x , n )) = ( − ( r − n (cid:18) rn (cid:19) where we denote by χ ( Gr ( G x , n )) the topological Euler characteristic of Gr ( G x , n ). General case.
Recall that by hypothesis, C ∩ C ′ ∩ D = ∅ . Now C and C ′ intersects in q points x , x , . . . , x q . Put a i = ( − ( r − i (cid:0) ri (cid:1) .The intersection between L (1 n ) C ( G ) and L (1 n ) C ′ ( s ) is, set-theoretically, the set of framedsheaves F such that F ∨∨ ∼ = G and the length of the quotient sheaf A := G / F is n . Notethat A is a quotient of M x ∈ C ∩ C ′ G x .Consider a subdivision ν = ( ν , ν , . . . , ν q ) of n , i.e. an ordered q-pla of non-negativeintegers ν , . . . , ν q such that P qi =1 ν i = n . Let x = P i ν i [ x i ] be a cycle with x i ∈ C ∩ C ′ .Let W ν be the subset of L (1 n ) C ( G ) ∩ L (1 n ) C ′ ( s ) formed by all framed sheaves F that arekernels of morphisms G → A , where A is an Artin sheaf supported at the cycle x . Thus W ν is isomorphic to a products of a Grassmanians Q x i ∈ C ∩ C ′ Gr ( G x i , ν i ), hence it is airreducible closed subset of L (1 n ) C ( G ) ∩ L (1 n ) C ′ ( s ) (note that W ν does not dipend from thefixed cycle x but only from the subdivision).Moreover, by definition of W ν follows L (1 n ) C ( G ) ∩ L (1 n ) C ′ ( s ) = [ ν subdivision W ν and therefore the subsets W ν are the irreducible components of L (1 n ) C ( G ) ∩ L (1 n ) C ′ ( s ). Toeach W ν we can associate a excess bundle V as before.Since all sheaves split into direct products over the individual Grassmanians, we getthat V is direct sum of the excess bundles associates to each individual Grassmanian.Thus the top Chern class of V is equal to the product a ν · · · a ν q .By lemma 7.1 (a) and by definition 6.1.2 in [10], to get the intersection number between L (1 n ) C ( G ) and L (1 n ) C ′ ( s ) we need to sum up the products a ν · · · a ν q over all subdivisions ν = ( ν , ν , . . . , ν q ) of n . By a combinatorial argument and using the binomial formula,we get ∞ X n =0 z n < [ L (1 n ) C ( G )] , [ L (1 n ) C ′ ( s )] > = (1 + ( − r − z ) rq . References [1] Amar Abdelmoubine Henni. Monads for torsion-free sheaves on multi-blow-ups of the projectiveplane. 2009, 0903.3190.[2] Luis F. Alday, Davide Gaiotto, and Yuji Tachikawa. Liouville correlation functions from 4-dimensional gauge theories. 2010, 0906.3219.[3] Vladimir Baranovsky. Moduli of sheaves on surfaces and action of the oscillator algebra.
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Mathematical Physics Sector, SISSA, Via Bonomea 265, 34100, Trieste (ITALY)Laboratoire Paul Painlev´e, Universit´e Lille 1, Cit´e Scientifique, 59655, Villeneuve D’AscqCedex (FRANCE)
E-mail address : [email protected] Mathematical Physics Sector, SISSA, Via Bonomea 265, 34100, Trieste (ITALY)Laboratoire Paul Painlev´e, Universit´e Lille 1, Cit´e Scientifique, 59655, Villeneuve D’AscqCedex (FRANCE)
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