Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves

Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves

FRANCESCOBOTTACIN(*)

ABSTRACT- Let X be a smoothn-dimensional projective variety, and let Y b e a moduli space of stable sheaves onX. By using the local Atiyah class of a universal family of sheaves onY, which is well defined even when such a universal family does not exist, we are able to construct natural maps

f :Hⁱ(X;V_X^j) !H^{k‡i n}(Y;V_Y^{k‡j n});

for anyi;jˆ1;. . .;nand anykmaxfn i;n jg. In particular, forkˆn i, the mapf associates a closed differential form of degreej i on the moduli spaceY to any element ofHⁱ(X;V_X^j). This method provides a natural way to construct closed differential forms on moduli spaces of sheaves. We remark that no smoothness hypothesis is made on the moduli spaceY. As an application, we describe the construction of closed differential forms on the Hilbert schemes of points ofX.

Introduction.

Let X be a smooth projective variety, defined over an algebraically closed field k of characteristic 0, and let Y be a moduli space of stable sheaves onX. UsuallyYwill be a singular quasi-projective variety, whose points parametrize isomorphism classes of stable coherent sheaves ofO_X- modules with a fixed Hilbert polynomial.

It is by now well known that many geometric properties of the moduli spaceY are determined by similar properties of the varietyX. One of the first examples was discovered by S. Mukai in [Mu1], where he proved that

(*) Indirizzo dell'A.: Dipartimento di Matematica Pura e Applicata, UniversitaÁ di Padova, Via Trieste, 63 - 35121 Padova.

E-mail: bottacin@math.unipd.it

2000Mathematics Subject Classification. Primary: 14J60, Secondary: 14D20, 58A10.

ifXis a symplectic algebraic surface (i.e., an abelian or K3 surface), one can construct, in a natural way, a non-degenerate 2-form on the moduli spaceY of stable sheaves onX. The closedness of this 2-form was later proved by the same author in [Mu2], but only for stable vector bundles. A proof of the closedness of Mukai's 2-form valid in general was finally given in [HL].

Mukai's result was later generalized by S. Kobayashi [K] to the case of the moduli space of simple vector bundles over a compact KaÈhler manifold Xendowed with a holomorphic symplectic structure.

A different proof of the closedness of these so-called ``trace forms'' can be found in [R]. In this case too, the proof is given only for locally free sheaves.

Tyurin [Ty] gave another example of the relations between geometric properties of the moduli spaceYand those of the spaceX, by proving that the choice of a Poisson structure on an algebraic Poisson surface X de- termines a Poisson structure on the moduli spaceY of stable vector bun- dles on X. Actually Tyurin did not prove that the Poisson bracket con- structed onYsatisfies the Jacobi identity; this is nevertheless true, as we proved later in [B1].

In the present paper we shall prove the following result:

THEOREM. Let nˆdimX. For any i;jˆ1;. . .;n and any k maxfn i;n jg, there is a natural map

f :Hⁱ(X;V_X^j) !H^{k‡i n}(Y;V_Y^{k‡j n}):

Moreover, for anys2Hⁱ(X;V_X^j), the element f(s)is d-closed.

As a corollary, we obtain, forkˆn iand anyij, a natural map f :Hⁱ(X;V_X^j) !H⁰(Y;V_Y^{j i});

i.e., we can construct closed differential forms of degreej ion the moduli spaceYstarting from elements inHⁱ(X;V_X^j). In particular, ifXhas trivial canonical bundle, we obtain a (natural) closedn-formf(1)2H⁰(Y;Vⁿ_Y) on any moduli space of stable sheaves onX. This result may be considered as a higher dimensional generalization of Mukai's result.

After a first version of this paper was completed, we came across the paper [KM], where the authors construct closed 2-forms on the smooth locus of some moduli spaces of sheaves via the Atiyah class of a universal family, using a construction that is essentially a specialization of our con- struction of closedn-forms to the casenˆ2. In some cases they are also

able to prove that the resulting 2-forms are non-degenerate, hence they define a symplectic structure on the moduli spaceY.

However, the results concerning the existence of closed holomorphic 2- forms contained in [KM] are stated and proved only for the smooth locus of the moduli spaces of stable sheaves. This is, in general, a very strong lim- itation; in fact, moduli spaces of stable sheaves on an algebraic varietyXof dimension greater than 2 are almost always singular. Moreover, the use of the Atiyah class of a universal family of sheaves is problematic, since usually there does not exist such a universal family of sheaves. As we shall prove in the sequel, these difficulties can be overcome by using a local version of the Atiyah class of some (locally defined) universal family of sheaves.

Let us briefly describe now the organization of the paper. In Section 1 we start by recalling some basic results about cup-products and trace maps for coherent sheaves. Then we recall the definition of the Atiyah class of a co- herent sheaf and we introduce the notion of the local Atiyah class of a flat family of coherent sheaves. In a relative setting, the local Atiyah class turns out to be the natural object to consider. In fact, we shall prove that, when we consider a moduli spaceYof stable sheaves on a smooth projective varietyX, the local Atiyah class of a universal family of sheaves onYis well defined even when such a universal family does not exist (which is usually the case).

Finally, we introduce some cohomology classes, called Newton poly- nomials, constructed by taking the trace of repeated compositions of the Atiyah class with itself. In this case too, in a relative setting, there exists a local versionoftheNewton polynomials, constructed byusingthe localAtiyahclass.

In Section 2 we use the local Newton polynomials of a universal family of sheaves to define the map

f :Hⁱ(X;V_X^j) !H^{k‡i n}(Y;V_Y^{k‡j n}):

Then we prove that the elements in the image off ared-closed.

As an application, we show how these results can be applied in order to construct closed differential forms on Hilbert schemes of points of a smooth projective variety.

1. Preliminaries.

1.1 ±Cup-product and trace maps.

In this section we shall briefly recall some standard results about trace maps and cup-products. For more details (and proofs) we refer to [A] or [HL].

LetXbe a scheme of finite type over a fieldkof characteristic 0. In the sequel, whenever we speak of a sheaf onXwe shall always mean a sheaf of O_X-modules. LetEbe a locally free sheaf of finite rank onX. The usual trace map

tr:End(E)! O_X induces natural maps

tr:Extⁱ(E;E)!Hⁱ(X;O_X);

for anyi0. More generally, for any coherent sheafFonX, we can define a trace map

tr:Hom(E;E F)! F: In this case the induced maps on cohomology are

tr:Extⁱ(E;E F)!Hⁱ(X;F):

IfE,FandGare three locally free sheaves onX, there is a cup-product (or Yoneda composition) map

Extⁱ(F;G)Ext^j(E;F) ! Ext^i‡j(E;G):

In the special case EˆFˆG, we find that the composition of the cup- product and trace maps is graded commutative in the following sense: if a2Extⁱ(E;E) andb2Ext^j(E;E), then

tr(ab)ˆ( 1)^ijtr(ba) (1:1)

as an element ofH^i‡j(X;OX).

Analogous maps can be easily defined when the locally free sheavesE, F and G are replaced by finite complexes E, F and G of locally free sheaves of finite rank onX. This fact can be used to define cup-products and trace maps also for coherent sheavesE,F andGonX, provided that these sheaves admit finite resolutions by locally free sheaves. This is the case, for instance, ifXis a smooth projective variety.

There is also a relative version of the preceding constructions.

LetXandYbe two schemes of finite type overkandE,F andGbeY- flat families of coherent sheaves onX, i.e., coherent sheaves onXY, flat overY. We shall always assume that all these sheaves admit finite locally free resolutions. We shall denote byp:XY!Xandq:XY!Ythe canonical projections and byExt_qⁱ(E;F) thei-th relative Ext-sheaf, i.e., the i-th derived funtor ofqHom_OXY(E;F).

In this situation there is a relative trace map tr:Ext_qⁱ(E;E F)!RⁱqF and a relative cup-product

Ext_qⁱ(F;G) Ext_q^j(E;F) ! Ext_q^i‡j(E;G);

satisfying the analogue of (1.1) whenE ˆ F ˆ G.

1.2 ±The Atiyah class.

LetXbe a scheme of finite type overkandEbe a coherent sheaf onX which admits a finite locally free resolution. Let us denote by P¹_X(E) the sheaf of principal parts of order 1 of sections ofE(cf. [EGA4, 4]). There is a natural exact sequence

0! E V¹_X ! P¹_X(E)! E !0;

(1:2)

henceP¹_X(E) can be regarded as an extension ofEbyE V¹_X. DEFINITION1.1. The Atiyah class ofEis the class

a(E)2Ext¹(E;E V¹_X) corresponding to the extension (1.2).

By taking the trace, we get an element tr(a(E))2H¹(X;V¹_X): this is just the opposite of the first Chern classc₁(E).

IfY is another scheme of finite type overkandEis aY-flat family of coherent sheaves onX, admitting a finite locally free resolution, we have a global Atiyah class

a(E)2Ext¹_XY(E;E V¹_XY);

defined as above. However, in this relative situation we are more inter- ested in a ``local'' version of the Atiyah class ofE, which we shall denote by a(E).~

DEFINITION1.2. The local Atiyah class ofEis the section

~

a(E)2H⁰(Y;Ext¹_q(E;E V¹_XY))

defined as the image ofa(E) under the natural map

Ext¹_XY(E;E V¹_XY)!H⁰(Y;Ext¹_q(E;E V¹_XY)) (1:3)

coming from the local-to-global spectral sequenceHⁱ(Y;Extq^j))Ext^i‡j_XY. We shall now explain why, in a relative setting, the ``correct'' object to consider is not the global Atiyah classa(E) but the local one~a(E).

First of all, let us recall that twoY-flat familiesE andF of coherent sheaves on X are said to be equivalent ifF  E OXYqL, for some in- vertible sheafLonY. Since the Atiyah class of a tensor product of sheaves is given by

a(E₁ E₂)ˆa(E₁)id_E2‡id_E1a(E₂);

it follows that a(F)ˆa(E OXYqL)6ˆa(E), in general. However, if we consider the local Atiyah classes, we have:

LEMMA1.3. LetEandFbe two Y-flat families of coherent sheaves on X.

IfEandFare equivalent, then the local Atiyah classesa(E)~ anda(F~ )are identified under the natural isomorphism

H⁰(Y;Ext¹_q(E;E V¹_XY))H⁰(Y;Ext¹_q(F;F V¹_XY)):

(1:4)

PROOF. IfEandFare equivalent we can writeF  E qL, for some invertible sheafLonY. SinceLis locally free of rank one, there is a natural isomorphism of sheaves

Ext¹_q(E;E V¹_XY) Ext¹_q(F;F V¹_XY);

(1:5)

hence, from now on, we shall identify these two sheaves.

By taking the global Atiyah class ofF, we have a(F)ˆa(E)id_qL‡id_Ea(qL):

The difference betweena(F) anda(E), under the natural identification (1.4) induced by (1.5), is then given by idE a(qL)ˆidE q(a(L)), where

a(L)2Ext¹_Y(L;LV¹_Y)H¹(Y;V¹_Y):

The standard exact sequence associated to the local-to-global spectral se- quence for the Ext groups is

H¹(Y;qHom(E;E V¹_XY))!Ext¹(E;E V¹_XY)!H⁰(Y;Ext¹_q(E;E V¹_XY)):

Now it can be checked that idEa(qL) b elongs to H¹(Y;qHom(E;E V¹_XY)), hence its image inH⁰(Y;Ext¹_q(E;E V¹_XY)) is zero. It follows that

a(F~ )ˆ~a(E) as stated. p

1.3 ±The Newton polynomials.

LetXbe a scheme of finite type overkandEbe a coherent sheaf onX that admits a finite resolution by locally free sheaves. By repeatedly composing the Atiyah class a(E)2Ext¹(E;E V¹_X) with itself, we obtain classes in Extⁱ(E;E (V¹_X)ⁱ), for anyi1. Then, by composing with the map induced by the natural map (V¹_X)ⁱ!Vⁱ_X, we get classes a(E)ⁱ2 2Extⁱ(E;E Vⁱ_X).

We now give the following definition:

DEFINITION 1.4. The i-th Newton polynomial of E is the cohomology class

gⁱ(E)ˆtr(a(E)ⁱ)2Hⁱ(X;Vⁱ_X):

The Newton polynomials ofE coincide, up to a scalar factor, with the components of the Chern character ofEand, as the Chern classes, they are d-closed. More precisely:

LEMMA1.5. Let d:Hⁱ(X;Vⁱ_X)!Hⁱ(X;V^i‡1_X )be the map induced by the exterior differential d:Vⁱ_X !V^i‡1_X . Then dgⁱ(E)ˆ0.

For the proof of this result we refer the reader to [HL, Sect. 10.1.6].

In a relative situation, whereEis aY-flat family of coherent sheaves on X, we have the global Newton polynomials gⁱ(E)2Hⁱ(XY;Vⁱ_XY) and also their local versions, constructed by using the local Atiyah class ofE,

~gⁱ(E)2H⁰(Y;Rⁱq(Vⁱ_XY)):

We note that~gⁱ(E) is the image ofgⁱ(E) under the natural map Hⁱ(XY;Vⁱ_XY)!H⁰(Y;Rⁱq(Vⁱ_XY))

sending a global section of a presheaf to a global section of the associated sheaf.

Exactly as for the global Newton polynomials, the local ones too ared- closed, i.e., we haved~gⁱ(E)ˆ0.

2. Closed Forms on Moduli Spaces.

From now on we shall denote byXa smooth projective variety defined overkand we take asY a moduli space of stable sheaves onX.

For the definition and the construction of moduli spaces of stable sheaves onXwe refer to [S]: what we need to know, for the moment, is that moduli spaces of stable sheaves exist and they are quasi-projective vari- eties (usually singular).

In order to apply the constructions described in the previous section we need to know that finite locally free resolutions of coherent sheaves exist:

this is indeed the case sinceXis a smooth projective variety.

Finally, there is another technical problem we have to deal with: the non- existence of universal families of sheaves. In general, on a moduli spaceYof stable sheaves onXthere does not exist a universal family of sheavesE, not even locally for the Zariski topology. However, universal families onYexist locally in the eÂtale topology (or in the usual complex topology, if the base field kis the complex field) [S, Theorem 1.21]. These local universal families are not uniquely determined, they are defined only up to tensoring with the pull- back of an invertible sheaf onY. Usually these ambiguities prevent the local universal families to glue together to a globally defined one.

Let us denote byE_aa collection of local universal families, one for each open setUaof a suitable covering of the moduli spaceY(in the appropriate topology). Even if the sheavesEa do not glue to define a global universal familyE, the relative Ext-sheavesExtⁱ_q(E_a;E_a) andExtⁱ_q(E_a;E_aVⁱ_XYj_XUa) do indeed glue to define global sheaves onY. For this reason, by abusing notation, we shall denote these sheaves byExtⁱ_q(E;E) andExtⁱ_q(E;E Vⁱ_XY), even if the universal familyEdoes not exist globally onY.

Since eachEais defined only up to tensoring with the pull-back of an in- vertible sheaf onUa, the global Atiyah classa(Ea) is not well defined. How- ever, by recalling the result of Lemma 1.3, we see that the local Atiyah class

~a(Ea)2H⁰(Ua;Ext¹_q(Ea;EaV¹_XYj_XUa))ˆH⁰(Ua;Ext¹_q(E;E V¹_XY)) is well defined, since it depends only on the equivalence class ofE_a. Now it is easy to check that all these local sections~a(Ea) glue together to define a global section of the sheafExt¹_q(E;E V¹_XY). Again, by abuse of notation, we shall denote this global section by

~a(E)2H⁰(Y;Ext¹_q(E;E V¹_XY));

even if the universal familyEdoes not exist.

Analogous considerations hold also for the Newton polynomials: while the global Newton polynomialsgⁱ(E) are not defined, there are well defined local Newton polynomials

~gⁱ(E)2H⁰(Y;Rⁱq(Vⁱ_XY)):

Now, for any open subset UY we have a direct sum decomposition V¹_XUˆpV¹_XqV¹_U. SinceXis a smooth variety, it follows that there is a KuÈnneth decomposition

Hⁿ(XU;Vⁿ_XU)ˆMⁿ

i;jˆ0

Hⁱ(X;V_X^j)H^{n i}(U;V^{n j}_U );

for everyn0. The sheaf onY associated to the presheaf U7!Hⁿ(XU;Vⁿ_XU)

is the higher direct image Rⁿq(Vⁿ_XY) and, from the above KuÈnneth de- composition, we obtain an analogous decomposition of sheaves

Rⁿq(Vⁿ_XY)ˆMⁿ

i;jˆ0

Hⁱ(X;V_X^j) H^{n i}(Y;V^{n j}_Y );

whereH^{n i}(Y;V^{n j}_Y ) is the sheaf associated to the presheaf U7!H^{n i}(U;V^{n j}_U )

and where, by abuse of notation, we have writtenHⁱ(X;V_X^j) to denote the corresponding constant sheaf onY.

By using this direct sum decomposition, we obtain a decomposition of the local Newton polynomials:

DEFINITION2.1. For a Newton polynomial~gⁿ(E)2H⁰(Y;Rⁿq(Vⁿ_XY)), we shall write~gⁿ(E)ˆP

i;j~gⁿ_i;j(E), where~gⁿ_i;j(E) is a global section of the sheaf Hⁱ(X;V_X^j) H^{n i}(Y;V^{n j}_Y ).

We note that the restriction of~gⁿ_i;j(E) to a sufficiently small open subset UY gives a section

~

gⁿ_i;j(E)j_U 2Hⁱ(X;V_X^j)H^{n i}(U;V^{n j}_Y j_U):

We are now able to prove the following result:

THEOREM 2.2. Let nˆdimX. For any i;jˆ1;. . .;n and any kmaxfn i;n jg, there is a natural map

f :Hⁱ(X;V_X^j) !H^{k‡i n}(Y;V_Y^{k‡j n}):

Moreover, for anys2Hⁱ(X;V_X^j), the element f(s)is d-closed.

PROOF. The mapf is defined by composing the isomorphism Hⁱ(X;V_X^j) ! H^{n i}(X;V^{n j}_X )

given by Serre duality, with the map

H^{n i}(X;V^{n j}_X ) !H^{k‡i n}(Y;V^{k‡j n}_Y )

induced by multiplication by the section ~g^k_{n i;n j}(E) of the sheaf H^{n i}(X;V^{n j}_X ) H^{k‡i n}(Y;V^{k‡j n}_Y ). It only remains to show that, for any s2Hⁱ(X;V_X^j), we haved(f(s))ˆ0. The proof is an adaptation of the one given in [HL] for 2-forms.

Let us write ~g^k_{n i;n j}(E)ˆP

l a_lb_l, for some a_l2H^{n i}(X;V^{n j}_X ) and some sections b_l of H^{k‡i n}(Y;V_Y^{k‡j n}). Then, since the Newton poly- nomials, and also their KuÈnneth components, ared-closed, we can write

0ˆd~g^k_{n i;n j}(E)ˆX

l

dXalb_l‡aldYb_l :

Now, since X is a smooth projective variety, we have dXalˆ0, hence it follows that

X

l

a_ld_Yb_l ˆ0:

(2:1)

By recalling the definition of the mapf, we can write f(s)ˆP

l hs;a_lib_l, where

hs;ali ˆ Z

X

s^al

is the Serre duality pairing. It follows, by (2.1), that d(f(s))ˆ

ˆP

l hs;alidb_lˆ0. p

We note, in particular, that forkˆn iwe obtain a natural map f :Hⁱ(X;V_X^j) !H⁰(Y;V_Y^{j i});

(2:2)

for anyij. It follows that we can construct closed holomorphicp-forms onY by starting with elements inHⁱ(X;V^i‡p_X ), for anyi0.

We remark that a similar construction is used in [KM] to define holo- morphic symplectic structures on certain moduli spaces of sheaves onX, in some cases whenXdoes not possess any non-zero holomorphic 2-forms.

A more elementary construction of closed forms on moduli spaces of stable sheaves (valid only on the smooth locus of the moduli space) was given in [B3].

As an application, we will now show how these results can be applied in order to construct closed differential forms on Hilbert schemes of points.

Let X be a smooth projective variety and let us denote by X^[d] ˆ

ˆHilb^d(X) the Hilbert scheme parametrizing 0-dimensional subschemes of X of length d. X^[d] is a projective scheme; it is smooth if d3 or dimX2.

For any 0-dimensional subschemeZ2X^[d], let us denote byI_Zits sheaf of ideals: we have the standard exact sequence

0! I_Z! O_X! O_Z!0:

(2:3)

The Hilbert schemeX^[d] can then be identified with the moduli spaceM parametrizing ideal sheavesI_Zof colengthd. IfZ2X^[d], it is well known that the tangent spaceT_ZX^[d]is given by

T_ZX^[d]Hom(I_Z;O_Z):

By applying the functor Hom(;O_Z) to the exact sequence (2.3) and obser- ving that Ext¹(O_X;O_Z)H¹(X;O_Z)ˆ0 and Hom(O_X;O_Z)Hom(O_Z;O_Z), since any O_X-linear homomorphism from O_X to O_Z vanishes on I_Z, we obtain a canonical isomorphism

Hom(IZ;OZ)Ext¹(OZ;OZ):

The identification ofX^[d] with a moduli space of sheaves onXallows us to apply, to the Hilbert scheme of points ofX, the preceding results about the construction of closed differential forms. It follows that, given a p-form s2H⁰(X;V^p_X), there is a naturally defined closedp-form~sonX^[d]. At a point Z2X^[d]whose support consists ofddistinct pointsP1;. . .;P_dofX, the map

s(Z)~ :TZX^[d] TZX^[d]!k

can be easily described. First of all, note that such aZis a smooth point of X^[d], and we have

T_ZX^[d]ˆM^d

iˆ1

T_PiX:

For any pointPiin the support ofZwe have the map s(P_i):T_PiX T_PiX!k:

Then the map~s(Z) coincides (possibly up to a non-zero constant factor) with the product map Q^d

iˆ1s(P_i).

Note that this explicit description allows us to define thep-form~sonly on the open subsetUofX^[d]parametrizing subschemes ofXwhose support consists ofddistinct points. What our method shows is that thep-form~s thus defined on U actually extends as a closed differential form on the whole ofX^[d](see also [Be] for the construction of closed 2-forms onX^[d], whenXis a K3 surface).

Acknowledgments. We would like to thank the referee for some valu- able comments and suggestions.

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Manoscritto pervenuto in redazione il 21 gennaio 2008.