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Local Structure of Brill-Noether Strata in the Moduli Space of Flat Stable Bundles

ELENAMARTINENGO(*)

ABSTRACT- We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact KaÈhler manifold, proving that these strata have quadratic algebraic singularities.

1. Introduction.

LetXbe a compact complex KaÈhler manifold of dimensionn. LetMbe the moduli space of locally free sheaves of OX-moduleson X which are stable and flat. It is known that a coarse moduli space of these sheaves can be constructed and that it is a complex analytic space (see [Nor] or [LePoi]

for the algebraic case). Moreover the following result, proved in [N] and [G-M88], determines the type of singularities of this moduli space:

THEOREM 1.1. The moduli spaceMhas quadratic algebraic singu- larities.

In his article [N], Nadel constructs explicitly the Kuranishi family of deformations of a stable and flat locally free sheaf ofOX-modulesEonX and he proves that the base space of this family has quadratic algebraic singularities. Whereas the proof given by Goldman and Millson in [G-M88]

is based on the study of a germ of analytic space which prorepresents the functor of infinitesimal deformations of a sheafE. They find out thisana- lytic germ and prove that it hasquadratic algebraic singularities.

(*) Indirizzo dell'A.: Dipartimento di Matematica ``Guido Castelnuovo'', UniversitaÁ di Roma ``La Sapienza''; P.le Aldo Moro, 5 - I-00185 Roma.

E-mail: martinengo@mat.uniroma1.it

1991Mathematics Subject Classification: 14H60, 14D20, 13D10, 17B70.

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This paper is devoted to the local study of the strata of the Brill-Noe- ther stratification of the moduli space Mand in particular, in the same spirit as Theorem 1.1, to the determination of their type of singularities.

In section 2 we study an equivalence relation between germs of analytic spaces under which they are said to have the same type of singularities. We prove that this relation is formal (Proposition 2.14) and that the set of germs with quadratic algebraic singularities is closed under this relation (Theorem 2.16).

In section 3 we introduce the Brill-Noether stratification of the moduli spaceM, we are interested in. The subsets of this stratification are defined in the following way. We fix integers hi2N, for all iˆ0. . .n, and we consider the subspace N(h0. . .hn) M of stable and flat locally free sheaves of OX-moduleson X, with cohomology spaces dimension fixed:

dimHiˆhi, for alliˆ0. . .n. Our aim is to study the local structure of these strataN(h0. . .hn), proving the following

THEOREM1.2 (Main Theorem). The Brill-Noether strataN(h0. . .hn) have quadratic algebraic singularities.

In sections 3 and 4, we define and study the functor DefE0 of in- finitesimal deformations of a stable and flat locally free sheaf of OX- modulesEonX, such thatHi(X;E)ˆhifor alliˆ0. . .n, which preserve the dimensions of cohomology spaces.

In section 5, we find out another functor linked to DefE0 by smooth morphisms and for which it is easy to find a germ which prorepresents it.

Then the Main Theorem followsfrom the formal property of the relation of having the same type of singularities and from the closure of the set of germs with quadratic algebraic singularities with respect to this relation.

2. Singularity type.

LetAnbe the category of analytic algebrasand letAn^ be the category of complete analytic algebras. We recall that ananalytic algebra isaC- algebra which can be written in the formCfx1. . .xng=Iand a morphism of analytic algebrasisa local homomorphism ofC-algebras.

DEFINITION2.1. Ahomomorphism of ringsc:R!S is calledformally smooth if, for every exact sequence of local Artinian R-algebras:

0!I!B!A!0, such that I is annihilated by the maximal ideal of B, the induced mapHomR(S;B)!HomR(S;A)is surjective.

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We recall some facts about formally smooth morphisms of analytic al- gebras, which we use in this section. We start with the following equiva- lence of conditions (see [Ser], Proposition C.50):

PROPOSITION 2.2. Let c:R!S be a local homomorphism of local NoetherianC-algebras conteining a field isomorphic to their residue field C. Then the following conditions are equivalent:

- cis formally smooth,

- ^S is isomorphic to a formal power series ring overR,^

- the homomorphismc^ : ^R!^S induced bycis formally smooth.

Furthermore, we recall two Artin's important results (see [A], Theorem 1.5a and Corollary 1.6):

THEOREM2.3. Let R and S be analytic algebras and let R and^ S be^ their completions. Letc :R!S be a morphism of analytic algebras, then,^ for all n2N, there exists a morphism of analytic algebrascn :R!S, such that the following diagram is commutative:

COROLLARY 2.4. With the notation of Theorem2.3, if in addition c induces an isomorphismc^ : ^R!S, then^ cnis an isomorphism, provided

n2.

Using these results, we can prove the following

PROPOSITION2.5. Let R and S be analytic algebras and letR and^ ^S be their completions. Letc^: ^R!S be a smooth morphism, then there exists^ a smooth morphism R!S.

PROOF. By Thereom 2.2, there exists an isomorphism^f: ^R[[x]]!S,^ Corollary 2.4 implies that there exists an isomorphism f:Rfxg !S, which is obviously smooth by Theorem 2.2. Thus the morphism

fi:R,!Rfxg !S issmooth. p

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To complete our study of analytic algebras, we prove the following PROPOSITION2.6. Let R and S be analytic algebras, such that - dimCmR=m2RˆdimCmS=m2Sand

- Rfz1;. . .;zNg Sfz1;. . .;zMg, for some N and M, then R and S are isomorphic.

PROOF. The first hypothesis implies that, in the isomorphism Rfz1;. . .;zNg Sfz1;. . .;zMg,NˆM. Moreover, proving the proposition by induction on N, the first hypothesis makes the inductive step trivial.

Thus it is sufficient to prove the proposition forNˆ1.

Let RˆCfx1;. . .;xng=I and SˆCfy1;. . .;ymg=J be analytic alge- bras, with I(x1;. . .;xn)2 and J(y1;. . .;ym)2. Let f:Cfxgfzg=I!

!Cfygfzg=J be an isomorphism and let c itsinverse. Let f(z)ˆ

ˆaz‡b(y)‡g(y;z) and let c(z)ˆaz‡b(x)‡c(x;z), where a;a2C are costant,b, g,b and c are polynomial, g and c do not contain degree one terms and, with a linear change of variables, we can suppose thatfandc do not contain constant term.

If at least one betweenaandaisdifferent from zero, then the thesis followseasely. For example, ifa6ˆ0, the imagef(z) satisfies the hypoth- esis of Weierstrass Preparation Theorem and so it can be written as f(z)ˆ(z‡h(y))u(y;z), whereuisa unit andh(y) isa polynomial. Thenf is well defined and induces an isomorphism on quotients: f:Cfxg=I!

!Cfygfzg=J(z‡h(y))Cfyg=J.

Let'snow analyse the caseaˆaˆ0. Letn:Cfxgfzg=I!Cfxgfzg=I be a homomorphism defined byn(xi)ˆxi, for alli, andn(z)ˆz‡b(x). It is obviously an isomorphism and the composition fn is an isomorphism fromCfxgfzg=ItoCfygfzg=J, such thatfn(z) containsa linear term in z, thus, passing to the quotient, it induces an isomorphism Cfxg=I

Cfyg=J. p

Now we consider the following relation between analytic algebras:

R/S iff 9 R !S formally smooth morphism,

let be the equivalence relation between analytic algebrasgenerated by /. We define another equivalence relation:

RS iff Rfx1. . .xng Sfy1. . .ymgare isomorphic, for somenandm:

The relationisthe same asthe relation. Infact, ifRS, there exists a chain of formally smooth morphismsR!T1 T2!. . .!Tn S, that,

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by Theorem 2.2 and Corollary 2.4, gives an isomorphism Rfxg Sfyg, thenRS. Viceversa, ifRS, there exists an isomorphismRfxg Sfyg which is formally smooth, thus we have the chain of formally smooth morphismsR!Rfxg !Sfxg SandRS.

We consider the following relation between complete analytic algebras:

R^ /S^ iff 9R^ !^S formally smooth morphism,

letbe the equivalence relation between analytic algebrasgenerated by/.

We define an other equivalence relation:

R^ S^ iffR[[x^ 1. . .xn]]S[[x^ 1. . .xm]] are isomorphic, for somen andm:

Asbefore, the relationisthe same asthe relation. Furthermore, the equivalence relationon completionsof analytic algebrascoincideswith the relationbetween the analytic algebras themselves, because obviously the two relationsandare the same.

The opposite category of the category of analytic algebrasAnoiscalled the category ofgerms of analytic spaces. The geometrical meaning of this definition isthat a germAocan be represented by (X;x;a), whereXisa complex space with a distinguished pointxandais a fixed isomorphism of C-algebras OX;xA. Two triples, (X;x;a) and (Y;y;b), are equivalent if there exists an isomorphism from a neighborhood of x in X to a neigh- borhood of yinY which sendsxinyand which induces an isomorphism OX;x OY;y.

Let (X;x) and (Y;y) be germsof analytic spaces, given by the analytic algebras S and R respectively, let C:(X;x)!(Y;y) be a morphism of germsof analytic spacesand letc:R!Sbe the corresponding morphism of analytic algebras.

DEFINITION2.7. The morphismC:(X;x)!(Y;y)is calledsmoothif the morphismc:R!S is formally smooth.

We consider the following relation between germs of analytic spaces:

(X;x)/(Y;y) iff 9 (X;x) !(Y;y) smooth morphism and we defineto be the equivalence relation between germsof analytic spaces generated by the relation/. It isobviousthat the relationdefined between germs of analytic spaces is the same as the relation defined between their corresponding analytic algebras. As in [V], we give the fol- lowing

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DEFINITION2.8. The analytic spaces(X;x)and(Y;y)are said to have the sametype of singularitiesif they are equivalent under the relation. Our aim isto prove that the property that two germsof analytic spaces have the same type of singularitiesisformal, that isthat it can be con- trolled at the level of functors.

In all thispaper we consider covariant functorsF :ArtC !Set from the category of local Artinian C-algebraswith residue fieldC to the ca- tegory of sets, such thatF(C)ˆone point set. The functors of this type are calledfunctors of Artin rings. In the following we recall some basic notions about these functors.

Let F be a functor of Artin rings. Thetangent space toF isthe set F(C[e]), whereC[e] isthe local ArtinianC-algebra of dual numbers, i.e.

C[e]ˆC[x]=(x2) . It can be proved thatF(C[e]) hasa structure ofC-vector space (see [S], Lemma 2.10).

Anobstruction theory(V;ve) forF isthe data of a C-vector space V, called obstruction space, and, for every exact sequence in the category ArtC:

e:0 !I !B !A !0;

such that I isannihilated by the maximal ideal of B, a map ve:F(A)!VCIcalledobstruction map. The data (V;ve) have to satisfy the following conditions:

- ifj2 F(A) can be lifted toF(B), thenve(j)ˆ0,

- (base change) for every morphismf :e1!e2of small extensions, i.e.

for every commutative diagram

thenve2(fA(j))ˆ(IdVfI)(ve1(j)), for everyj2 F(A1).

An obstruction theory (V;ve) forFiscalledcompleteif the converse of the first item above holds, i.e. the lifting ofj2 F(A) toF(B) exists if and only if the obstructionve(j) vanishes.

For morphisms of functors we have the following notion of smoothness:

DEFINITION2.9. Letn:F ! Gbe a morphism of functors, it is said to be smoothif, for every surjective homomorphism B!Ain the category ArtC, the induced mapF(B)! G(B)G(A)F(A)is surjective.

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LetFbe a functor of Artin rings, acoupleforF isa pair (A;j), where A2ArtCandj2 F(A). A couple (A;j) forFinducesan obviousmorphism of functors, Hom(A; )! F, which associates, to every B2ArtC and f2Hom(A;B), the elementf(j)2 F(B).

We can extend the functorF to the categorydArtC of local Noetherian complete C-algebraswith residue field C by the formula Fb(A)ˆ

ˆlimF(A=mn). A procouplefor F isa pair (A;j), where A2dArtC and j2F(A). It induces an obvious morphism of functors: Hom(A;b )! F.

DEFINITION 2.10. Aprocouple(A;j)for a functor F is called apro- representable hull of F, or just a hull of F, if the induced morphism Hom(A; )! F is smooth and the induced map between tangent spaces Hom(A;C[e])! F(C[e])is bijective.

Afunctor F is called prorepresentable by the procouple(A;j)if the induced morphismHom(A; )! F is an isomorphism of functors.

The existence, for a functor of Artin ringsF, of a procouple which isa hull of it or which prorepresents it is regulated by the well known Schlessinger conditions(see [S], Theorem 2.11). Moreover a functor which satisfies the two of the Schlessinger's conditions (H1) and (H2) is said to have agood deformation theory. We do not precise this concept, because all functorsthat appear in the following are functorswith a good deformation theory and all the onesinvolved in the proof of the Main Theorem have hull.

With the above notions, we can state the following Standard Smooth- ness Criterion (see [Man99], Proposition 2.17):

THEOREM2.11. Letn:F ! Gbe a morphism of functors with a good deformation theory. Let(V;ve)and(W;we)be two obstruction theories for F andGrespectively. If:

- (V;ve)is a complete obstruction theory, - nis injective between obstructions, - nis surjective between tangent spaces, thennis smooth.

For morphisms between Hom functors the following proposition holds (see [S], Proposition 2.5):

PROPOSITION2.12. Letc^ : ^R!S be a local homomorphism of local^ Noetherian complete C-algebras, let f^:Hom(^S; )!Hom(R;^ ) be the

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morphism of functors induced byc. Then^ f^is smooth if and only if^S is isomorphic to a formal power series ring overR.^

We consider the following relation between functors:

F / G iff 9 F ! G smooth morphism and we defineto be the equivalence relation generated by/.

DEFINITION2.13. The functorsFandGare said to have the sametype of singularity if they are equivalent under the relation.

Now we want to link Definitions2.8 and 2.13 in the case the functors considered have hulls. LetFandGbe two functorswith hulls, given by the germsof analytic spaces(X;x) and (Y;y), defined by the analytic algebrasS andRrespectively. If (X;x)(Y;y), or equivalently, ifRS, there exists a chain of smooth morphisms of analytic algebrasS T1!T2 . . .!R that inducesa chain of smooth morphismsof functorsF Hom(^S; )!

!Hom(T^1; ) Hom(T^2; ). . .Hom(R;^ )! G, by Propositions 2.2 and 2.12 and by definition of hull, thusF G. For the other implication we need the following

PROPOSITION2.14. LetFandGbe two functors with hulls given by the germs of analytic spaces(X;x)and(Y;y)respectively and letf:F ! Gbe a smooth morphism. Then there exists a smooth morphism between the two germs(X;x)and(Y;y).

PROOF. Let S and R be the analytic algebrasthat define the germ (X;xˆ0) and (Y;yˆ0) respectively. By hypothesis, we have the following diagram:

where, by definition of hull, aand b are smooth morphism and they are bijective on tangent spaces. Then, by smoothness, there exists a morphism

^f:Hom(S;^ )!Hom(R;^ ) that makesthe diagram commutative. By hypothesis ona,bandf, it is surjective on tangent spaces and it is injective on obstruction spaces. Thus f^ is a smooth morphism, by the Standard Smoothness Criterion (Theorem 2.11).

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The morphism ^f determinesuniquely an homomorphism c^ : ^R!S,^ which is formally smooth, by Propositions 2.2 and 2.12. Now, by Proposi- tion 2.5, there exists a formally smooth morphism c:R!S and so a smooth morphism between the germs (X;x) and (Y;y). p Now, if F G, there exists a chain of smooth morphisms of functors F H1! H2 . . .! G. ThenHinecessary have hulls, we indicate with Tithe complete analytic algebra that isan hull forHi. By Proposition 2.14, the chain of smooth morphisms of functors gives a chain of smooth morphisms of complete analytic algebras S^!T1 T2!. . . R, thus^ S^R, so, as we have observed,^ SRand (X;x)(Y;y).

Now we return to germsof analytic spacesand we concentrate our interests on quadratic algebraic singularities.

DEFINITION2.15. Let X be a complex affine scheme, it is said to have quadratic algebraic singularitiesif it is defined by finitely many quadratic homogeneous polynomials. LetXbe an analytic space, it is said to have quadratic algebraic singularitiesif it is locally isomorphic to complex af- fine schemes with quadratic algebraic singularities.

For germsof analytic spaceswe want to prove the following

THEOREM2.16. Let(X;0)and(Y;0)be two germs of analytic spaces and let f:(X;0) !(Y;0)be a smooth morphism. Then(X;0)has quad- ratic algebraic singularities if and only if(Y;0)has quadratic algebraic singularities.

We need the following

LEMMA2.17. Let(X;0)and(Y;0)be two germs of analytic spaces and let f:(X;0) !(Y;0) be a smooth morphism. Let XCN and let Hˆ fx2CNjh(x)ˆ0gbe an hypersurface ofCN, such that:

- dh(0)6ˆ0 - TH6Tf 1(0),

then:fjX\H:(X\H;0) !(Y;0)is a smooth morphism.

PROOF. Let (X;0) and (Y;0) be defined by Cfx1;. . .;xng=I and Cfy1;. . .;ymg=J respectively. Since f issmooth, OX;0 isa power series ring over OY;0, i.e. OX;0 OY;0ft1;. . .;tsg, for somes.

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Let X0ˆX\H be the intersection, then OX0;0 OX;0=(h). If g cor- responds to h by the isomorphism OX;0 OY;0ft1;. . .;tsg, then OX0;0

 OY;0ft1;. . .;tsg=(g).

The hypothesis dh(0)6ˆ0 becomes dg(0)6ˆ0, that impliesthat there exists an indeterminate betweenyiandti, such that the partial derivative ofgwith respect to this indeterminate calculated in zero does not vanish.

Moreover, the hypothesis TH6Tf 1(0) impliesthat thisindeterminate must be one of theti, for exampleti.

Thus, using the Implicit Function Theorem, we obtain OX0;0

 OY;0ft1;. . .;tsg=(g) OY;0ft1;. . .;tbi;. . .;tsg and fjX0 isa smooth

morphism. p

Now we can prove Theorem 2.16:

PROOF. We start by assuming that (Y;0) hasquadratic algebraic sin- gularities, so (Y;0) isdefined by the analytic algebra Cfy1;. . .;ymg=J, whereJisan ideal generated by quadratic polynomials. Sincefissmooth, we haveOX;0 OY;0ft1;. . .;tsg Cfy1;. . .;ymgft1;. . .;tsg=J, for some s, andXhas quadratic algebraic singularities.

Now we prove the other implication. Let Cfx1;. . .;xng=I and Cfy1;. . .;ymg=J be the analytic algebras, that define the germs (X;0) and (Y;0) respectively, where I isan ideal generated by quadratic polynomials. We can assume that f is not an isomorphism, otherwise the theorem istrivial. Since f issmooth, OX;0 OY;0ft1;. . .;tsg, for some s>0.

Now we can intersectXCNx with hyperplanesh1;. . .;hsofCNx, which correspond, by the isomorphism OX;0 OY;0ft1;. . .;tsg, to the hyper- planesof equationst1ˆ0;. . .;ts ˆ0 ofCm‡sy;t and we call the intersection X0. Then (X0;0) has quadratic algebraic singularities. Moreover, by Lemma 2.17,frestricted to (X0;0) is a smooth morphism and it is bijective because OX0;0 OX;0=(h1;. . .;hs) OY;0ft1;. . .;tsg=(t1;. . .;ts) OY;0. Thus(Y;0)

has quadratic algebraic singularities. p

This theorem assures that the set of germs of analytic spaces with quadratic algebraic singularities is closed under the relationand so it is a union of equivalent classes under this relation. Moreover we know that the relationdefined between functorswith hullsisthe same asthe relation defined between their germs. Thus is natural to introduce the following definition for functors:

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DEFINITION2.18. LetFbe a functor with hull the germ of analytic space (X;x). It is said to have quadratic algebraic singularities if (X;x) has quadratic algebraic singularities.

Thisdefinition isindipendent by the choice of the germ of analytic space which is a hull of F, because the isomorphism class of a hull is un- iquely determined.

3. Brill-Noether stratification ofM.

LetXbe a compact complex KaÈhler manifold of dimensionnand letM be the moduli space of locally free sheaves ofOX-modulesonXwhich are stable and flat. In the first part of this section we concentrate our interests to the study of subspaces of the moduli spaceM, defined globally assetsin the following way:

DEFINITION3.1. Let hi2Nbe fixed integers, for all iˆ0. . .n, we de- fine:

N(h0. . .hn)ˆ fE02 M jdimHi(X;E0)ˆhig:

It isobviousthat, for a generic choice of the integershi2N, the sub- spaceN(h0. . .hn) isempty, from now on we fix our attention on non empty ones.

LetE 2 Mbe one fixed stable and flat locally free sheaf ofOX-modules onXand lethiˆdimHi(X;E). LetU !MXbe the universal Kuranishi family of deformationsofE, parametrized by the germ of analytic spaceM, which isisomorphic to a neighbourhood of E in M(see [N] for the con- struction). Letn:MX!Mbe projection, thus, for allE02M, we have n 1(E0)XandUjn 1(E0)ˆ UjE0  E0.

Now let'sdefine the germ of the strata N(h0. . .hn) at its point E.

Since, for all iˆ0. . .n, the functionE02 M !dimHi(X;E0)2N isup- per semicontinuos, for Semicontinuity Theorem (see [Hart], Theorem 12.8, ch.III), the set Uiˆ fE02 M jdimHi(X;E0)hig and the inter- section Uˆ T

iˆ0...nUiˆ fE02 M jdimHi(X;E0)hi; foriˆ0. . .ng are open subsets ofM.

For alliˆ0. . .n, let

Ni(E)ˆV(Fhi 1(RinU))ˆ fE02MjdimRinU OMk(E0)>hi 1g be the closed subschemes ofMdefined by the sheaf of idealsFhi 1(RinU),

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which isthe sheaf of (hi 1)-th Fitting idealsof the sheaf ofOM-modules RinU. LetN(E)ˆ T

iˆ0...nNi(E) be the closed subscheme ofMgiven by the intersection of the previous ones.

DEFINITION3.2. The germ of the strata N(h0. . .hn) at its pointE is given by:

U\N(E)ˆ fE02 M jdimHi(X;E0)hi; 8iˆ0. . .ng\ \

iˆ0...n

V(Fhi 1(RinU)):

REMARK3.3. We observe that the support of the germ of the strata N(h0. . .hn) atE, defined in 3.2, coincide with a neighbourhood ofEin the set given in definition 3.1. Infact, for the Theorem of Cohomology and Base Change (see [Hart], Theorem 12.11, ch.III), we have RnnU OM k(E0

Hn(X;E0), then the condition which defines Nn(E) becomes dimHn(X;E0)hn and the oneswhich define the intersectionU\Nn(E) become dimHn(X;E0)ˆhn and dimHi(X;E0)hi, for all iˆ0. . .n 1.

Applying iteratively the Theorem of Cohomology and Base Change, we obtainU\N(E)ˆ fE02MjdimHi(X;E0)ˆhi; foriˆ0. . .ngaswe want.

Now we prove the following:

PROPOSITION3.4. The germ of the strataN(h0. . .hn)atE is the base space of a Kuranishi family of deformations of E which preserve the dimensions of cohomology spaces.

PROOF. LetFbe a locally free sheaf ofOTX-module onTXwhich is a deformation of the sheafEover the analytic spaceTthat preserve the dimensions of cohomology spaces. If the morphism g:T!M such that (gIdX)U  F, whose existence is assured by the universality ofU, can be factorized asin the following diagram:

then F (gIdX)U (hIdX)(iIdX)U (hIdX)Uj(N(h0...hn)\M)X and the restriction of U to (N(h0. . .hn)\M)X satisfies the universal property.

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Let'sanalise the pullback via the mapgof the sheaf of idealsFk(RinU), foriˆ0. . .n, which defineslocallyN(h0. . .hn)\M. Since Fitting ideals commute with base change, for all i and k we have g(Fk(RinU))ˆ

ˆFk(gRinU). Let'sconsider the diagram:

using the Theorem of Cohomology and Base Change, since for all iˆ0. . .n the functions E02ImgM!hi(X;E0)2N are costant, we have Fk(gRinU)Fk(Rim(gIdX)U)Fk(RimF), and since F isa deformation which preserves the dimensions of cohomology spaces, the sheavesRimF are locally free and so the Fitting ideals Fk(RimF) are equal to zero. ThengFk(RinU) isequal to zero, aswe want. p In the following part of this section we study infinitesimal deformations ofEthat preserve the dimensions of its cohomology spaces, explaining this condition and defining precisely the deformation functor associated to this problem.

DEFINITION 3.5. A deformation of the sheaf E on the manifold X parametrized by an analytic space S with a fixed point s0is the data of a locally free sheafE0of(OXS)-modules on XS and a morphismE0! E inducing an isomorphism betweenE0jXs0 andE.

Let p:XS!S be the projection, then, for all s2S, E0jp 1(s) ˆ

ˆ E0jXs ˆ E0sisa locally free sheaf onXand so it makes sense to calculate the cohomology spaces of these sheaves,Hi(E0s).

By the Theorem of Cohomology and Base Change, the condition that, for alli2N, dimHi(E0s) iscostant whensvariesinS, isequivalent to the con- dition that, for alli2N, the direct imageRipE0isa locally free sheaf onS, and in thiscase we have that the fibreRipE0k(s) isisomorphic toHi(E0s).

DEFINITION 3.6. An infinitesimal deformation of the sheaf E on the manifold X over a local ArtinianC-algebra Awith residue fieldCis the data of a locally free sheafEAof(OXA)-modules on XSpecAand a morphismEA! Einducing an isomorphismEAAC E.

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In the case of an infinitesimal deformation EA, we can replace the condition that the dimensions of the cohomology spaces are costant along the fibresof the projectionp:XSpecA!SpecA, with the condition that the direct imagesRipEAare locally free sheaves, and in this case we have isomorphisms RipEACHi(E). We observe also that Hi(EA

RipEA(SpecA).

Our aim is to study this type of infinitesimal deformations of the sheaf E onXand the functor associated to them, defined in the following

DEFINITION 3.7. Let Def0E :ArtC!Set be the covariant functor de- fined, for all A2ArtC, by:

Def0E(A)ˆ EA

EA is a deformation of the sheaf Eover A

RipEA is a locally free sheaf on SpecAfor all i2N

) (

=: We note that, sinceN(h0. . .hn) isa moduli space, the functor of de- formationsof the sheafE which preserve the dimensions of cohomology spaces is prorepresented byN(h0. . .hn).

To give now two equivalent interpretationsof Def0E, which are useful in the following, we start by defining a deformationof aC-vector spaceV over a local ArtinianC-algebraAwhich isthe data of a flatA-moduleVA, such that the projection onto the residue field induces an isomorphism VAACV. It is easy to see that every deformation of a vector spaceV overAistrivial, i.e. it isisomorphic toVA.

IfEAisan infinitesimal deformation ofEoverA, it belongsto Def0E(A), asdefined in (3.7), if and only if it issuch thatHi(EA) are freeA-modules, that isthe same asflat A-modulessince A islocal Artinian, and Hi(EA)ACHi(E). Thuswe have the two following equivalent defini- tionsfor the functor Def0E:

DEFINITION3.8. The functorDef0E is defined, for all A2ArtC, by:

Def0E(A)ˆ EA

EA is a deformation of the sheaf Eover A

Hi(EA)is a deformation of Hi(E)over Afor all i2N

) (

=:

or equivalently by:

Def0E(A)ˆ EA

EA is a deformation of the sheaf Eover A Hi(EA)is isomorphic to Hi(E)Afor all i2N

) (

=: (1)

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4. Definition of Def0E using DGLAs.

We are interested in the study of the functor DefE0and in thissection we link it with the theory of deformationsvia DGLAs.

We start with some reminds about this theory. LetLbe a differential graded Lie algebra (DGLA), then it isdefined a deformation functor DefL:ArtC!Setcanonically associated to it (see [Man99], section 3).

DEFINITION4.1. For all(A;mA)2ArtC, we define:

DefL(A)ˆMCL(A) gauge ; where:

MCL(A)ˆ x2L1mAjdx‡1

2[x;x]ˆ0

and the gauge action is the action ofexp (L0mA)onMCL(A), given by:

eaxˆx‡X‡1

nˆ0

([a; ])n

(n‡1)!([a;x] da):

We recall that the tangent and an obstruction space to the deformation functor DefLare the first and the second cohomology spaces of the DGLA L,H1(L) and H2(L). Moreover the deformation functor DefLisa functor with good deformation theory and, if H1(L) isfinite dimensional, it hasa hull, but in general it isnot prorepresentable.

If the functor of deformationsof a geometrical object X isisomorphic to the deformation functor associated to a DGLA L, then we say that L governsthe deformationsofX.

Let x:L!M be a morphism of DGLAs which respects the DGLA structures of L and M, then it isdefined a deformation functor Defx:ArtC!Setcanonically associated to it (see [Man07], section 2).

DEFINITION4.2. For all(A;mA)2ArtC, we define:

Defx(A)ˆMCx(A) gauge ; where:

MCx(A)ˆ (x;ea)2(L1mA)exp (M0mA)jdx‡1

2[x;x]ˆ0;eax(x)ˆ0

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and the gauge action is the action ofexp (L0mA)exp (dM 1mA)on MCx(A), given by:

(el;edm)(x;ea)ˆ(elx;edmeae x(l)):

To write the tangent space and an obstruction space of a deformation functor associated to a morphism of DGLAs, we start by recalling that, if x:L!Mis a morphism of DGLAs, thesuspension of the mapping cone ofxisdefined to be the complexCxiˆLiMi 1with differentiald(l;m)ˆ

ˆ(dLl;x(l) dMm).

Then the tangent space to the functor Defx isthe first cohomology space of the suspension of the mapping cone of the morphismx,H1(Cx), an obstruction space of Defxis the second cohomology space,H2(Cx), and the obstruction theory for Defxiscomplete.

Moreover Defxisa functor with good deformation theory and, ifH1(Cx) is finite dimensional, it has a hull, but in general it is not prorepresentable.

We also observe that every commutative diagram of differential graded Lie algebras

(2)

inducesa morphism between the conesCx!Chand a morphism of functors Defx!Defh, for which the following Inverse Function Theorem (see [Man07], Theorem 2.1) holds:

THEOREM 4.3. If the diagram (2) induces a quasi isomorphism between the cones Ch!Cx, then the induced morphism of functor Defx!Defhis an isomorphism.

Now we return to our situation, so letXbe a compact complex KaÈhler manifold of dimensionnand letEbe a stable and flat locally free sheaf of OX-modulesonX, with dimHi(X;E)ˆhi, for alliˆ0. . .n.

LetA(0;)X (EndE) be the DGLA of the (0;)-formsonX with valuesin the sheaf of the endomorphisms ofE. It can be proved the following result (see [F], Theorem 1.1.1):

PROPOSITION4.4. The deformation functorDefA(0;)

X (EndE)is isomorphic to the functor of deformations of the sheafE,DefE. The isomorphism is

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given, for all A2ArtC, by DefA(0;)

X (EndE)(A) ! DefE(A) x ! ker(@‡x)

Let (A(0;)X (E);@) be the complex of the (0; )-formsonXwith valuesin the sheafEwith the Dolbeault differential and let Hom(A(0;)X (E);A(0;)X (E)) be the DGLA of the homomorphisms of this complex.

We recall that adeformationof a complex of vector spaces (Vi;d) over a local ArtinianC-algebraAwith residue fieldCisa complex ofA-modules of the form (ViA;dA), such that the projection onto the residue field induces an isomorphism between (ViA;dA) and (Vi;d).

It is easy to prove the following result (see [Man05], pages 3-4):

PROPOSITION 4.5. The deformation functor DefHom(A(0;)

X (E);A(0;)X (E)) is isomorphic to the functor of deformations of the complex (A(0;)X (E);@), Def(A(0;)

X (E);@). The isomorphism is given, for all A2ArtC, by:

DefHom(A(0;)X (E);A(0;)X (E))(A) ! Def(A(0;) X (E);@)(A) x ! (A(0;)X (E)A;@‡x):

Letx:A(0;)X (EndE)!Hom(A(0;)X (E);A(0;)X (E)) be the natural inclusion of DGLAsand let Defx be the deformation functor associated to x. Let (x;ea)2MCx(A), forA2ArtC. Sincex2A(0;1)X (EndE)mA satisfies the Maurer-Cartan equation, it givesa deformationEAˆker(@‡x) ofEover A. While ea2exp (Hom0(A(0;)X (E);A(0;)X (E))mA) givesa gauge equiva- lence between x(x)ˆx and zero in the DGLA Hom(A(0;)X (E);A(0;)X (E)).

Thuseais an isomorphism between the two correspondent deformations of the complex (A(0;)X (E);@) or equivalently eais an isomorphism between the cohomology spacesHi(EA) andHi(E)A, for alli2N. Thus:

Defx(A)

ˆ (EA;fAi) EA isa deformation of the sheaf Eover A

fAi is the isomorphismfAi :Hi(EA)!Hi(E)Afor alli2N

:

Now letFbe the morphism of functors given, for allA2ArtC, by:

F: Defx(A) ! DefA(0;)

X (EndE)(A) (x;ea) ! x;

with the above geometric interpretationsof the functorsDefx and

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DefA(0;)

X (EndE)(A), the morphismFis the one which associates to every pair (EA;fAi)2Defx(A) the element EA2DefA(0;)

X (EndE)(A). Thuswe have the following characterization of Def0Eusing DGLAs point of view (see [Man07], Lemma 4.1).

PROPOSITION4.6. The subfunctorDef0E is isomorphic to the image of the morphismF:Defx!DefA(0;)

X (EndE). 5. Proof of the Main Theorem.

Thissection isdevoted to the proof of the Main Theorem. With the above notations

THEOREM5.1 (Main Theorem). The Brill-Noether strataN(h0. . .hn) have quadratic algebraic singularities.

PROOF. The local study of the strataN(h0. . .hn) at one of itspointE, corresponds to the study of a germ of analytic space which prorepresents the functor Def0E. Our proof isdivided into four stepsin which we find out a chain of functors, linked each other by smooth morphisms, from the functor DefE0to a deformation functor for which it isknown that the germ of analytic space that prorepresents it has quadratic algebraic singularities. Then, we conclude, using properties proved in section 2.

First Step. We prove that the morphism F:Defx!DefE0 issmooth.

Then, given a principal extension in ArtC, 0!J!B!a A!0, and an element (EA;fAi)2Defx(A), we have to prove that, if itsimageEA2Def0E(A) hasa liftingEB2DefE0(B), it hasa lifting in Defx(B).

Since EA2DefE0(A) and EB 2DefE0(B), their cohomology spaces are deformationsof Hi(E) over A and B respectively and so Hi(EA

Hi(E)AandHi(EB)Hi(E)B. ThusHi(EB) isa lifting ofHi(EA) and it isa polynomial algebra overB. It followsthat in the diagram

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there exists a homomorphismfBi :Hi(EB)!Hi(E)B, which liftsfAi. Then also the following diagram commutes:

and sofBi is an isomorphism, for alli2N.

Second Step.SinceXisa KaÈhler manifold andEisa hermitian sheaf, the operators @E, adjoint of@E, and the Laplacian pE ˆ@E@E‡@E@E can be defined between formson X with valuesin the sheaf E. Let H(0;)X (E)ˆkerpEbe the complex of (0;)-harmonic formsonXwith values in E and let Hom(H(0;)X (E);H(0;)X (E)) be the formal DGLA of the homo- morphisms of this complex.

Also for the sheaf EndE the operator @End E, adjoint of @EndE, and the Laplacian pEndE ˆ@EndE@EndE‡@End E@EndE can be defined. Let H(0;)X (EndE)ˆkerpEndE be the complex of the (0;)-harmonic forms on X with valuesin EndE.

Siu proved (see [Siu]) that, for a flat holomorphic vector bundleLon a KaÈhler manifoldX, the two Laplacian operatorspLandpLcoincide. Then a (0;)-form onXwith valuesinLisharmonic if and only if it anhilates@, which iswell defined becauseLisflat.

Since EndEisflat, these factsimply that the complexH(0;)X (EndE) is a DGLA with bracket given by the wedge product on formsand the com- position of endomorphisms.

Moreover, we can define a morphism V:H(0;)X (EndE)!

!Hom(H(0;)X (E);H(0;)X (E)). Every element x2 A(0;)X (EndE) gives naturally an homomorphismV(x) from A(0;)X (E) in itself, defined locally to be the wedge product between formsand the action of the en- domorphism on the elements of E. If we defined it on an open cover of X on which both the sheaves E and EndE have costant transition functions, whenx2 H(0;)X (EndE) andV(x) isrestricted to the harmonic forms H(0;)X (E), it givesasa result an harmonic form. Let V :H(0;)X (EndE)!Hom(H(0;)X (E);H(0;)X (E)) be the DGLAsmorphism just defined and let DefV be the deformation functor associated to it.

We want to prove that the two functorsDefV and Defxare isomorphic.

Then we consider the following commutative diagram:

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whereMˆ

W2Hom(A(0;)X (E);A(0;)X (E))jW(H(0;)X (E)) H(0;)X (E) . The morphism b is a quasiisomorphism, infact it is injective and cokerbˆHom(A(0;)X (E);A(0;)X (E))=M Hom(A(0;)X (E);A(0;)X (E)=H(0;)X (E)) isan acyclic complex. Thenaandbinduce a quasiisomorphism between the conesCh!Cxand so, by the Inverse Function Theorem (Theorem 4.3), an isomorphism between the functors Defh!Defx.

Also the morphismdis a quasiisomorphism, infact it is surjective and itskernel iskerdˆ

W2Hom(A(0;)X (E);A(0;)X (E))jW(H(0;)X (E))ˆ0 that isomorphic to the acyclic complex Hom(A(0;)X (E)=H(0;)X (E);A(0;)X (E)). Then ganddinduce a quasiisomorphism between the conesCh!CV and so an isomorphism between the functors Defh!DefV.

Third Step.LetH~X(EndE) be the DGLA equal to zero in zero degree and equal toH(0;)X (EndE) in positive degrees, with zero differential and bracket given by wedge product on formsand composition of en- domorphisms.

Let V~ : ~H(0;)X (EndE)!Hom(H(0;)X (E);H(0;)X (E)) be the DGLAs morphism defined as in the previous step and let DefV~ be the deformation functor associated to it.

The inclusion H~X(EndE),! HX(EndE) and the identity on Hom(H(0;)X (E);H(0;)X (E)) induce a morphism of functorsC:DefV~ !DefV. We note that the morphism induced byCbetween the cohomology spaces of conesofVandV~ respectively is bijective in degree greater equal than 2 and it is surjective in degree 1. Thus, using the Standard Smootheness Criterion (Theorem 2.11), we conclude thatCissmooth.

Fourth Step. Let'swrite explicitly DefV~. The functor MCV~, for all A2ArtC, isgiven by:

MCV~(A)

ˆ (x;ea)2(L1mA)exp (M0mA)jdx‡1

2[x;x]ˆ0;eaV(x)~ ˆ0

where LˆH~(0;)X (EndE) and MˆHom(H(0;)X (E);H(0;)X (E)). Since the

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differential in the DGLA H~(0;)X (EndE) iszero and since the equation eaV(x)~ ˆ0 can be written as V(x)~ ˆe a0ˆ0, we obtain, for all A2ArtC:

MCV~(A)ˆx2kerV~mAj[x;x]ˆ0

exp (Hom0(H(0;)X (E);H(0;)X (E))mA):

Moreover exp (H~(0;0)X (EndE)mA)exp (dHom 1(H(0;)X (E);H(0;)X (E))mA) is equal to zero, thus there isn't gauge action. Thus, for allA2ArtC, we have:

DefV~(A)ˆ

x2kerV~mAj[x;x]ˆ0gexp (Hom0(H(0;)X (E);H(0;)X (E))mA):

SinceV~ isa DGLAsmorphism, kerV~ isa DGLA and it isdefined the de- formation functor DefkerV~associated to it. Now, for allA2ArtC, we obtain:

DefV~(A)ˆDefkerV~(A)exp (Hom0(H(0;)X (E);H(0;)X (E))mA):

The DGLA kerV~ haszero differential, so the functor DefkerV~ ispro- represented by the germ in zero of the quadratic cone (see [G-M90], The- orem 5.3):

Xˆ fx2ker1V~ j[x;x]ˆ0g;

that has quadratic algebraic singularities. Then also the functor DefV~ is prorepresented by a germ of analytic space with quadratic algebraic sin- gularities.

Conclusion. Since now we have constructed smooth morphisms be- tween the functor DefE0and the functor DefV~:

By Proposition 2.14, there exists a smooth morphism between the germsof analytic spaceswhich are hullsof the two functorsDefE0and DefV~. Moreover, by Theorem 2.16, since the germ which is a hull of DefV~ has quadratic algebraic singularities, the same is true for the hull of Def0E.

REFERENCES

[A] M. ARTIN,On solutions to analytic equation,Invet. Math., vol.5(1968), pp. 277-291.

[F] K. FUKAYA,Deformation theory, homological algebra and mirror sym- metry,Geometry and physics of branes (Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., IOP Bristol (2003), pp. 121-209.

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[G-M88] W. GOLDMAN- J. MILLSON,The deformation theory of representations of foundamental groups of compact KaÈhler manifolds, Publ. Math.

I.H.E.S.,67(1988), pp. 43-96.

[G-M90] W. GOLDMAN- J. MILLSON,The homotopy invariance of the Kuranishi space,IllinoisJournal of Math.,34(1990), pp. 337-367.

[Hart] R. HARTSHORNE,Algebraic Geometry,Graduate text in mathematics,52 (Springer, 1977).

[LePoi] J. LE POITIER, Lectures on vector bundles, Cambridge studies in advanced mathematics,54(Cambridge University Press, 1997).

[Man99] M. MANETTI,Deformation theory via differential graded Lie algebras, Seminari di Geometria Algebrica 1998-1999, Scuola Normale Superiore (1999).

[Man05] M. MANETTI,Differential graded Lie algebras and formal deformation theory, notes of a course at AMS Summer Institute on Algebraic Geometry, Seattle, (2005).

[Man07] M. MANETTI, Lie description of higher obstructions to deforming sub- manifolds,Ann. Scuola Norm. Sup. Pisa, (5), Vol.VI(2007), pp. 631-659.

[N] A. M. NADEL,Singularities and Kodaira dimension of the moduli space of flat Hermitian-Yang-Mills connections, Comp. Math.,67(1988), pp.

121-128.

[Nor] A. NORTON, Analytic moduli of complex vector bundles,Indiana Univ.

Math. J.,28(1979), pp. 365-387.

[S] M. SCHLESSINGER,Functors of Artin rings, Trans. Amer. Math. Soc.,130 (1968), pp. 205-295.

[Ser] E. SERNESI,Deformation of algebraic schemes,Springer, (2006).

[Siu] Y. T. SIU, Complex-analyticity of harmonics maps, vanishing and Lefschetz theorems, J. Diff. Geom.,17(1982), pp. 55-138.

[V] R. VAKIL,Murphy's law in algebraic geometry: badly-behaved deforma- tion spaces,Preprint, arXiv: math.AG/0411469.

Manoscritto pervenuto in redazione il 7 luglio 2008.

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