Mixed Hodge structures and vector bundles on the projective Plane I

MIXED HODGE STRUCTURES AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I O.PENACCHIO Abstract. We describe an equivalence of categories between the category of mixed Hodge structures and a category of vector bundles on the toric complex projective plane which verify some semistability condition. We then apply this correspondence to dene an invariant which generalises the notion of R -split mixed Hodge structure and compute extensions in the category of mixed Hodge structures in terms of extensions of the corresponding vector bundles. We also give a relative version of this correspondence and apply it to dene stratications of the bases of the variations of mixed Hodge structure. Introduction The purpose of this note is to give a geometric equivalent of the notion of mixed Hodge structure. To this end, following Simpson, we adopt the Rees' philosophy of associating a graded ring to a ring ltered by a chain of ideals. A mixed Hodge structure is roughly speaking the data of a vector space endowed with three ordered ltrations which are in a certain position called opposed. We dene a functor, named the Rees functor, which converts each pair of ltrations into a graded module, and, next, we look at the associated coherent sheaf on the ane plane. It turns out that these sheaves are locally free and equivariant for the standard action by translation of C∗×C∗ . We can glue these local descriptions to form an equivariant locally free sheaf on the projective plane P2 C . Starting with such a sheaf associated to a 3 -ltered vector space, the action of the torus allows us to recover the ltrations on the same vector space. We thus get an equivalence of categories between ltered vector spaces and equivariant vector bundles. As in the classic Rees' construction, one deforms a ltered object into its corresponding graded object. Indeed, the bre over the points of the dense open orbit of the torus action of a bundle associated to three ltration is the underlying vector Date : 2004. 1991 Mathematics Subject Classication. 14D,32G20,14J60,14M25. Key words and phrases. Hodge theory, toric geometry, period matrices, equivariant vector bundle. 1 2 O.PENACCHIO space, whereas the bres over the origins of the ane charts are naturally isomorphic to the bigraded objects associated to each pair of ltrations. The idea of associating a vector bundle on the complex projective plane P2 C to the three ltrations which form a mixed Hodge structure has a a double origin: Simpson's construction [21],[22] of mixed twistor structures on the complex projective line P1 C associated to mixed Hodge structures, which involves the Hodge ltration and its conjugate, and Sabbah's construction [19] of Frobenius manifolds starting from families of vector bundles over P1 C in which such families are constructed using the weight and the Hodge ltrations of a variation of mixed Hodge structure. There are two motivations for handling the three ltration simultaneously. On the one hand, the relative position of the Hodge ltration and its conjugate plays an important role in the study of degenerations of mixed Hodge structure. On the other hand, the weight ltration contains all the extension data of a mixed Hodge structure. Although they were rstly stated without reference to toric geometry in [16], the formalism derived by Perling in [17] after Klyachko [12],[13] for describing equivariant sheaves over toric varieties in terms of ltered vector spaces seems to be appropriate for describing the equivalences between categories of vector bundles and the categories involved in Hodge theory we are interested in. The paper is organised in four parts. In section 1 we rst dene the Rees functor and its inverse for 2 -ltered vector spaces and next introduce Perling's formalism [17], which, given a fan ∆ , allows us to describe the category of ∆(1) -families of complete ltrations in terms of equivariant locally free sheaves on the corresponding toric variety X∆ . It turns out that the locally free sheaves on P2 k which are associated to 3 -ltered k -vector space whose ordered ltrations are opposed verify a semistability condition. They are P1 0 -semistable, namely their restrictions to the divisor corresponding to the rst of the three ltrations are of degree 0 and µ -semistable, or, equivalently, a direct sum of line bundles of the same slope 0 . Notice that this notion of semistability is stronger than the µ -semistability. The principal consequence of this geometric characterisation is the fact that the category of vector spaces endowed with three opposed ltrations is abelian. Indeed, when one denes the cokernel of a morphism in the category of vector bundles of degree 0 verifying this semistability condition to be the reexivization of the cokernel in the category of coherent sheaves, which corresponds to the cokernel in the category of 3 -ltered vector spaces by the equivalence, one gets an abelian category [15]. The section 2 consists of an application of the previous section to Hodge theory. The ltered vector spaces involved here arise from Hodge theory. MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 3 We recover in a geometric way the fact that the category of mixed Hodge structures is abelian, which was proved by Deligne in [4] using linear algebra. After dening the R -split level α(H) of a mixed Hodge structure H to be the second Chern class of its associated Rees bundle on the projective plane P2 C , we give some properties this invariant veries. In particular, it generalises the notion of R -split mixed Hodge structure which is important when one studies degenerations of Hodge structure [3]. A mixed Hodge structure is R -split if and only if α(H) = 0 . We then give some computations of the R -split level of the mixed Hodge structures on the cohomology of possibly non complete and singular curves of genus 0,1 . Finally we study extensions in the abelian category of mixed Hodge structures in terms of extensions of the corresponding equivariant sheaves. A relative version of the Rees construction is proposed in section 3 . Starting from a vector bundle on an algebraic variety S endowed with three ltrations by subbundles, one builds a sheaf on P2 k×S called the relative Rees sheaf. This sheaf is reexive, S -at and equivariant for the action of the torus in the direction of the projective plane. However, its bres over points of S do not in general directly provide the Rees bundles corresponding to the expected 3 -ltered vector spaces. Once again we have to consider the reexivization of these coherent sheaves to recover the right object. In section 4 , we apply the relative Rees construction to vector bundles endowed with three ltrations provided by Hodge theory. After recalling the denition of a variation of mixed Hodge structure on a variety S and constructing the corresponding classifying space M [3], one would like to dene a universal relative Rees sheaf on it. Since the conjugate of the Hodge ltration do not vary algebraically nor holomorphically, in order to perform the construction, we have to consider an open subvariety Mopp of the product M×M which parametrizes pairs of opposed ltrations. On this variety we have a canonical 3 -ltered vector bundle formed by the weight ltration, which is constant since it is at for the canonical connection, and two ltrations which come from the universal Hodge ltration on the rst and second factor of M×M . Once we have this 3 -ltered vector bundle we can build the universal relative Rees sheaf ξMopp on P2 C×Mopp . A variation of mixed Hodge structure on S gives rise to a morphism from its universal covering to the classifying space ϕ:˜ S→M . One recovers the Rees bundle associated to the mixed Hodge structure over s∈˜ S by looking at the bre over s of the pullback on P2 Cט S of the universal relative Rees sheaf (idP2 C×ϕ×ϕ)∗ξMopp . This sheaf is a real algebraic, or real analytic, coherent sheaf of C∞ ˜ SOP2 -module whose bres are coherent sheaves of OP2 - modules by GAGA, and are equivariant, of degree 0 and P1 0 -semistable. Finally, we show that the R -split level is upper semi-continuous and hence 4 O.PENACCHIO denes a stratication of the base of the variation by real algebraic or real analytic closed subsets. In this correspondence between variations of mixed Hodge structure and families of P1 0 -semistable equivariant sheaves, two notions are missing to be able to understand degenerations of mixed Hodge structure in terms of the compactication of the related moduli space of semistable Rees coherent sheaves: an equivalent in the category of relative Rees sheaves of the connection underlying a variation of mixed Hodge structure and a translation of the polarizations. I would like to thank C.Simpson, my advisor, for his help and encouragement. I would also like to thank C.Sabbah for suggesting me the approach in the rst section, C.Sorger for pointing out a mistake in the semistability condition in a preliminary version and M.Perling for explaining me his work. I gratefully acknowledges the support of the european comission (Marie Curie Intra-European Fellowship MEIF-CT-2003-501550). I would like to thank the sta of the CRM for its hospitality. 1. Rees construction and toric vector bundles Let k be an algebraically closed eld of characteristic zero. By an algebraic variety over k we understand a separated scheme of nite type over Spec k . 1.1. Rees construction. The technical tool we will use to geometrize multi-ltrate vector spaces is the Rees construction. This construction allows us to associate a coherent sheaf on the n -dimensional ane space An k to a vector space endowed with n ltrations. Let V be a nite dimensional k -vector space endowed with n decreasing ltrations F• 1, F• 2, ..., F• n . The object (V, F• 1, F• 2, ..., F• n) is called a n - ltered vector space. All the ltrations we will consider in this section are complete ; a ltration F• of a vector space V is said to be complete if there exists two integers m and n , m≤n , such that Fm=V and Fn={0} . A morphism between two n -ltered vector spaces f: (V, F• 1, F• 2, ..., F• n)→(V0, G• 1, G• 2, ..., G• n) is a morphism between the underlying vector spaces that is compatible with the ltrations , or ltered , that is, for any integers i and p , f(Fp i)⊂Gp i . We will denote by Cn filtr the category whose objects are n -ltered vector spaces and morphisms are ltered morphisms. Consider the k[u1, u2, ..., un] -module R(V, F• 1, F• 2, ..., F• n) generated by the elements of the form (Y i∈[1,n] upi i)⊗v, where v∈Fp1 1∩Fp2 2∩... ∩Fpn n. MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 5 The module R(V, F• 1, F• 2, ..., F• n) is called the Rees module associated with the n -ltered vector space (V, F• 1, F• 2, ..., F• n) . Since V is nite dimensional and the ltrations are complete, the Rees module is a nitely generated torsion free k[u1, u2, ..., un] -module. We generalise the denition and the construction of Rees coherent sheaves given in [22] for one ltration to n -tuple of ltrations. Denition 1.1. The Rees coherent sheaf ξ(V, F• 1, F• 2, ..., F• n) associated to a n -ltered vector space V is the coherent sheaf on the ane space An k= Spec k[u1, u2, ..., un] associated to the Rees k[u1, u2, ..., un] -module R(V, F• 1, F• 2, ..., F• n) , ξ(V, F• 1, F• 2, ..., F• n) = ˜ R(V, F• 1, F• 2, ..., F• n). Let G be an algebraic group which acts on an algebraic variety X and let σ:G×X→X denote the action and p2 the projection on the second factor. A sheaf of OX -module E is said to be equivariant if there exists an isomorphism Φ : σ∗E∼ =p∗ 2E satisfying the cocycle condition (π⊗idX)∗Φ = p∗ 12 ◦(idG⊗σ)∗Φ , where π is the group multiplication and p12 :G×G×X→ G×X the projection on the two latest factors. Consider the standard action on the ane space An k of the torus (Gm)n= Spec B where B=k[t±1 1, t±1 2, ..., t±1 n] . The morphism σ: (Gm)n×An k→An k corresponds to a structure of B -comodule on k[u1, u2, ..., un] given by the morphism σ]:k[u1, u2, ..., un]→B⊗kk[u1, u2, ..., un] ui7→ ti⊗ui . Then, this morphism induces a structure of B -comodule on the Rees module dened by the morphism σ] R:R(V, F• 1, F• 2, ..., F• n)→B⊗kR(V, F• 1, F• 2, ..., F• n) (Qi∈[1,n]upi i)⊗v7→ (Qi∈[1,n]tpi i)⊗upi i⊗v . This morphism endows the corresponding Rees coherent sheaf with a (Gm)n - equivariant structure. Thus, the Rees construction yields a functor φR from the category of n - ltered nite dimensional vector spaces Cn filtr to the category of equivariant coherent sheaves on An k for the standard action of the torus (Gm)n . As we will see in the following sections the functor ΦR has an inverse. We will explicit it here in a particular case we will be interested in for the applications. Let us focus on the case of coherent sheaves on A2 k associated to vector spaces endowed with two ltrations. In that case, the associated coherent sheaves are locally free. Starting with a locally free sheaf E on A2 k which is equivariant for the action of (Gm)2 , then, we get two ltrations on the bre V=E(1,1) over (1,1) in the following manner: all vector bundles on 6 O.PENACCHIO the plane being trivial, E corresponds to a free k[u1, u2] -module R=E(A2 k) . The action of the group gives a trivialisation over (Gm)2⊂A2 k which yields an isomorphism of k[u±1 1, u±1 2] -modules k[u±1 1, u±1 2]⊗k[u1,u2]R∼ =k[u±1 1, u±1 2]⊗kV . By taking the quotient by the ideal (u1−1, u2−1) we show that V can be canonically identied with R/(u1−1, u2−1)R , which is the bre E(1,1) . This gives R⊂k[u±1 1, u±1 2]⊗kV . We can dene two complete decreasing ltrations by taking Fp 1∩Fq 2 to be the subspace of vectors v such that u−p 1u−q 2⊗v∈R . This construction denes a functor ΦI from the category of equivariant vector bundles on the projective plane to C2filtr . This construction is inverse to the Rees construction. Starting with a 2 - ltered vector space (V, F•, G•) and applying it to the associated Rees bundle ξ(V, F•, G•) , we recover the same ltrations on the same vector space. It can be seen by taking a basis adapted to both ltrations corresponding to a splitting of the two ltrations V=⊕p,qVp,q such that Fp=⊕p0≥p,q0Vp0,q0 and Gq=⊕p0,q0≥qVp0,q0 . Hence the module in the above discussion is the k[u1, u2] -module B=⊕p,qu−p 1u−q 2Vp,q and, applying the functor ΦI , we recover the ltrations on the same vector space V . Proposition 1.2. ( [16] , [22] ) The Rees functor ΦR and the inverse functor ΦI establish an equivalence of categories between the category C2filtr and the category of equivariant vector bundles on A2 k . Let (V, F•)∈ C1filtr . We denote by GrF•V=⊕pFp/Fp+1 the associated graded object whose pth piece is Grp F•V=Fp/Fp+1 . If (V, F•, G•)∈ C2filtr , then G• induces a decreasing complete ltration on each graded piece of GrF•V and gives rise to a bigraded object GrG•GrF•V . We refer to [4] for a background on graded objects. Note that, according to Zassenhaus' lemma, the objects Grp F•Grq G•V and Grq G•Grp F•V are canonically isomorphic. We denote by E(s) the bre of the coherent sheaf E over s . By taking the quotient of the corresponding Rees modules we get: Lemma 1.3. All the following isomorphisms are canonical. (i) Let (V, F•)∈ C1filtr . Then: (a) ξ(V, F•)(s)∼ =V if s∈A1 k\{0} . (b) ξ(V, F•)(0) ∼ =GrF•V . (ii) Let (V, F•, G•)∈ C2filtr . Then: MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 7 (a) ξ(V, F•, G•)|A1 k×{s}=f∗ sξ(V, F•, G•)∼ =ξ(V, F•) as Gm -equivariant locally free sheaves on the ane line, where s6= 0 and fs:A1 k×{s} → A2 k is the inclusion morphism. (b) ξ(V, F•, G•)((s, t)) ∼ = ∼ =   V if (s, t)∈A2 k\(A1×{0}∪{0}×A1), GrF•V if s= 0 and t6= 0, GrF•GrG•V∼ =GrG•GrF•V if (s, t) = (0,0). Rees sheaves could be considered as total spaces of deformations of ltered vector spaces into the associated graded vector spaces. 1.2. Rees bundles and Toric bundle. Let X be a toric variety, that is, a normal variety which contains an algebraic torus T as a dense open subset such that the torus multiplication extends to an action of the algebraic group T on X . We refer to [9] for a background on toric varieties. The variety X is dened by a fan ∆ contained in the real vector space NR=N⊗ZR associated with a lattice N∼ =Zn and will be denoted by X∆ . Let M be the lattice dual to N and h,i:M×N→Z the canonical pairing. The elements of the abelian group M are denoted by m, m0 if written additively and by χ(m), χ(m0) if written multiplicatively. M is the natural group of characters of the torus T= Hom gr(M, k∗) . A cone σ of the fan ∆ is a convex rational polyhedral cone contained in NR . The cones will be denoted by Greek small letters σ, ρ, τ and etc., the order relation among cones is denoted by < . Let us recall the following standard notations (see [9], [17]): (i) σ(i) = {τ < σ| dim τ=i} , the elements of ∆(1) are called the rays , (ii) n(ρ) the primitive lattice element spanning the ray ρ∈∆(1) , (iii) the conedual to σ is dened by ˇσ={m∈MR|hm, ni≥0 forall n∈σ} , (iv) σ⊥={m∈MR|hm, ni= 0 for all n∈σ} is the orthogonal cone to σ , (v) σM= ˇσ∩M is the subsemigroup of M associated with σ . The category of σ -families. Let σ be a cone. We introduce the notion of σ -family following [17]. Let E be a quasicoherent sheaf on a toric variety X∆ . The torus action on the T -invariant ane open sets Uσ= Spec k[σM] , σ∈∆ , gives an isotypical decomposition of the OUσ -modules of sections of E into T -eigenspaces Γ(Uσ,E) = M m∈M Γ(Uσ,E)m. The module structure over k[σM] induces, for each m, m0∈M , a map Γ(Uσ,E)m→Γ(Uσ,E)m0 dened by e7→ χ(m0−m).e provided m0−m∈ 8 O.PENACCHIO σM . This induces a natural preorder on M associated with σM by setting m≤σm0 if m0−m∈σM . Since we want to work with decreasing ltrations, which is more convenient for the applications to Hodge theory, the notations and conventions we adopt here dier from the one used in [17]. We denote by Eσ the k[σM] - module Γ(Uσ,E) and by Fm σ the direct summand Γ(Uσ,E)−m . Consider the preceding isotypical decomposition Eσ=Lm∈MFm σ . Note that the torus T acts by multiplication by χ(−m) on Fm σ and that there is a morphism Fm σ→Fm0 σ given by the multiplication by χ(m−m0) provided m−m0∈σM . To recover Perling's convention it suces to consider the increasing ltration F• associated to each decreasing ltration F• by letting, for each integer p , Fp=F−p . The family of vector spaces Fm σ and characters χ(m) form a direct family of vector spaces associated with the preorder imposed by σM . Following [17], such a data is called a σ -family: Denition 1.4. Let {Fm σ}m∈M be a family of k -vector spaces. Suppose that for every m and m0 such that m0≤σm we have a vector space homomorphism χm,m0 σ:Fm σ→Fm0 σ such that for all m , χm,m σ=id , and, for every triple (m, m0, m00) such that m00 ≤σm0≤σm we have χm,m00 σ=χm0,m00 σ◦χm,m0 σ . Such a data is a σ -family . A σ -family is said to be nite if all the vector spaces Fm σ are nitedimensional, if for each ascending chain of characters ... ≤σmi+1 ≤σmi≤σ m−1≤σ... there exists an integer i0 such that Fmi σ= 0 for each i < i0 and nally if there is only a nite number of vector spaces Fm σ such that the map ⊕m≤σm0Fm0 σ→Fm σ dened by the summation of the χm0,m σ is not surjective. Denition 1.5. Suppose given two σ -families {Fm σ}m∈M and {Gm σ}m∈M with respective vector space homomorphisms χm,m0 σ and γm,m0 σ . A morphism of σ -family φσ from the rst to the second σ -family is a set of vector space homomorphisms {φm σ:Fm σ→Gm σ}m∈M such that for all m, m0 verifying m0≤σm , φm0 σ◦χm,m0 σ=γm,m0 σ◦φm σ . Consider a M -graded module Eσ with associated decomposition Eσ= ⊕m∈MFm σ . We dene χm,m0 σ:Fm σ→Fm0 σ to be the morphism induced by the multiplication by χ(m−m0) in the structure of M -graded module. Then the family of vector spaces Fm σ with the morphisms χm,m0 yields a σ -family. Each graded morphism provides a morphism of σ -families by its decomposition into homogenous components. Reciprocally, consider a σ -family and let Eσ=⊕m∈MFm σ . Dene an M - graded structure by letting, for each m∈σM and each v∈Fm0 σ , χ(m).v = χm0,m0+m σ(v) . Morphisms of σ -families give rise to graded morphisms. MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 9 We can state a rst correspondence between families of vector spaces and equivariant sheaves: Proposition 1.6. [17] Let Uσ be a T -invariant ane open scheme of X∆ . The following categories are equivalent: (i) equivariant quasicoherent sheaves over Uσ , (ii) M -graded k[σM] -modules with morphisms of degree 0 , and (iii) σ -families. Moreover, nite σ -families correspond to equivariant coherent sheaves. The category of ∆ -families. Let ∆ be a fan. Suppose given a σ - family for each σ∈∆ . We obtain a system of quasicoherent sheaves Eσ over each Uσ . If certain conditions of compatibility between the σ -families are fullled, these sheaves glue together to form a quasicoherent sheaf E on X∆ . These conditions are that for each pair τ≤σ , with associated inclusion iτ σ:Uτ,→Uσ , the τ -family associated with Eτ and the τ -family associated with the pullback iτ σ∗Eσ are isomorphic and that for each triple ρ≤τ≤σ we have an equality ηρσ =ηρτ ◦iρ τ∗ησ τ . Such families are called ∆ - families . They are nite if all the underlying σ -families are nite. The ∆ -families we will consider are of the following form: Denition 1.7. A ∆(1) -family of complete ltrations is the data of a vector space V and, for each ρ∈∆(1) , a complete decreasing ltration F• ρ , the vector space homomorphisms χm,m0 ρ being given by the inclusion Fm0 ρ,→Fm ρ for m0≥σm . . Such a family will be denoted by (V, {F• ρ}ρ∈∆(1)) . A morphism between ∆(1) -family of complete ltrations from (V, {F• ρ}ρ∈∆(1)) to (V0,{G• ρ}ρ∈∆(1)) is a morphism between the underlying vector spaces that respects the ltrations. This denition corresponds to the notion of vector space with full ltrations associated with each ray of ∆(1) in [17]. Note that a morphism between ∆(1) -families of complete ltrations is simply a morphism of ∆ -families between two ∆(1) -families of complete ltrations. Consider an equivariant reexive sheaf E on X∆ . Since it is normal, it is completely determined by its restriction to an open set whose complementary is at least 2 -codimensional. So, if we let ∆0= ∆(0) ∪∆(1) , we have Γ(X∆,E) = Γ(X∆0,E) . This implies Γ(Uσ,E) = ∩ρ∈σ(1)Γ(Uρ,E) on each ane toric variety Uσ , and hence, Fm σ=∩ρ∈σ(1)Fm ρ , where Fm ρ=Fm0 ρ if m−m0∈ρ⊥ M . For each 16 O.PENACCHIO in each bundle ξE(Grr F• 0V, F• 0, F• 1, L•, F• 2) , we can therefore split it into a direct sum of line bundles. This gives ch (ξE(V, F• 0, F• 1, L•, F• 2)) = Pr,p,q dim k(Grq F• 2Grp F• 1Grr F• 0V) ch (ξE(k, L[r]•, L[p]•, L•, L[q]•)), where L[i]• is the ltration L• shifted by i , dened by L[i]k=Lk−i . Denote by e Di and e DE the divisors on e P2 k corresponding respectively to ρi , i∈ {1,2,3} , and ρE . The Poincaré dual of a divisor D is denoted by ηD . Let eω∈H2(e P2 k,Z) be the cohomology class of a hyperplane. We have c 1(ξE(k, L[r]•, L[p]•, L•, L[q]•)) = rη e D0+pη e D1+qη e D2 and so ch (ξE(k, L[r]•, L[p]•, L•, L[q]•) = 1+(r+p+q)eω+1 2(r2+2rp+2rq)ew2 , which gives, the ltrations being opposed, ch (ξE(V, F• 0, F• 1, L•, F• 2)) = −1 2Pp,q hp,q(p+q)2ew2 . Let us now compute the Chern character of TE . This coherent sheaf is supported on E and do not depend on F• 0 . Thus, we can rewrite the exact sequence (1.1) using F• 1 (or F• 2 ) instead of F• 0 0→ξE(V, F• 1, F• 1, L•, F• 2)→π∗ξ(V, F• 1, F• 1, F• 2)→ TE→0. Both vector bundles ξE(V, F• 1, F• 1, L•, F• 2) and π∗ξ(V, F• 1, F• 1, F• 2) split into a sum of line bundles. The Chern character of the rst is given by the above formula ch (ξE(V, F• 1, F• 1, L•, F• 2)) = dim kV+Pp,q sp,q ¡(2p+q)eω+1 2(3p2+ 2pq)ew2¢ . For the second, ch (π∗ξ(V, F• 1, F• 1, F• 2)) = =π∗ ch (⊕p,qξ(V, L[p]•, L[p]•, L[q]•)sp,q ) = dim kV+X p,q sp,q µ(2p+q)eω+1 2(4p2+q2+ 4pq)ew2¶. This gives ch (TE) = 1 2Pp,q sp,q(p+q)2ew2 , which allows us to conclude since the Chern character is additive. ¤ Remark 1.19 . The three ltration do not play the same role in the formula giving the Chern classes. The reason for this asymmetry is the one explained above: the condition for three ltrations to be opposed is not symmetric. The following theorem gives the geometric property which characterises the equivariant vector bundles corresponding to opposed ltrations among the Rees bundles on the projective plane associated to triple of ltrations. Theorem 1.20. The Rees construction establishes an equivalence of categories between the category of nite dimensional 3 -ltered k -vector spaces MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 17 whose ordered ltrations are opposed and the category of T -equivariant P1 0 - semistable vector bundles of degree 0 on the projective plane: C3filtr,opp //BunP1 0− semistable ,µ=0(P2 k/T) oo. Proof. According to the preceding proposition a Rees bundles associated to a vector space endowed with opposed ltrations is P1 0 -semistable and has zero rst Chern class. Reciprocally, suppose given a P1 0 -semistable degree 0T -equivariant vector bundle E on P2 k . Let (V, F• 0, F• 1, F• 2) be the associated element in C3filtr . Suppose there exists a triple (r0, p0, q0)∈Z3 such that r0+p0+q0>0 and Grp0 F• 1Grq0 F• 2Grr0 F• 0V6={0} . Then, there exists a one-dimensional subvector space V0⊂V whose projection on Grp0 F• 1Grq0 F• 2Grr0 F• 0V is not zero. Let (V0, L•[r0], L•[p0], L•[q0]) be the triltered vector space whose ltrations are dened by Lp[i] = V0 if p≤i and Lp[i] = 0 otherwise. The monomorphism of Rees k[u0, u1, u2] -modules induces an injective map of locally free sheaves 0//ξ(V0, L•(r0), L•(p0), L•(q0)) //E. By the formula given in the proof of Proposition 1.18 c1(ξ(V0, L•(r0), L•(p0), L•(q0))) = r0+p0+q0>0 , which contradicts the µ -semistability and hence the P1 0 -semistability of E . Moreover, c 1(E) = Pr,p,q dim k(Grq F• 2Grp F• 1Grr F• 0V)(r+p+q)ω . The preceding fact proves that there is no positive contribution to the rst Chern class of E in this formula. Every triple (r0, p0, q0) such that r0+p0+q0<0 and Grp0 F• 1Grq0 F• 2Grr0 F• 0V6= 0 will now give a negative contribution to the rst Chern class which is zero; such a space does therefore not exist. This proves the theorem. ¤ 1.5. Semistable reexive sheaves. Let Reflµ(X) be the category whose objects are equivariant µ -semistable reexive sheaves on an algebraic variety X . Let F be a coherent sheaf on X and consider the canonical morphism to its double dual ν:F → F∗∗ . Recall from [10] that F∗∗ is reexive. It is called the reexive sheaf associated to F and is canonically isomorphic to it when F is reexive. The kernel, cokernel, image and coimage of a morphism in the category of reexive sheaves are dened to be reexive sheaves respectively associated with the kernel, cokernel, image and coimage 18 O.PENACCHIO in the category of coherent sheaves. In fact, the kernel of a morphism f:E → F in Reflµ(X) is still reexive. We get (1.2) 0//Ker(f)∼ =Ker(f)∗∗ //E//F// $$ I I I I I I I I I ICoker(f)// ν ²² 0 Coker(f)∗∗ //0, where the horizontal sequence is exact in the category of coherent sheaves and the other is exact in Reflµ(X) . Suppose now X is endowed with the action of an algebraic group G and denote by Reflµ(X/G) the subcategory of Reflµ(X) whose objects and morphisms are G -equivariant. This category is clearly additive. To prove that it is exact we use the fact that µ -semistability allows us to exhibit an isomorphism between the image and the coimage of a morphism in the complement of a subvariety of codimension at least 2 . This provides an isomorphism since reexive sheaves are normal. Theorem 1.21. [15] Reflµ(X/G) is an abelian category. Remark 1.22 . Note that the abelian category Reflµ(X) is not a sub-abelian category of the abelian category of coherent sheaves, the cokernels in both categories do not agree. Since reexive sheaves on curves and surfaces are locally free, considering the action of the trivial group, we recover the classical result (see [18] for example): Corollary 1.23. The category of µ -semistable sheaves on a curve or a surface is abelian. Recall that an ane toric variety Uσ is nonsingular if and only if σ is nonsingular, which means that σ is generated by part of a basis for the lattice. A fan is said to be nonsingular if all of its cones are nonsingular. Corollary 1.24. Let ∆ be a nonsingular fan. The category of ∆(1) -families of complete ltrations is abelian. Let us turn back to consider the projective plane P2 k dened above. One can show that when one imposes a stronger semistabily condition the category of semistable vector bundles of degree 0 on P2 k is still abelian (For a proof of this fact see [15], Theorem 3.1 ). Proposition 1.25. The category of T -equivariant P1 0 -semistable vector bundles of degree 0 on the projective plane BunP1 0− semistable ,µ=0(P2 k/T) is abelian. MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 19 Thus, by Theorem 1.20, we recover in a geometric way the following result (see [4]): Corollary 1.26. The category C3filtr,opp of nite dimensional triltered k -vector spaces whose ordered ltrations are opposed is abelian. 1.6. Real structures. In this section we take k=C . Consider the antiholomorphic involution of P2 C , τ: (u0, u1, u2)7→ (u0, u2, u1) . Let E be an OP2 C -module. We dene the sheaf τ∗(E) by letting, for each Zariski open set U , τ∗E(U) = E(τ(U)) . It is canonically endowed with an OP2 C -module structure by setting, for each e∈τ∗E(U) and each f∈ OP2 C(U) , f.e =τ∗(f)e . Note that P1 0 is globally invariant by τ . A τ -equivariant coherent sheaf on the complex projective plane is the data of a coherent sheaf E and of a morphism of OP2 C -modules f:E → τ∗E such that τ∗(f)◦f=idE . A Tτ -equivariant coherent sheaf is a coherent sheaf that is both T and τ -equivariant. Tτ -equivariant sheaves are naturally associated with nite dimensional triltered complex vector spaces with underlying real structure whose both latest ltrations are conjugated one another, that is, for each integer p , Fp 2=Fp 1 . We denote by C3filtr,opp,R the category whose objects are of this form and whose morphisms are morphisms of real vector spaces which induce ltered morphisms when passing to the complex structure. There is a forgetful functor from C3filtr,opp,R to C3filtr,opp which consists in forgetting the real structure. Theorem 1.27. The Rees construction establishes an equivalence of categories: C3filtr,opp,R//BunP1 0− semistable ,µ=0(P2 C/Tτ) oo. Moreover, the category C3filtr,opp,R is abelian. Proof. One immediately veries that τ -equivariant objects correspond bijectively to triples of ltrations whose ltrations are conjugate. The fact that the category is abelian is a direct consequence of the fact that BunP1 0− semistable ,µ=0(P2 C/T) is and that the functor τ∗ is exact since τ is an homeomorphism. ¤ 2. Application to Hodge theory In this section we apply the correspondence stated before to ltered vector spaces arising from Hodge theory. We take k=C . Let us recall some 20 O.PENACCHIO denitions and results about Hodge structures. Notice that we do not consider mixed Hodge structures dened on Z nor Q . Denition 2.1. A pure R -Hodge structure of weight r is a triple (HR, F•, F•) consisting of a nite dimensional R -vector space HR and two decreasing ltrations of HC=HR⊗RC , the Hodge ltration F• and the conjugate ltration with respect to the underlying real structure F• , such that F• and F• are r -opposed. Denition 2.2. A R -mixed Hodge structure is a quadruple (HR ,W• ,F• ,F•) which consists of a nite dimensional R -vector space HR , an increasing ltration of this real vector space W• , called the weight ltration , two ltrations F• and F• of HC=HR⊗RC , conjugate to each other with respect to the underlying real structure, such that F• and F• induce a pure Hodge structure of weight r on each quotient GrW r=Wr/Wr−1 , or, equivalently, such that the three ordered ltrations (W•, F•, F•) of HC are opposed; here the weight ltration is viewed as a ltration of HC . We will denote by R -MHS the category whose objects are R -mixed Hodge structures and morphisms are morphisms between vector spaces compatible with the ltrations. In the same way we can dene the category of complex mixed Hodge structures, denoted by C -MHS, by only requiring the objects to consist of a nite dimensional complex vector space endowed with three ordered opposed ltrations, the rst being increasing and the other decreasing. An element H∈C -MHS corresponds to a quadruple (HC, W•, F•,ˆ F•) . When a denition or a result concerns both categories, we do not specify any of them. Denition 2.3. The length of a mixed Hodge structure is the length of the largest interval [a, b] such that GrW r6= 0 for each r∈ {a, b} . The level of a mixed Hodge structure is the length of the largest interval [a, b] such that GrF p6= 0 for each r∈ {a, b} . A Tate Hodge structure of weight l , denoted by Thli , is the unique Hodge structure of rank 1 and of pure type (l, l) . Real mixed Hodge structures of length 0 are pure real Hodge structures and mixed Hodge structures of level 0 are extensions of Tate's Hodge structures. A bigrading of a real mixed Hodge structure H= (HR, W•, F•, F•) is a direct sum decomposition HC=⊕p,qVp,q of the underlying vector space which veries Wr=⊕j+l≤rVj,l and Fp=⊕j≥p,lVj,l . Following Deligne, one obtains an analogue of the Hodge decomposition for mixed Hodge structures: MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 21 Lemma 2.4. [4] Let H= (HR, W•, F•, F•) be a mixed Hodge structure. Then, there exists a unique bigrading of H , denoted by {Ip,q}p,q , such that Ip,q =Iq,p mod ⊕j<p,l<q Ij,l. The following notion is important when one considers degeneration of mixed Hodge structures (see [3]). Denition 2.5. A R -mixed Hodge structure H= (HR, W•, F•, F•) is said to be R -split if for each p, q the Ip,q vector spaces verify Iq,p =Ip,q . In this case the Ip,q spaces give a decomposition of HC which is compatible with the three ltrations. By Lemma 2.4 every mixed Hodge structure whose length is lower than 2 is R -split. In particular, every pure Hodge structure is R -split. 2.1. Vector bundles associated with mixed Hodge structures. We associate to each mixed Hodge structure H∈R -MHS (resp. C -MHS) an object in C3filtr,opp,R (resp. C3filtr,opp ). For this purpose, we take the decreasing ltration associated with the weight ltration W• , denoted by W• , by setting, for each r∈Z , Wr=Wr . Then, we associate to each object H∈ C3filtr,opp,R (resp. C3filtr,opp ) the corresponding Rees bundle on the toric complex projective plane as described in section (1.3), or, equivalently, the equivariant vector bundle corresponding to the ∆(1) -family of complete ltrations that H represents. Denition 2.6. Let H be a mixed Hodge structure in R -MHS (resp. in C - MHS). The Rees bundle associated to H is the Rees bundle associated with the triltered vector space (HC, W•, F•, F•) (resp. (HC, W•, F•,ˆ F•) ): ξP2 C(H) = ξP2 C(HC, W•, F•, F•) ( resp. ξP2 C(HC, W•, F•,ˆ F•)). Remark that to give a splitting of a mixed Hodge structure is equivalent to show that the ∆(1) -family of complete ltrations veries Klyachko's condition. As a direct outcome of Theorem 1.20 and Theorem 1.27 we get: Theorem 2.7. The category of complex mixed Hodge structures C -MHS is equivalent to the category of T -equivariant P1 0 -semistable vector bundles of degree 0 on the projective plane C -MHS //BunP1 0− semistable ,µ=0(P2 C/T) oo. The category of real mixed Hodge structures R -MHS is equivalent to the category of Tτ -equivariant P1 0 -semistable vector bundles of degree 0 on the projective plane R -MHS //BunP1 0− semistable ,µ=0(P2 C/Tτ) oo. 22 O.PENACCHIO By Corollary 1.26, we thus recover Deligne's result in a geometric way: Corollary 2.8. [4] The category of real and complex mixed Hodge structures are abelian. Consider an exact sequence of mixed Hodge structures in R -MHS 0//Ai//Hπ//B//0, and the associated exact sequence in BunP1 0− semistable ,µ=0(P2 C/Tτ) 0//ξP2 C(A)i//ξP2 C(H)π//ξP2 C(B)//0 . By construction, ξP2 C(B) is the reexive sheaf associated with the cokernel of i in the category of coherent sheaves, Coker(i) , whose singularity set is included in P12 . Let TP12 be the cokernel of the canonical injective morphism Coker(i)→Coker(i)∗∗ ∼ =ξP2 C(B) . In the category of coherent sheaves we thus have the exact sequence (2.1) 0//ξP2 C(A)i//ξP2 C(H)˜π//ξP2 C(B)//TP12 //0. 2.2. New Hodge numbers and R -split level. The fact that the Rees bundle associated to a cokernel in the category of mixed Hodge structures is not a cokernel in the category of coherent sheaves is due to the behaviour of some integers similar to the Hodge numbers, classically denoted by hp,q and given by hp,q H= dim CGrp FGrq FGr−p−q WHC= dim CGrp FGr−p−q WHC= dim CGrq FGr−p−q WHC . These integers, which we denote by sp,q , measure in some sense the relative position of the ltrations but are not additive contrary to the Hodge numbers. Denition 2.9. Let H be a mixed Hodge structure. We let sp,q H= dim CGrp FGrq FHC . We make the identication H4(P2 C,Z) = Z . Denition 2.10. Let H= (HR, W•, F•, F•) be a mixed Hodge structure. We dene the R -split level of H to be the integer α(H) = c 2(ξP2 C(H)). Remark 2.11 . The denition makes sense for C -MHS too. The term split level would be, however, more appropriate for the objects of C -MHS. Since the Rees bundles associated with mixed Hodge structures are of degree 0 , one easily veries that for each mixed Hodge structures H, H0 and each k∈Z : (i) α(H⊕H0) = α(H) + α(H0) . MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 23 (ii) α(H∗) = α(H) , where H∗= Hom R−MHS(H, Th0i) . (iii) α(H⊗H0) = dim CH0 Cα(H) + dim CHCα(H0) . (iv) α(H⊗Thki) = α(H) . Next we give an explicit formula for the R -split level. It is a direct consequence of Proposition 1.18. Proposition 2.12. The R -split level is expressed by (2.2) α(H) = 1 2X p,q (p+q)2(hp,q H−sp,q H). We can dene extensions in the abelian categories R -MHS and C -MHS (see [1] for example). Theorem 2.13. The R -split level is sub-additive that is, for A and B two mixed Hodge structures in R -MHS (resp. C -MHS)and H∈ Ext 1 R−MHS(A, B) (resp. Ext 1 C−MHS(A, B)), α(H)≥α(A) + α(B). Proof. The exact sequence given by the extension leads to the exact sequence (2.1) in the category of coherent sheaves. We thus have c 2(ξP2 C(H))+ c 2(TP12 ) = c 2(ξP2 C(A)) + c 2(ξP2 C(B)) . Since the support of TP12 is at least 2 -codimensional, c 2(TP12 )≤0 , which allows us to conclude. ¤ Since every mixed Hodge structure can be written as a successive extension of pure Hodge structures whose R -split level is 0 , we have: Corollary 2.14. For each H∈R -MHS, α(H)≥0 . In particular, it means that if A→H (resp. H→B ) is an injective (resp. surjective) morphism of mixed Hodge structures we have α(H)≥α(A) (resp. α(H)≥α(B) ). Moreover, the R -split level generalises the notion of R -split mixed Hodge structure: Proposition 2.15. A mixed Hodge structure is R -split if and only if its R -split level is 0 . Proof. We only have to prove the only if part. It is a consequence of Donaldson's theory on vector bundles. Consider a mixed Hodge structure whose R -split level is zero. The associated Rees bundle has vanishing Chern classes and hence, according to [7], is trivial. The action of the torus gives then a splitting compatible with all the ltrations. ¤ 24 O.PENACCHIO 2.3. Examples of calculation of the R -split level. According to Deligne (see [4], [5]), the cohomology groups of algebraic varieties, here separated schemes of nite type over C , are endowed with natural and functorial mixed Hodge structures. Let X be such an algebraic variety. We will consider the R -split levels of X , αj(X) = α((Hj(X, C), W•, F•, F•)) , where j∈Z , αj(X) being associated to the mixed Hodge structure on the jth group of cohomology. They are invariants of the variety depending on its algebraic (or analytic) class. In this section we explicit the R -split level of the mixed Hodge structures on the cohomology of some algebraic curves. For certain varieties these invariants are trivial. The mixed Hodge structures of length lower than 2 are indeed R -split. It is the case for pure Hodge structures and hence for the Hodge structures on the cohomology groups of smooth projective algebraic varieties or of compact Kähler varieties. The lengths of the mixed Hodge structures on the cohomology of weighted projective spaces (see [6]) and of varieties with logarithmic singularities (see [23]) are lower than 2 , these mixed Hodge structures are therefore R -split. The construction of mixed Hodge structures is functorial. Let X be an algebraic variety, π:X0→X be a resolution of singularities, j:X→X be a compactication and S=X\X . By [4], [5] we can nd a compatible smooth compactication j:X0→X0 and a morphism π:X0→X making the square below cartesian: X0j// π ²² X0 π ²² Xj//XS. i oo i __@ @ @ @ @ @ @ @ By [5], Proposition (8.2.6), π induces an epimorphism of mixed Hodge structures on the cohomology, and j induces a monomorphism. We thus have, for each integer l , αl(X)≥αl(X0) and αl(X)≥αl(X) . Let us focus our attention on curves. The variety X0 of the preceding construction is now the normalization of X . The mixed Hodge structures on 0th and second cohomology groups are pure. Let us describe it on the rst cohomology groups. This cohomology is given by the hypercohomology of the complex [OX d →π∗Ω1 X0( log S)] . MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 25 The weight and Hodge ltrations on H1(X, C) = H1([OX d →π∗Ω1 X0( log S)]) are described by: Lemma 2.16. ( [5] , Lemme ( 10.3.11)) (i) W1(H1(X, R)) = Im (H1(X, R)→H1(X, R)) and W0(H1(X, R)) = Ker (H1(X, R)→H1(X0,R)). (ii) The spectral sequence dened by the naive tration of [OX d → π∗Ω1 X0( log S)] degenerates into the Hodge ltration of H∗(X, C) . In order to compute the R -split level of the mixed Hodge structures on the cohomology groups we will compute the period matrix using the preceding lemma and then look at the intersection between the sub-vector space this matrix determines and its conjugate with respect to the underlying real structure. Curves of genus 0 . Let us consider a rst non trivial example: a non complete nodal curve of genus 0 . Let (m1, m2, p1, q1) be four distinct points of P1 C and X be the non complete singular curve obtained by gluing p1 and q1 together and removing the mi (for a justication of the identication of the points see [20], chapter IV, part 3). We have S=∪i{mi} . Let u be a coordinate on P1 C\{∞} . Lemma 2.17. F1H1(X, C) =< ω = ( 1 u−m1−1 u−m2)du > . Since the mixed Hodge structure on the rst cohomology group of such a curve is an extension of a Tate Hodge structure of weight 1 by a Tate Hodge structure of weight 0 (we have h0,0=h1,1= 1 and the other Hodge numbers are zero), the R -split level is completely determined by the integer s1,1 . To know s1,1 we have to compute the dimension of the intersection of the subspace F1 with its conjugate with respect to the real structure on H1(X, R) . The Hodge ltration induces a ltration on the dual of the rst cohomology group using the isomorphism H1 DR(X, C)∼ =H1 Betti(X, C)∼ =H1(X, C)∗= (H1(X, R)⊗C)∗ . Let us choose a basis γ0, γ1 of H1(X, R) . Let γ0 be a positively oriented loop whose homology class is nonzero in X but vanish in X∪m1 and γ1 be the loop formed by a path from p1 to q1 in X0 by identifying these points. 32 O.PENACCHIO The associated torsion free coherent sheaf on A2×S is called the relative Rees sheaf associated with the 2 -ltered vector bundle (V,F•,G•) and is denoted by ξS(V,F•,G•) . Note that the denition coincides with the denition of the Rees sheaf on the ane plane associated with a pair of ltrations when S= Spec (k) is reduced to a point. The relative Rees sheaf ξS(V,F•,G•) is naturally equivariant for the action of the torus (Gm)2 on A2×S which reduces to the standard action on the rst factor and is trivial on the second. In terms of comodule this action can be described as follows: if we let (Gm)2= Spec B , where B= k[u±1, v±1] , the structure of B -comodule on A[u, v] is deduced by extension of scalars from the B -comodule structure on k[u, v] made explicite in the rst section. It is clear that, similarly to the Rees module case, it induces a B -comodule structure on the relative Rees module RS(V,F•,G•) and hence an action of the torus on the corresponding coherent sheaf. Proposition 3.2. The coherent sheaf of OA2×S -modules ξS(V,F•,G•) is reexive and, moreover, its at least 3 -codimensional singularity locus is an at least 1 -codimensional subset of the 2 -codimensional variety {(0,0)}×S . To prove this statement, we rst need to dene the saturation of a module. Denition 3.3. Let A be a regular local ring, L a free A -module and M a torsion free submodule of L . The saturation of M in L with respect to the ideal a of A is dened by Msat a={l∈L|aj.l ⊂M for jÀ0}. Lemma 3.4. Let A be a regular local ring with maximal ideal m , and M be a torsion free nitely generated A -module. Let Msat m be its saturated in a free A -module with respect to m , then, Msat m=M⇒ depth mM≥2. Proof. Consider the short exact sequence associated to the embedding of M in a free module L . The associated long exact sequence of local cohomology at m and the fact that L is free, and hence Hi m(L) for i≥0 , implies that Hi m(M)∼ =Hi−1 m(L/M) for all i≥1 . Since Msat/M =H0 m(L/M) , we can deduce the result from the equivalence ( depth m(M)≥j)⇔(∀i < j, Hi m(M) = 0) (see [11], III, Ex.3.4). ¤ Proof. (of Proposition 3.2) Recall that a coherent sheaf is reexive if and only if it is torsion free and normal (see [10]). Since the A[u, v] -module RS(V,F•,G•) is torsion free, according to Lemma 3.4, it will be sucient MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 33 to prove that its local rings are saturated. Let us consider the maximal ideal m=a.(u, v) . The Rees module is a submodule of the free module L=A[u±1, v±1]⊗V . Let l=upvq⊗b∈RS(V,F•,G•)sat m be a generator. There is an integer n such that mj.l ∈RS(V,F•,G•) for all j≥n . In particular, this means that un.l =un+pvq⊗b∈RS(V,F•,G•) and hence b∈Fp+n∩Gq . In the same way there exists an integer n0 such that b∈Fp∩Gq+n0 . So, b∈Fp∩Gq , and thus l∈RS(V,F•,G•) , which proves the rst statement. Remark now that the localisation of the Rees module at a geometric point that do not belong to {(0,0)}×S is isomorphic to the localisation of a Rees module dened by only one of the ltration, the other being trivial. The proof of this fact is similar to the proof of Lemma 1.3 (ii)(a): let s∈A1\{0} , let p be the prime ideal corresponding to the closed subset {s}×A1 of A2 and fs:{s}×A1×S→A2×S the morphism it determines. Note that, for each integer q , Gq/Gq+1 is a vector bundle on S . Then, ξS(V,F•,G•)|{s}×A1×S=f∗ sξS(V,F•,G•) = ˜ R , where R is an A[v] -module given by R=RS(V,F•,G•)⊗A[u,v](A[u, v]/p.A[u, v]) ∼ =RS(V,G•) = ⊕q(v−q⊗kk rk (Gq/Gq+1)) . This proves that ξS(V,F•,G•)|{s}×A1×S is locally free. The restriction of the Rees sheaf to (A2\ {(0,0)})×S is thus locally free, which completes the proof. ¤ 3.2. Localisation. Let (V,F•,G•) be a 2 -ltered vector bundle on a smooth ane algebraic variety S= Spec A and s∈S be a point whose corresponding maximal ideal is p . We let X=A2×S and denote by f:X→S the projection onto the second factor. To each s∈S is associated a morphism fs:Xs→X , where Xs= Spec (k(s)) ×SX . We want to compare the bre over s of the family of coherent sheaves given by the relative Rees construction, that is the equivariant coherent sheaf on A2 , ξS(V,F•,G•)|Xs=f∗ sξS(V,F•,G•) , with the "expected" - bre, that is, the Rees equivariant locally free sheaf associated with the 2 - ltered vector space obtained by localising at s the 2 -ltered vector bundle, ξ(V(s),F•(s),G•(s)) . To shorten the notations, we will denote by R(p) = R(V(s),F•(s),G•(s)) the k[u, v] -module corresponding to locally free sheaf on A2 , ξ(V(s),F•(s), G•(s)) , and by RS(p) the k[u, v] -module Γ(f∗ sξS(V,F•,G•)) corresponding to the coherent sheaf f∗ sξS(V,F•,G•) . 34 O.PENACCHIO Lemma 3.5. With the above notations, there exists a short exact sequence of k[u, v] -modules, (3.2) 0//RS(p)//R(p)//R0(p)//0, in which R0(p) is a torsion module. Moreover, if p corresponds to a point s∈V , then R0(p) = 0 . Proof. Let us rst remark that for each pair (p, q)∈Z2 we have an injective morphism of k -vector spaces (Fp∩Gq)(s)→ Fp(s)∩Gq(s) . (Fp∩Gq)(s) is indeed the kernel of the morphism of bre over s given by the morphism of stalks Fp s→(V/Gq)s , when Fp(s)∩Gq(s) is the kernel of the morphism of vector spaces Fp(s)→ V(s)/Gq(s) . Then, we have RS(p) = RS⊗A[u,v] (A[u, v]/pA[u, v]) ∼ =RS/pRS . The k[u, v] -module RS(p) is hence isomorphic to the Rees module generated by the elements of the form u−pv−q⊗kmp,q where mp,q ∈Mp,q/pMp,q ∼ =(Fp∩Gq)(s) . This yields, using the previous remark, an injective morphism of k[u, v] -modules from RS(p) to R(p) which maps the piece u−pv−q⊗kMp,q/pMp,q to u−pv−q⊗k(Fp(s)∩Gq(s)) . Let us show that the cokernel of this morphism, R0(p) , is a torsion k[u, v] -module if its localisation at (0) , R0(p)0= 0 , is zero. Let m0∈ R0(p) , and let m=Pp,q u−pv−q⊗kmp,q be a preimage of m0 in R(p) , where the sum is nite. For each pair of integers (p, q) there exists a pair (rp,q, sp,q) , rp,q ≤p , sp,q ≤q , such that mp,q ∈Mrp,q,sp,q /pMrp,q,sp,q . Then up−rp,q vq−sp,q .(u−pv−q⊗kmp,q)∈RS(p) . Let us consider the element P(u, v) = Qp,q up−rp,q vq−sp,q , which is well-dened since the product is nite. Then P(u, v).m ∈RS(p) and hence P(u, v).m0= 0 in R0(p) . Let p be the maximal ideal corresponding to a point s∈V . Then there exists an open set containing s on which, for each (p, q)∈Z , the rank of the morphism between vector bundles Fp→ V/Gq is constant. The restrictions of the coherent sheaves Fp∩Gq are hence locally free, and thus (Fp∩Gq)(s)∼ =Fp(s)∩Gq(s) . The k[u, v] -modules RS(p) and R(p) are then isomorphic. ¤ Lemma 3.6. With the above notations, for each s∈S with corresponding ideal p , we have RS(p)∗∗ ∼ =R(p) , where the isomorphism is an isomorphism of k[u, v] -modules compatible with the structure of B -comodule. Proof. We prove the statement for the canonically associated coherent sheaves on A2 k . Let 0//FS(p)//F(p)//T(p)//0 the exact sequence associated to (3.2). By Proposition 3.2, we know that Supp (T(p)) ⊂ {(0,0)} . This yields an isomorphism FS(p)|U∼ =F(p)|U MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 35 where U=A2\{(0,0)} , and hence an isomorphism FS(p)∗∗|U∼ =F(p)∗∗|U . Since the dual of each coherent sheaf is reexive, thus normal, it is determined by its restriction to each open whose complementary is at least 2 - codimensional. In consequence FS(p)∗∗ ∼ =F(p)∗∗ ; the corresponding modules are then isomorphic and all the isomorphisms are compatible with the action. ¤ 3.3. Gluing and relative Rees sheaves on the projective plane. Consider now a 3 -ltered vector bundle (V,(F• i)i∈{0,1,2}) on an algebraic variety S as above. Let S=qβ∈BSβ be a covering of the base space by ane schemes. Let i∈ {0,1,2} and β∈B . We make, on the open set A2 i×Sβ , the Rees construction involving the 2 -ltered vector bundle on Sβ , (Vβ,F• j,F• l) , with {i, j, l}={0,1,2} and j < l , where Vβ denotes the restriction of the vector bundle V to Sβ . Lemma 3.7. With the above notations, the T -equivariant reexive sheaves on A2 i×Sβ , ξSβ(Vβ,F• j,F• l) , can be glued together to form a T -equivariant reexive sheaf on P2 k×S denoted by ξS(V,F• 0,F• 1,F• 2) and called the Rees sheaf associated to the 3 -ltered vector bundle (V,(F• i)i∈{0,1,2}) . Proof. Let (i, j) , i < j , and (i0, j0) , i0< j0 , be two dierent pairs of integers of {0,1,2} , and let l be the integer such that {l}={i, j} ∩ {i0, j0} . h (respectively h0 ) is the integer such that {i, j, h}={0,1,2} (respectively {i0, j0, h0}={0,1,2} ). Then, the restrictions of the relative Rees sheaves ξSβ(Vβ,F• i,F• j) and ξSβ(Vβ,F• i0,F• j0) to the open set (A2 h×Sβ)∩(A2 h0× Sβ)∼ =Gm×A1×Sβ are isomorphic, via ϕl h and ϕl h0 , respectively, as equivariant coherent sheaves, to the pullback f∗ξSβ(Vβ,F• l) of the relative Rees bundle on A1×Sβ associated to the ltered vector bundle on Sβ , (Vβ,F• l) , by the projection f:Gm×A1×Sβ→A1×Sβ . This gives a transition map Φl hh0 , where {h, h0, l}={0,1,2} , on the open intersection (A2 h∩A2 h0)×Sβ . The three transition maps satisfy the cocycle relation since all the restrictions coincide on the intersection of the three standard open sets covering P2 k×Sβ . This yields an equivariant sheaf ξSβ(V,F• 0,F• 1,F• 2) on P2 k×Sβ . One can clearly glue these sheaves to form a sheaf on P2 k×S . ¤ As a consequence of Proposition 3.2 we have: Proposition 3.8. The Rees sheaf associated to a 3 -ltered vector bundle V on a smooth algebraic variety S , ξS(V,F• 0,F• 1,F• 2) , is a T -equivariant reexive sheaf whose singularity locus is included in {P01, P02, P12}×S . Since the relative Rees sheaves are reexive, they form at families of torsion free sheaves. 36 O.PENACCHIO Corollary 3.9. The Rees sheaf ξS(V,F• 0,F• 1,F• 2) on P2 k×S associated to a 3 -ltered vector bundle V on a smooth algebraic variety S is S -at. Proof. Denote by ξS the sheaf ξS(V,F• 0,F• 1,F• 2) and by E its restriction to U×S , where U=P2 k\ {P01, P02, P12} ; recall that E is locally free. Let f:X=P2 k×S→S and let OX(1) be a line bundle on X whose restriction to any bre Xs is ample. It suces to prove that for all m suciently large the sheaves f∗(ξS(m)) are locally free. Let V be an open set of S , then, for each integer m , since ξS is reexive and hence determined by its restrictions to open sets whose complementary is at least 2 -codimensional, f∗(ξS(m))(V) = ξS(m)(P2 k×V)∼ =ξS(m)(U×V) = f∗E(m)(V). Since f∗E(m)(V) is a free OS(V) -module, so is f∗(ξS(m))(V) . ¤ We are now able to compare the bres of the relative Rees family to the Rees vector bundles obtained at each point from the 3 -ltered vector spaces deduced from the 3 -ltered vector bundle. Proposition 3.10. Let (V,(F• i)i∈{0,1,2}) be a 3 -ltered vector bundle on a smooth algebraic variety S . Let ξS(V,F• 0, F• 1,F• 2) be the associated relative Rees sheaf on P2 k×S . Then for each s∈S we have an isomorphism of T -equivariant locally free sheaves on P2 k ξS(V,F• 0, F• 1,F• 2)(s)∗∗ ∼ =ξ(V(s),F• 0(s),F• 1(s),F• 2(s)). Moreover, there exists an open set V⊂S , such that for each s∈V ξS(V,F• 0,F• 1,F• 2)(s)∼ =ξ(V(s),F• 0(s),F• 1(s),F• 2(s)). Proof. According to Lemma 3.5, by gluing the local descriptions of the bres of the relative Rees sheaves and the Rees locally free sheaves of the bres of the ltered vector bundle as in Lemma 3.7, we have, for each s∈S , the exact sequence 0//ξS(V,F• 0, F• 1,F• 2)(s)//ξ(V(s),F• 0(s),F• 1(s),F• 2(s)) // //T(s)//0, where Supp (T(s)) ⊂ {P01, P02, P12} . Then, Lemma 3.6 gives the rst isomorphism. Next, the relative Rees sheaf being S -at by Corollary 3.9, the set V={s∈S|ξS(V,F• 0, F• 1,F• 2)(s) is a locally free sheaf } is an open subset of S . According to Lemma 3.5, for each s∈V , T(s) = 0 . For each point of V we thus have the required isomorphism. ¤ MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 37 Example 3.11. Let V be a 2 -dimensional vector space over k , {e, f} be a basis of V and V be the trivial vector bundle associated to V on A2= Spec k[λ, µ] . Consider the three nite, complete and decreasing ltrations of V given by F−2 0=V , F−1 0=F0 0=< f > , F1 0={0} , F0 1=V , F1 1=< e+P(λ, µ)f > , F2 1={0} , and, F0 2=V , F1 2=< e +Q(λ, µ)f > , F2 2= {0} , where P, Q are two polynomials in λ and µ . One verifes that at each point of the base space the given ltrations are opposed. On A2 0×A2 , the Rees k[u, v, λ, µ] -module RA2(V,F• 1,F• 2) is generated by {u−1⊗(e+ P(λ, µ)f), v−1⊗(e+Q(λ, µ)f), f} (with u=u1 u0 and v=u2 u0 ). Consider the closed subset of A2 , Z={(λ, µ)∈A2|P(λ, µ) = Q(λ, µ)} . The singularity set of the relative Rees sheaf on P2 k×A2 is P12 ×Z . This is due to the fact that the dimension of the intersection of the two subvector spaces of V given at each point by F1 1 and F1 2 jumps at the points which belong to Z . Indeed, for each maximal ideal p which corresponds to a point s∈Z , we have an exact sequence of k[u, v, λ, µ] -modules 0→RA2(V,F• 1,F• 2)(p)→RA2(V,F• 1,F• 2)(p)∗∗ →R0(p)→0, where RA2(V,F• 1,F• 2)(p)∗∗ is isomorphic to the free k[u, v] -module generated by {u−1v−1⊗e, f} , which is exactly R(V(s),F• 1(s),F• 2(s)) , and R0(p) is the torsion k[u, v] -module u−1v−1k[u, v]/(u−1, v−1)k[u, v] . Since the dimensions of the intersections of the subspaces given by the ltrations do not jump on the other charts A2 i×A2 , i∈ {1,2} , the associated relative Rees sheaves are locally free. So, for each s∈A2\Z , ξA1(V,F• 0,F• 1,F• 2)(s) is locally free, and, if s∈Z , we have 0→ξA2(V,F• 0,F• 1,F• 2)(s)→ξ(V(s),F• 0(s),F• 1(s),F• 2(s)) → TP12 →0, where TP12 is a skyscraper sheaf of rank one supported at P12 ⊂P2 k . 3.4. Analytic relative Rees sheaves. Let (V,(F• i)i∈{0,1,2}) be a 3 -ltered holomorphic vector bundle on a complex manifold S . One can perform the construction of the relative Rees sheaf on P2 C×S in an analytic context; here P2 C= Proj C[u0, u1, u2] is considered as a complex manifold. The coherent sheaves Fp j∩Fq l , j6=l , can be dened in the same manner and are analytic torsion free coherent sheaves. Next, on C2 i×S , where C2 i= Spec C[u, v] is considered as a complex manifold, with j < l , {i, j, l}={0,1,2} , one denes the OS[u, v] -module Pp,q u−pv−qFp j∩Fq l . This module gives a reexive coherent sheaf on C2 i×S which is equivariant for the action of (C∗)2= Spec C[t±1 j, t±1 l] on the rst factor. One can glue the local descriptions to get a T -equivariant analytic reexive Rees sheaf ξS(V,F• 0, F• 1,F• 2) on P2 C×S . Propositions 3.8 and 3.10 translate directly in this context. 38 O.PENACCHIO Note that, by GAGA, the coherent sheaves parametrized by S are algebraic coherent sheaves on the projective plane. Indeed, for each s∈S , ξS(V,F• 0, F• 1,F• 2)|P2 C×{s} is an analytic coherent sheaf on P2 C , hence algebraic. 4. Variation of mixed Hodge structure 4.1. Rees sheaf associated to a variation of mixed Hodge structure. The aim of this section is to codify variations of mixed Hodge structure in terms of families of Rees bundles, namely relative Rees sheaves. We do not give an analogue of the connections underlying these objects nor take into account the polarizations. In this context, we dene a variation of mixed Hodge structure to be: Denition 4.1. A variation of R -mixed Hodge structure on a complex manifold S is an ordered triple (HR,W•,F•) consisting of a local system HR of free R -modules of nite rank over the variety S and two ltrations: an increasing ltration W• of HR by local subsystems and a decreasing ltration F• by subbundles of the holomorphic vector bundle HO=HR⊗ OS such that: (i) For each point s∈S , the bre (HR(s),W•(s),F•(s),F•(s)) ∈R - MHS, namely is a R -mixed Hodge structure. (ii) The canonical connection ∇ corresponding to the local system HR satises, for each integer p , the condition ∇Fp⊂Ω1 S⊗Fp−1. This denition, and all the results of this section, can be stated in a complex algebraic context. In that case, the base of the variation is a smooth scheme over C and the Hodge ltration is algebraic. A variation of R -mixed Hodge structure can be described in terms of a morphism from the base to an appropriate classifying space M of mixed Hodge structures. Before giving its construction, after [3], we need to recall some facts about R -split mixed Hodge structures. There is a canonical way to deform a mixed Hodge structure into one which is split over R . Consider (HR, W•, F•, F•)∈R -MHS and its canonically associated splitting {Ip,q}p,q . We associate to this Hodge structure the nilpotent Lie algebra l−1,−1={δ∈ End (HC)|δ(Ip,q)⊂ ⊕j<p,l<qIj,l}. Since l−1,−1=l−1,−1 , this algebra admits the real form l−1,−1 R= End (HR)∩ l−1,−1 . Proposition 4.2. [3] Let (HR, W•, F•, F•) be a R -mixed Hodge structure, there exists a unique δ∈l−1,−1 R such that the mixed Hodge structure (HR, W•, e−iδF•, eiδF•) is R -split. MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 39 Moreover, the functor (HR, W•, F•, F•)7→ ((HR, W•, e−iδF•, eiδF•), δ) establishes an equivalence of categories between the category of R -mixed Hodge structures and the category whose objects are pairs formed by a R - split mixed Hodge structure and an endomorphism of its associated real nilpotent Lie algebra δ∈l−1,−1 R , and whose morphisms are morphisms of mixed Hodge structures commuting with the endomorphisms δ . Consider a variation of mixed Hodge structure (HR,W•,F•) on S as above and let fp= rk C(Fp) and fp r= rk C(Fp r) , where the vector bundle Fp r is the pth stage of the Hodge ltration of the variation of pure Hodge structures of weight r induced by Fp on Wr/Wr−1 . From Lemma 2.4, a R -mixed Hodge structure is completely determined by its splitting {Ip,q}p,q . The classifying space of R -mixed Hodge structures M can therefore be viewed as the set of equivalences of splittings. Let V be a bre of HC and VR be the underlying real vector space. Let W0GL(V) be the group of automorphisms of V which preserve the weight ltration of V . Consider the kernel W−1GL(V) of the canonical morphism of groups W0GL(V)π →QrGL(GrW rV). The set of real splittings is a principal homogenous space under W0GL(V)R= W0GL(V)∩GL(VR) , and, according to Proposition 4.2, the set of all possible splittings is a principal homogenous space under H=W0GL(V)R.W−1GL(V). Let us choose a R -split splitting and let F• 0 be the corresponding ltration. Let P be the subgroup of H which preserves F• 0 , then, M∼ =H/P, the isomorphism depending on the choice of the reference R -split mixed Hodge structure. The variation of mixed Hodge structure (HR,W•,F•) on S gives rise to a morphism ϕ:˜ S→M from the universal covering of S to the classifying space which gives a morphism ϕ0:S→Γ\M, where Γ = Im (π1(S, ?)→GL(VR)) is the image of the monodromy homomorphism. By analogy with the case of variations of pure Hodge structure, the morphism ϕ0 is often called the period mapping . 40 O.PENACCHIO Each point of s∈M gives a Hodge ltration F• s such that (VR, W•, F• s, F• s) is a R -mixed Hodge structure. There is a universal ltered vector bundle (V,F• univ) on M such that, for each integer p and each point s∈M , Fp univ(s) = Fp s, and such that the Hodge ltration of the variation of mixed Hodge structure is given by F•=ϕ∗F• univ . Next, each point (s, t)∈M×M gives an ordered 3 -ltered vector space (V, W•, F• s, F• t) and we have a universal 3 -ltered vector bundle on M×M of the form (V⊗COM×M, W•⊗COM×M,F• univ,1,F• univ,2) , where F• univ,i , i= 1,2 is the pullback on M×M of the universal Hodge ltration on M by the projection on the rst and second factors. We construct the relative Rees sheaf associated to this 3 -ltered vector bundle and get a T -equivariant reexive sheaf on P2 C×M×M , ξM×M(V⊗COM×M, W•⊗COM×M,F• univ,1,F• univ,2). Next, we focus on the locus of opposed ltrations Mopp ⊂M×M which is dened to be the set of points (s, t)∈M×M such that (V, W•, F• s, F• t)∈C - MHS, namely such that (W•, F• s, F• t) forms an ordered triple of opposed ltrations on V . This space can be considered as the classifying space of complex mixed Hodge structures associated to the data {(V, W•),{fp r}r,p} . Notice that (s, t)∈Mopp ⇔(t, s)∈Mopp . Lemma 4.3. The locus of opposed ltrations Mopp is an open submanifold of M×M . Before giving the proof, let us remark that the property to be opposed for the ltrations, namely, for each integer r , the fact that F• s and F• t are r -opposed on GrW rV , is detected on the graded pieces of the graded object GrWV . For each r there is a surjective map πr:M→Fl(Vr) , where Fl(Vr) = Fl(GrW rV, fn r, ..., f1 r) is the ag manifold which parametrizes ltrations by subvector spaces, GrW rV=F0⊃F1⊃... ⊃Fn⊃ {0} , such that, for each p, dim CFp=fp r . The ag manifold Fl(Vr) is a projective variety. This map sends a point s , corresponding to F• s , to the Hodge ltration F• s induces on GrW rV . Let nr= dim C(GrW rV) and remark that fp r=nr−fr−p+1 r . Proof. Consider the morphism MHS AND VECTOR BUNDLES ON THE PROJECTIVE PLANE I 41 M×M Q r(πr×πr) −−−−−−−→ Qr(Fl(Vr)×Fl(Vr)) . Let (s, t)∈M×M . Then, F• s and F• t are r -opposed on GrW rV if and only if for each p , (4.1) Fp s⊕Fr−p+1 t=GrW rV. This condition is open in Qr(Fl(Vr)×Fl(Vr)) . Indeed, to each ag manifold Fl(Vr) is associated a natural injection into a product of Grassmanians Fl(Vr),→QpG(GrW rV, fp r) , the image of which is a compact manifold. For each integers p, r , and each point s∈M , the image of s in G(GrW rV, fp r) can be described as a nr×fp r matrix of maximal rank fp r . In this setting, (4.1) amounts to the fact that the nr×nr matrix formed by the nr×fp r matrix which represents Fp s in G(GrW rV, fp r) and the nr×fr−p+1 r which represents Fr−p+1 in G(GrW rV, fr−p+1 r) has maximal rank. Let Up r be the open submanifold of points in G(GrW rV, fp r)×G(GrW rV, fp r) which verify this condition. Then, the set Ur= (Fl(Vr)×Fl(Vr)) ∩(∩pUp r) is an open submanifold of Fl(Vr)×Fl(Vr) . We thus get an open inclusion (4.2) j:Mopp =∩r(πr×πr)−1(Ur),→M×M. ¤ The construction of Mopp allows us to dene a universal Rees sheaf on P2 C×Mopp which parametrizes triples of opposed ordered ltrations, namely objects of C -MHS corresponding to the data of linear algebra of the variation of mixed Hodge structure (HR,W•,F•) on S , ξMopp =j∗ξM×M(V⊗COM×M, W•⊗COM×M,F• univ,1,F• univ,2). Consider now the morphism ϕ×ϕ:˜ S→M×M , where the conjugation is taken with respect to the real structure on VC which comes from VR . This morphism is real analytic and takes values in Mopp ; the image by ϕ×ϕ of a point s∈˜ S is the R -mixed Hodge structure (VR, W•, F• s, F• s) .