DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 1 0 SERGIO CONSOLE AND ANNA FINO 0 2 n a J Abstract. LetM =G/Γbeacompactnilmanifoldendowedwithaninvariant complexstructure. Weprovethat,onanopensetofanyconnectedcomponent 3 ofthemodulispaceC(g)ofinvariantcomplexstructuresonM,theDolbeault 2 cohomologyofM isisomorphictotheoneofthedifferentialbigradedalgebra associated to the complexification gC of the Lie algebra of G. To obtain this ] G result,wefirstprovetheaboveisomorphismforcompactnilmanifoldsendowed with a rational invariant complex structure. This is done using a descending D seriesassociatedtothecomplexstructureandtheBorelspectralsequencesfor . thecorrespondingsetofholomorphicfibrations. Thenweapplythetheoryof h Kodaira-Spencer fordeformationsofcomplexstructures. t a m 1. Introduction [ Let M be a compact nilmanifold of real dimension 2n. It follows from a result 2 ofMal’ˇcev[16]thatM =G/ΓwhereGisasimplyconnected(s+1)-stepnilpotent v Lie group admitting a basis of left invariant 1-forms for which the coefficients in 5 3 the structure equations are rational numbers, and Γ is a lattice in G of maximal 1 rank(i.e.,adiscreteuniformsubgroup,cf. [23]). We willletΓactonGonthe left. 3 It is well known that such a lattice Γ exists in G if and only if the Lie algebra g 0 of G has a rational structure, i.e. if there exists a rational Lie subalgebra g such 8 Q 9 that g∼=gQ⊗R. / The de Rham cohomologyof a compact nilmanifold can be computed by means h of the cohomology of the Lie algebra of the corresponding nilpotent Lie group t a (Nomizu’s Theorem [21]). m We assume that M has an invariant complex structure J, that is to say that J : comes from a (left invariant) complex structure J on g. v Our aim is to relate the Dolbeault cohomology of M with the cohomology ring i X H∗,∗(gC)ofthedifferentialbigradedalgebraΛ∗,∗(gC)∗,associatedtogCwithrespect ∂ r to the operator ∂ in the canonical decomposition d=∂+∂ on Λ∗,∗(gC)∗. a ThestudyoftheDolbeaultcohomologyofnilmanifoldswithaninvariantcomplex structureismotivatedbythefactthatthelatterprovidedthefirstknownexamples ofcompactsymplecticmanifoldswhichdonotadmitanyKa¨hlerstructure[1,5,28]. Since there exists a natural map i:H∗,∗(gC)→H∗,∗(M) ∂ ∂ which is always injective (cf. Lemma 7), the problem we will study is to see for which complex structure J on M the above map gives an isomorphism Hp,q(M)∼=Hp,q(gC). (1) ∂ ∂ 1991 Mathematics Subject Classification. 53C30,53C35. ResearchpartiallysupportedbyMURSTandCNRofItaly. 1 2 SERGIO CONSOLE AND ANNA FINO NotethatH∗,∗(gC)canbeidentifiedwiththecohomologyoftheDolbeaultcomplex ∂ of the forms on G which are invariant by the left action of G (we shall call them briefly G-invariant forms) and H∗,∗(M) with the cohomology of the Dolbeault ∂ complex of Γ-invariant forms on G. We shall use these identifications throughout this note. Our main result is the following TheoremATheisomorphism(1)holdsonanopensetofanyconnectedcomponent of the moduli space C(g) of invariant complex structures on M. To obtain Theorem A we first consider the case of complex structures J which arerational,i.e. theyarecompatiblewiththerationalstructureofG(J(g )⊆g ). Q Q Theorem B For any rational complex structure J, the isomorphism (1) holds. It is an open problem whether the isomorphism (1) holds for any compact nil- manifold endowed with an arbitrary invariant complex structure. We do not know examples for which (1) does not hold. Theorem A will follow from Theorem B using the theory of deformations of complex structures [13, 26]. Indeed by [24] the set C(g) of complex structures on g is at least infinitesimally a complex variety. Using the theory of deformations of complex structures, we are able to prove that for any small deformation of a rational complex structure J, the isomorphism (1) holds (Lemma 8). IfM is a compactcomplex parallelisablenilmanifold,i.e., Gis anilpotent complex LiegroupandJ isalsorightinvariant,wehavethatgi =gi andTheoremBfollows J from [25, Theorem 1]. An important class of complex structures is given by the abelian ones (i.e. those satisfying the condition [JX,JY] = [X,Y], for any X,Y ∈ g [3, 8]). The nil- manifolds with an abelian complex structure are to some extent dual to complex parallelisable nilmanifolds: indeed, in the complex parallelisable case dλ1,0 ⊂λ2,0, and in the abelian case dλ1,0 ⊂λ1,1, where λp,q denotes the space of (p,q)-forms ong. In this last case we will compute the minimal model of the Dolbeault cohomology of M and prove that the isomor- phism(1)holdsforanyabeliancomplexstructure. In[6]wasprovedasimilarresult for the Dolbeault cohomology of M endowed with a nilpotent complex structure, which is a slight generalization of the abelian one. Note that, however,if M is a complex solvmanifold G/Γ (with G a solvable not nilpotentLiegroup),theisomorphism(1)doesnotholdingeneral,asshownin[15]; a discussion of the behaviour of the Dolbeult cohomology of homegeneous mani- folds under group actions can be found in [2]. If G is a compact even dimensional (semisimple) Lie group endowed with a left invariant complex structure, the Dol- beault cohomology of G does not arise from just invariant classes, as the example in [22] shows. This paper is organized as follows. In Section 2, following [24], we define a descending series of subalgebras {gi} J (with g0 = g and gs+1 = {0}) for the Lie algebra g associated to the complex J J DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 3 structure J. In general, the subalgebra gi is not a rational subalgebra of gi−1. If J J J is rational, any gi is rational in gi−1. J J The importance of this series is twofold. First(Section3),inthecasethatthesubalgebragi isrationalingi−1 (inparticular J J ifJ isrational),itallowsustodefineasetofholomorphicfibrationsofnilmanifolds: p˜ :M =G/Γ→G0,1/p (Γ), with standard fibre G1/Γ1, 0 J 0 J ... p˜ :Gs−1/Γs−1 →Gs−1,s/p (Γs−1) with standard fibre Gs/Γs. s−1 J J s−1 J For the above fibrations we will consider the associated Borel spectral sequence (E ,d ) ([12, Appendix II by A. Borel, Theorem 2.1], [10, 14]), which relates the r r Dolbeault cohomology of each total space with the Dolbeault cohomology of each base and fibre. Secondly (Section 4) following [24], we will prove that one can choose a basis of (1,0)-forms (and also (0,1)-forms) on g which is compatible with the above de- scending series. This basis will give a basis of Gi−1,i-invariant (1,0)-forms on the J nilmanifoldsGi−1,i/p (Γi−1)andofGi-invariant(1,0)-formsonthenilmanifolds J i−1 J Gi/Γi, i=0,...,s. J Next, in Section 5, we consider a spectral sequence (E˜ ,d˜) concerning the Gi−1- r r J invariantDolbeaultcohomologyofeachtotalspaceGi−1/Γi−1,theGi−1,i-invariant J J DolbeaultcohomologyeachbaseGi−1,i/p (Γi−1)andtheGi-invariantDolbeault J i−1 J cohomology of each fibre Gi/Γi. In this way (E˜ ,d˜) is relative to the Dolbeault J r r cohomologiesofthe Liealgebrasgi,gi−1 andgi−1/gi. Notethatthelatterarethe J J J J underlying Lie algebras of the fibre, the total space and the base, respectively, of the above holomorphic fibrations. In Section 6 we compare the spectral sequence (E˜ ,d˜) with the Borel spectral r r sequence (E ,d ). Inductively (starting with i = s) these two spectral sequences r r allow us to give isomorphisms between the Dolbeault cohomologies of the total spaces and the one of the correspondingLie algebras. The laststep gives Theorem B. Note that our construction of this set of holomorphic fibrations is in the same vein as principal holomorphic torus towers, introduced by Barth and Otte [4]. In some cases, like the one of abelian complex structures, G/Γ is really a principal holomorphic torus tower. In Section 7 we will give a proof of Theorem A. InSection8wegiveexamplesofcompactnilmanifoldswithnonrationalcomplex structures. We wish to thank Prof. Simon Salamon for the great deal of suggestions he gave us and his constant encouragement. We are also grateful to Prof. Isabel Dotti for useful conversationsand the hospitality at the FaMAF of C´ordoba (Argentina). 2. A descending series associated to the complex structure We recall that, since G is (s+1)-step nilpotent, one has the descending central series {gi} , where i≥0 g=g0 ⊇g1 =[g,g]⊇g2 =[g1,g]⊇...⊇gs ⊇gs+1 ={0}. (D) 4 SERGIO CONSOLE AND ANNA FINO We define the following subspaces of g gi :=gi+Jgi. J Note that gi is J -invariant. J Lemma 1. (1) gi is an ideal of gi−1. J J (2) gi−1/gi is an abelian algebra. J J (3) gs is an abelian ideal of gs−1. J J Proof. (1) For any X = X +JX ∈ gi−1 and Y = Y +JY ∈ gi (with X ∈ gi−1 and 1 2 J 1 2 J l Y ∈gi), we have that k [X,Y]=[X ,Y ]+[X ,JY ]+[JX ,Y ]+[JX ,JY ]. 1 1 1 2 2 1 2 2 Wecaneasilyseethat[X ,Y ],[X ,JY ]and[JX ,Y ]belongtogi bydefinitionof 1 1 1 2 2 1 thedescendingcentralseries. Moreover[JX ,JY ]belongstogi becauseJ satisfies 2 2 J an integrability condition, namely the Nijenhuis tensor N of J, given by N(Z,W)=[Z,W]+J[JZ,W]+J[Z,JW]−[JZ,JW], Z,W ∈g, must be zero [20]. (2) For any X =X +JX , Y =Y +JY elements of gi−1 we have that 1 2 1 2 J [X +gi,Y +gi]=[X ,Y ]+[X ,JY ]+[JX ,Y ]+[JX ,JY ]+gi. J J 1 1 1 2 2 1 2 2 J Then using the same argument as in (1) it follows that [X +gi,Y +gi]=gi. J J J (3) Using the fact that gs is central (i.e. [gs,g]= 0) and that N = 0 it is possible to prove that [X,Y] vanishes for any X,Y ∈gs. J Observe moreoverthat any gi is nilpotent. J Hence we have the descending series g=g0 ⊃g1 ⊇g2 ⊇...⊇gs ⊇gs+1 ={0}. (DJ) J J J J J Remark 1. The first inclusion g1 ⊂g is always strict [24, Corollary 1.4]. J Observe also that in case of complex parallelisable nilmanifolds [25], the filtration {gi}coincideswiththedescendingcentralseries{gi}andthenthe{gi}arerational. j j In general, given a rational structure g for g, we say that a R-subspace h of g is Q rationalifhistheR-spanofh =h∩g . Ingeneralg1 isnotarationalsubalgebraof Q Q J g. When J isrational,itis possibleto provethatgi isrationalingi−1. Indeed, we J J havethat gi is rationalin gi−1. Then gi =R−span{gi∩gi−1}. Since Jgi−1 ⊆gi−1 Q Q Q itfollowsthatgi =R−span{gi ∩gi−1}. MoreoverwhenJ isabelian,gi isanideal J J Q J of g, for any i and the center g ={X ∈g|[X,g]=0} 1 is a rational J-invariant ideal of g. DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 5 3. Holomorphic fibrations and Borel spectral sequences In this section we suppose that the complex structure J is rational and we associate a set of holomorphic fibrations to the above descending series. We recall that a holomorphic fibre bundle π : T → B is a a holomorphic map between the complex manifolds T and B, which is locally trivial, whose typical fibre F is a complex manifold and such that the transition functions are holomorphic. By definitionthe structuregroup(i.e. the groupofholomorphicautomorphismsofthe typical fibre) is a complex Lie group. To define the above fibrations, we consider first the surjective homomorphism p :gi−1 →gi−1/gi, i−1 J J J for each i = 1,... ,s. If Gi−1 and Gi−1,i denote the simply connected nilpotent J J Lie group corresponding to gi−1 and gi−1/gi respectively, we have the surjective J J J homomorphism p :Gi−1 →Gi−1,i. i−1 J J We define inductively Gi to be the fibre of p . Remark that the Lie algebra of J i−1 Gi is gi. J J Given the uniform discrete subgroup Γ of G = G0, we consider the continuous surjective map p˜ :G/Γ→G0,1/p (Γ). 0 J 0 Since J is rational,g1 is a rationalsubalgebraof g, then Γ1 :=Γ∩G1 is a uniform J J discrete subgroupof G1 [7, Theorem 5.1.11]. Then, by [7, Lemma 5.1.4 (a)], p (Γ) J 0 is a a uniform discrete subgroup of G0,1 (i.e. G0,1/p (Γ) is compact, cf. [23]). J J 0 NotemoreoverthatG1 issimplyconnected. Thisfollowsfromthe homotopyexact J sequence of the fibering p . Indeed we have 0 ...→π (G0,1)=(e)→π (G1)→π (G)=(e)→... 2 J 1 J 1 Finally it is not difficult to see that G1 is connected. Indeed, if C is the connected J component of the identity in G1, id: G → G induces a covering homomorphism J G/C →G/G1 ∼=G0,1 whichmustbetheidentity,sinceG0,1 ∼=RN0. ThusC =G1. J J J J Now one can repeat the same construction for any i, since gi is a rational ideal J of gi−1. So, for any i=1,... ,s we have a map J p˜ :Gi−1/Γi−1 →Gi−1,i/p (Γi−1). i−1 J J i−1 Lemma 2. p˜ : Gi−1/Γi−1 → Gi−1,i/p (Γi−1) is a holomorphic fibre bundle. i−1 J J i−1 Proof. Observefirstthatp˜ istheinducedmapofp takingquotientsofdiscrete i−1 i−1 subgroups. The tangent map of p˜ i−1 gi−1 →gi−1/gi J J J is J-invariant. Thus p˜ is a holomorphic submersion. In particular it is a holo- i−1 morphic family of compact complex manifolds in the terminology of [13] (see also [27]). The fibres of p˜ are all holomorphically equivalent to Gi/Γi (the typical i−1 J fibre). Thus a theorem of Grauert and Fisher [9] applies, implying that p˜ is a i−1 holomorphic fibre bundle. Note that Gi−1/Γi−1, Gi/Γi, Gi−1,i/p (Γi−1) are compact connected nilmani- J J J i−1 folds. 6 SERGIO CONSOLE AND ANNA FINO Givenaholomorphicfibrebundle itis possibleto constructthe associatedBorel spectralsequence,that relates the Dolbeault cohomologyof the totalspace T with that of the basis B and of the fibre F. We will need the following Theorem(which follows from [12, Appendix II by A. Borel, Theorem 2.1] and [10]). Theorem3. Letp:T →Bbeaholomorphicfibrebundle,withcompactconnected fibre F and T and B connected. Assume that either (I) F is K¨ahler or (I’) the scalar cohomology bundle Hu,v(F)= Hu,v(p−1(b)) [ ∂ b∈B is trivial. Then there exists a spectral sequence (E ,d ), (r ≥0) with the following prop- r r erties: (i) E is 4-gradedby the fibre degree,the basedegree andthe type. Letp,qEu,v be r r the subspace of elements of E of type (p,q), fibre degree u and base degree v. We r have p,qEu,v = 0 if p+q 6= u+v or if one of p,q,u,v is negative. The differential r d maps p,qEu,v into p,q+1Eu+r,v−r+1. r r r (ii) If p+q =u+v p,qEu,v ∼= Hk,u−k(B)⊗Hp−k,q−u+k(F). 2 X ∂ ∂ k (iii) The spectral sequence converges to H (T). ∂ 4. An adapted basis of (1,0)-forms In this section we prove that one can choose a basis of (1,0)-forms on g which is compatible with the descending series (DJ). We consider, like in [24], some subspacesV (i=0,... ,s+1)ofV :=(T G)∗ ∼=g∗, thatdetermine a series,which i e is related to the descending central series (D). Indeed we define: V ={0} 0 V ={α∈V |dα=0} 1 ... V ={α∈V |dα∈Λ2V } i i−1 ... V =V. s+1 Note that V is the annihilator (gi)o of the subspace gi and that {0}=V ⊆ V ⊆ i 0 1 ...⊆V =V [24, Lemma 1.1]. s+1 If we now let (gi)0∩λ1,0 =:V1,0, by [24, Lemma 1.2] we have that there exists a J i basisof(1,0)-forms{ω ,...ω }suchthatifω ∈V1,0 thendω belongsto the ideal 1 n l i l (in (gC)∗) generated by V1,0 [24, Theorem 1.3]. In particular there exists at least i−1 a closed (1,0)-form (this implies Remark 1). Moreoverwe have the following isomorphisms: gi−1/gi C ∼=V1,0/V1,0 ⊕V0,1/V0,1, i=1,... ,s, (cid:0) J J(cid:1) i i−1 i i−1 where V0,1 is the conjugate of V1,0. i i DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 7 With respect to the subspaces V1,0 the above basis can be ordered as follows (we i let n :=dim gi): i C J ω ,... ,ω are elements of V1,0 (such that dω = 0) or g/g1 is the real vector 1 n−n1 1 l J space underlying V1,0; 1 ω ,... ,ω are elements of V1,0\V1,0 or g1/g2 is the real vector space n−n1+1 n−n2 2 1 J J underlying the quotient V1,0/V1,0; 2 1 ... ω ,... ,ω are elements of V1,0\V1,0; n−ns−1+1 n−ns s s−1 ω ,... ,ω are elements of λ1,0\V1,0. n−ns+1 n s Here V1,0\V1,0 denotes a complement of V1,0 in V1,0 (which corresponds to the i i−1 i−1 i choice of a complement of gi in gi−1). J J Hence,bydefinition, the elementsofV1,0\V1,0 and,by identification,the elements i i−1 of the quotient V1,0/V1,0, are (1,0)-forms on g which vanish on gi. So they may i i−1 J be identified with forms on the quotient gi−1/gi. J J In this way we can consider: -theelementsofλ1,0/V1,0 =V1,0/V1,0⊕V1,0/V1,0⊕...⊕λ /V1,0 as(1,0)-forms 1 2 1 3 2 1,0 s on g1, J ... - the elements of λ1,0/V1,0 =V1,0/V1,0 ⊕λ1,0/V1,0 as (1,0)-forms on gs−1 and s−1 s s−1 s J - the elements of λ1,0/V1,0 as (1,0)-forms on gs. s J Thus we canprovea Lemma onthe existence of a basis of(1,0)forms on g related to the series (DJ). Lemma 4. It is possible to choose a basis of (1,0)-forms on g such that (with respect to the order of before) {ω ,...ω ,...ω } is a basis of (1,0)- n−ni−1+1 n−ni n forms on gi−1. Moreover we can consider (up to identifications) {ω ,...ω } J n−ni+1 n as forms on gi and {ω ,...ω } as forms on gi−1/gi. J n−ni−1+1 n−ni J J Proof. By the above arguments, for any i = 1,... ,s+1, it is possible to choose a basis of (1,0)-forms on gi−1 as elements of λ1,0/V1,0 = λ1,0/V1,0 ⊕V1,0/V1,0. J i−1 i i i−1 With respect to the above decomposition the forms on λ1,0/V1,0 can be identified i with forms ongi extended by zero ongi−1 and the forms on V1,0/V1,0 with forms J J i i−1 on gi−1/gi, because these forms vanish on gi. J J J Remark 2. dω , i = n−n +1,... ,n−n belongs to the ideal generated by i i−1 i {ω, l =1,... ,n−n }. l i−1 Remark3. IfJ isabelianitispossibletochooseabasisof(1,0)-forms{ω ,... ,ω } 1 n on g such that dω ∈∧2hω ,... ,ω ,ω ,... ,ω i∩ λ1,1. i 1 i−1 1 i−1 5. A spectral sequence for the complex of invariant forms We construct a spectral sequence p,qE˜u,v for the complexes of Gs−1-invariant r J forms on Gs−1/Γs−1 whose Dolbeault cohomology identifies with H ((gs−1)C). To J ∂ J do this, we give a filtration of the complex Λ =⊕Λp,q of differential forms of type t t (p,q) on t=gi−1. J We know from Lemma 4 that there exists a basis of (1,0)-forms ωt on t (and of h (0,1)-formsωt)suchthatpartofthemare(1,0)-formsωb onb=gi−1/gi andpart h j J J 8 SERGIO CONSOLE AND ANNA FINO are forms ωf on f=gi. We define k J L˜ := {ωt ∈Λ |ωt is a sum of monomials k t ωb∧ωb ∧ωf ∧ωf in which |I|+|J|≥k}, I J I′ J′ where |A| denotes the number of elements of the finite set A. Note that L˜ =Λ and that 0 t L˜ =0 for k >dim b, k R L˜ ⊃L˜ , ∂L˜ ⊆L˜ , k ≥0 k k+1 k k Theaboveshowsthat{L˜ }definesaboundeddecreasingfiltrationofthedifferential k module (Λ ,∂). t Of course L˜ = p,qL˜ , where p,qL˜ =L˜ ∩Λp,q k X k k k t p,q and the filtration is compatible with the bigrading provided by the type (and also with the total degree). Recall that, by definition (see e.g. [11]) p,qZu,v p,qE˜u,v = r , r p,qZu+1,v−1+p,qBu,v r−1 r−1 where p,qZu,v =p,qL˜ (Λu+v)∩ker∂(p,q+1L˜ (Λu+v+1)) r u t u+r t p,qBu,v =p,qL˜ (Λu+v)∩∂(p,q−1L˜ (Λu+v−1)) r u t u+r t Moreover(cf. [11]) p,qL˜ p,qE˜u = u , 0 p,qL˜ u+1 wherewedenotebyp,qE˜u,v andp,qE˜u thespacesofelementsoftype(p,q)andtotal r r degree u+v and degree u respectively in the grading defined by the filtration. Note also that an element of L˜k identifies with an element of Λa,b⊗Λc,d. Pc+d≥k f b Lemma 5. Given the holomorphic fibration Gi−1/Γi−1 →Gi−1,i/p (Γi−1), J J i−1 with standard fibre Gi/Γi, the spectral sequence (E˜ ,d˜) (r ≥ 0) converges to J r r H ((gi−1)C) and ∂ J p,qE˜u,v ∼= Hk,u−k((gi−1/gi)C)⊗Hp−k,q−u+k((gi)C). (I˜i) 2 X ∂ J J ∂ J k Proof. The fact that (E˜ ,d˜) converges to H ((gi−1)C) is a general property of r r ∂ J spectral sequences associated to filtered complexes (cf. [11]). Let [ω] ∈ p,qE˜u = p,qL˜u . We compute the differential d˜ : p,qE˜u → p,q+1E˜u 0 p,qL˜u+1 0 0 0 defined by d˜[ω] = [∂ω]. We can write (up the above identifications) ω = ωb∧ 0 I ωb∧ωf ∧ωf where|I|+|J|=u(becauseweoperate mod L˜ ),and|I′|+P|J′|= J I′ J′ u+1 p+q−u, since |I′|+|J′|+|I|+|J|=p+q. Moreover,using the fact that ∂ sends forms in b on forms that either are in b or vanish on b (cf. Remark 2) and that b is abelian, we get ∂ω = ∂ (ωb∧ωb)∧(ωf ∧ωf )+(−1)s(ωb∧ωb)∧∂(ωf ∧ωf )= = P(−1)bs(ωIb∧ωJb)∧∂I(′ωf ∧Jω′ f ) mod L˜I ,J I′ J′ I J f I′ J′ u+1 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 9 where ∂ and ∂ denote the differential on the complexes Λ and Λ , respectively. b f b f Thus d˜[ω]=[∂ ω], 0 f which implies p,qE˜u ∼= Hp−k,q−u+k(fC)⊗Λk,u−k. (k1) 1 X ∂ b k Performing the same proof as in [12, Appendix II by Borel, Section 6] and using the fact that b is abelian, it is possible to prove that d˜ identifies with ∂ via the 1 above isomorphism and that we have p,qE˜u ∼= Hk,u−k(bC)⊗Hp−k,q−u+k(fC)(∼= Λk,u−k⊗Hp−k,q−u+k(fC)). (k2) 2 X ∂ ∂ X b ∂ k k 6. Proof of Theorem B First we note that Theorem B is trivially true if the nilmanifold comes from an abelian group, i.e., it is a complex torus. Namely, if A/Γ is a complex torus we have H (A/Γ)∼=H (aC). (a) ∂ ∂ We consider the holomorphic fibrations Gs/Γs ֒→Gs−1/Γs−1 →Gs−1,s/p (Γs−1) J J J s−1 ... G1/Γ1 ֒→G/Γ→G0,1/p (Γ). J J 0 The aim is to obtain informations about the Dolbeault cohomology of G/Γ induc- tively through the Dolbeault cohomologies of Gi/Γi (the nilmanifolds Gi/Γi play J J alternately the rˆoles of fibres and total spaces of the above fibre bundles). Note that since the bases are complex tori, H∂(GJi,i−1/pi−1(Γi−1)∼=H∂((gi−1/gi)C). To this purpose we will associate to these fibrations two spectral sequences. Thefirstis aversionofthe Borelspectralsequence(consideredinSection3)which relatestheDolbeaultcohomologiesofthetotalspaceswiththoseoffibresandbases. The second is the spectral sequence (E˜ ,d˜) (constructed in the previous Section) r r relative to the Dolbeault cohomologies of the Lie algebras gi,gi−1,gi−1/gi. Wewillproceedinductivelyontheindexiinthedescendingseries(DJ),starting from i=s. First inductive step. Let us use the holomorphic fibre bundle p˜ :Gs−1/Γs−1 →Gs−1,s/p (Γs−1) s−1 J J s−1 with typical fibre Gs/Γs. J Recall (cf. Lemma 1) that Gs/Γs and Gs−1,s/p (Γs−1) are complex tori. Thus J J s−1 by (a) H (Gs/Γs)∼=H ((gs)C) (s) ∂ J ∂ J H∂(GsJ−1,s/ps−1(Γs−1))∼=H∂((gJs−1/gsJ)C). (s,s−1) Applying Theorem3 (since the fibreGs/Γs is K¨ahler)andusing (s) and (s,s−1), J we get p,qEu,v ∼= Hk,u−k((gs−1/gs)C)⊗Hp−k,q−u+k((gs)C). (Is) 2 X ∂ J J ∂ J k 10 SERGIO CONSOLE AND ANNA FINO Next we use the spectral sequence (E˜ ,d˜). Note that the inclusion between r r the Dolbeault complex of Gs−1-invariant forms on Gs−1/Γs−1 and the forms on J J Gs−1/Γs−1, induces an inclusion of each term in the spectral sequences J p,qE˜u,v ⊆p,q Eu,v, r r (which is actually a morphism of spectral sequences). By Lemma 5, for i=s, we have that (E˜ ,d˜) converges to H ((gs−1)C) and r r ∂ J p,qE˜u,v ∼= Hk,u−k((gs−1/gs)C)⊗Hp−k,q−u+k((gs)C). (I˜s) 2 X ∂ J J ∂ J k Comparing (Is) with (I˜s), we get that E = E˜ , hence the spectral sequences 2 2 (E ,d ) and (E˜ ,d˜) converge to the same cohomologies. Thus r r r r H (Gs−1/Γs−1)∼=H ((gs−1)C). (s−1) ∂ J ∂ J General inductive step. We use the holomorphic fibre bundle p˜ :Gi−1/Γi−1 →Gi−1,i/p (Γi−1) i−1 J J i−1 with typical fibre Gi/Γi. We assume inductively that J H (Gi/Γi)∼=H ((gi)C). (i) ∂ J ∂ J Lemma 6. The scalar cohomology bundle Hu,v(Gi/Γi)= Hu,v(p˜−1 (b)) J [ ∂ i−1 b∈GiJ−1,i/pi−1(Γi) is trivial. Proof. By [13, Section 5, formula 5.3] there exists a locally finite covering {U} of l Gi−1,i/p (Γi−1) such that the action of the structure group of the holomorphic J i−1 fiberbundleonU ∩(Gi/Γi)isthedifferentialofthechangeofcomplexcoordinates l J onthefibre,soonecanrestrictoneselftoconsiderthelefttranslationbyelementsof Gi−1 aschangeofcoordinates. Thenthe scalarcohomologybundleHu,v(Gi/Γi)is J J trivialsinceanyofitsfibresiscanonicallyisomorphictoH ((gi)C). Moreexplicitly, ∂ J a global frame for Hu,v(Gi/Γi) is given as follows: for any cohomology class J α∈Hu,v(p˜−1 (1))=Hu,v(Gi/Γi)∼=Hu,v((gi)C), ∂ i−1 ∂ J ∂ J (1: identity element of Gi−1,i/p (Γi)) one can take the corresponding cohomol- J i−1 ogy class ω ∈ Hu,v((gi)C) and regard it as a Gi−1-invariant differential form on ∂ J J GiJ−1/Γi−1. Thus b7→ω|p˜−i−11(b) gives a globalholomorphic section of Hu,v(GiJ/Γi). Taking a basis of H ((gi)C) one gets a global holomorphic frame of Hu,v(Gi/Γi). ∂ J J Thus the assumption (I’) in Theorem 3 is fulfilled. Observe moreoverthat, since Gi−1,i/p (Γi−1) is a complex torus, by (a), J i−1 H∂(GJi−1,i/pi−1(Γi−1))∼=H∂((giJ−1/giJ)C). (i,i−1) Hence, by Theorem 3, we have p,qEu,v ∼= Hk,u−k((gi−1/gi)C)⊗Hp−k,q−u+k((gi)C). (Ii) 2 X ∂ J J ∂ J k