11 S. CONSOLE AND A. FINO compact symplectic manifolds which do not admit any Kahler structure [Ab, CFG, Th]. Since there exists a natural map i : H (

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1 Transformation Groups, Vol. 6, No., 001, pp. 111{14 cbirkhauser Boston (001) DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS S. CONSOLE Dipartimento di Matematica Universita ditorino Via Carlo Alberto 10 I-1013 Torino, Italy sergio.consoleunito.it A. FINO Dipartimento di Matematica Universita ditorino Via Carlo Alberto 10 I-1013 Torino, Italy anna.nounito.it Abstract. Let M = G=; be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component ofthe moduli space C(g) of invariant complex structures on M, the Dolbeault cohomology of M is isomorphic to the cohomology of the dierential bigraded algebra associated to the complexication g C of the Lie algebra of G. To obtain this result, we rst prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic brations. Then we apply the theory of Kodaira{Spencer for deformations of complex structures. Introduction Let M be a compact nilmanifold of real dimension n, i.e., the quotient of a connected simply connected (s + 1)-step nilpotent real Lie group G by a uniform discrete subgroup ;. It follows from a result of Mal'cev [Ma] that, for any simply connected real nilpotent Lie group G, for which the coecients in the structure equations are rational numbers, there is a lattice ; in G of maximal rank (i.e., a discrete uniform subgroup, cf. [Ra]). We will let ; act on G on the left. It is well known that such a lattice ; exists in G if and only if the Lie algebra g of G has a rational structure, i.e., if there exists a rational Lie subalgebra g Q such that g = gq R. The de Rham cohomology of a compact nilmanifold can be computed by means of the cohomology of the Lie algebra of the corresponding nilpotent Lie group (Nomizu's Theorem [No]). We assume that M has an invariant complex structure, that is that comes from a (left invariant) complex structure on g. Our aim is to relate the Dolbeault cohomology of M with the cohomology ring H (gc ) of the dierential bigraded algebra (g C ), associated to g C with respect to the operator in the canonical decomposition d = + on (g C ). The study of the Dolbeault cohomology of nilmanifolds with an invariant complex structure is motivated by the fact that the latter provided the rst known examples of Research partially supported by MURST and CNR of Italy. Research partially supported by MURST and CNR of Italy. Received September 3, Accepted anuary 17, 000.

2 11 S. CONSOLE AND A. FINO compact symplectic manifolds which do not admit any Kahler structure [Ab, CFG, Th]. Since there exists a natural map i : H (gc )! H (M) thatisalways injective (cf. Lemma 9), the problem we will study is to see for which complex structure on M the above map gives an isomorphism H p q (M) = H p q (gc ): (1) Note that H (gc ) can be identied with the cohomology of the Dolbeault complex of the forms on G that are invariant by the left action of G (we shall call them briey G-invariant forms) and H (M) with the cohomology of the Dolbeault complex of ;-invariant forms on G. We shall use these identications throughout. Our main result is the following theorem. Theorem 1. The isomorphism (1) holds on an open set of any connected component of the moduli space C(g) of invariant complex structures on M. To obtain Theorem 1 we rst consider the case of complex structures that are rational, i.e., they are compatible with the rational structure of G ((g Q ) g Q ). Theorem. For any rational complex structure, the isomorphism (1) holds. It is an open problem as to whether the isomorphism (1) holds for any compact nilmanifold endowed with an arbitrary invariant complex structure. We donotknow examples for which (1) does not hold. Theorem 1 will follow from Theorem using the theory of deformations of complex structures [KS, Su]. Indeed by [Sa] the set C(g) of complex structures on g is at least in- nitesimally 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, the isomorphism (1) holds (Lemma 10). If M is a compact complex parallelizable nilmanifold, i.e., G is a nilpotent complex Lie group and is also right invariant, we have that i g = gi and Theorem follows from [Sak, Theorem 1]. An important class of complex structures is given by the abelian ones (i.e., those satisfying the condition [X Y]=[X Y ], for any X Y g [BD, DF]). The nilmanifolds with an abelian complex structure are to some extent dual to complex parallelizable nilmanifolds: indeed, in the complex parallelizable case d and in the abelian case d where p q denotes the space of (p q)-forms on g. In this last case we will compute the minimal model of the Dolbeault cohomology of M and prove that the isomorphism (1) holds for any abelian complex structure. In [CFGU] a similar result was proved for the Dolbeault cohomology of M endowed with a nilpotent complex structure this is a slight generalization of the abelian one. Note that, if M is a complex solvmanifold G=; (with G asolvable not nilpotent Lie group), the isomorphism (1) does not hold in general, as shown in [Le] a discussion of the behavior of the Dolbeault cohomology of homogeneous manifolds under group actions can be found in [Ak]. If G is a compact even dimensional (semisimple) Lie group endowed with a left invariant complex structure, the Dolbeault cohomology of G does not arise from just invariant classes, as the example in [Pi] shows. This paper is organized as follows: In Section 1, following [Sa], we dene a descending series of subalgebras i fgg (with g 0 = g and gs+1 = f0g) for the Lie algebra g associated

3 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 113 to the complex structure. In general, the subalgebra i g is not a rational subalgebra of i;1 g. If is rational, any i g is rational in gi;1. The importance of this series is twofold. First (Section ), in case the subalgebra i g is rational in i;1 g (in particular if is rational), it allows us to dene a set of holomorphic brations of nilmanifolds: ~p 0 : M = G=;! G 0 1 =p 0(;) with standard bre G 1 =;1 ::: ~p s;1 : G s;1 =; s;1! G s;1 s =p s;1 (; s;1 ) with standard bre G s =;s : For the above brations we will consider the associated Borel spectral sequence (E r d r ) ([Hi, Appendix II by A. Borel, Theorem.1], [FW, LeP]), which relates the Dolbeault cohomology of each total space with the Dolbeault cohomology of each base and bre. Second (Section 3), following [Sa], 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 descending series. This basis will give a basis of G i;1 i -invariant (1 0)-forms on the nilmanifolds G i;1 i =p i;1 (; i;1 ) and of G i -invariant (1 0)-forms on the nilmanifolds Gi =;i, i =0 ::: s. Next, in Section 4, we consider a spectral sequence ( Er ~ d ~ r ) concerning the G i;1 - invariant Dolbeault cohomology of each total space G i;1 =; i;1, the G i;1 i -invariant Dolbeault cohomology of each baseg i;1 i =p i;1 (; i;1 ) and the G i -invariant Dolbeault cohomology of each bre G i =;i. In this way (~ Er d ~ r ) is relative tothedolbeault cohomologies of the Lie algebras i g, gi;1 and i;1 g =g i. Note that the latter are the underlying Lie algebras of the bre, the total space and the base, respectively, ofthe above holomorphic brations. In Section 5we compare the spectral sequence ( Er ~ d ~ r ) with the Borel spectral sequence (E r d r ). Inductively (starting with i = s) these two spectral sequences allow us to give isomorphisms between the Dolbeault cohomologies of the total spaces and the cohomology of the corresponding Lie algebras. The last step gives Theorem. Note that our construction of this set of holomorphic brations is in the same vein as principal holomorphic torus towers, introduced by Barth and Otte [BO]. In some cases, like the case of abelian complex structures, G=; is really a principal holomorphic torus tower. In Section 6 we will give a proof of Theorem 1. In Section 7 we give examples of compact nilmanifolds with non-rational complex structures. We wish to thank Simon Salamon for the many suggestions he gave us and his constant encouragement. We are also grateful to Isabel Dotti for useful conversations and the hospitality at the FaMAF of Cordoba (Argentina) and to the referee for useful comments. 1. A descending series associated to the complex structure We recall that, since G is (s + 1)-step nilpotent, one has the descending central series fg i g i0,where g = g 0 g 1 =[g g] g =[g 1 g] ::: g s g s+1 = f0g: (D)

4 114 S. CONSOLE AND A. FINO We dene the following subspaces of g g i := g i + g i : Note that g i is -invariant. Lemma 3. (1) i g is an ideal of gi;1 () i;1 g =g i is an abelian algebra. (3) s g is an abelian ideal of gs;1.. Proof. (1) For any X = X 1 + X g i;1 and Y = Y 1 + Y i g (with X l g i;1 and Y k i g ), we have that [X Y ] = [X 1 Y 1 ]+[X 1 Y ]+[X Y 1 ]+[X Y ]: We can easily see that [X 1 Y 1 ], [X 1 Y ] and [X Y 1 ] belong to i g by denition of the descending central series. Moreover [X Y ] belongs to i g because satises an integrability condition, namely, the Nijenhuis tensor N of, given by N(Z W )= [Z W ]+[Z W]+[Z W ] ; [Z W] Z W g must be zero [NN]. () For any X =X 1 +X, Y =Y 1 +Y elements of i;1 g,wehave that i [X+g Y+gi ]= [X 1 Y 1 ]+[X 1 Y ]+[X Y 1 ]+[X Y i ]+g: Then using the same argument asin (1) it follows that [X + i g Y + gi ]= gi : (3) Using the fact that g s is central (i.e., [g s g] = 0) and that N = 0 it is possible to prove that[x Y ]vanishes for any X Y g s. Observe moreover that any i g is nilpotent. Hence we have the descending series g = g 0 g 1 g ::: g s g s+1 = f0g: (D) Remark 1. The rst inclusion g 1 g is always strict [Sa, Corollary 1.4]. Observe that in case of complex parallelizable nilmanifolds [Sak], the ltration i fgg coincides with the descending central series i fg g and then the i fgg are rational. In general, given a rational structure g Q for g, wesay that a R-subspace h of g is rational if h is the R-span of h Q = h \ g Q. In general g 1 is not a rational subalgebra of g. When is rational, it is possible to prove that i g is rational in gi;1. Indeed, we have that i g is rational in g i;1. Then i g = i R;spanfg \ i;1 gq g. Since i;1 gq i;1 gq,itfollows that i g = R;spanfgi \ g i;1 Q g. Moreover when is abelian, i g is an ideal of g, for any i and the center g1 = fx g j [X g] =0g is a rational -invariant ideal of g.. Holomorphic brations and Borel spectral sequences In this section we assume that the complex structure is rational and we associate a set of holomorphic brations to the above descending series. We recall that a holomorphic bre bundle : T! B is a a holomorphic map between the complex manifolds T and B, which is locally trivial, and whose typical bre F is a complex manifold such that the transition functions are holomorphic. By denition the structure group (i.e., the group of holomorphic automorphisms of the typical bre) is a complex Lie group.

5 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 115 To dene the above brations, we rst consider the surjective homomorphism p i;1 : g i;1! i;1 g =g i for each i =1 ::: s. If G i;1 and G i;1 i denote the simply connected nilpotent Lie group corresponding to g i;1 respectively, wehave the surjective homomorphism and g i;1 =g i p i;1 : G i;1! G i;1 i : We dene inductively G i to be the bre of p i;1. Remark that the Lie algebra of G i is g i. Given the uniform discrete subgroup ; of G = G 0,we consider the continuous surjective map ~p 0 : G=;! G 0 1 =p 0(;): Since is rational, g 1 is a rational subalgebra of g, and then ;1 := ; \ G 1 is a uniform discrete subgroup of G 1 [CG, Theorem ]. Then, by [CG,Lemma (a)], p 0 (;) is a a uniform discrete subgroup of G 0 1 (i.e., G 0 1 =p 0(;) is compact, cf. [Ra]). Note moreover that G 1 is simply connected. This follows from the homotopy exact sequence of the bering p 0. Indeed we have :::! (G 0 1 )=(e)! 1(G 1 )! 1 (G) = (e)! :::: Finally it is not dicult to see that G 1 is connected. Indeed, if C is the connected component of the identity in G 1, id: G! G induces a covering homomorphism G=C! G=G 1 = G 0 1 which must be the identity, since G 0 1 = R N0. Thus C = G 1. Now one can repeat the same construction for any i, since i g is a rational ideal of. So, for any i =1 ::: s we have a map g i;1 Lemma 4. ~p i;1 : G i;1 ~p i;1 : G i;1 =; i;1! G i;1 i =p i;1 (; i;1 ): =; i;1! G i;1 i =p i;1 (; i;1 ) is a holomorphic bre bundle. Proof. Observe rst that ~p i;1 is the induced map of p i;1 taking quotients of discrete subgroups. The tangent mapof~p i;1 i;1 g! i;1 g =g i is -invariant. Thus ~p i;1 is a holomorphic submersion. In particular, it is a holomorphic family of compact complex manifolds in the terminology of [KS] (see also [Sun]). The bres of ~p i;1 are all holomorphically equivalent to G i =;i (the typical bre). Thus a theorem of Grauert and Fisher [FG] applies, implying that ~p i;1 is a holomorphic bre bundle. Note that G i;1 =; i;1, G i =;i, G i;1 i =p i;1 (; i;1 ) are compact connected nilmanifolds. Given a holomorphic bre bundle it is possible to construct the associated Borel spectral sequence that relates the Dolbeault cohomology of the total space T with that of the basis B and the bre F. We will need the following Theorem (which follows from [Hi, Appendix II by A. Borel, Theorem.1] and [FW]).

6 116 S. CONSOLE AND A. FINO Theorem 5. Let p : T! B be a holomorphic bre bundle with compact connected bre F and T and B connected. S Assume that either (I) F is Kahler or (I 0 ) the scalar cohomology bundle H u v (F )= bb Hu v (p ;1 (b)) is trivial. Then there exists a spectral sequence (E r d r )(r0) with the following properties: (i) E r is 4-graded by the bre degree, the base degree, and the type. Let p q Er u v be the subspace of elements of E r of type (p q), bredegree u, and base degree v. We have p q Er u v =0if p + q 6= u + v or if any p q u v is negative. The dierential d r maps p q Er u v into p q+1 Er u+r v;r+1. (ii) If p + q = u + v, then p q E u v = P k Hk u;k (B) H p;k q;u+k (F ): (iii) The spectral sequence converges to H (T ). 3. An adapted basis of (1 0)-forms In this section we prove that one can choose a basis of (1,0)-forms on g that is compatible with the descending series (D). We consider, as in [Sa], some subspaces V i (i =0 ::: s+1)ofv := (T e G) = g, that determine a series, which is related to the descending central series (D). Indeed, we dene V 0 = f0g V 1 = f V j d =0g ::: V i = f V j d V i;1 g ::: V s+1 = V: Note that V i is the annihilator i (g ) o of the subspace i g and that f0g = V 0 V 1 ::: V s+1 = V [Sa, Lemma 1.1]. If we now let i (g )o \ 1 0 =: V 1 0 i,by [Sa, Lemma 1.], we have that there exists a basis of (1,0)-forms f! 1 :::! n g such that if! l V 1 0 i,thend! l belongs to the ideal (in C (g ) ) generated by V 1 0 i;1 [Sa, Theorem 1.3]. In particular, there exists at least a closed (1 0)-form (this implies Remark 1). ; C Moreover we have the following isomorphisms: i;1 g =g i = V 1 0 i =V 1 0 i;1 V 0 1 i =V 0 1 i;1 i =1 ::: s where V 0 1 is the conjugate of V 1 0. With respect to the subspaces V 1 0 i the above basis can be ordered as follows (we let n i := dimc gi ):! 1 :::! n;n1 are elements of V (such thatd! l =0)or g=g is the real vector space underlying V1! n;n1+1 :::! n;n are elements of V 1 0 nv or g 1 =g is the real vector space underlying the quotient V 1 0 =V :::! n;ns;1+1 :::! n;ns are elements of Vs 1 0 nv 1 0 s;1! n;ns+1 :::! n are elements of 1 0 nvs 1 0. Here V 1 0 i nv 1 0 i;1 denotes a complement of V i;1 in Vi (which corresponds to the choice of a complement of i g in gi;1 ). Hence, by denition, the elements of V 1 0 i nv 1 0 i;1 and, by identication, the elements of the quotient V 1 0 i =V 1 0, are (1,0)-forms on i;1 g which vanish on gi. So they may be identied with forms on the quotient g i;1 =g i. In this way we can consider: the elements of =V 1 =V =V 1 V 3 =V ::: 1 0 =V s as (1 0)-forms on g 1,... the elements of 1 0 =V 1 0 = V 1 0 s;1 s =V 1 0 s;1 1 0 =Vs 1 0 as (1 0)-forms on s;1 g and the elements of 1 0 =Vs 1 0 as (1,0)-forms on s g. Thus we can prove a lemma on the existence of a basis of (1 0) forms on g related to the series (D). i i

7 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 117 Lemma 6. It is possible to choose abasis of (1 0)-forms on g such that (with respect to the previous order) f! n;ni;1+1 :::! n;ni :::! n g is a basis of (1 0)-forms on g i;1. Moreover we can consider (up to identications) f! n;ni+1 :::! n g as forms on g i and f! n;ni;1+1 :::! n;ni g as forms on g i;1 =g i. Proof. By the above arguments, for any i =1 ::: s+ 1, it is possible to choose a basis of (1,0)-forms on i;1 g as elements of 1 0 =V 1 0 = i;1 1 0 =V 1 0 i V 1 0 i =V 1 0 i;1. With respect to the above decomposition the forms on 1 0 =V 1 0 i can be identied with forms on i g extended by zero on i;1 g and the forms on V 1 0 i =V 1 0 i;1 with forms on gi;1=g i, because these forms vanish on i g. Remark. d! i, i = n ; n i;1 +1 ::: n; n i belongs to the ideal generated by f! l l = 1 ::: n; n i;1 g. Remark 3. If is abelian, it is possible to choose a basis of (1 0)-forms f! 1 :::! n g on g such that d! i ^ h! 1 :::! i;1! 1 :::! i;1 i\ 1 1 : 4. A spectral sequence for the complex of invariant forms We construct a spectral sequence p q E ~ u v r for the complexes of G s;1 -invariant forms on G s;1 =; s;1 whose Dolbeault cohomology identies with H s;1 ((g ) C ). To do this, we give a ltration of the complex t = p q of dierential forms of type (p q) on t t = i;1 g. We know from Lemma 6 that there exists a basis of (1 0)-forms! h t on t (and of (0 1)-forms! t h )such that part of them are (1 0)-forms!b j on b = gi;1=g i and part are forms! f k on f = gi. We dene ~L k := f! t t j! t is a sum of monomials! I b ^! b ^! f ^ I 0!f 0 in which jij + jjkg where jaj denotes the number of elements of the nite set A. Note that L0 ~ = t and that Lk ~ = 0 for k>dimr b Lk ~ Lk+1 ~ Lk ~ Lk ~ k0: The above shows that f~ Lk g denes a bounded decreasing ltration of the dierential module ( t ). Of course ~L P k = p q p q Lk ~ p q Lk ~ = Lk ~ \ p q and the ltration is compatible with the bigrading t provided by the type (and also with the total degree). Recall that, by denition (see e.g., [GH]) p q E ~ u v r = p q Zr u v =( p q Z u+1 v;1 r;1 + p q B u v ) r;1 where p q Zr u v p q Br u v = p q ~ Lu ( u+v t = p q ~ Lu ( u+v t ) \ ker ( p q+1 Lu+r ~ ( u+v+1 t ) \ ( p q;1 Lu+r ~ ( u+v;1 )): t Moreover (cf. [GH]), p q E ~ u 0 = p q Lu ~ = p q Lu+1 ~ where we denoteby p q E ~ u v r and p q E ~ u r the spaces of elements of type (p q) and total degree u + v and degree u, respectively, in the grading dened by the ltration. Note also that an element of~ L k is identied with an element of P c+dk a b f )) c d b. Lemma 7. Given the holomorphic bration (with standard bre G i =;i ) G i;1 =; i;1! G i;1 i =p i;1 (; i;1 ) the spectral sequence ( Er ~ d ~ r ) (r 0) converges to H i;1 ((g ) C ) and p q ~ E u v = X k H k u;k ((g i;1 i =g) C ) H p;k q;u+k i ((g ) C ): (~ Ii)

8 118 S. CONSOLE AND A. FINO Proof. The fact that ( Er ~ d ~ r )converges to H i;1 ((g ) C ) is a general property of spectral sequences associated to ltered complexes (cf. [GH]). Let [!] p q E ~ u 0 = p q L ~ u. We compute the dierential d ~ p q L ~ u+1 0 : p q u E ~ 0! p q+1 u E ~ 0 dened by ~ d 0 [!] =[!]. We can write (up to the above identications)! = P! b I ^! b ^! f I 0 ^!f 0 where jij + jj = u (because we operate mod Lu+1 ~ ), and ji 0 j + j 0 j = p + q ; u, since ji 0 j + j 0 j + jij + jj = p + q. Moreover, using the fact that sends forms in b on forms P that either are in b or vanish on b (cf. Remark ) and that b is abelian, we get! = b (! I b ^! b ) ^ (! f ^ I 0!f 0)+(;1)s (! I b ^! b ) ^ (! f ^ I 0!f 0)=(;1)s (! I b ^! b ) ^ f (! f ^ I 0!f 0)mod Lu+1 ~ where b and f denote the dierential on the complexes b and f, respectively. Thus d ~ 0 [!] =[ f!] which implies p q ~ E u 1 = X k H p;k q;u+k C (f ) k u;k : (k1) b Performing the same proof as in [Hi, Appendix II by Borel, Section 6] and using the fact that b is abelian, it is possible to prove that d ~ 1 identies with via the above isomorphism and that we have p q E ~ u X X = H k u;k C (b ) H p;k q;u+k C (f )( = k u;k H p;k q;u+k C (f )): (k) b k k 5. Proof of Theorem First we note that Theorem is trivially true if the nilmanifold comes from an abelian group, i.e., it is a complex torus. Namely, ifa=; is a complex torus, we have H (A=;) = H (a C ): (a) We consider the holomorphic brations G s =; s,!g s;1 =p s;1 (; s;1 ) ::: ::: G 1 =;1,! G=;! G 0 1 =p 0(;): The aim is to obtain information about the Dolbeault cohomology of G=; inductively through the Dolbeault cohomologies of G i =;i (the nilmanifolds G i =;i play alternately the roles of bres and total spaces of the above bre bundles). Note that since the bases are complex tori, H (G i i;1 =p i;1 (; i;1 )) = H ((g i;1 i =g ) C ): To this end we will associate to these brations two spectral sequences. The rst is a version of the Borel spectral sequence (considered in Section ) which relates the Dolbeault cohomologies of the total spaces with those of bres and bases. The second is the spectral sequence ( Er ~ d ~ r ) (constructed in the previous section) relative to the Dolbeault cohomologies of the Lie algebras i g g i;1 g i;1 i =g. We will proceed inductively on the index i in the descending series (D), starting from i = s. =;s;1!g s;1 s First inductive step. Let us use the holomorphic bre bundle ~p s;1 : G s;1 =; s;1! =p s;1 (; s;1 ) with typical bre G s =;s. Recall (cf. Lemma 3) that G s =;s and =p s;1 (; s;1 ) are complex tori. Thus by (a) G s;1 s G s;1 s H (G s =; s ) = H ((g s ) C ) (s)

9 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 119 H (G s;1 s =p s;1 (; s;1 )) = H ((g s;1 =g s ) C ): (s s ; 1) Applying Theorem 5 (since the bre G s =;s is Kahler) and using (s) and(s s ; 1), we get X p q E u v = H k u;k ((g s;1 s =g) C ) H p;k q;u+k s ((g ) C ): (Is) k Next we use the spectral sequence ( Er ~ d ~ r ). Note that the inclusion between the Dolbeault complex of G s;1 -invariant forms on G s;1 =; s;1 and the forms on G s;1 =; s;1 induces an inclusion of each term in the spectral sequences p q E ~ u v r actually a morphism of spectral sequences). By Lemma 7, for i = s, wehave that ( ~ Er ~ d r ) converges to H ((g s;1 p q E u v r ) C )and (which is p q ~ E u v = X k H k u;k ((g s;1 s =g) C ) H p;k q;u+k s ((g ) C ): (~ Is) Comparing (Is)with(~ Is), we get that E = ~ E, hence the spectral sequences (E r d r ) and ( ~ Er ~ d r )converge to the same cohomologies. Thus H (G s;1 =; s;1 ) = H ((g s;1 ) C ): (s ; 1) General inductive step. We use the holomorphic bre bundle ~p i;1 : G i;1 =; i;1! =p i;1 (; i;1 ) with the typical bre G i =;i. We assume inductively that G i;1 i H (G i =; i ) = H ((g i ) C ): (i) Lemma 8. The scalar cohomology bundle is trivial. H u v (G i =; i )= [ bg i;1 i =p i;1(; i ) H u v (~p ;1 (b)) i;1 Proof. By [KS, Section 5, Formula 5.3] there exists a locally nite covering fu l g of G i;1 i =p i;1 (; i;1 ) such that the action of the structure group of the holomorphic ber bundle on U l \ (G i =;i ) is the dierential of the change of complex coordinates on the bre, so one can restrict oneself to considering the left translation by elements of G i;1 as change of coordinates. Then the scalar cohomology bundle H u v (G i =;i ) is trivial since any of its bres is canonically isomorphic to H i ((g )C ). More explicitly, a global frame for H u v (G i =;i ) is given as follows: for any cohomology class H u v (~p ;1 (1)) = i;1 H u v (Gi =;i ) = H u v i ((g )C ) (1: identity element ofg i;1 i =p i;1 (; i )) one can take the corresponding cohomology class! H u v i ((g )C ) and regard it as a G i;1 -invariant dierential form on G i;1 =; i;1. Thus b 7!!j ~p ;1 gives a global holomorphic section i;1 (b) of H u v (G i =;i ). Taking a basis of H i ((g )C ) one gets a global holomorphic frame of H u v (G i =;i ).

10 10 S. CONSOLE AND A. FINO Thus the assumption (I 0 ) in Theorem 5 is fullled. =p i;1 (; i;1 ) is a complex torus, by (a), G i;1 i Observe moreover that, since Hence, by Theorem 5, we have H (G i;1 i =p i;1 (; i;1 )) = H ((g i;1 =g i ) C ): (i i ; 1) p q E u v = X k H k u;k ((g i;1 i =g) C ) H p;k q;u+k i ((g ) C ): (Ii) Then we proceed exactly as in the rst inductive step. Namely, we consider the spectral sequence p q E ~ u v r for the complexes of G i;1 -invariant forms on G i;1 =; i;1 whose Dolbeault cohomology identies with H i;1 ((g ) C ). By Lemma 7 we have that ( Er ~ d ~ r ) converges to H ((g i;1 ) C )and p q ~ E u v = X k H k u;k ((g i;1 i =g) C ) H p;k q;u+k i ((g ) C ): (~ Ii) Using (Ii) and proceeding as in the rst inductive step, we get the proof of Theorem. Remark 4. (On abelian complex structures) Iftheinvariant structure is abelian, the centre g1 is a rational -invariant idealofg (cf. Remark 1 in Section 1). We give analternative (and simpler) proof of that given in [CFGU] in the abelian case. In the same vein of [No] we consider the principal holomorphic bre bundle G=;! G=(;G 1 ) with typical bre G 1 =(; \ G 1 ) = ;G1 =G 1 (where G 1 is the simply connected Lie group corresponding to g1). First we can consider the Borel spectral sequence (E r d r ) associated to the Dolbeault complex (G=;) (cf. Theorem 5) and a spectral sequence ( Er ~ d ~ r ) associated to (g) C and constructed as in Section 4. As to the latter spectral sequence ( Er ~ d ~ r ), observe that it is possible to prove that there exists a basis f! 1 :::! n g of (1 0)-forms on g such thatf! 1 :::! n;k g is a basis on g=g1 (with dim g1 =k) andf! n;k+1 :::! n g is a basis on g1. So one can perform the same construction as in Section 4 (with t = g, b = g=g1 and f = g1), using the fact that f is abelian. Thus one gets p q E ~ u v = Pk Hk u;k ((g=g1) C ) H p;k q;u+k ((g1) C ) and ( Er ~ d r ) converges to H ((g) C ). Moreover, since G 1 =; \ G 1 and G=(;G 1 ) are complex tori, H (G 1 =; \ G 1 ) = H ((g1) C ) and H (G=(;G 1 ) = H ((g=g1) C ), by Theorem 5 we have p q E u v = P k Hk u;k (G=(;G 1 )) H p;k q;u+k ((g1) C ): This implies that Theorem holds also for complex abelian structures. Next we construct a minimal model for the Dolbeault cohomology of a nilmanifold endowed with an abelian complex structure. Recall that a model for the Dolbeault cohomology of M is a dierential bigraded algebra (M ) for which there exists a homomorphism : M! (M) of dierential bigraded algebras inducing an isomorphism on the respective Dolbeault cohomologies [NT]. Suppose M is free on a vector space V. Then is called decomposable if there is an ordered basis of V such that the dierential of any generator v of V can be expressed

11 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 11 in terms of the elements of the basis preceding v. Amodelis called minimal if M is free and is decomposable [Su]. By [Sa], in the abelian case, there exists a basis f! 1 :::! n g of (1 0)-forms on g such that X d! i = A ijk! j ^! k i =1 ::: n: (m) j<k<i Thus, by Theoremand(m), ( (g C ) ) is a minimal model for the Dolbeault cohomology of G=;. 6. Proof of Theorem 1 Let C(g) = f End(g) j = ;id [X Y] = [X Y ]+[X Y]+[X Y ]g denote the set of complex structures on g. We will use the same notation as in [Sa]. If M = G=; is a nilmanifold associated to g, then 0! Hom( )! Hom( )! :::! Hom( n )! 0 is a subcomplex of the Dolbeault complex of M tensored with the holomorphic tangent bundle T 1 0 M. The kernel K of acting on Hom( ) can be identied with the subspace of invariant classes in the sheaf cohomology space H 1 (M O(T )). By [Sa, Proposition 4.1] if is a smooth point ofc(g), then the tangent space T C(g) toc(g) is contained in the complex subspace of T C (where C = GL(n R)=GL(n C )istheset of all almost complex structures on g) determined by K =ker. By the above section we know that if 0 is a rational complex structure, then H p q (M) = H p q (gc ). Given 0 C(g), we know by [KS] that there exists a complete complex analytic family fm t =(M t ) j t Bg. Let t and t be respectively the -operator and the Laplacian determined by the global inner product g t induced by aninvariant Hermitian metric on M compatible with t. More precisely, t = t t + t t where t is the adjoint of t with respect to g t. Lemma 9. (i) t sends G-invariant forms of type (p q) to G-invariants forms. Moreover, the orthogonal complement with respect to g t of the invariant forms on the space p q t of ;-invariant forms of type (p q) on (M t ) is preserved by t. (ii) H p q (gc ) is a subspace ofh p q (M) for any invariant complex structure on M. Proof. (i) follows given that t and t preserve G-invariant forms. (ii) We have to prove that there exists an injective homomorphism : H p q (gc )! H p q (M): By the decomposition p q = H p q ImIm where H p q denotes the space of ;-invariant harmonic forms of type (p q) onm, wehave similarly, for the G-invariant forms, the decomposition p q inv = Hp q inv Imj inv Im j inv where H p q (gc ) = H p q inv. We can dene (!) as the orthogonal projection of! on H p q =(Im Im )?, for any [!] H p q (gc ). To prove that is injective, suppose that (!) =0onH p q (M). Then (!) ='. We may assume that ' belongs to the orthogonal complement to the G-invariant forms. Since ' is G-invariant, we have alsothat (') is invariant and then the inner product of ' with (') is zero and thus ' must be G-invariant, i.e., [!] =0inH p q (gc ).

12 1 S. CONSOLE AND A. FINO Lemma 10. For any small deformation of the rational complex structure, the isomorphism (1) holds. Proof. Using the same proof as in [KS, pp. 67{68] it is possible to show that for any (p q) and on any (M t ) there exists a complete orthonormal set of forms of type (p q) orthogonal to those that are G-invariant fe p q p q t1 ::: ft1 ::: gp q t1 :::g such that (i) t e p q tj = 0, for any j = 1 ::: d q, d q = dim H p q?, where Hp q? is the space of harmonic forms of type (p q) orthogonal to those that are G-invariant (ii) t f p q tj = a p q j (t)f p q tj (iii) t g p q tj = a p q;1 j (t)g p q tj withap q j (t) > 0. Let c be a positive constant and denote by p q (t) thenumber of eigenvalues a p q j (t) of t such that a p q j (t) <c. Let p q t (c)? be the subspace spanned by the eigenforms of t (orthogonal to the G-invariant ones) such that the corresponding eigenvalues are less than c. Thus we have dim p q t (c)? = h p q (t)? + p q (t)+ p q;1 (t) where h p q (t)? is the dimension of the orthogonal complement H p q (M t)? of the space of G-invariant forms H p q (gc ) in H p q (M t). Given a point t0 C(g), we can choose c such that 0 <c<a p q j (t 0 )forany p q =0 ::: n and for any j. Moreover let U be a suciently small neighbourhood of t 0 in C(g). Since any eigenvalue a p q j (t) is a continuous function of t [KS, Theorem, p.47] wehave that the dimension of p q t (c)? is independentoft U and p q t0 (c)? = H p q (M t0)?. Thus we have h p q (t)? + p q (t) + p q;1 (t) =h p q (t 0 )? for any t U. This means that the function h p q (t)? is upper semicontinuous. Since, for any rational complex structure 0,wehave that h p q (t 0 )? =0,thenh p q (t)? =0in some neighbourhood of t 0. Consequently the set ft C(g) j h p q (t)? =0g = ft C(g) j H (M t) = H (gc )g is an open set in any connected component ofc(g) with respect to the induced topology of gl(n R) on C(g). 7. Examples of compact nilmanifolds with non-rational complex structures Let M =;n G be the Iwasawa manifold. Recall that it can be constructed by taking as a nilpotent Lie group G the complex Heisenberg group G = f4 1 z 1 z z : zi C i =1 3g and, as a lattice ;, the subgroup of G consisting of those matrices whose entries are Gaussian integers. It is known that the 1-forms! 1 = dz 1! = dz! 3 = dz 3 + z 1 dz are left invariant ong. G has structure equations d! 1 = d! =0 d! 3 =! 1 ^! : If one regards G as a real Lie group and sets! 1 =: e 1 + ie! =: e 3 + ie 4! 3 =: e 5 + ie 6 () then (e i ) is a real basis of g such that 8 < : de i =0 1 i 4 de 5 = e 1 ^ e 3 + e 4 ^ e de 6 = e 1 ^ e 4 + e ^ e 3 :

13 DOLBEAULT COHOMOLOGY OF COMPACT NILMANIFOLDS 13 By [Na], the bi-invariant complex structure 0 dened by () on the complex Heisenberg group has a deformation space of positive dimension. Then the complex structure associated to the subspace h! 1!! 3 + t! 1 i of g C belongs to the orbit of 0 in C(g) with respect to the group of automorphisms of g (see [Sa, Section 4]) and it is not rational for appropriate t C. In the same way one can see that every compact nilmanifold M = G=; of real dimension n, with at least one (rational) complex structure 0 having a non-trivial deformation, has a non-rational complex structure. Indeed, by [Sa, Theorem 1.3], given the complex structure 0 it is possible to construct a basis of left invariant (1 0)-forms f! 1 :::! n g such that d! i+1 belongs to the ideal generated by the set f! 1 :::! i g (i = 0 ::: n; 1) in the complexied exterior algebra. Then the complex structure associated to the subspace h! 1 :::! n + t! 1 i is a not rational complex structure on M for appropriate t C. [Ab] [Ak] [BD] [BO] [CFG] References E. Abbena, An example of an almost Kahler manifold which is not Kahlerian, Boll. Un. Mat. Ital. A 6 (3) (1984), 383{39. D. Akhiezer, Group actions on the Dolbeault cohomology of homogeneous manifolds, Math. Z. 6 (1997), 607{61. M. L. Barberis, I. Dotti Miatello, Hypercomplex structures on a class of solvable Lie groups, Quart.. Math. Oxford Ser. () 47 (1996), 389{404. W. Barth, M. Otte, Uber fast-uniforme Untergruppen komplexer Liegruppen und auosbare komplexe Mannigfaltigkeiten, Comment. Math. Helv. 44 (1969), 69{81. L. A. Cordero, M. Fernandez, A. Gray, Symplectic manifolds with no Kahler structure, Topology 5 (1986), 375{380. [CFGU] L. A. Cordero, M. Fernandez, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc., to appear. [CG] [DF] [FG] [FW] [GH] [Hi] L.. Corwin, F. P. Greenleaf, Representations of nilpotent Lie groups and their applications, Cambridge Studies in Advanced Mathematics 18, Cambridge, New York, I. Dotti, A. Fino, Abelian hypercomplex 8-dimensional nilmanifolds, Ann. Global Anal. Geom. 18 (000), 47{59. G. Fischer, H. Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Gottingen Math.{Phys. Kl. II (1965), 89{94. H. R. Fischer, F. L. Williams, The Borel spectral sequence: some remarks and applications, in: Dierential Geometry, Calculus of Variations and Their Applications, Dedic. Mem. L. Euler 00th Anniv. Death., Lect. Notes Pure Appl. Math. 100 (1985), 55{66. P. Griths,. Harris, Principles of Algebraic Geometry, Wiley{Interscience, New York, Russian transl.: F. Griffits, D. Harris, Principy algebraiqesko geometrii, tt. 1 i, Mir, M, 198. F. Hirzebruch, Topological Method in Algebraic Geometry, Springer Grundl. Math. Wissen. 131, Berlin, Heidelberg, New York, Russian transl.: F. Hircebruh, Topologiqeskie metody v algebraiqesko geometrii, Mir, M., 1973.

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