Dolbeault cohomology of compact nilmanifolds

LetM=G/ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification of the Lie algebra ofG. to obtain this result, we first 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 fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.Research partially supported by MURST and CNR of Italy.Research partially supported by MURST and CNR of Italy.     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products ${\mathbb{C}^{n}\ltimes_{\phi}N}$ of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product ${\mathbb{C}^{n}\times N}$ . As a corollary of this computation, we can compute the Dolbeault cohomology H ^p,q (G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.     

Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology is canonically isomorphic to the -cohomology of the bigraded complex of complex valued left invariant differential forms on the nilpotent Lie group .

     

Rational cohomology of the free loop space of a simply connected 4-manifold

The purpose of this paper is to calculate the rational cohomology ${H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}$ of the free loop space for a simply connected closed 4-manifold X. We use minimal models, so the starting point is the cohomology algebra ${H^{\ast}(X; \mathbb{Q})}$ which depends only on the second Betti number b ₂ and the signature of X itself. Calculations of ${H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}$ for b ₂ ≤ 2 are known. We study the case b ₂ > 2. We obtain an explicit formula for Poincaré series of the space ${X^{{S}^{1}}}$ , with the second Betti number b ₂ as a parameter.     

The free loop space equivariant cohomology algebra of some formal spaces

Let \mathbbK{\mathbb{K}} be a field of characteristic p > 0 and S ¹ the unit circle. We construct a model for the negative cylic homology of a commutative cochain algebra with two stages Sullivan minimal model. Using the notion of shc-formality introduced in Bitjong and Thomas (Topology 41:85–106), the main result of Bitjong and El Haouari (Math Ann 338:347–354) and techniques of Vigué-Poirrier (J Pure Appl Algebra 91:347–354) we compute the S ¹-equivariant cohomology algebras of the free loop spaces of the infinite complex projective space \mathbbCP(￥){\mathbb{CP}(\infty)} and the odd spheres S ^2q+1.     

On the cohomology of the free loop space of a 4- dimensional manifold

In this paper we consider a theorem of Feigin and Tsygan that is about the gereralized free product of a pushout diagram (Theorem 2 below) and try to show that it can be generalized, i e., the theorem is satisfied in the more general case than that they have stated. In order to do this, we formulate a conjecture stating the validity of such generalizatzon. To support the conjecture we use it in two different computations, and show that the results of these computations coincide with the results that C. Löfwall has proved in a different way.

I wish to thank J. E. Roos who called my attention to this problem. I am grateful to him because of his great help and good advice during the work on this paper. I wish to thank Ralf Fröberg, L Lambe, and C Löfwall for stimulating discussions.     

A separation theorem and Serre duality for the Dolbeault cohomology

LetX be a complex manifold with finitely many ends such that each end is eitherq-concave or (n−q)-convex. If , then we prove thatH ^pn−q (X) is Hausdorff for allp. This is not true in general if (Rossi’s example withn=2 andq=1). If all ends areq-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for ). Moreover the result was already known in the case when theq-concave ends can be ‘filled in’ (again also for ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree (p, q) from (n−q)-convex boundaries, . This research was partially supported by TMR Research Network ERBFMRXCT 98063.     

Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

Annals of Global Analysis and Geometry - In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete...     

Dolbeault cohomology of G /( P,P )

Let G be a complex connected semi-simple Lie group, with parabolic subgroup P. Let (P,P) be its commutator subgroup. The generalized Borel-Weil theorem on flag manifolds has an analogous result on the Dolbeault cohomology . Consequently, the dimension of is either 0 or . In this paper, we show that the Dolbeault operator has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or . For the case when P is maximal, we apply our result to compute the Dolbeault cohomology of certain examples, such as the punctured determinant bundle over the Grassmannian. Received: September 2, 1997; in final form February 9, 1998     

The Dolbeault complex in infinite dimensions III. Sheaf cohomology in Banach spaces

We prove that the sheaf cohomology groups H ^q (Ω,?) vanish if Ω is a pseudoconvex open subset of a Banach space with unconditional basis, and q≥1. Oblatum 23-XII-1999 & 12-V-2000?Published online: 16 August 2000     

Topological group cohomology with loop contractible coefficients

We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.