Relations Between the Cohomology Groups of Dolbeault and Topological Invariants

Abstract

For complex-analytic manifolds there are the invariants of the complex structure on one side and the topological invariants of the underlying manifold on the other. It seems that few relations between them are known, except in the special case of Kihler manifolds.In this note a spectral sequence is defined which relates the cohomology groups of Dolbeault as invariants of the complex structure to the groups of de Rham as topological invariants (Theorem 3). Two applications are given: the Euler characteristic of the manifold is expressed by means of the dimensions of the Dolbeault groups (Theorem 1); for the Betti numbers, corresponding inequalities are obtained (Theorem 2).1. Let M be a complex-analytic manifold, m its complex dimension. Qjq shall denote the sheaf of germs of holomorphic differential forms of degree q, HP(M, £l7) the pth cohomology group with coefficients in this sheaf.1 THEOREM 1. Between the Euler characteristic x of M and the dimensions bp, of the Dolbeault cohomology groups HP(M, Q2) there holds the equality m x = E (-1)P+ b, q.(1) P,q =0 THEOREM 2. For the Betti number b, of M the following inequality holds: br < i bp,,q (r = 0, 1, 2, ...,2m).(2) P+q =T Remark: Using a result of Serre,2 which states that bp, q = bmp, m-q, one can reduce thp number of terms in equality (1) and inequality (2). Hence m odd implies x even;(it is known that x ia even for any manifold of real dimension 4k -2).2. Before stating Theorem 3 and; giving the outlines of the proofs, it will be useful to state some general facts on spectral sequences. All notations used here are from the appendix of the work cited in note 3. Proofs can be found there3 or in the original paper.4 The spectral sequence is a sequence of groups and homomorphisms which can be associated to any K-module T (K denotes a principal ring) which has a decreasing filtration {PT} (i.e., a sequence of submodules PT with P+1T c PT), a differential operator d (i.e., an endomorphism of T with d2 -

The Frölicher spectral sequence can be arbitrarily non-degenerate

“…In order to produce relations between these Frölicher studied in [7] a spectral sequence connecting Dolbeault cohomology and de Rham cohomology: if we denote by (A k (X ), d) the complex valued de Rham complex then the decomposition of the exterior differential d = ∂ +∂ gives rise to a decomposition…”

Complex-Foliated Structures. I. Cohomology of the Dolbeault-Kostant Complexes

“…In addition, independently of any assumptions about ellipticity of dF, we are now going to establish the Frölicher spectral sequence relating the "Dolbeault" cohomologies H'(M, &') and the (de Rham) cohomology H'(M, C) with complex coefficients (cf. [5] for the complex-analytic case); it is interesting to observe that the first terms E0, Ex and E2 of the spectral sequence have exactly the same interpretations as they do classically. We obtain all this by elementary computations, the key being Lastly, the single grading (by total degree) of C and the filtration are compatible: if x = "Zxk E Cp for some/?, then xk E Cp for every k. Thus, the usual conditions are satisfied and the bigrading of C defines a spectral sequence (Ep,r) which will be regular.…”

“…The last two propositions are the analogues of Frölicher's two theorems in [5]. Recall that if M is compact, then the Hq(M, âp) are finite-dimensional; thus, in this case Proposition 12 will hold.…”

Comparison of de Rham and Dolbeault cohomology for proper surjective mappings

“…As before we consider the diagram (X, E) and ψ e &*-* nq (X, E*), then one can give a meaning to and interpret φ A ψ as a scalar-valued differential form on X of type (n, n) (cf. Serre [7] One can check that this definition of φ A ψ is independent of the local frame used, and does give a globally defined scalar-valued differential form. Using this we are able to interpret, for φe ϊ?…”

“…Thus X is an elliptic surface in Kodaira's class VΠ 0 (Kodaira [11] We remark that any elliptic surface in Kodaira's class VΠ 0 has the same Hodge numbers as in (4.7). Frδhlicher proved in [7] that for any compact complex manifold X Using the Hodge numbers from (4.7), along with the fact that the χ(X) = 0 vanishes, we obtain easily from (4.8) that h lί (X) = 0 (cf. Kodaira-Spencer [13-Π]).…”