THE NON-VANISHING OF FIRST COHOMOLOGY GROUPS FOR CERTAIN INFINITE-DIMENSIONAL COMPLEX MANIFOLDS

Abstract. Here, using the ideas of an old paper by S. Dineen (1976), we give large classes of pairs (X,E) such that X is an infinite-dimensional complex space very far from a Banach manifold, E is a holomorphic vector bundle on X and H1(X,E) is infinite-dimensional. 2000 Mathematics Subject Classification: 32L05, 32K05. Key words and phrases: Infinite-dimensional complex manifold, first Cousin problem, additive Cousin problem, non-vanishing of cohomology. Let V be a complex locally convex and Hausdorff topological vector space. A sequence {xn}n≥1 ⊂ V is called a nontrivial very strongly convergent sequence if the sequence {λnxn}n≥1 converges to 0 ∈ V for all λn ∈ C and xn 6 = 0 for all n. For instance, if V = CN, then the sequence (1, 0, 0,...), (0, 1, 0,...),... is a nontrivial very strongly convergent sequence. By [4], Th. 2.6.13, a Fréchet space contains CN if and only if it has no continuous norm. Hence a Fréchet space has a continuous norm if and only if it has a nontrivial strongly convergent sequence. The aim of this short note is to give the following generalization o