Comparison of de Rham and Dolbeault cohomology for proper surjective mappings

Abstract

In this paper it is shown that if :X->X is a proper holomorphic surjection of equidimensional complex manifolds then the induced mapping *: H q (X, \) - H Q (X, \) on Dolbeault groups is injective. As a consequence one obtains the inequality h p ' 9 (X) g h p -9 (X) for the Hodge numbers of X and X. This result is valid also in the case of vector bundle coefficients, and can be generalized to the case of nondiscrete fibres of the mapping (non equidimensional case) by the imposition of a Kahlerian condition on X. Corresponding results for differentiate mappings are formulated and proved. Illustrative examples are provided to show the necessity of the various assumptions made.