Vol. 53, No. 1, 1974

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Comparison of de Rham and Dolbeault cohomology for proper surjective mappings

Raymond O’Neil Wells, Jr

Vol. 53 (1974), No. 1, 281–300

In this paper it is shown that if π : X X is a proper holomorphic surjection of equidimensional complex manifolds then the induced mapping π;Hq(X,ΩXp) Hq(X,ΩXp) on Dolbeault groups is injective. As a consequence one obtains the inequality hp,q(X) hp,q(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 π (non equidimensional case) by the imposition of a Kählerian condition on X. Corresponding results for differentiable mappings are formulated and proved. Illustrative examples are provided to show the necessity of the various assumptions made.

Mathematical Subject Classification 2000
Primary: 32J25
Received: 25 June 1973
Published: 1 July 1974
Raymond O’Neil Wells, Jr