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.