Comparison of de Rham and Dolbeault cohomology for proper surjective mappings

Abstract

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*: H q (X, Ω\) -» H Q (X, Ω\) 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 π (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.

On the blow-up formula of twisted de Rham cohomology

“…Moreover the blow-up morphism π has finite generic fiber of cardinal 1, i.e., of degree 1, one can show that π * j π * = j by following the totally same step as [14,Lemma 2.3] or [5,Theorem 12.9]. The fact is essentially due to the condition that π is biholomorphic outside sets of Lebesgue measure zero.…”

On the Morse–Novikov Cohomology of blowing up complex manifolds

“…One needs a proposition of Meng [7,Corollary 2.9] by the projection formula and here we give another proof following [15], which is to be postponed in the next section. Proposition 3.2.…”

Quotients by 𝐶*×𝐶* actions

“…Since our T action is meromorphic, F is a finite subgroup of K. Then U/F is a principal T/F bundle over U/T. A basic fact [18] is that A* and B* are injective on rational cohomology since A and B are generically finite to one surjective map between compact complex manifolds.…”

“…Theorem (1.3) holds for any meromorphic action of T on a compact complex space and any T-invariant open U e X with a geometric quotient U/T. Theorem (1.5) needs in addition that A" is a rational homology manifold so that the result of [18] will hold.…”