Dolbeault cohomologies of blowing up complex manifolds II: Bundle-valued case

Abstract

We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata-Viehweg and Green-Lazarsfeld in a uniform manner and study the blow-up invariance of some classical holomorphic invariants.

On the blow-up formula of twisted de Rham cohomology

“…Claim 3.5 (cf. [11,Lemma 4.5]). For any 0 ≤ p, q ≤ n, the pullback of differential forms induces an isomorphism π * : H q (X, K p X,Z (H)) −→ H q ( X, K p X,E ( H)).…”

“…It is of importance to notice that many important invariants of complex manifolds hold such a similar formula, for example, the Deligne cohomology [2], the Chern classes [6] and the Dolbeault cohomology of holomorphic vector bundles [10,11], etc.…”

On the Morse–Novikov Cohomology of blowing up complex manifolds

“…Explicit isomorphisms between cohomologies. The construction of the morphism φ is inspired by [12]. In view of Proposition 3.4, we get…”

“…Moreover, the relative de Rham cohomology is isomorphic to the de Rham cohomology with compact support, while it does not hold true for the Morse-Novikov case anymore (see [11,17]). So we are not able to apply the relative cohomological method via the compactly supported cohomology directly as [11,12].…”

The Double Complex of a Blow-up

“…Independently, the Dolbeault case is also established in [4] in the situation that Z admits a neighborhood with a holomorphic retract to Z, using a Thom-isomorphism for Dolbeault cohomology. In [13], the case of Morse-Novikov cohomology is considered and in [16], an explicit isomorphism is given for Dolbeault cohomology with values in a vector bundle.…”