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

TL;DR

This paper develops a sheaf-theoretic method to derive a blow-up formula for bundle-valued Dolbeault cohomology groups on complex manifolds, with applications to vanishing theorems and invariance of holomorphic invariants.

Contribution

It introduces a new sheaf-theoretic approach to compute bundle-valued Dolbeault cohomology under blow-ups, extending previous results to the bundle-valued case.

Findings

Provides a blow-up formula for bundle-valued Dolbeault cohomology groups.

Demonstrates examples related to classical vanishing theorems.

Studies invariance of holomorphic invariants under blow-up operations.

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.