Dolbeault cohomologies of blowing up complex manifolds

TL;DR

This paper establishes a blow-up formula for Dolbeault cohomologies of compact complex manifolds, providing new insights into bimeromorphic invariance and stability of spectral sequences, with applications to complex geometry.

Contribution

Introduces a relative Dolbeault cohomology framework to prove a blow-up formula and demonstrates bimeromorphic invariance of certain Hodge numbers.

Findings

Proves a blow-up formula for Dolbeault cohomologies.

Shows bimeromorphic invariance of $(ullet,0)$- and $(0,ullet)$-Hodge numbers.

Establishes stability of the Fr"olicher spectral sequence at $E_1$ for certain complex manifolds.

Abstract

We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of $(\bullet,0)$- and $(0,\bullet)$-Hodge numbers on a compact complex manifold, and obtain the equality for the numbers of the blow-ups and blow-downs in the weak factorization of the bimeromorphic map between two compact complex manifolds with equal $(1,1)$-Hodge number or equivalently second Betti number. Many examples of the latter one are listed. Inspired by these, we obtain the bimeromorphic stability for degeneracy of the Fr\"olicher spectral sequences at $E_1$ on compact complex threefolds and fourfolds.