Dolbeault cohomologies of blowing up 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 (•, 0)-and (0, •)-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 Frölicher spectral sequences at E1 on compact complex threefolds and fourfolds.

On the blow-up formula of twisted de Rham cohomology

“…Moreover, for the pair (X, H), we also have the (twisted) Hodge-de Rham (or Frölicher) spectral sequence This result is a natural generalization of [10,Theorem 1.6] where the birational invariance of the non-twisted E 1 -degeneracy is also obtained for compact complex threefolds and fourfolds.…”

Relative Čech–Dolbeault homology and applications

“…[15]). In fact, the Kähler hypothesis can be dropped and several formulas for the de Rham, Dolbeault and Bott-Chern cohomologies can be found in [10], [16], [8], [8], [7], [2]. Here we discuss a blow-up formula for the Dolbeault cohomology in terms of the relative cohomology groups which follows directly by the previous results and which finds further applications in [2].…”

“…This blow-up formula for the Dolbeault cohomology will find further applications in [2]. For other formulas of this kind we refer the reader to [16], [8], [8], [10], [7].…”

“…Notice that in [8] and [9] the authors prove a Dolbeault blow-up formula using a sheaftheoretic approach. However their definition of relative Dolbeault cohomology is different from Suwa's one.…”

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

“…In Section 4, we introduce the notation of relative Dolbeault sheaves associated to the blow-up morphism. In Section 5, we use the preparations of Sections 2-4 to give the first proof of the main result (Theorem 1.2), following the full approach of [32] v3 . Finally, Section 6 contains some applications of Theorem 1.2.…”