The heredity and bimeromorphic invariance of the ∂∂ ¯-lemma property

Lingxu Meng

doi:10.5802/crmath.203

Abstract

We give a simple proof of a result on the $\partial\bar{\partial}$-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of spaces of forms and currents. At last, we discuss the heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property.

Highlights

In non-Kähler geometry, the heredity and bimeromorphic invariance of the ∂∂-lemma property are two interesting problems, extensively studied in [2, 3, 6, 7, 12, 15,16,17] especially in the recent days
The ∂∂-lemma on a compact complex manifold X refers to that for every pure-type d closed form on X, the properties of d -exactness, ∂-exactness, ∂-exactness and ∂∂-exactness are equivalent while a compact complex manifold is called a ∂∂-manifold if the ∂∂-lemma holds on it
Alessandrini [2] posed a question in its inverse direction: if X satisfies the ∂∂-lemma, so does X ? We can prove that, Question 2 is equivalent to Alessandrini’s one. It is true on complex surfaces by the classical results that each compact complex surface with even first Betti number is Kähler

Summary

Introduction

In non-Kähler geometry, the heredity and bimeromorphic invariance of the ∂∂-lemma property are two interesting problems, extensively studied in [2, 3, 6, 7, 12, 15,16,17] especially in the recent days. Yang [12, Theorem 1.6] investigated the bimeromorphic invariance of the degeneracy of Frölicher spectral sequence at E1 by their Dolbeault blow-up formula and pointed out that these results are applicable to Question 2 in the remarks after [12, Question 1.2]. Tomassini [3, Theorem 13, Questions 22-24] studied this equivalence by the Cech–Dolbeault cohomology with additional hypotheses and generalized their results to compact complex orbifolds. In his PhD thesis, by Angella–Tomassini’s characterization [4, Theorems A and B], J.

Preliminaries

Some notations

Blow-up cases

Projective bundle cases

A proof of Theorem 4