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Preprints, Working Papers, ... Year : 2020

$\partial\bar\partial$-lemma, double complex and $L^2$ cohomology

Abstract

Inspired by the recent works of J. Stelzig and M. Khovanov-Y. Qi on the structure of bounded double complex, we improve further this theory with more emphasis on the bounded double complexes possibly without real structures by calculating out six types of indecomposable double complexes in the complex decomposition, to characterize various (weak) $\partial\bar\partial$-lemmata by de Rham, Dolbeault, Bott-Chern, Appeli and Varouchas' cohomology groups. In particular, we obtain new proofs of D. Angella-A. Tomassini's characterization of $\partial\bar\partial$-lemma and Frölicher-type inequalities, a weak $\partial\bar\partial$-lemma for logarithmic forms with a unitary local system to confirm a conjecture of X. Wan as a direct corollary, and also several new injectivity results on complex manifolds. From the double complex perspective, one can easily see the mechanism behind the validity or invalidity of the (weak) $\partial\bar\partial$-lemmata and injectivity.
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Dates and versions

hal-02904394 , version 1 (22-07-2020)

Identifiers

  • HAL Id : hal-02904394 , version 1

Cite

Sheng Rao, Yongpan Zou. $\partial\bar\partial$-lemma, double complex and $L^2$ cohomology. 2020. ⟨hal-02904394⟩
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