$E_1$-degeneration and $d'd''$-lemma

TL;DR

This paper proves that for certain double complexes satisfying the $d'd''$-lemma, the associated spectral sequence degenerates at $E_1$, with applications to Hodge-de Rham spectral sequences on specific manifolds.

Contribution

It establishes a new criterion linking the $d'd''$-lemma to spectral sequence degeneration at $E_1$ for double complexes.

Findings

Spectral sequence degenerates at E_1 under $d'd''$-lemma conditions.

Application to bi-generalized Hermitian manifolds.

Extension of degeneration results to new geometric contexts.

Abstract

For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma.