Higher-Page Bott-Chern and Aeppli Cohomologies and Applications

Abstract

For every positive integer $r$, we introduce two new cohomologies, that we call $E_r$-Bott-Chern and $E_r$-Aeppli, on compact complex manifolds. When $r=1$, they coincide with the usual Bott-Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when $r\ge 2$. They provide analogues in the Bott-Chern-Aeppli context of the $E_r$-cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-$(r-1)$-$\partial \bar{\partial }$-manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott-Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

37 pages. Originally part of arXiv:2001.02313. Final version. To appear in J. Reine Angew. Math

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• Deformed Aeppli cohomology: canonical deformations and jumping formulas
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