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arXiv · 2303.17449

On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms

Lorenzo Sillari·Adriano Tomassini
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Abstract

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott-Chern and Aeppli cohomologies defined using the operators $d$, $d^c$. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to $d$, $d^c$, showing their relation with Bott-Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott-Chern cohomology of $1$-forms is finite-dimensional on compact manifolds and provides an almost complex invariant $h^1_{d + d^c}$ that distinguishes between almost complex structures. On almost K\"ahler $4$-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.

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Lorenzo Sillari, Adriano Tomassini. 2023-03-30. On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms. https://doi.org/10.1016/j.exmath.2023.09.001

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