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

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 $h_{d+d^c}^1$ that distinguishes between almost complex structures. On almost Kähler $4$-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.

34 pages

References in corpus (4)

• Autour de la cohomologie de Bott-Chern
• Maximally non-integrable almost complex structures: an $h$-principle and cohomological properties
• Hodge-de Rham numbers of almost complex 4-manifolds
• Rank of the Nijenhuis tensor on parallelizable almost complex manifolds

Cited by in corpus (1)

• Invariant and non-invariant almost complex structures on compact quotients of Lie groups