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, dc. 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, dc, 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 hd+dc1 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.