Cohomology of Almost-Complex Manifolds

Let X be a differentiable manifold endowed with an almost-complex structure J. Note that if J is not integrable, then the Dolbeault cohomology is not defined. In this chapter, we are concerned with studying some subgroups of the de Rham cohomology related to the almost-complex structure: these subgroups have been introduced by T.-J. Li andW. Zhang in (Comm. Anal. Geom. 17(4): 651683, 2009), in order to study the relation between the compatible and the tamed symplectic cones on a compact almost-complex manifold, with the aim to throw light on a question by S. K. Donaldson, (Two-forms on four-manifolds and elliptic equations, Inspired by S. S. Chern, Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 153-172, Question 2) (see Sect. 4.4.2), and it would be interesting to consider them as a sort of counterpart of the Dolbeault cohomology groups in the non-integrable (or at least in the non-Kahler) case, see Dr. aghici et al. (Int. Math. Res. Not. IMRN 1:1-17, 2010, Lemma 2.15, Theorem 2.16). In particular, we are interested in studying when they let a splitting of the de Rham cohomology, and their relations with cones of metric structures.