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Cohomology of Almost-Complex Manifolds

  • Daniele Angella
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2095)

Abstract

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 and W. Zhang in (Comm. Anal. Geom. 17(4):651–683, 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-Kähler) case, see Drǎghici 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.

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Daniele Angella
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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