Abstract
In this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.
Highlights
Let (M, J ) be a 2m-dimensional almost complex manifold
Motivated by the comparison between the J -tamed symplectic cone KtJ and the J -compatible symplectic cone KcJ of an almost complex manifold, defined as the projection in cohomology of the space of symplectic forms taming J, respectively calibrating J, Li and Zhang introduced in [13] two cohomology groups: the J -invariant cohomology groups, respectively J -anti-invariant cohomology groups, of an almost complex manifold (M, J ) denoted with H +, respectively H −, are formed by 2nd -de Rham classes represented by closed J -invariant forms, respectively J -anti-invariant forms, with respect to the natural action of J on the space of 2-forms
In this paper we study the relation between the complex cohomology group HC+ of J -invariant complex forms and the Dolbeault cohomology group HD1,o1l on almost complex manifolds
Summary
Let (M, J ) be a 2m-dimensional almost complex manifold. the almost complex structure J induces a bigrading on the bundle of differential forms on M. Given any solvmanifold endowed with a left-invariant almost complex structure, we prove that the left-invariant spectral sequence satisfies Serre duality at every page and that the left-invariant Dolbeault cohomology groups are isomorphic to the kernel of a suitable Laplacian (Theorem 6.1). Calculations of left-invariant spectral sequence and J -invariant cohomology are performed on almost complex manifolds and solvmanifolds endowed with a left-invariant almost complex structure to give concrete applications. We provide computations of the left-invariant spectral sequence on 4-dimensional solvmanifolds that do not admit any integrable almost complex structure. For such examples the Dolbeault cohomology theory for almost complex manifolds becomes the main tool to investigate their geometry
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