On non-pure forms on almost complex manifolds

TL;DR

This paper investigates the decomposition of cohomology groups into J-invariant and J-anti-invariant parts on almost complex manifolds, proving an analytic continuation result for anti-invariant forms.

Contribution

It extends the study of cohomology decompositions to possibly non-compact almost complex manifolds and establishes an analytic continuation theorem for anti-invariant forms.

Findings

Any almost complex structure on a 4-dimensional compact manifold is extit{ extbf{ extless}Cpf extgreater}.

Proved an analytic continuation result for anti-invariant forms.

Analyzed J-invariant and J-anti-invariant cohomology subgroups on non-compact manifolds.

Abstract

T.-J. Li and W. Zhang defined an almost complex structure $J$ on a manifold $X$ to be {\em \Cpf}, if the second de Rham cohomology group can be decomposed as a direct sum of the subgroups whose elements are cohomology classes admitting $J$-invariant and $J$-anti-invariant representatives. It turns out (see T. Draghici, T.-J. Li and W. Zhang) that any almost complex structure on a 4-dimensional compact manifold is \Cpf. We study the $J$-invariant and $J$-anti-invariant cohomology subgroups on almost complex manifolds, possibly non compact. In particular, we prove an analytic continuation result for anti-invariant forms on almost complex manifolds.