Symplectic forms and cohomology decomposition of almost complex 4-manifolds

Draghici, Tedi
Li, Tian-Jun
Zhang, Weiyi
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For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by $J$-invariant, respectively, $J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of $H^2(M, \mathbb{R})$. This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.
Comment: v2. This is the published version of some of the results of v1; Other parts of v1 have been considerably extended and included in arXiv:1104.2511
Mathematics - Symplectic Geometry, Mathematics - Differential Geometry