Symplectic forms and cohomology decomposition of almost complex 4-manifolds

Abstract

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.

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

References in corpus (3)

• Taming symplectic forms and the Calabi-Yau equation
• The Calabi-Yau equation, symplectic forms and almost complex structures
• Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

Cited by in corpus (10)

• Almost Kähler structures on four dimensional unimodular Lie algebras
• The Calabi-Yau equation on the Kodaira-Thurston manifold
• Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds
• Almost Kahler forms on rational 4-manifolds
• Moduli space of $J$-holomorphic subvarieties
• The curve cone of almost complex 4-manifolds
• From Smooth to Almost Complex
• Intersection of almost complex submanifolds
• A note on exact forms on almost complex manifolds
• On second non-HLC degree of closed symplectic manifold