On the J-anti-invariant cohomology of almost complex 4-manifolds

Abstract

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H_J^{\pm }$ of $H^2(M,\mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^{+}=1$, we show that for generic almost complex structures on $M$, the subgroup $H_J^{-}$ is trivial. Computations of the subgroups and their dimensions $h_J^{\pm }$ are obtained for almost complex structures related to integrable ones. We also prove semi-continuity properties for $h_J^{\pm }$.

30 pages