On Taming and Compatible Symplectic Forms

Abstract

Let \((X,J)\) be an almost-complex manifold. The almost-complex structure \(J\) acts on the space of \(2\)-forms on \(X\) as an involution. A \(2\)-form \(\alpha \) is \(J\)-anti-invariant if \(J\alpha =-\alpha \). We investigate the anti-invariant forms and their relation to taming and compatible symplectic forms. For every closed almost-complex manifold, in contrast to invariant forms, we show that the space of closed anti-invariant forms has finite dimension. If \(X\) is a closed almost-complex manifold with a taming symplectic form, then we show that there are no non-trivial exact anti-invariant forms. On the other hand, we construct many examples of almost-complex manifolds with exact anti-invariant forms, which are therefore not tamed by any symplectic form. In particular, we use our analysis to give an explicit example of an almost-complex structure which is locally almost-Kahler but not globally tamed. The non-existence of exact anti-invariant forms, however, does not in itself imply that there exists a taming symplectic form. We show how to construct examples in all dimensions.