R. Hind, C. Medori, A. Tomassini
Oct 1, 2015
The Journal of Geometric Analysis
Let $$(X,J)$$(X,J) be an almost-complex manifold. The almost-complex structure $$J$$J acts on the space of $$2$$2-forms on $$X$$X as an involution. A $$2$$2-form $$\alpha $$α is $$J$$J-anti-invariant if $$J\alpha =-\alpha $$Jα=-α. 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 $$X$$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-Kähler 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.