On Some Cohomological Properties of Almost Complex Manifolds

Abstract

We study a special type of almost complex structures, called pure and full and introduced by T.J. W. Zhang (arXiv:0708.2520, 2007), in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.

Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

“…Together with Lemma 2.8, we obtain [12] that if J is C ∞ pure and full, then it is pure. Fino and Tomassini further show that if J is a C ∞ pure and full almost complex structure on a 2n-dimensional manifold, then J is also pure and full under any of the following three conditions:…”

“…We believe that they are important invariants of almost complex structures and deserve further study. We would like to mention that there are two recent papers [11,12] that are closely related to this work (see Remark 2.3). In particular, it is shown in [11] that any 4-dimensional almost complex structure is C ∞ pure and full.…”

“…We also appreciate F. Zheng, S. Morgan, V. Tosatti, B. Weinkove for discussions, S. Donaldson, M. Furuta, R. Hind, K. Ono for their interest, A. Fino, A. Tomassini for sending to us their preprint [12]. We are grateful to T. Drȃghici for his interest and pointing out reference [10].…”

“…It is shown in [11,12] that if J is almost Kähler, then it is C ∞ pure. While in [12] a left-invariant almost complex structure on a 6-dimensional nilmanifold is found to be not C ∞ pure.…”

“…Many families of pure and full (non-Kähler) almost complex structures on compact nilmanifolds and solvmanifolds are also constructed this way in [12].…”