We study the space of closed anti-invariant forms on an almost complex manifold, possibly non-compact. We construct families of (non-integrable) almost complex structures on \({{\mathbb {R}}}^4\), such that the space of closed J-anti-invariant forms is infinite dimensional, and also 0- or 1-dimensional. In the compact case, we construct 6-dimensional almost complex manifolds with arbitrary large anti-invariant cohomology and a 2-parameter family of almost complex structures on the Kodaira–Thurston manifold whose anti-invariant cohomology group has maximum dimension.

中文翻译：

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关于几乎复杂流形的反不变同调性

我们研究了几乎复杂的流形上可能是非紧凑型的闭合反不变形式的空间。我们在\（{{\ mathbb {R}}} ^ 4 \）上构造（不可积）几乎复杂的结构族，使得闭合J-反不变形式的空间是无限维的，并且也是0-或一维。在紧凑的情况下，我们在Kodaira-Thurston流形上构造具有任意大的反不变同调性和2参数族几乎复杂结构的6维几乎复杂的流形，其反不变同调群具有最大的维数。