Invariants of almost complex and almost symplectic manifolds

We study invariants of almost complex, almost symplectic and almost Hermitian manifolds. First, we use the Nijenhuis tensor to measure how far an almost complex structure is from being integrable. We prove that the space of maximally non-integrable structures is open and dense or empty in the space of almost complex structures. Then, we provide a technique that allows to produce almost complex structures whose Nijenhuis tensor has arbitrary prescribed rank on parallelizable manifolds, and we use the rank to classify invariant structures on 6-dimensional nilmanifolds. From a cohomological point of view, we give definitions of Bott–Chern and Aeppli cohomologies of almost complex manifolds, based on the operators $d$ and $d^c$, and of their almost symplectic counterparts, based on the operators $d$ and $d^\Lambda$. These cohomologies generalize the usual notions of Bott–Chern and Aeppli cohomologies and of symplectic cohomologies of Tseng and Yau. We also explain the importance of the operators $\delta$ and $\bar \delta$, a suitable generalization of the complex operators $\partial$ and $\bar \partial$. In the non-integrable setting, these cohomologies do not admit a natural bigrading. However, they naturally have a $\mathbb{Z}_2$-splitting induced by the parity of forms. Finally, we deal with spaces of harmonic forms on compact almost Hermitian manifolds. We describe a series of Laplacians that generalize the complex Bott–Chern and Aeppli Laplacians and the symplectic Laplacians of Tseng and Yau. We formulate a general version of Kodaira–Spencer’s problem on the metric-independence of the dimensions of the kernels of such Laplacians. It turns out that these invariants are especially well-behaved on 4-manifolds. If the 4-manifold admits an almost K ̈ahler metric, we find a series of metric-independent invariants, solving the generalized Kodaira–Spencer’s problem. Motivated by the complex case, we show that the invariants we defined have a strong link with topological invariants. On almost K ̈ahler 4-manifolds, they essentially reduce to topological numbers and to the almost complex invariants $h_1^{d+d^c}$ and $h^-_J$. Motivated by a conjecture of Li and Zhang on the generic vanishing of $h^-_J$, we conjecture that $h_1^{d+d^c}$ generically vanishes. We are able to confirm our conjecture on high-dimensional manifolds. We complement the theoretical results with a large number of examples on locally homogeneous manifolds of dimension 4 and 6, where we explicitly compute the rank of the Nijenhuis tensor, the almost complex cohomologies and the spaces of harmonic forms.