Generally, if we have any space $X$, then multiplication by any odd class $\eta$ makes $H^*(X)$ a complex (usually called Aomoto complex) $A_{\eta}$ because cup product is graded commutative. It is pretty interesting structure: for example, for space $M$ with 1-formal $\pi_1$ tangent cone at 0 of $V \subset H^1(M, \mathbb C)$, $V := \{\alpha ; \,H^1(A_{\alpha}) \neq 0 \}$ coincides with tangent cone at 1 of $R \subset Hom(\pi_1, \mathbb C^{\times})$, where $R := \{ \chi \in Hom(\pi_1, \mathbb C^{\times}); \, \mathrm{dim} \, H^1(M, \mathbb C_{\chi}) \neq 0 \}$. Also, Betti numbers of $p$-torsion local systems are bounded by Aomoto Betti numbers with $\mathbb Z/p$ coefficients.

But if we have a distinguished closed $2k+1$-form $\eta$ on a manifold, then we have one special complex $A_{\eta}$. There are a few examples: locally conformally Kähler/symplectic manifolds with Lee form, Lie groups with their Chern-Simons 3-form, Cartan 1-form, canonical 3-cocycle of ss group, deforming 3-form for generalized complex manifold and so on.

So the quesion is: are there nice geometric interpretations of $H^*(A_{\theta})$ or $H^*(A_{d\omega})$ for LCK manifold $M$? Are they isomorphic to cohomology of particular sheaf on $M$? Which conditions imply or implied by their vanishing in some degree? (I'm mainly interested in LCK/LCS case, but happy to see aswers for other geometric structures.)

Also, as $(H^*(M), \wedge \eta)$ is in some sense linearization of twisted de Rham $(\Omega^*(M), d + \wedge \eta)$, we may ask whether twisted cohomology can be recovered via some kind of spectral sequence from Aomoto complex (and if so, we can adjust some known things about Morse-Novikov cohomology to initial data).