Kunneth formula for Hessian manifolds

Abstract: We study Dolbeault--Koszul cohomology $H^{p,q}(M)$ of flat affine manifolds. We proove a Künneth formula [ H^{p,q}(M\times N) \cong \bigoplus_{i,j} H^{i,j}(M)\otimes H^{p-i,q-j}(N) ] for flat affine manifolds $M,N$ with at least one compact. For compact manifolds we also give a proof via Hodge theory on flat affine manifolds, analogous to the classical Künneth formula for Dolbeault cohomology. We apply this formula to Hessian manifolds. A Hessian metric $g$ defines a class $[g]\in H^{1,1}(M)$, and metrics in the same class differ by $Dα$ for a closed $1$-form $α$. Using the Künneth formula we describe all Hessian metrics on products, on products with hyperbolic manifolds, and on manifolds admitting a flat Riemannian metric.