Kunneth formula for Hessian manifolds

TL;DR

This paper establishes a K"unneth formula for Dolbeault--Koszul cohomology on flat affine manifolds, with applications to classifying Hessian metrics on product manifolds.

Contribution

It proves a K"unneth formula for cohomology of flat affine manifolds and applies it to characterize Hessian metrics on product and hyperbolic manifolds.

Findings

K"unneth formula for flat affine manifolds with at least one compact component

Hessian metrics classified via cohomology classes and differences by closed 1-forms

Application to product manifolds and manifolds with flat Riemannian metrics

Abstract

We study Dolbeault--Koszul cohomology $H^{p,q}(M)$ of flat affine manifolds. We proove a K\"unneth 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 K\"unneth 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 $D\alpha$ for a closed $1$-form $\alpha$. Using the K\"unneth formula we describe all Hessian metrics on products, on products with hyperbolic manifolds, and on manifolds admitting a flat Riemannian metric.