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A manifold is said to be Hessian if it admits a flat affine connection $D$ and a Riemannian metric $g$ such that $g=D^{2}u$ where $u$ is a local function. We study cohomology for Hessian manifolds, and prove a duality theorem and vanishing theorems.

A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form $g_{ij}=\frac{{\partial }^2\Phi }{\partial x^i\partial x^j}$ where $\Phi$ is a $C^{\infty }$-function and $\{x^1,...,x^n\}$ is an affine local coordinate system. Let $M$ be a Hessian manifold. We show that if $M$ is homogeneous, the universal covering manifold of $M$ is a convex domain in ${\mathbf{R}}^n$ and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of ${\mathbf{R}}^n$.