Vanishing theorems of \(L^2\)-cohomology groups on Hessian manifolds (Q2414888)

Let $(M,D)$ be a flat manifold, i.e., a manifold $M$ equipped with a flat affine connection $D$. Then a Riemannian metric $g$ on the flat manifold $(M,D)$ is called a Hessian metric if $g$ can be locally expressed in the Hessian form with respect to an affine coordinate system $\{x^1,\ldots,x^n\}$ and a potential function $\varphi$, that is \[ g_{ij}=\frac{\partial^2\varphi}{\partial x^i\partial x^j}. \] In this case, the triple $(M,D,g)$ is said to be a Hessian manifold. In the first part of the paper under review, the author proves a vanishing theorem of $L^2$-cohomology group of Kodaira-Nakano type on a complete Hessian manifold (Theorem 2). Next, this result is applied to regular convex cones with the Cheng-Yau metric in order to obtain stronger vanishing theorems (Theorem 3 and Theorem 4).