Infinitesimal locally trivial deformation spaces of compact complex surfaces with ordinary singularities

Abstract

Let $S$ be a compact complex surface with ordinary singularities. We denote by $\Theta_S$ the sheaf of germs of holomorphic tangent vector fields on $S$. In this paper we shall give a description of the cohomology $H^1(S, \Theta_S)$, which is called the infinitesimal locally trivial deformation space of $S$, using a 2-cubic hyper-resolution of $S$ in the sense of F. Guillén, V. Navarro Aznar et al. ([1]). As a by-product, we shall show that the natural homomorphisim $H^1(S, \Theta_S)\rightarrow H^1(X, \Theta_X(-\log D_X))$ is injective under some condition, where $X$ is the (non-singular) normal model of $S$, $D_X$ the inverse image of the double curve $D_S$ of $S$ by the normalization map $f\colon X\rightarrow S$, and $\Theta_X(-\log D_X)$ the sheaf of germs of logarithmic tangent vector fields along $D_X$ on $X$. Note that the cohomology $H^1(X, \Theta_X(-\log D_X))$ is nothing but the infinitesimal locally trivial deformation space of a pair $(X, D_X)$.