Infinitesimal parameter spaces of locally trivial deformations of compact complex surfaces with ordinary singularities

In this paper we shall give a description of the cohomology H-1(S, Theta (S)) for a compact complex surface S with ordinary singularities, using a 2-cubic hyper-resolution of S in the sense of F. Guillen, V. Navarro Aznar et al. ([2]), where Os denotes the sheaf of germs of holomorphic tangent vector fields on S. As a by-product, we shall show that the natural homomorphism H-1(S, Theta (S)) --> H-1(X, Theta (X)(-log D-X)) is injective under some condition, where X is the (nonsingular) normal model of S, Dr the inverse image of the double curve D-S of S by the normalization map f : X --> S, and Theta (X)(-log D-X) the sheaf of germs of logarithmic tangent vector fields along D-X on X.