Infinitesimal Parameter Spaces of Locally Trivial Deformations of Compact Complex Surfaces with Ordinary Singularities

In this paper we shaD give a description of the cohomology H¹ (S, Θ _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 Θ _s 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, Θ _s ) → H¹(X, Θ_x(-logD_x)) is infective 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 : X → S and Θ _x (–log D_x) the sheaf of germs of logarithmic tangent vector fields along D_x on X.