About the d-bar-equation at isolated singularities with regular exceptional set

Abstract

Let Y be a pure dimensional analytic variety in C^n with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of this paper is to present a technique which allows to determine obstructions to the solvability of the d-bar-equation in the $L^2$ respectively $L^\infty$ sense on Y^*=Y-{0} in terms of certain cohomology classes on X. More precisely, let D be a Stein domain, relatively compact in Y, containing the origin, D^*=D-{0}. We give a sufficient condition for the solvability of the d-bar-equation in the $L^2$-sense on D^*; and in the $L^\infty$ sense, if D is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Hoelder estimates for solutions of the d-bar-equation.