ABOUT THE $\bar{\partial}$-EQUATION AT ISOLATED SINGULARITIES WITH REGULAR EXCEPTIONAL SET

Abstract

Let Y be a pure dimensional analytic variety in ℂn with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of the present paper is to present a technique which allows us to determine obstructions to the solvability of the [Formula: see text] equation in the L2, respectively L∞, sense on Y* = Y\{0} in terms of certain cohomology classes on X. More precisely, let Ω ⊂⊂ Y be a Stein domain with 0 ∈ Ω, Ω* = Ω\{0}. We give a sufficient condition for the solvability of the [Formula: see text] equation in the L2-sense on Ω*; and in the L∞ sense, if Ω is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Hölder estimates for solutions of the [Formula: see text] equation.