Remarks on the $L^2$-Dolbeault Cohomology Groups of Singular Algebraic Surfaces and Curves

Abstract

Let V be a complex algebraic variety of dimension <2 and its singularity set Sing F is assumed to be consisted of isolated singular points. Restrict the Fubini-Study metric of the ambient projective space containing V to the smooth part F—Sing V. Then the purpose of this paper is to investigate the relationships among various L-Dolbeault cohomology defined on the incomplete Kahler manifold F—Sing V. As is well-known, if V is nonsingular, the so-called Dolbeault cohomology groups are defined naively, with no need of care of its metric, by J^'*(F)=Ker S^/Range d~, where d' is the d-operator acting on smooth (p, q)-forms on F. However, if F is singular and one must consider those cohomology groups on F—Sing F, the situation changes greatly. That is, the d-operator §*•* is not permitted to be used so roughly as in the nonsingular case. For example, the operators d or the exterior derivative d acting on the following forms would define different kinds of cohomology groups: