LOCAL L² RESULTS FOR $\bar\partial$: THE ISOLATED SINGULARITIES CASE

Abstract

Abstract. Let X be a pure n-dimensional complex analytic set in C N with an isolated singularity at 0. We obtain some L 2 -existence and regularity results for the ∂-operator on a deleted neighborhood of the singular point 0 in X. IntroductionLet X be a pure n-dimensional complex analytic set in C N with an isolated singularity at 0 and letand identify i ∂∂ z 2 with the euclidean metric in C N . Then X * inherits a Kähler metric from its embedding in C N , which we call the ambient metric. Due to the incompleteness of the metric, there are many possible closed L 2 -extensions of the ∂-operator originally acting on smooth forms on X * . We consider the maximal (distributional) ∂-operator. The pointwise norm of a form f defined in X * with respect to the ambient metric will be denoted by |f |, its L 2 -norm by f and the volume element by dV . Let R be a positive number. We set for 0 < r < R, B r := {z ∈ C N ; z < r}, B r := {z ∈ C N ; z ≤ r}, X r := X ∩ B r , X * r := X * ∩ B r and X r * := X * ∩ B r . We shall choose R small enough, so that bB r intersects X transversally for all 0 < r < R.In this paper we address the question of whether it is possible to solve the equation ∂u = f in X * r with L 2 -estimates for f ∈ L 2 p,q (X r ), ∂f = 0 in X * r . This problem was studied in [5] for conic singularities and ∂-closed, square integrable (0, 1) forms and later on in [4], for generic surfaces with an isolated singularity at 0 and for the same type of forms. Global analogues of this question for projective surfaces with isolated singularities were studied by Haskell [9], Nagase [12], Pardon [15], Pardon and Stern [16], and for compact Kähler spaces with isolated singularities by Ohsawa [13,14].There seems to be a "dichotomy of results" for our problem, depending on whether p + q < n or p + q > n. More precisely we have: Theorem 1.1. Let p + q < n, q > 0. There exists a closed subspace H of finite codimension in

Koppelman formulas on the A1-singularity

“…Solvability of the∂-equation on singular varieties has been studied in various articles in recent years, for example describing in certain senses explicitly the obstructions to solving the∂-equation in L 2 , see [FØV,ØV,R5]. Among these and other results, one can find examples when the∂-equation is not always locally solvable in L p , for example when p = 1 or p = 2.…”

Semiglobal results for $\overline∂$ on complex spaces with arbitrary singularities, Part II

“…Here the right-hand side distance is defined using the Euclidean metric in C When X is a pure n-dimensional complex analytic set in C N with an isolated singularity at 0, a version of Theorem 1.2 was obtained in [5]. In Proposition 3.1 in loc.cit., we obtained L 2 -solvability results for square-integrable ∂-closed (p, q)-forms with p + q < n, q > 0, compactly supported in a small neighborhood of the singular point.…”

Compactness of the ∂¯-Neumann operator on singular complex spaces

“…After the first period of intensive research on the L 2 -theory for the ∂-operator on singular spaces (see [27,31,14,26,32,11]), there has been good progress in this subject recently due to Pardon and Stern (see [33]), Diederich, Fornaess, Øvrelid and Vassiliadou (see [8,6,10,29,30]), Ruppenthal and Zeron (see [36][37][38][39]). On the other hand, the ∂-Neumann operator has not been studied on singular complex spaces yet.…”