Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Canonical sheaves at isolated canonical Gorenstein singularities

Jean Ruppenthal

Abstract

It is well known that the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$ of holomorphic square-integrable $n$-forms is a central tool in $L^2$-theory for the $\overline\partial$-operator on a singular complex space $X$ of pure dimension $n$. It was shown a few years ago that a comprehensive $L^2$-theory requires also the study of the sheaf $\mathcal{K}_X^s$ of holomorphic square-integrable $n$-forms with a Dirichlet boundary condition at the singular set of $X$. In the present paper, we describe and classify the behaviour of $\mathcal{K}_X^s$ in isolated canonical Gorenstein singularities, and give applications to the $L^2$-theory for the $\overline\partial$-operator on such spaces.

Canonical sheaves at isolated canonical Gorenstein singularities

Abstract

It is well known that the Grauert-Riemenschneider canonical sheaf of holomorphic square-integrable -forms is a central tool in -theory for the -operator on a singular complex space of pure dimension . It was shown a few years ago that a comprehensive -theory requires also the study of the sheaf of holomorphic square-integrable -forms with a Dirichlet boundary condition at the singular set of . In the present paper, we describe and classify the behaviour of in isolated canonical Gorenstein singularities, and give applications to the -theory for the -operator on such spaces.

Paper Structure

This paper contains 18 sections, 7 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Two $\overline{\partial}$-complexes on singular complex spaces
  3. The $\overline{\partial}_w$-$L^2$-complex
  4. The $\overline{\partial}_s$-$L^2$-complex
  5. On the domain of the $\overline{\partial}_s$-operator
  6. Coherence of $\mathcal{K}_X^s$ at isolated singularities
  7. $K_X^s$ in canonical Gorenstein singularities
  8. Cut-off functions
  9. Resolution of singularities
  10. Smooth representation of $\overline{\partial}$-cohomology
  11. $L^2$-forms and resolution of singularities
  12. Takegoshi vanishing and first part of Theorem \ref{['thm:main02']}
  13. $L^2$-Serre duality and second part of Theorem \ref{['thm:main02']}
  14. Singularities of the second kind
  15. A vanishing theorem of Kodaira--Grauert--Riemenschneider--Takegoshi type
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $X$ be a complex space of pure dimension with only isolated singularities. Then $\mathcal{K}_X^s$ is coherent.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Theorem 6.1