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Kodaira vanishing theorems for Kahler Lie algebroids

Tengzhou Hu

Abstract

A Lie algebroid is a generalization of Lie algebra that provides a general framework to describe the symmetries of a manifold. In this paper, we generalize the Kodaira vanishing theorem, which is a basic result in complex geometry, to Kahler Lie algebroids. The generalization of the Kodaira vanishing theorem states that the kernel of the Lie algebroid Laplace operator on Lie algebroid positive line bundle-valued (p,q)-forms vanishes when p+q is sufficiently large. The most difficult part of the proof of the generalized Kodaira vanishing theorem is the generalization of the Kahler identities to Lie algebroids. In this paper, we provide an approach by using local coordinate calculation.

Kodaira vanishing theorems for Kahler Lie algebroids

Abstract

A Lie algebroid is a generalization of Lie algebra that provides a general framework to describe the symmetries of a manifold. In this paper, we generalize the Kodaira vanishing theorem, which is a basic result in complex geometry, to Kahler Lie algebroids. The generalization of the Kodaira vanishing theorem states that the kernel of the Lie algebroid Laplace operator on Lie algebroid positive line bundle-valued (p,q)-forms vanishes when p+q is sufficiently large. The most difficult part of the proof of the generalized Kodaira vanishing theorem is the generalization of the Kahler identities to Lie algebroids. In this paper, we provide an approach by using local coordinate calculation.
Paper Structure (27 sections, 53 theorems, 365 equations)

This paper contains 27 sections, 53 theorems, 365 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kähler Lie algebroid
  4. The Lie bracket and the Lie algebroid differential in terms of a local frame
  5. Lie algebroid Kodaira vanishing theorem
  6. Lie algebroid Kähler identities
  7. Lie algebroid Bochner–Kodaira–Nakano identity
  8. Main theory
  9. Application of a b-manifold
  10. Appendix I
  11. Calculation of
  12. Calculation of
  13. Calculation of
  14. Calculation of
  15. Proof of
  16. ...and 12 more sections

Key Result

Theorem 1

Let $(M,\mathfrak{g},\omega)$ be a Kähler Lie algebroid and $E\to M$ be a positive line bundle. The kernel of vanishes when $p+q>n$. The same result holds on a negative line bundle under the condition that $p+q<n$.

Theorems & Definitions (124)

  • Theorem : Lie algebroid Kodaira–Nakano vanishing theorem
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • proof
  • Definition 7
  • ...and 114 more