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Lie groupoid Riemann-Roch-Hirzebruch theorem and applications

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 introduce Lie algebroid index theory and study the Lie algebroid Dolbeault operator. We also introduce Connes' index theory on regular foliated manifolds to obtain a generalized Riemann-Roch theorem on manifolds with regular foliation. We show that the topological side of Connes' index theory can be identified with the topological side of Lie algebroid index theory. Finally, we introduce Lie algebroid Kodaira vanishing theorem, and provide some applications and examples. The Lie algebroid Kodaira vanishing theorem can be used on the analytic side of Connes' theorem to attest to the criterion of a positive line bundle from its topological information.

Lie groupoid Riemann-Roch-Hirzebruch theorem and applications

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 introduce Lie algebroid index theory and study the Lie algebroid Dolbeault operator. We also introduce Connes' index theory on regular foliated manifolds to obtain a generalized Riemann-Roch theorem on manifolds with regular foliation. We show that the topological side of Connes' index theory can be identified with the topological side of Lie algebroid index theory. Finally, we introduce Lie algebroid Kodaira vanishing theorem, and provide some applications and examples. The Lie algebroid Kodaira vanishing theorem can be used on the analytic side of Connes' theorem to attest to the criterion of a positive line bundle from its topological information.
Paper Structure (15 sections, 15 theorems, 90 equations)

This paper contains 15 sections, 15 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie groupoids and Kähler Lie algebroids
  4. Lie algebroid characteristic classes and Chern-Weil theory
  5. Index theory on Lie algebroids
  6. G-invariant differential operator and index theory
  7. Application of index theory to the Lie algebroid Dolbeault operator
  8. Riemann-Roch formula for regular foliation
  9. Introduction of foliated manifolds
  10. Riemann-Roch formula for regular foliation
  11. Difference between Riemann-Roch formula for regular foliation and Lie algebroid
  12. Applications and examples
  13. Lie algebroid Kodaira vanishing theorem
  14. Application of index theory and the vanishing theorem for foliated Kähler manifolds
  15. An example of Kähler foliated manifolds

Key Result

Theorem 1

Let $\mathfrak{g}\to M$ be an integrable complex Lie algebroid and $E\to M$ be a holomorphic vector bundle. If we set $\Phi_{\mathfrak{g}}(v)=\Omega_{\mathfrak{g}}\in\mathrm{H}^{0}_{}(\mathfrak{g};L_{\mathfrak{g}})$ on the analytic side, then the index formula for the Lie algebroid Dolbeault operato

Theorems & Definitions (34)

  • Theorem
  • Proposition
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • proof
  • Theorem 1
  • ...and 24 more