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Weitzenböck Remainder Spectrum on Rational Homogeneous Varieties

Eder M. Correa, Lucas Almeida, Samuel Wainer

Abstract

In this paper, we precisely describe the spectrum of closed invariant $(1,1)$-forms viewed as an operator acting on complex spinor bundles over rational homogeneous varieties. Using this result, we describe the spectrum of the Weitzenböck remainder of ${\rm{Spin}}^{c}$ Dirac operators on rational homogeneous varieties. In particular, we present an explicit formula for their smallest eigenvalue. As a byproduct, we obtain a new lower bound for the eigenvalues of the ${\rm{Spin}}^{c}$ Dirac operator, expressed in terms of Lie-theoretic data. Additionally, combining the Atiyah-Singer index theorem with the Borel-Weil-Bott theorem, we provide a complete classification of ${\rm{Spin}}^{c}$ structures on rational homogeneous varieties which admit harmonic spinors. In this last setting, we present an explicit formula for the index of the associated ${\rm{Spin}}^{c}$ Dirac operator in terms of Lie theory.

Weitzenböck Remainder Spectrum on Rational Homogeneous Varieties

Abstract

In this paper, we precisely describe the spectrum of closed invariant -forms viewed as an operator acting on complex spinor bundles over rational homogeneous varieties. Using this result, we describe the spectrum of the Weitzenböck remainder of Dirac operators on rational homogeneous varieties. In particular, we present an explicit formula for their smallest eigenvalue. As a byproduct, we obtain a new lower bound for the eigenvalues of the Dirac operator, expressed in terms of Lie-theoretic data. Additionally, combining the Atiyah-Singer index theorem with the Borel-Weil-Bott theorem, we provide a complete classification of structures on rational homogeneous varieties which admit harmonic spinors. In this last setting, we present an explicit formula for the index of the associated Dirac operator in terms of Lie theory.
Paper Structure (13 sections, 13 theorems, 111 equations)

This paper contains 13 sections, 13 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Spin$^c$-structures and Dirac Operators
  3. Generalities on Spin$^c$-structures
  4. Spin$^{c}$ Dirac operator and Weitzenböck remainder
  5. Atiyah-Singer index theorem
  6. Rational homogeneous varieties
  7. The Picard group of flag varieties
  8. The first Chern class of flag varieties
  9. Borel-Weil-Bott theorem
  10. Proof of main results
  11. Proof of Theorem A
  12. Proof of Theorem B
  13. Proof of Theorem C

Key Result

Theorem A

Let $(X_{P},\omega)$ be a rational homogeneous variety, such that $\omega$ is a $G$-invariant Kähler metric, and let ${\bf{L}} \in {\rm{Spin}}^{c}(X_{P})$ be a ${\rm{Spin}}^{c}$ structure with associated spinor bundle $\mathcal{S}(X_{P})$. Then, the spectrum of the operator $\theta \colon \Gamma^{\i such that $\phi([\theta]), \phi([\omega]) \in \Lambda_{P} \otimes \mathbbm{R}$.

Theorems & Definitions (29)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4: Schrödinger-Lichnerowicz formula
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • ...and 19 more