Weitzenb\"{o}ck Remainder Spectrum on Rational Homogeneous Varieties

TL;DR

This paper characterizes the spectrum of Weitzenböck remainder operators on rational homogeneous varieties, providing explicit eigenvalue formulas, bounds, and classifications of structures admitting harmonic spinors using Lie theory.

Contribution

It offers a precise spectral description of invariant forms and Dirac operators on rational homogeneous varieties, including explicit eigenvalues and classifications based on Lie-theoretic data.

Findings

Explicit spectrum formulas for invariant forms and Dirac operators.

A new lower bound for eigenvalues of the ${\rm{Spin}}^{c}$ Dirac operator.

Complete classification of ${\rm{Spin}}^{c}$ structures admitting harmonic spinors.

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 Weitzenb\"{o}ck 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.