Weitzenböck Remainder Spectrum on Rational Homogeneous Varieties

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 ${\mathrm{Spin}}^c$ 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 ${\mathrm{Spin}}^c$ 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 ${\mathrm{Spin}}^c$ structures on rational homogeneous varieties which admit harmonic spinors. In this last setting, we present an explicit formula for the index of the associated ${\mathrm{Spin}}^c$ Dirac operator in terms of Lie theory.

22 pages. Affiliations updated. Comments are welcome!