Spectral bundles on Abelian varieties, complex projective spaces and Grassmannians

TL;DR

This paper analyzes the spectral properties of Laplacians on Abelian varieties, projective spaces, and Grassmannians, using a physics-inspired method to explicitly describe eigensections and their dimensions.

Contribution

It introduces a novel method to convert eigensections into holomorphic sections, providing explicit formulas and structures for spectral bundles on these complex manifolds.

Findings

Explicit expression of eigensections via theta functions

Holomorphic structure on spectral bundles over dual Abelian varieties

Dimension formulas for eigensections on projective spaces

Abstract

In this paper we study the spectral analysis of Bochner-Kodaira Laplacians on an Abelian variety, complex projective space $\mathbb{P}^{n}$ and a Grassmannian with a holomorphic line bundle. By imitating the method of creation and annihilation operators in physics, we convert those eigensections (of the \textquotedblleft higher energy" level) into holomorphic sections (of the \textquotedblleft lowest energy" level). This enables us to endow these spectral bundles, which are defined over the dual Abelian variety, with natural holomorphic structure. Using this conversion expressed in a concrete way, all the higher eigensections are explicitly expressible using holomorphic sections formed by theta functions. Moreover, we give an explicit formula for the dimension of the space of higher-level eigensections on $\mathbb{P}^{n}$ through vanishing theorems and the Hirzebruch-Riemann-Roch theorem. These give a theoretical study related to some problems newly discussed by string theorists using numerical analysis. Some partial results on Grassmannians are proved and some directions for future research are indicated.