On the Spectrum of geometric operators on K\"ahler manifolds

TL;DR

This paper investigates the spectral properties of geometric operators on compact Kähler manifolds, revealing conditions under which quantum ergodicity holds when symmetries are considered, especially in negatively curved cases.

Contribution

It determines the asymptotic distribution of irreducible representations of a Lie-superalgebra on eigenforms and establishes quantum ergodicity under certain symmetry and ergodicity conditions.

Findings

Asymptotic distribution of representations is characterized.

Quantum ergodicity holds when symmetries are accounted for.

Results apply to negatively curved Kähler manifolds of odd complex dimension.

Abstract

On a compact K\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin^c-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved K\"ahler manifolds of odd complex dimension.