Lefschetz decompositions for eigenforms on a K\"ahler manifold

TL;DR

This paper demonstrates that eigenspaces of the Laplacian on k-forms in a compact Kähler manifold admit Hodge and Lefschetz decompositions, revealing spectral relationships between different form degrees.

Contribution

It establishes the existence of Hodge and Lefschetz decompositions for Laplacian eigenspaces on Kähler manifolds and uncovers spectral inclusion properties.

Findings

Eigenspaces of the Laplacian have Hodge and Lefschetz decompositions.

The positive spectrum of $ riangle_k$ is contained in that of $ riangle_{k+1}$.

Spectral relationships between different degrees of forms are characterized.

Abstract

We show that the eigenspaces of the Laplacian $\Delta_k$ on $k$-forms on a compact K\"ahler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of $\Delta_k$ lies in the spectrum of $\Delta_{k+1}$.