$L^2$ Fr\"olicher inequalities

TL;DR

This paper establishes a new inequality relating $L^2$ Betti and Hodge numbers on complex manifolds, using spectral projector techniques, and offers a novel proof of the classical Fr"olicher inequality without spectral sequences.

Contribution

It introduces an $L^2$ Fr"olicher inequality for normal coverings of compact complex manifolds and provides a spectral projector-based proof of the classical inequality.

Findings

Proved an $L^2$ Fr"olicher inequality for complex manifolds.

Developed a spectral projector method for the proof.

Provided a new proof of the classical Fr"olicher inequality.

Abstract

We prove a Fr\"olicher inequality between $L^2$ Betti and $L^2$ Hodge numbers on normal coverings of compact complex manifolds. This is achieved by building an injection using suitable spectral projectors associated to the self adjoint operators $(D_h)^2:=(\overline\partial+\overline\partial^*+h\partial+h\partial^*)^2$ for $h\in[0,1]$. As a by-product, we find a new proof of the classical Fr\"olicher inequality on compact complex manifolds which does not rely at all on spectral sequences.