The $L^2-$Atiyah-Bott-Lefschetz theorem on manifolds with conical singularities. A heat kernel approach

TL;DR

This paper proves an Atiyah-Bott-Lefschetz theorem for $L^2$-Lefschetz numbers on manifolds with conical singularities using heat kernel methods, extending classical fixed point formulas to singular spaces.

Contribution

It introduces a heat kernel approach to establish an $L^2$-Atiyah-Bott-Lefschetz theorem for elliptic complexes on singular manifolds, including the de Rham complex.

Findings

Established an $L^2$-Lefschetz fixed point formula for conical singularities.

Applied the theorem specifically to the de Rham complex.

Extended classical fixed point theorems to singular geometric settings.

Abstract

Using an approach based on the heat kernel we prove an Atiyah-Bott-Lefschetz theorem for the $L^2-$Lefschetz numbers associated to an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.