Hyperbolic localization and Lefschetz fixed point formulas for higher-dimensional fixed point sets

TL;DR

This paper extends Lefschetz fixed point formulas to higher-dimensional fixed point sets using hyperbolic localization, confirming a conjecture for smooth components and providing explicit examples.

Contribution

It introduces a new approach to compute local contributions in Lefschetz formulas for higher-dimensional fixed points, confirming a conjecture of Goresky-MacPherson.

Findings

Local contributions are expressed by constructible functions via hyperbolic localizations.

The conjecture of Goresky-MacPherson is affirmed for smooth fixed point components.

Explicit examples demonstrate the effectiveness of the method.

Abstract

We study Lefschetz fixed point formulas for constructible sheaves with higher-dimensional fixed point sets. Under fairly weak assumptions, we prove that the local contributions from them are expressed by some constructible functions associated to hyperbolic localizations. This gives an affirmative answer to a conjecture of Goresky-MacPherson in particular for smooth fixed point components. In the course of the proof, the new Lagrangian cycles introduced in our previous paper will be effectively used. Moreover we show various examples for which local contributions can be explicitly determined by our method.