Hirzebruch class and Bialynicki-Birula decomposition

TL;DR

This paper explores the relationship between homological and geometric localization methods for algebraic torus actions on complex varieties, focusing on Hirzebruch genus decompositions and their extensions to singular varieties.

Contribution

It establishes a connection between homological and geometric localization approaches for torus actions, providing a unified perspective on Hirzebruch genus decompositions.

Findings

Homological and geometric decompositions are connected via a limit process.

Results extend to singular algebraic varieties.

The study clarifies the structure of Hirzebruch $\

Abstract

Suppose an algebraic torus acts on a complex algebraic variety $X$. Then a great part of information about global invariants of $X$ are encoded in some data localized around the fixed points. The goal of this note is to present a connection between two approaches to localization for $C^*$-action. The homological results are related to $S^1$-action, while from $R^*_{>0}$-action we obtain a geometric decomposition. We study the resulting decompositions of Hirzebruch $\chi_y$-genus and their relative versions. We show that via a limit process the second decomposition is obtained from the first one. The results are also valid for singular varieties.