Hirzebruch class and Białynicki-Birula decomposition (Q2410924)

Suppose an algebraic torus \(\mathbb C^\ast = \mathbb C -\{0\}\) acts on a complex algebraic variety \(X\). We assume that the action is algebraic. Then a great part of the information about global invariants of \(X\) is encoded in some data localized around the fixed points. The aim of this paper is to present a connection between two approaches to localization for \(\mathbb C^\ast\)-action. The homological results are related to the circle \(\mathbb S^1\)-action, while from positive \(\mathbb R^\ast\)-action we obtain a geometric decomposition. From this point of view what really matters are the invariants of fixed point set components and information about the characters of the torus acting on the normal bundles. All the data can be deduced from Bia lynicki-Birula decomposition of the fixed point sets for the finite subgroups of \(\mathbb C^\ast\). The main goal is to express a relation between invariants of the Bia lynicki Birula cell and the localized Hirzebruch class. The procedure works for smooth algebraic varieties. A part of the construction can be carried out for singular varieties. He discuss two decompositions of the Hirzebruch \(\chi_y\)-genus: the first one related to the circle \(\mathbb S^1\)-action, the second one related to positive \(\mathbb R^\ast\) flow. The author shows that via a limit process the second decomposition is obtained from the first one.