  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.
