Calculations of the Hirzebruch ${\chi }_y$ genera of symmetric products by the holomorphic Lefschetz formula

Abstract

We calculate the Hirzebruch ${\chi }_y$ and ${\hat{\chi }}_y$-genera of symmetric products of closed complex manifolds by the holomorphic Lefschetz formula of Atiyah and Singer \cite{Ati-Sin}. Such calculation rederive some formulas proved in an earlier paper \cite{Zho} by a different method.