Calculations of the Hirzebruch $\chi_y$ genera of symmetric products by the holomorphic Lefschetz formula

TL;DR

This paper computes the Hirzebruch $\chi_y$ and $\hat{\chi}_y$-genera of symmetric products of complex manifolds using the holomorphic Lefschetz formula, providing an alternative derivation of known formulas.

Contribution

It introduces a new method based on the holomorphic Lefschetz formula to calculate these genera, offering an alternative to previous approaches.

Findings

Derived formulas for Hirzebruch $\chi_y$-genera of symmetric products

Reproduced earlier results through a different method

Enhanced understanding of the connection between Lefschetz formula and complex manifold invariants

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.