The $\chi_y$-genus, Chern number inequalities and signature

Abstract

This article has two parts. In the first part we introduce two positivity conditions for the modified $\chi_y$-genus on almost-complex manifolds and show that each of them implies a family of optimal Chern number inequalities. It turns out that many important K\"{a}hler and symplectic manifolds satisfy either of the two positivity conditions, and hence these Chern number inequalities hold true on them. In the second part we focus on the signature, a special value of the $\chi_y$-genus, of symplectic manifolds equipped with symplectic circle actions and give applications. Our results in this part unify and generalize various related results in the existing literature.