The ${\chi }_y$-genus, Chern number inequalities and signature

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 Kähler 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.