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

TL;DR

This paper introduces positivity conditions for the modified $hi_y$-genus on almost-complex manifolds, deriving optimal Chern number inequalities, and explores the signature of symplectic manifolds with circle actions, unifying previous results.

Contribution

It establishes new positivity conditions leading to optimal Chern number inequalities and generalizes results on signatures of symplectic manifolds with circle actions.

Findings

Positivity conditions imply optimal Chern number inequalities.

Many Ka4hler and symplectic manifolds satisfy these conditions.

Results unify and extend existing theorems on signatures.

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.