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The $χ_y$-genus, Chern number inequalities and signature

Ping Li, Yibo Ren

Abstract

This article has two parts. In the first part we introduce two positivity conditions for the modified $χ_y$-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 $χ_y$-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.

The $χ_y$-genus, Chern number inequalities and signature

Abstract

This article has two parts. In the first part we introduce two positivity conditions for the modified -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 -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.

Paper Structure

This paper contains 23 sections, 18 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Positivity conditions on the $\chi_y$-genus and Chern number inequalities
  3. The Hirzebruch $\chi_y$-genus
  4. The modified version
  5. Positivity conditions and Chern number inequalities
  6. Proof of Theorem \ref{['main']}
  7. The $-1$-phenomenon
  8. Proof of Theorem \ref{['main']}
  9. Applications to Kähler manifolds
  10. Kähler manifolds with $\chi$-positivity
  11. Signed $\chi$-positive Kähler manifolds
  12. Applications to symplectic manifolds
  13. Almost-complex $S^1$-manifolds and the $\chi_{-y}$-genus
  14. Symplectic and Hamiltonian $S^1$-actions
  15. Applications to symplectic $S^1$-manifolds
  16. ...and 8 more sections

Key Result

Theorem 2.2

Let $M$ be an almost-complex manifold, which is either $\chi$-positive or signed $\chi$-positive. Take $\epsilon=1$ (resp. $\epsilon=-1$) if $M$ is $\chi$-positive (resp. signed $\chi$-positive). Then $M$ satisfies $[\frac{n}{2}]+1$ optimal Chern number inequalities where $A_i(c_1,\ldots,c_n)$ can be determined by a recursive algorithm and the first three Chern number inequalities read as follows

Theorems & Definitions (43)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Example 4.3
  • Theorem 4.4
  • ...and 33 more