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The Alexandrov-Fenchel type inequalities, revisited

Ping Li

Abstract

Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by comparatively investigating them in several aspects. \emph{The principal result} in this article is a complete solution to the equality characterization problem of various Alexandrov-Fenchel type inequalities for intersection numbers of nef and big classes on compact Kähler manifolds, extending some earlier related results. In addition to this central result, we also give a geometric proof of the complex version of the Alexandrov-Fenchel inequality for mixed discriminants and a determinantal generalization of various Alexandrov-Fenchel type inequalities.

The Alexandrov-Fenchel type inequalities, revisited

Abstract

Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by comparatively investigating them in several aspects. \emph{The principal result} in this article is a complete solution to the equality characterization problem of various Alexandrov-Fenchel type inequalities for intersection numbers of nef and big classes on compact Kähler manifolds, extending some earlier related results. In addition to this central result, we also give a geometric proof of the complex version of the Alexandrov-Fenchel inequality for mixed discriminants and a determinantal generalization of various Alexandrov-Fenchel type inequalities.

Paper Structure

This paper contains 9 sections, 17 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and background materials
  3. Main results
  4. Proof of Theorem \ref{['AF complex mixed discriminant']}
  5. Proofs of Theorem \ref{['determinantal']} and its corollaries
  6. Proof of Theorem \ref{['determinantal']}
  7. Proofs of Corollaries \ref{['coro1']} and \ref{['coro2']}
  8. Proof of Theorem \ref{['equality case1']}
  9. Appendix

Key Result

Theorem 2.1

Suppose that $K_1,\ldots,K_n\in\mathcal{K}^n$. Then we have

Theorems & Definitions (28)

  • Theorem 2.1: AF inequality for mixed volumes
  • Remark 2.2
  • Corollary 2.3: General BM theorem for mixed volumes
  • Theorem 2.4: AF inequality for mixed discriminants
  • Remark 2.5
  • Proposition 2.6: AF type inequality for Kähler/nef classes
  • Remark 2.7
  • Theorem 3.1: Complex version of AF type inequality for mixed discriminants
  • Remark 3.2
  • Corollary 3.3
  • ...and 18 more