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A note on Teissier problem for nef classes

Yashan Zhang

Abstract

Teissier problem aims to characterize the equality case of Khovanskii-Teissier type inequality for $(1,1)$-classes on a compact Kähler manifold. When each of the involved $(1,1)$-classes is assumed to be nef and big, this problem has been solved by the previous works of Boucksom-Favre-Jonsson, Fu-Xiao and Li. In this note, we shall settle the case that the involved $(1,1)$-classes are just assumed to be nef. We also extend the results to some settings where some of the $(1,1)$-classes are not necessarily nef. By constructing examples, it is shown that our results are optimal.

A note on Teissier problem for nef classes

Abstract

Teissier problem aims to characterize the equality case of Khovanskii-Teissier type inequality for -classes on a compact Kähler manifold. When each of the involved -classes is assumed to be nef and big, this problem has been solved by the previous works of Boucksom-Favre-Jonsson, Fu-Xiao and Li. In this note, we shall settle the case that the involved -classes are just assumed to be nef. We also extend the results to some settings where some of the -classes are not necessarily nef. By constructing examples, it is shown that our results are optimal.

Paper Structure

This paper contains 15 sections, 12 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Khovanskii-Teissier inequalities and Teissier problem
  3. Results on Teissier Problem \ref{['teissier-prob']}
  4. Organization
  5. Examples
  6. limits of pairs satisfying Hodge index theorem
  7. A generalized $m$-positivity
  8. Hodge index theorems
  9. Boundary elements in $\overline{\mathscr{H}}$
  10. Proofs of the results
  11. A general result
  12. Proof of Theorem \ref{['thm1']}
  13. Proof of Theorem \ref{["thm1'"]}
  14. Extensions
  15. Results on Problem \ref{["teissier-prob'"]}

Key Result

Theorem 1.2

Let $X$ be an $n$-dimensional compact Kähler manifold, and $[\alpha_1],...,[\alpha_{n-2}],[\alpha],[\beta]\in H^{1,1}(X,\mathbb R)$. Assume either of the followings is satisfied. Then if and only if $[\alpha_1\wedge...\wedge\alpha_{n-2}\wedge\alpha]$ and $[\alpha_1\wedge...\wedge\alpha_{n-2}\wedge\beta]$ are proportional.

Theorems & Definitions (35)

  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 2.1: Theorem \ref{['thm1']} is optimal
  • Remark 2.2
  • Example 2.3
  • ...and 25 more