A note on Teissier problem for nef classes

TL;DR

This paper investigates the equality conditions of the Khovanskii-Teissier inequality for nef (1,1)-classes on compact Kähler manifolds, extending previous results to cases where classes are only assumed to be nef and not necessarily big.

Contribution

It establishes the equality characterization for nef (1,1)-classes and extends the results to non-nef classes, demonstrating optimality through constructed examples.

Findings

Characterization of equality cases for nef (1,1)-classes

Extension of results to non-nef classes

Examples showing optimality of the results

Abstract

Teissier problem aims to characterize the equality case of Khovanskii-Teissier type inequality for $(1,1)$-classes on a compact K\"ahler 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.