On the Non-Semipositivity of a Nef and Big Line Bundle on Grauert's Example

TL;DR

This paper investigates the relationship between nefness, bigness, and semipositivity of line bundles on compact Kähler manifolds, providing explicit examples and computations that challenge previous assumptions, especially in dimension two.

Contribution

It presents the first explicit example of a nef and big line bundle on a surface that is not semipositive, confirming conjectures and extending understanding of line bundle properties.

Findings

Constructed an explicit nef and big line bundle on Grauert's example that is not semipositive.

Computed the first obstruction class to demonstrate non-semipositivity.

Confirmed the existence of such line bundles in dimension two, previously known only in higher dimensions.

Abstract

We study the relation between semipositivity, nefness, and bigness of line bundles on compact K\"ahler manifolds. Every nef and big line bundle on a compact K\"ahler manifold $X$ is positive when ${\rm dim}\,X = 1$. Kim constructed an explicit example of a nef and big line bundle that is not semipositive in the case ${\rm dim}\,X \ge 3$. Motivated by a conjecture of Filip and Tosatti, we then focus on the case of dimension two. In this talk, we show that the line bundle on Grauert's example is nef and big but not semipositive, by explicitly computing its first obstruction class, which was originally introduced by Koike as a generalization of the Ueda class.