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

TL;DR

Addresses whether nef and big line bundles must be semipositive on compact Kähler manifolds; uses Grauert’s two-dimensional example and Koike's first obstruction class $u_1(Y,X,L)$ to test semipositivity. The paper shows that the line bundle $L=p^*F\otimes[Y]$ on Grauert’s surface is nef and big but not semipositive by explicit obstruction calculation, thereby affirming Filip and Tosatti’s conjecture in dimension two. This work illustrates how obstruction theory can separate nef/bigness from semipositivity and motivates further study of currents in nef-and-big classes with bounded potentials. Overall, it provides a concrete, computable instance where nef and big do not imply semipositivity, enriching the landscape of positivity notions on compact Kähler manifolds.

Abstract

We study the relation between semipositivity, nefness, and bigness of line bundles on compact Kähler manifolds. Every nef and big line bundle on a compact Kähler 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.

Theorems & Definitions (9)

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• proof : Proof of Theorem \ref{['Theorem 3.2']}