Smooth and Rough Positive Currents

TL;DR

This paper investigates various notions of semipositivity for (1,1) classes on K3 surfaces, showing conditions under which classes admit smooth representatives and providing counterexamples for irrational nef classes.

Contribution

It establishes that big and nef classes on K3 surfaces are semiample with smooth semipositive representatives, and constructs examples of nef classes lacking smooth positive currents.

Findings

Big and nef classes are semiample and contain smooth semipositive representatives.

Existence of irrational nef classes with no smooth positive current outside a proper analytic subset.

Counterexamples of nef R-divisors on projective K3 surfaces that are not semipositive.

Abstract

We study the different notions of semipositivity for (1,1) cohomology classes on K3 surfaces. We first show that every big and nef class (and every nef and rational class) is semiample, and in particular it contains a smooth semipositive representative. By contrast, we show that there exist irrational nef classes with no closed positive current representative which is smooth outside a proper analytic subset. We use this to answer negatively two questions of the second-named author. Using a result of Cantat and Dupont, we also construct examples of projective K3 surfaces with a nef R-divisor which is not semipositive.