Parabolic nef currents on hyperkaehler manifolds

TL;DR

This paper investigates nef (1,1)-currents on hyperkähler manifolds with parabolic cohomology classes, showing that their Lelong sets are coisotropic and exploring implications for generic and Picard rank 1 cases.

Contribution

It establishes that all Lelong sets of nef currents with parabolic classes are coisotropic and demonstrates the existence of non-trivial coisotropic subvarieties in certain hyperkähler manifolds.

Findings

Lelong sets of nef currents are coisotropic

All Lelong numbers vanish on generic hyperkähler manifolds

Existence of non-trivial coisotropic subvarieties in Picard rank 1 hyperkähler manifolds

Abstract

Let M be a compact, holomorphically symplectic Kahler manifold, and $\eta$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $\eta$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $\eta$ are coisotropic. When M is generic, this is used to show that all Lelong numbers of $\eta$ vanish. We prove that any hyperkahler manifold with Pic(M) of rank 1 has non-trivial coisotropic subvarieties, if a generator of Pic(M) is parabolic.