A note on the cone of mobile curves

TL;DR

This paper explores the duality of the cone of mobile curves in projective and Kähler manifolds, showing that a conjecture implies a specific cone equality, and proves this equality at the level of degree functions without assuming the conjecture.

Contribution

It demonstrates that the duality conjecture implies a cone equality and proves this equality at the degree function level independently of the conjecture.

Findings

The duality conjecture implies the cone of mobile curves equals the cone of classes represented by positive smooth forms.

The cone equality holds at the level of degree functions even without assuming the conjecture.

The paper extends the understanding of the cone of mobile curves in Kähler geometry.

Abstract

S. Boucksom, J.-P. Demailly, M. Paun and Th. Peternell proved that the cone of mobile curves ME(X) of a projective complex manifold X is dual to the cone generated by classes of effective divisors and conjectured an extension of this duality in the Kaehler set-up. We show that their conjecture implies that ME(X) coincides with the cone of integer classes represented by closed positive smooth (n-1,n-1)-forms. Without assuming the validity of the conjecture we prove that this equality of cones still holds at the level of degree functions.