Sufficient Bigness Criterion for Differences of Two Nef Classes

TL;DR

This paper proves a key part of Demailly's conjecture on Morse inequalities for differences of nef classes on compact Kähler manifolds, improving previous bounds and introducing new estimation techniques.

Contribution

It establishes the qualitative part of Demailly's conjecture with an optimal constant, extending the Boucksom-Demailly-Paun-Peternell cone duality theorem to transcendental classes.

Findings

Proved the qualitative part of Demailly's conjecture with optimal constant n.

Extended cone duality theorem to Kähler and transcendental classes.

Developed new methods for estimates in Monge-Ampère equations.

Abstract

We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact K\"ahler (and even more general) manifold. The result improves on an earlier one by J. Xiao whose constant $4n$ featuring in the hypothesis is now replaced by the optimal and natural $n$. Our method follows arguments by Chiose as subsequently used by Xiao up to the point where we introduce a new way of handling the estimates in a certain Monge-Amp\`ere equation. This result is needed to extend to the K\"ahler case and to transcendental classes the Boucksom-Demailly-Paun-Peternell cone duality theorem if one is to follow these authors' method and was conjectured by them.