Sufficient bigness criterion for differences of two nef classes

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 Kähler (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–Ampère equation. This result is needed to extend to the Kähler 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.