A note on Demailly's transcendental Morse inequalities conjecture

TL;DR

This paper provides a partial solution to Demailly's transcendental Morse inequalities conjecture by showing that under certain conditions, a specific cohomology class contains a Kähler current on a compact Hermitian manifold.

Contribution

It proves that if a nef class admits a bounded quasi-plurisubharmonic potential, then the difference of two nef classes contains a Kähler current, advancing understanding of Demailly's conjecture.

Findings

Establishes a condition for a class to contain a Kähler current.

Provides a partial proof of Demailly's transcendental Morse inequalities.

Connects the existence of potentials to geometric properties of classes.

Abstract

Let $(X,\omega)$ be an $n$-dimensional compact Hermitian manifold with $\omega$ a pluriclosed Hermitian metric, i.e. $dd^c\omega=0$. Let $\{\alpha\},\{\beta \}\in H^{1,1}_{BC}(X,\mathbb R)$ be two nef classes, such that $\alpha^n-n\alpha^{n-1}\cdot\beta>0$. In this short note, we prove that if there is a bounded quasi-plurisubharmonic potential $\rho$, such that $\alpha+dd^c\rho\geq 0$ in the weak sense of currents, then the class $\{\alpha-\beta\}$ contains a K\"ahler current. This gives a partial solution of Demailly's transcendental Morse inequalities conjecture.