A Note on Demailly’s Transcendental Morse Inequalities Conjecture

Abstract

Let (X, ω) be an n-dimensional compact Hermitian manifold with ω a pluri-closed Hermitian metric, i.e., dd^cω = 0. Let $\left\{\alpha \right\},\left\{\beta \right\} \in H_{BC}^{1,1}\left({X,\mathbb{R}} \right)$ be two nef classes, such that αⁿ − nαⁿ⁻¹ · β > 0. In this short note, we prove that if there is a bounded quasi-plurisubharmonic potential ρ, such that α + dd^cρ ≥ 0 in the weak sense of currents, then the class {α − β} contains a Kähler current. This gives a partial solution of Demailly’s transcendental Morse inequalities conjecture.

Keywords

Demailly’s transcendental Morse inequalities conjecture / Hermitian manifold / pluri-closed metric / Kähler current / nef class / complex Monge–Ampère equation / 32U05 / 32W20 / 53C55

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Peking University