Degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds and applications II

TL;DR

This paper develops solutions to degenerate complex Monge-Ampère equations on compact Hermitian manifolds within a specific cohomology space, and applies these results to address longstanding conjectures in complex geometry.

Contribution

It establishes existence and stability of solutions to degenerate Monge-Ampère equations on Hermitian manifolds in the Bott-Chern space, advancing understanding of complex geometric PDEs.

Findings

Solutions to degenerate Monge-Ampère equations are constructed within the Bott-Chern space.

Stability results for these solutions are derived.

Partial resolutions to the Tosatti-Weinkove and Demailly-Pun conjectures are provided.

Abstract

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$, equipped with a Hermitian metric $\omega$. Let $\beta$ be a possibly non-closed smooth $(1,1)$-form on $X$ such that $\int_X\beta^n>0$. Assume that there is a bounded $\beta$-plurisubharmonic function $\rho$ on $X$ and $\underline{\mathrm{Vol}}(\beta) > 0$. In this paper, we establish solutions to the degenerate complex Monge-Amp\`ere equations on $X$ within the Bott-Chern space of $\beta$ (as introduced by Boucksom-Guedj-Lu) and derive stability results for these solutions. As applications, we provide partial resolutions to the extended Tosatti-Weinkove conjecture and Demailly-P\u aun conjecture.