On a fully nonlinear elliptic equation with differential forms

TL;DR

This paper introduces a unified fully nonlinear PDE involving differential forms that encompasses key equations in Kähler geometry, providing new solvability criteria and confirming a conjecture for the deformed Hermitian Yang-Mills equation.

Contribution

It formulates a general nonlinear PDE unifying several important equations in Kähler geometry and establishes new analytical and algebraic criteria for its solvability.

Findings

Unified several key equations in Kähler geometry into one PDE.

Provided criteria for solvability under positivity conditions.

Proved a conjecture for the deformed Hermitian Yang-Mills equation with small phase.

Abstract

We introduce a fully nonlinear PDE with a differential form $\Lambda$, which unifies several important equations in K\"ahler geometry including Monge-Amp\`ere equations, J-equations, inverse $\sigma_{k}$ equations, and the deformed Hermitian Yang-Mills (dHYM) equation. We pose some natural positivity conditions on $\Lambda$, and prove analytical and algebraic criterions for the solvability of the equation. Our results generalize previous works of G.Chen, J.Song, Datar-Pingali and others. As an application, we prove a conjecture of Collins-Jacob-Yau for the dHYM equation with small global phase.