On the Solvability of General Inverse $\sigma_k$ Equations

TL;DR

This paper proves the existence and uniqueness of solutions for a broad class of inverse $\sigma_k$ equations under certain conditions, confirming a conjecture related to the deformed Hermitian--Yang--Mills equation.

Contribution

It establishes a general solvability result for inverse $\sigma_k$ equations with a $C$-subsolution, covering many classical equations and confirming a key conjecture.

Findings

Existence and uniqueness of solutions under $C$-subsolution condition

Unified framework covering classical inverse $\sigma_k$ equations

Confirmation of the Collins--Jacob--Yau conjecture for the deformed Hermitian--Yang--Mills equation

Abstract

We prove that if there exists a $C$-subsolution to a constant coefficients strictly $\Upsilon$-stable general inverse $\sigma_k$ equation, then there exists a unique solution. As a consequence, this result covers all the analytical results of the classical strictly $\Upsilon$-stable general inverse $\sigma_k$ equations, for example, the complex Monge--Amp\`ere equation, the complex Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills equation, etc. Hence, we confirm an analytical conjecture by Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed Hermitian--Yang--Mills equation. Their conjecture states that the existence of a $C$-subsolution to a supercritical phase deformed Hermitian--Yang--Mills equation gives the solvability.