On the Solvability of General Inverse ${\sigma }_k$ Equations

Abstract

We prove that if there exists a $C$-subsolution to a constant coefficients strictly ${\rm Y}$-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 ${\rm Y}$-stable general inverse ${\sigma }_k$ equations, for example, the complex Monge--Ampère 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.

45 pages, 3 figures