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

We prove that if there exists a $C$-subsolution to a constant coefficients strictly $\mathrm{{\rm Y}}$-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 $\mathrm{{\rm Y}}$-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.