On the Solvability of General Inverse $\sigma_k$ Equations

Chao-Ming Lin

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arXiv · 2310.05339

On the Solvability of General Inverse $\sigma_k$ Equations

Chao-Ming Lin

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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.

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Chao-Ming Lin. 2023-10-09. On the Solvability of General Inverse $\sigma_k$ Equations. https://arxiv.org/abs/2310.05339

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