On the Solvability of General Inverse $\sigma_k$ Equations
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--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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2023
- DOI:
- 10.48550/arXiv.2310.05339
- arXiv:
- arXiv:2310.05339
- Bibcode:
- 2023arXiv231005339L
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Algebraic Geometry;
- 32Q15;
- 32W50;
- 35J60;
- 52A05
- E-Print:
- 45 pages, 3 figures