\((1,1)\) forms with specified Lagrangian phase: \textit{a priori} estimates and algebraic obstructions (Q1988447)

The deformed Hermitian Yang-Mills equation is the mirror object for special Lagrangian sections of a semiflat SYZ fibration. This paper studies the deformed HYM equation under certain phase angle constraints, and the main result is to prove the existence of solution once there is a given subsolution. This is done via a two-stage continuity method, and the most difficult part is to obtain a priori \(C^2\)-estimate for the prescribed Lagrangian phase equation, which is rather delicate because the phase operator is not necessarily concave. The techniques are inspired by related works in Kähler geometry, such as the study of the \(J\)-equation. The paper also notices that the existence of the subsolution imposes cohomological constraints, which curiously resemble Bridgeland stability conditions. It then conjectures that the cohomological constraints are also sufficient for the solvability of the equation. The reader needs to be aware that the field is developing rapidly, and in particular a variant version of the conjecture is claimed in [\textit{G. Chen}, ``On \(J\)-equation'', Preprint, \url{arXiv:1905.10222}; ``Supercritical deformed Hermitian-Yang-Mills equation'', Preprint, \url{arXiv:2005.12202}].