On a fully nonlinear elliptic equation with differential forms

We introduce a fully nonlinear PDE with a differential form $\mathrm{\Lambda }$, which unifies several important equations in Kähler geometry including Monge-Ampère equations, J-equations, inverse ${\sigma }_k$ equations, and the deformed Hermitian Yang-Mills (dHYM) equation. We pose some natural positivity conditions on $\mathrm{\Lambda }$, and prove analytical and algebraic criterions for the solvability of the equation. Our results generalize previous works of G.Chen, J.Song, Datar-Pingali and others. As an application, we prove a conjecture of Collins-Jacob-Yau for the dHYM equation with small global phase.