On a fully nonlinear elliptic equation with differential forms

We introduce a fully nonlinear PDE with a differential form, which unifies several important equations in Kähler geometry including Monge-Ampère equations, J-equations, inverse ${\sigma }_k$ equations, and deformed Hermitian Yang-Mills (dHYM) equations. We pose some natural positivity conditions on Λ, and prove analytical and algebraic criterion 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 dHYM equations with small global phase.