A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge-Amp`ere PDE with “non-constant coefficients” thus improving a result of Sun in the K"ahler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins-Sz`ekelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau.