A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler 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–Szèkelyhidi on toric varieties and use it to address a conjecture of Collins–Jacob–Yau.