A note on the deformed Hermitian Yang-Mills PDE

TL;DR

This paper establishes a priori estimates for a generalized Monge-Ampère PDE, applies these to the deformed Hermitian Yang-Mills equation to prove existence results, and extends related theorems to toric varieties, addressing a key conjecture.

Contribution

It improves existing estimates for a generalized PDE, applies these to the dHYM equation, and generalizes a theorem to toric varieties, advancing understanding of complex geometric PDEs.

Findings

Established a priori estimates for a generalized Monge-Ampère PDE.

Proved existence results for the deformed Hermitian Yang-Mills equation under certain conditions.

Extended a theorem to toric varieties, addressing a significant conjecture.

Abstract

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.