The deformed Hermitian Yang-Mills equation on three-folds

TL;DR

This paper proves the existence of solutions to the deformed Hermitian Yang-Mills equation on compact complex three-folds for a full range of phase parameters, using a new continuity path and subsolution condition.

Contribution

It introduces a novel continuity method and extends existence results for the deformed Hermitian Yang-Mills equation to the entire admissible phase range on three-folds.

Findings

Established existence for all phase parameters in (π/2, 3π/2)

Developed a new continuity path involving a generalized Monge-Ampère equation

Provided conditions under which solutions exist on compact complex three-folds

Abstract

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hat{\theta} \in (\frac{\pi}{2},\frac{3\pi}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Amp\`ere equation with mixed sign coefficients.