The supercritical deformed Hermitian Yang--Mills equation on compact projective manifolds

TL;DR

This paper extends solvability results for the twisted deformed Hermitian Yang--Mills equation on compact projective manifolds, allowing for non-constant, slightly negative twisting functions, and establishes solutions under specific numerical conditions.

Contribution

It generalizes previous results to non-constant twisting functions and proves existence of solutions under new numerical criteria.

Findings

Extended solvability to non-constant, slightly negative twisting functions.

Proved existence of solutions on compact projective manifolds under numerical conditions.

Generalized Gao Chen's results to broader settings.

Abstract

In this paper, we extend a result of Gao Chen regarding the solvability of the twisted deformed Hermitian Yang-Mills equations on compact K\"ahler manifolds to allow for the twisting function to be non-constant and slightly negative in all dimensions. Using this result, we prove that the twisted supercritical dHYM equation on compact, projective manifolds can be solved provided certain numerical conditions are satisfied.