A note on the supercritical deformed Hermitian-Yang-Mills equation

TL;DR

This paper investigates the set of real (1,1)-classes on compact Kähler manifolds that admit solutions to the supercritical deformed Hermitian-Yang-Mills equation, showing it is both open and closed but not necessarily equal to the entire class set.

Contribution

It proves that the classes admitting solutions form a subset that is both open and closed, and provides examples that this subset can be strictly smaller, disproving a previous conjecture.

Findings

The solution set is both open and closed within the numerical condition set.

Counterexamples show the solution set can be a proper subset.

Disproves the conjecture by Collins-Jacob-Yau.

Abstract

We show that on a compact K\"ahler manifold all real $(1,1)$-classes admitting solutions to the supercritical deformed Hermitian-Yang-Mills equation form a both open and closed subset of those which satisfy the numerical condition proposed by Collins-Jacob-Yau. More importantly, we show by examples that it can be a proper subset. This disproves a conjecture made by Collins-Jacob-Yau.