The Deformed Hermitian-Yang-Mills Equation and Level Sets of Harmonic Polynomials

TL;DR

This paper establishes algebraic criteria for the existence of certain level set graphs of harmonic polynomials and applies these results to stability conditions for solutions to the deformed Hermitian-Yang-Mills equation on specific projective bundles.

Contribution

It introduces a new algebraic characterization for level set graphs of harmonic polynomials and links this to stability conditions for the deformed Hermitian-Yang-Mills equation.

Findings

Derived necessary and sufficient conditions for level set graphs of harmonic polynomials.

Constructed a Kempf-Ness functional related to the problem.

Established a stability criterion for solutions on projective bundles.

Abstract

Suppose $v(x,y):\mathbb C\rightarrow \mathbb R$ is an entire harmonic polynomial with no critical points in the right half plane. Let $z_1, z_2\in\mathbb C$ lie on a level set of $v$ , and assume ${\rm Re}(z_2)>{\rm Re}(z_1)\geq0$. We give a necessary and sufficient condition, depending only on algebraic properties of the polynomial $v$, for when there exists a smooth real function $f$ whose graph $x+if(x)$ lies on a level curve of $v$ connecting $z_1$ to $z_2$. Inspired by GIT, we construct a Kempf-Ness functional on an appropriate function space, and prove the functional is bounded from below and proper if and only if a such a graph exists. As an application, we find a stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation on the family of projective bundles $ X_{r,m}:=\mathbb P(\mathcal O_{\mathbb P^m}\oplus \mathcal O_{\mathbb P^m}(-1)^{\oplus (r+1)})$ with Calabi Symmetry.