The Deformed Hermitian--Yang--Mills Equation, the Positivstellensatz, and the Solvability

TL;DR

This paper proves the conjecture that in four-dimensional compact Kähler manifolds, the existence of a C-subsolution guarantees the solvability of the deformed Hermitian--Yang--Mills equation when the phase parameter is near the supercritical value.

Contribution

It confirms Collins--Jacob--Yau's conjecture for complex dimension four, extending the conditions under which the deformed Hermitian--Yang--Mills equation is solvable.

Findings

Confirmed the conjecture for complex dimension four.

Established solvability when phase parameter is close to the supercritical value.

Extended existence results beyond previously known conditions.

Abstract

Let $(M, \omega)$ be a compact connected K\"ahler manifold of complex dimension four and let $[\chi] \in H^{1,1}(M; \mathbb{R})$. We confirmed the conjecture by Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed Hermitian--Yang--Mills equation, which is given by the following nonlinear elliptic equation $\sum_{i} \arctan (\lambda_i) = \hat{\theta}$, where $\lambda_i$ are the eigenvalues of $\chi$ with respect to $\omega$ and $\hat{\theta}$ is a topological constant. This conjecture was stated in [arXiv:1508.01934], wherein they proved that the existence of a supercritical $C$-subsolution or the existence of a $C$-suboslution when $\hat{\theta} \in [ ( (n-2) + {2}/{n} ) {\pi}/{2}, n\pi/2 )$ will give the solvability of the deformed Hermitian--Yang--Mills equation. Collins--Jacob--Yau conjectured that their existence theorem can be improved when $\hat{\theta} \in ( (n-2 ) {\pi}/{2}, ( (n-2) + {2}/{n} ) {\pi}/{2} )$, where $n$ is the complex dimension of the manifold. In this paper, we confirmed their conjecture that when the complex dimension equals four and $\hat{\theta}$ is close to the supercritical phase $\pi$ from the right, then the existence of a $C$-subsolution implies the solvability of the deformed Hermitian--Yang--Mills equation.