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

Abstract

Let $(M,\omega )$ be a compact connected Kähler 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.

56 pages, 7 figures; in version 2, we fully resolve the conjecture when the complex dimension equals four