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

Abstract

Let $(M, ω)$ be a compact connected Kähler manifold of complex dimension four and let $[χ] \in H^{1,1}(M; \mathbb{R})$. 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 $\sum_{i} \arctan (λ_i) = \hatθ$, where $λ_i$ are the eigenvalues of $χ$ with respect to $ω$ and $\hatθ$ 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 $\hatθ \in [ ( (n-2) + {2}/{n} ) π/{2}, nπ/2 )$ will give the solvability of the deformed Hermitian--Yang--Mills equation. Collins--Jacob--Yau conjectured that their existence theorem can be improved when $\hatθ \in ( (n-2 ) π/{2}, ( (n-2) + {2}/{n} ) π/{2} )$, where $n$ is the complex dimension of the manifold. In this paper, we confirmed their conjecture that when the complex dimension equals four and $\hatθ$ is close to the supercritical phase $π$ from the right, then the existence of a $C$-subsolution implies the solvability of the deformed Hermitian--Yang--Mills equation.