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

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.