Collins–Jiang–Yau conjecture for the deformed Hermitian–Yang–Mills equation

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $[\alpha]\in H^{1,1}(X,\mathbb R)$ be a real cohomology class, and let $\hat\theta$ be a supercritical phase, meaning that it admits a lift to $\mathbb R$ in $((n-2)\frac{\pi}{2},n\frac{\pi}{2})$. The deformed Hermitian–Yang–Mills equation for a representative $\alpha\in[\alpha]$ is

$$\operatorname{Im}(e^{-i\hat\theta}(\omega+i\alpha)^n)=0.$$

For any analytic subvariety $V\subset X$, define its charge by

$$Z_V([\alpha])=-\int_V e^{i\omega+\alpha},$$

where only the term of order $\dim(V)$ is integrated, and write $Z(X)=Z_X([\alpha])$. Collins–Jiang–Yau conjecture. The class $[\alpha]$ admits a solution to the deformed Hermitian–Yang–Mills equation with supercritical phase if and only if $Z(X)\neq 0$ and, for every analytic subvariety $V\subset X$,

$$\operatorname{Im}\left(\frac{Z_V([\alpha])}{Z_X([\alpha])}\right)>0.$$

This conjecture proposes a numerical stability criterion, expressed through the charges of all analytic subvarieties, that is equivalent to solvability of the deformed Hermitian–Yang–Mills equation. The necessity of the condition was developed in the cited work of Collins, Jiang, and Yau under the supercritical-phase assumption; the supplied source does not state whether the sufficiency direction has been proved.