Strong-subsolution conjecture for the higher-rank deformed Hermitian Yang–Mills equation

Let $E\to(X,\omega)$ be a Hermitian vector bundle, and let $h$ be a Hermitian metric on $E$. A strong subsolution for the $Z$-critical equation is a metric satisfying

$$\operatorname{Im}\bigl(e^{-i\varphi(E)}\widetilde Z^{(p)}(h)\bigr)>0$$

as an $\operatorname{End}E$-valued $(n-p,n-p)$-form for every $p=1,\dots,n$, where $\widetilde Z^{(p)}(h)$ is the $p$th formal matrix derivative of $\widetilde Z(h)$ with respect to $iF(h)/(2\pi)$. Strong-subsolution conjecture. A subsolution $h$ of the higher-rank deformed Hermitian Yang–Mills equation is a strong subsolution.

The claim would extend the symmetry between subsolution conditions and stability criteria from line bundles to higher-rank bundles, despite the matrix-valued curvature. The authors explicitly describe it as possibly very strong and do not expect pointwise validity except when $Z$ has particular symmetries; its resolution is not specified.