The existence of dHYM solutions has algebraic obstructions. Let $V$ be a $p$-dimensional subvariety of $X^{\vee}$, with $1\leq p<n$. Under the large phase assumption $\hat{\theta}>\frac{(n-2)\pi}{2}$, a pointwise consideration of eigenvalues shows [18, Prop. 3.4]

$$\text{Im}\big(\int_{V}e^{-\sqrt{-1}(\hat{\theta}-(n-p)\frac{\pi}{2})}(\omega_{X^{\vee}}+\sqrt{-1}\alpha_{\phi})^{n}\big)>0.$$

More suggestively, denote

$$Z_{V}(E)=-\int_{V}e^{-\sqrt{-1}\omega_{X^{\vee}}}ch(E)\in\mathbb{R}_{>0}e^{\sqrt{-1}\phi_{V}(E)},\quad\phi_{V}(E)\in(0,\pi),$$

(13)

so the algebraic obstruction can be rewritten as

$$\text{Im}(\frac{Z_{V}(E)}{Z_{X^{\vee}}(E)})>0,\quad\emph{i.e.}\quad\phi_{V}(E)>\phi_{X^{\vee}}(E)=\hat{\theta}-\frac{(n-2)\pi}{2}.$$

This bears some resemblance to a Bridgeland stability condition with central charge $Z_{X^{\vee}}(E)$ , even though on the technical level there are some discrepancies [18, section 3.2]. In the simplest understood examples, such as the blow-up of $\mathbb{CP}^{2}$ in a point, the obstruction criterion for dHYM seems to refine Bridgeland stability, in the sense that every dHYM stable object is Bridgeland stable, but not conversely. It is not entirely clear how to interpret this (cf. Remark 2.10).