The large-slope special case of the modularity conjecture

Let $\mathcal{E}$ be a rank $m$ vector bundle on $X'$, let $\mathcal{E}'=\sigma^{*}\underline{\operatorname{Hom}}(\mathcal{E},\nu^{*}\mathfrak{L})$, and suppose the maximal slope of $\mathcal{E}$ satisfies

$$\mu_{\max}(\mathcal{E})<\deg\mathfrak{L}-\deg\omega_X.$$

Then $\mathcal{A}_{\mathcal{E}'}(k)$ contains only the zero Hermitian map. Large-slope special case. In $\operatorname{Ch}_{r(n-m)}(\operatorname{Sht}^{r}_{U(n),\mathfrak{L}})$, one has

$$\eta(\mathfrak{L})^{mn}q^{n(\deg\mathcal{E}-m\deg\mathfrak{L})}\sum_{a\in\mathcal{A}_{\mathcal{E}}(k)}\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E}}(a)]=\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E}'}(0)].$$

This is a further specialization of the split Lagrangian form of the modularity conjecture, obtained when the opposite Hermitian-map space has only its zero element; it remains a conjectural identity unless separately proved in the source.