Theorem 11.15.5E769519C862E

Proof. Let $\mathcal P$ be the prioritary stack with invariants $(2,4H,13H^2)$. It contains $E_0$ by proposition 11.14.2, and Walter's theorem makes it smooth and irreducible. Let $\mathcal A$ and $\mathcal S$ be the loci of ample and of $\mu _ P$-stable vector bundles in $\mathcal P$. They are open by the three preceding family lemmas, which apply on smooth local charts of $\mathcal P$; the first lemma allows restriction to locally free families near a point given by a vector bundle. The ample bundle of proposition 11.13.2 is prioritary by lemma 11.13.3, so $\mathcal A$ is nonempty. A nonsplit extension is stable and prioritary by proposition 11.14.1, proposition 11.14.2, so $\mathcal S$ is nonempty. By irreducibility, $\mathcal U=\mathcal A\cap \mathcal S$ is nonempty and dense.

Choose a smooth local chart $q:T\to \mathcal P$ of finite type over $\mathbb C$, with a point $t_0$ representing $E_0$, and let $\mathcal E$ be the corresponding flat family on $X\times T$. Fix an identification $\mathcal E_{t_0}\simeq E_0$. The variety $T$ is smooth because the stack and the chart are smooth. Shrink $T$ to an irreducible neighborhood of $t_0$ and use lemma 11.15.1 to make $\mathcal E$ a vector bundle on the whole product. The inverse image $q^{-1}(\mathcal U)$ is dense: every nonempty open $T'\subset T$ has nonempty open image $q(T')$ under the smooth map $q$, and this image meets $\mathcal U$. Apply lemma 11.15.4 to this dense open set. Each transported operator defines a bundle isomorphic to the corresponding ample stable fiber. The fixed invariants of the stack give its Chern classes, and the lemma also gives the convergence of the dual operators. $\square$