The canonical global quantization of symplectic varieties in characteristic $p$

Let $X$ be a smooth symplectic variety over a field $k$ of characteristic $p>2$ equipped with a restricted structure, which is a class $[\eta] \in H^0(X, \Omega^1_X/d\mathcal O_X)$ whose de Rham differential equals the symplectic form. In this paper we construct a functorial in $(X, [\eta])$ formal quantization of the category $\mathrm{QCoh}(X)$ of quasi-coherent sheaves on $X$. We also construct its natural extension to a quasi-coherent sheaf of categories $\mathrm{QCoh}_h$ on the product $X^{(1)} \times {\mathbb S}$ of the Frobenius twist of $X$ and the projective line ${\mathbb S}=\mathbb P^1$, viewed as the one-point compactification of $\mathrm{Spec}\ \! k[h]$. Its global sections over $X^{(1)} \times \{0\}$ is the category of quasi-coherent sheaves on $X$. If $X$ is affine, $\mathrm{QCoh}_h$, restricted to $X^{(1)}\times \mathrm{Spf} \ \! k[[h]]$, is equivalent to the category of modules over the distinguished "Frobenius-constant" quantization of $(X,[\eta])$ defined by Bezrukavnikov and Kaledin.