Resolutions with conical slices and descent for the Brauer group classes of certain central reductions of differential operators in characteristic $p$

Abstract

For a smooth variety $X$ over an algebraically closed field of characteristic $p$, to a differential 1-form $\alpha$ on the Frobenius twist $X^{(1)}$ one can associate an Azumaya algebra $\mathcal D_{X,\alpha}$, defined as a certain central reduction of the algebra $\mathcal D_X$ of "crystalline differential operators" on $X$. For a resolution of singularities $\pi:X\to Y$ of an affine variety $Y$, we study for which $\alpha$ does the class $[\mathcal D_{X,\alpha}]$ in the Brauer group $\mathrm{Br}(X^{(1)})$ descend to $Y^{(1)}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of non-commutative resolutions of $Y$. We prove that the classes $[\mathcal D_{X,\alpha}]$ descend \'etale locally for all $\alpha$ if $\mathcal O_Y\simeq \pi_* \mathcal O_X$ and $R^{1,2}\pi_*\mathcal O_X =0$. We also define a certain class of resolutions which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[\mathcal D_{X,\alpha}]$ descend to $Y^{(1)}$ globally for all $\alpha$. Finally we give some examples, in particular we show that Slodowy slices, Nakajima quiver varieties and hypertoric varieties are resolutions with conical slices.