Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristicp

Abstract

“Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ 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 $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. 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 $Y^{\textrm{(1)}}$ 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.

Hodge-to-de Rham degeneration for stacks

“…This is a partial generalization of results of Section 6 in [KT16] to the case when R 1 π * O X is not necessarily 0.…”

“…G = G m . In this case we essentially get the definition of a conical resolution (for more details see e.g [KT16]…”

“…), except for the connectedness assumption on X. Indeed, Φ + = {∅}, so there is no condition on G; A is a finitely generated Z-graded ring (note that in[KT16] A is assumed to be H 0 (X, O X )), π : X → Spec A is proper, X is smooth and that the G m -action on X agrees with the grading on A. Finally, there should exist h ∈ X * (G m ) ≃ Z•id Gm such that A h<0 = 0 and A h=0 is finite-dimensional.…”

On the Bezrukavnikov–Kaledin quantization of symplectic varieties in characteristic p

“…In particular, specializing we have a sheaf of -algebras over . We show that if the action of on is contracting, then is an Azumaya algebra over and using Theorem 1 compute its class in the Brauer group proving a conjecture of Kubrak and Travkin [KT19].…”

“…Examples of -equivariant quantizations arise in geometric representation theory (see e.g. [BK04b, BF14, BL21, KT19]).…”