English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

international mathematics research notices

Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic<i>p</i>

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