A note on affine cones over Grassmannians and their stringy $E$-functions

TL;DR

This paper computes the stringy E-function of affine cones over Grassmannians, revealing cases where noncommutative crepant resolutions exist even without traditional resolutions, and explores their implications.

Contribution

It provides explicit calculations of stringy E-functions for affine cones over Grassmannians and investigates the link to noncommutative crepant resolutions.

Findings

Stringy E-function can be a polynomial even without a crepant resolution.

Noncommutative crepant resolutions may exist when traditional resolutions do not.

Raises questions about the relationship between polynomial stringy E-functions and noncommutative resolutions.

Abstract

We compute the stringy $E$-function of the affine cone over a Grassmannian. If the Grassmannian is not a projective space then its cone does not admit a crepant resolution. Nonetheless the stringy $E$-function is sometimes a polynomial and in those cases the cone admits a noncommutative crepant resolution. This raises the question as to whether the existence of a noncommutative crepant resolution implies that the stringy $E$-function is a polynomial.