The arc space of horospherical varieties and motivic integration

TL;DR

This paper extends motivic integration techniques to horospherical varieties, providing a formula for their stringy E-function and a new smoothness criterion, generalizing known results from toric varieties.

Contribution

It introduces a formula for the stringy E-function of horospherical varieties and proposes a new smoothness criterion, broadening the scope beyond toric varieties.

Findings

Stringy E-function formula for horospherical varieties

Non-polynomial nature of the E-function in certain cases

A new smoothness criterion for locally factorial horospherical varieties

Abstract

For arbitrary connected reductive group G we consider the motivic integral over the arc space of an arbitrary Q-Gorenstein horospherical G-variety associated with a colored fan and prove a formula for the stringy E-function of a horospherical variety X which generalizes the one for toric varieties. We remark that in contrast to toric varieties the stringy E-function of a Gorenstein horospherical variety X may be not a polynomial if some cones in the fan of X have nonempty sets of colors. Using the stringy E-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties.