The arc space of horospherical varieties and motivic integration

Abstract

AbstractFor an arbitrary connected reductive group $G$, we consider the motivic integral over the arc space of an arbitrary $ \mathbb{Q} $-Gorenstein horospherical $G$-variety ${X}_{\Sigma } $ associated with a colored fan $\Sigma $ and prove a formula for the stringy $E$-function of ${X}_{\Sigma } $ 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}_{\Sigma } $ may be not a polynomial if some cones in $\Sigma $ 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.