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Singularités canoniques et actions horosphériques

Let $G$ be a connected reductive linear algebraic group. We consider the normal $G$-varieties with horospherical orbits. In this short note, we provide a criterion to determine whether these varieties have at most canonical, log canonical or terminal singularities in the case where they admit an algebraic curve as rational quotient. This result seems to be new in the special setting of torus actions with general orbits of codimension $1$. For the given $G$-variety $X$, our criterion is expressed in terms of a weight function $ω_{X}$ that is constructed from the set of $G$-invariant valuations of the function field $k(X)$. In the log terminal case, the generating function of $ω_{X}$ coincides with the stringy motivic volume of $X$. As an application, we discuss the case of normal $k^{\star}$-surfaces.