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The arc space of horospherical varieties and motivic integration

  • Victor Batyrev (a1) and Anne Moreau (a2)

For 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.

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V. Batyrev , Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), 533.

N. Bourbaki , Lie groups and Lie algebras, Chapters 4–6 (Springer, Berlin, 2002), translated from the 1968 French original by Andrew Pressley.

M. Brion , Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J. 58 (1989), 397424.

M. Brion , Sur la géométrie des variétés sphériques, Comment. Math. Helv. 66 (1991), 237262.

M. Brion , Spherical varieties and Mori theory, Duke Math. J. 72 (1993), 369404.

J. Denef and F. Loeser , Germs of arcs on singular varieties and motivic integration, Invent. Math. 135 (1999), 201232.

S. Ishii , The arc space of a toric variety, J. Algebra 278 (2004), 666683.

B. Kostant , The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 9731032.

D. Luna and T. Vust , Plongements d’espace homogènes, Comment. Math. Helv. 58 (1983), 186245.

A. L. Onishchik and E. B. Vinberg , Lie groups and algebraic groups, Springer Series in Soviet Mathematics (Springer, Berlin, 1990), translated from the Russian and with a preface by D. A. Leites.

F. Pauer , Normale Einbettungen von $G/ U$, Math. Ann. 257 (1981), 371396.

F. Pauer , Glatte Einbettungen von $G/ U$, Math. Ann. 262 (1983), 421429.

P. Sankaran and V. Uma , Cohomology of toric bundles, Comment. Math. Helv. 78 (2003), 540554.

D. Timashev , Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, vol. 138. Invariant theory and algebraic transformation groups VIII (Springer, Heidelberg, 2011).

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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