Frobenius splitting and geometry of G-Schubert varieties

• Autores: Xuhua He, Jesper Funch Thomsen
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 219, Nº 5, 2008, págs. 1469-1512
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.06.018
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)V, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G2. Finally we also extend the Frobenius splitting results to the more general class of -Schubert varieties