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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
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  • 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


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