Frobenius splitting and geometry of $G$-Schubert varieties

Xuhua He; Jesper Funch Thomsen

doi:10.1016/j.aim.2008.06.018

Frobenius splitting and geometry of $G$-Schubert varieties

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Abstract

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) dot operator

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.

Original language	English
Journal	Advances in Mathematics
Volume	219
Issue	5
Pages (from-to)	1469-1512
Number of pages	44
ISSN	0001-8708
DOIs	https://doi.org/10.1016/j.aim.2008.06.018
Publication status	Published - 2008

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title = "Frobenius splitting and geometry of $G$-Schubert varieties",

abstract = "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.",

author = "Xuhua He and Thomsen, {Jesper Funch}",

year = "2008",

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language = "English",

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TY - JOUR

T1 - Frobenius splitting and geometry of $G$-Schubert varieties

AU - He, Xuhua

AU - Thomsen, Jesper Funch

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

U2 - 10.1016/j.aim.2008.06.018

DO - 10.1016/j.aim.2008.06.018

M3 - Journal article

SN - 0001-8708

VL - 219

SP - 1469

EP - 1512

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 5

ER -

Frobenius splitting and geometry of $G$-Schubert varieties

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