Frobenius splitting and geometry of G-Schubert varieties

Xuhua He; Jesper Funch Thomsen

Frobenius splitting and geometry of G-Schubert varieties

Publikation: Working paper/Preprint › Working paper › Forskning

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 $\mathrm{diag}(G) \cdot V$ , where $V$ is a $B \times B$ -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 that compatibly splits all the $G$ -Schubert varieties. Moreover, any $G$ -Schubert variety admits stable Frobenius splittings along ample divisors in case X is projective. Although this indicates that $G$ -Schubert varieties have nice singularities we give an example, in the wonderful compactification of a group of adjoint type, which is not normal. Finally we also extend the Frobenius splitting results to the more general class of $R$ -Schubert varieties.

Originalsprog	Engelsk
Udgivelsessted	Århus
Udgiver	Department of Mathematical Sciences , University of Aarhus
Antal sider	32
Status	Udgivet - 2007

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abstract = "Let be an equivariant embedding of a connected reductive group over an algebraically closed field of positive characteristic. Let denote a Borel subgroup of . A -Schubert variety in is a subvariety of the form , where is a -orbit closure in . In the case where is the wonderful compactification of a group of adjoint type, the -Schubert varieties are the closures of Lusztig's -stable pieces. We prove that admits a Frobenius splitting that compatibly splits all the -Schubert varieties. Moreover, any -Schubert variety admits stable Frobenius splittings along ample divisors in case X is projective. Although this indicates that -Schubert varieties have nice singularities we give an example, in the wonderful compactification of a group of adjoint type, which is not normal. Finally we also extend the Frobenius splitting results to the more general class of -Schubert varieties.",

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T1 - Frobenius splitting and geometry of G-Schubert varieties

AU - He, Xuhua

AU - Thomsen, Jesper Funch

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N2 - Let be an equivariant embedding of a connected reductive group over an algebraically closed field of positive characteristic. Let denote a Borel subgroup of . A -Schubert variety in is a subvariety of the form , where is a -orbit closure in . In the case where is the wonderful compactification of a group of adjoint type, the -Schubert varieties are the closures of Lusztig's -stable pieces. We prove that admits a Frobenius splitting that compatibly splits all the -Schubert varieties. Moreover, any -Schubert variety admits stable Frobenius splittings along ample divisors in case X is projective. Although this indicates that -Schubert varieties have nice singularities we give an example, in the wonderful compactification of a group of adjoint type, which is not normal. Finally we also extend the Frobenius splitting results to the more general class of -Schubert varieties.

AB - Let be an equivariant embedding of a connected reductive group over an algebraically closed field of positive characteristic. Let denote a Borel subgroup of . A -Schubert variety in is a subvariety of the form , where is a -orbit closure in . In the case where is the wonderful compactification of a group of adjoint type, the -Schubert varieties are the closures of Lusztig's -stable pieces. We prove that admits a Frobenius splitting that compatibly splits all the -Schubert varieties. Moreover, any -Schubert variety admits stable Frobenius splittings along ample divisors in case X is projective. Although this indicates that -Schubert varieties have nice singularities we give an example, in the wonderful compactification of a group of adjoint type, which is not normal. Finally we also extend the Frobenius splitting results to the more general class of -Schubert varieties.

M3 - Working paper

BT - Frobenius splitting and geometry of G-Schubert varieties

PB - Department of Mathematical Sciences , University of Aarhus

CY - Århus

ER -

Frobenius splitting and geometry of G-Schubert varieties

Abstract

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