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

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

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.

Originalsprog	Engelsk
Tidsskrift	Advances in Mathematics
Vol/bind	219
Nummer	5
Sider (fra-til)	1469-1512
Antal sider	44
ISSN	0001-8708
DOI	https://doi.org/10.1016/j.aim.2008.06.018
Status	Udgivet - 2008

Adgang til dokumentet

10.1016/j.aim.2008.06.018

Citationsformater

@article{1ec5a9a0e5fd11dd8fc3000ea68e967b,

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",

doi = "10.1016/j.aim.2008.06.018",

language = "English",

volume = "219",

pages = "1469--1512",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press",

number = "5",

}

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

Abstract

Adgang til dokumentet

Fingeraftryk

Citationsformater