Frobenius splitting and hyperplane sections of flag manifolds

Niels Lauritzen; Jesper Funch Thomsen

doi:10.1007/s002220050147

Frobenius splitting and hyperplane sections of flag manifolds

Inventiones mathematicae ◽

10.1007/s002220050147 ◽

1997 ◽

Vol 128 (3) ◽

pp. 437-442 ◽

Cited By ~ 3

Author(s):

Niels Lauritzen ◽

Jesper Funch Thomsen

Keyword(s):

Flag Manifolds ◽

Frobenius Splitting ◽

Hyperplane Sections

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Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

Forum of Mathematics Pi ◽

10.1017/fmp.2021.5 ◽

2021 ◽

Vol 9 ◽

Author(s):

Syu Kato

Keyword(s):

Positive Characteristic ◽

Formal Model ◽

Flag Manifold ◽

Schubert Varieties ◽

Flag Manifolds ◽

Line Bundles ◽

Closed Field ◽

Flag Varieties ◽

Fixed Degree ◽

Frobenius Splitting

Abstract We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$ -model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$ , and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$ . Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

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Frobenius Splitting of Thick Flag Manifolds of Kac–Moody Algebras

International Mathematics Research Notices ◽

10.1093/imrn/rny174 ◽

2018 ◽

Vol 2020 (17) ◽

pp. 5401-5427 ◽

Cited By ~ 1

Author(s):

Syu Kato

Keyword(s):

Line Bundle ◽

Schubert Variety ◽

Flag Manifold ◽

Schubert Varieties ◽

Flag Manifolds ◽

Flag Varieties ◽

Moody Algebra ◽

Frobenius Splitting ◽

Projective Normality ◽

Plücker Relations

Abstract We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac–Moody algebra. This naturally transplants the result of Kumar–Mathieu–Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara–Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties.

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Invariant generalized complex geometry on maximal flag manifolds and their moduli

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2021.104108 ◽

2021 ◽

pp. 104108

Author(s):

Elizabeth Gasparim ◽

Fabricio Valencia ◽

Carlos Varea

Keyword(s):

Complex Geometry ◽

Flag Manifolds ◽

Generalized Complex

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On a conjecture of Pappas and Rapoport about the standard local model for GL_d

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0030 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Dinakar Muthiah ◽

Alex Weekes ◽

Oded Yacobi

Keyword(s):

Type A ◽

Positive Answer ◽

Local Model ◽

Schubert Varieties ◽

Shimura Varieties ◽

Local Models ◽

Full Generality ◽

Frobenius Splitting ◽

Affine Grassmannians ◽

Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

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