Frobenius splitting methods in geometry and representation theory

By M. Brion, Shrawan Kumar. 2004

501 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/20b2762f08022b1c9e163c4b9890ad616a53e5a0

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In Algebra and Number Theory

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Citations in the corpus

p −1-Linear Maps in Algebra and Geometry (2012, background): “are not OX-linear). However, the complex F∗Ω q X is a complex of OX-modules (notice that d(xp) = 0). We now state the Cartier isomorphism. We take this presentation from [Car57], [Kat70], [EV92], and [BK05]. Deﬁnition-Proposition 3.1.1. There is a natural isomorphism (of OX-modules): C−1 : Ωi X→h i(”
Igusa integrals and volume asymptotics in analytic and adelic geometry (2009, background): “Since the boundary divisor of a smooth toric variety is a divisor with strict normal crossings, it then follows from [10], Proposition 6.”
FIXED POINTS OF THE EM ALGORITHM AND NONNEGATIVE RANK BOUNDARIES (2013, background): “The intersection of Schubert varieties is reduced by [9], Section 2.”
Cartier modules: Finiteness results (2009, background): “From the viewpoint of Frobenius splittings (see [9]) Proposition 4.”
Local models of Shimura varieties, I. Geometry and combinatorics (2010, background): “ are normal and have only rational singularities. If khas positive characteristic, then all f 0-Schubert varieties contained in a given f -Schubert variety are compatibly Frobenius split. We refer to [BK] for the notion of a scheme X in characteristic pbeing Frobenius split, and for a family of clos”
ON THE BEHAVIOR OF TEST IDEALS UNDER FINITE MORPHISMS (2010, methodology): “les of such maps include Frobenius splittings (usually referred to as F-splittings), which have historically been very useful in applying positive characteristic methods in representation theory; see [BK05]. In Section 5.1, we will give a precise answer to question (1) when πis separable by associat”
Globally F-regular and log Fano varieties (2009, background): “p 1)) + p(pe 1) + D+ E pfL ˘ fp(pe 1)M+ D 26 which is anti-ample. Therefore, Hi(X;O X( epe+fL+ pf(p 1) + D+ E)) vanishes for i<dimXsince Xis globally F-regular, by [Smi00, Corollary 4.4], see also [BK05]. Because of the injection above, it follows that Hi(X;O X( L)) vanishes, and the proof is com”
ON THE GEOMETRY OF SPHERICAL VARIETIES (2012, methodology): “ the level of structural sheaves i.e. the morphism OX→ F∗OX has a section ϕ: F∗OX→ OX. A subvariety Y ⊂ Xis compatibly split for ϕif its sheaf of ideals IY satisﬁes ϕ(F∗IY) ⊂ IY. We will use the book [39] for further reference on Frobenius splittings. Let Xbe sphericalanddeﬁned overan algebraicallyc”