Monotone Lagrangians in Flag Varieties

International Mathematics Research Notices ◽

10.1093/imrn/rnz227 ◽

2019 ◽

Cited By ~ 1

Author(s):

Yunhyung Cho ◽

Yoosik Kim

Keyword(s):

Maslov Index ◽

Chern Number ◽

Flag Manifolds ◽

Flag Varieties ◽

Holomorphic Disk ◽

Relative Version ◽

Number Formula

Abstract In this paper, we give a formula for the Maslov index of a gradient holomorphic disk, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian fibers of Gelfand–Cetlin systems on partial flag manifolds.

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Chern number formula for ramified coverings

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/05210001 ◽

2000 ◽

Vol 52 (1) ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Takeshi IZAWA

Keyword(s):

Chern Number ◽

Number Formula ◽

Ramified Coverings

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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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