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1 November 2003 Schubert varieties and cycle spaces
Alan T. Huckleberry, Joseph A. Wolf
Duke Math. J. 120(2): 229-249 (1 November 2003). DOI: 10.1215/S0012-7094-03-12021-9

Abstract

Let $G_0$ be a real semisimple Lie group. It acts naturally on every complex flag manifold $Z=G/Q$ of its complexification. Given an Iwasawa decomposition $G_0 = K_0 A_0 N_0$, a $G_0$-orbit $γ⊂Z$, and the dual $K$-orbit $κ⊂Z$, Schubert varieties are studied and a theory of Schubert slices for arbitrary $G_0$-orbits is developed. For this, certain geometric properties of dual pairs $(γ,κ)$ are underlined. Canonical complex analytic slices contained in a given $G_0$-orbit γ which are transversal to the dual $K_0$-orbit γ κ are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space $Ω_W$($D$) is a Stein domain that contains the universally defined Iwasawa domain $Ω_I$. This is one of the main ingredients in the proof that $Ω_W(D)=Ω_{AG}$ for all but a few Hermitian exceptions. In the Hermitian case, $Ω_W(D)$ is concretely described in terms of the associated bounded symmetric domain.

Citation

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Alan T. Huckleberry. Joseph A. Wolf. "Schubert varieties and cycle spaces." Duke Math. J. 120 (2) 229 - 249, 1 November 2003. https://doi.org/10.1215/S0012-7094-03-12021-9

Information

Published: 1 November 2003
First available in Project Euclid: 16 April 2004

zbMATH: 1048.32005
MathSciNet: MR2019975
Digital Object Identifier: 10.1215/S0012-7094-03-12021-9

Subjects:
Primary: 14M15, 22E30, 32E10
Secondary: 14C25, 32M10, 43A85

Rights: Copyright © 2003 Duke University Press

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Vol.120 • No. 2 • 1 November 2003
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