Duke Mathematical Journal

Schubert varieties and cycle spaces

Alan T. Huckleberry and Joseph A. Wolf

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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.

Article information

Duke Math. J., Volume 120, Number 2 (2003), 229-249.

First available in Project Euclid: 16 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32E10: Stein spaces, Stein manifolds 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
Secondary: 32M10: Homogeneous complex manifolds [See also 14M17, 57T15] 43A85: Analysis on homogeneous spaces 14C25: Algebraic cycles


Huckleberry, Alan T.; Wolf, Joseph A. Schubert varieties and cycle spaces. Duke Math. J. 120 (2003), no. 2, 229--249. doi:10.1215/S0012-7094-03-12021-9. https://projecteuclid.org/euclid.dmj/1082138583

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