Duke Mathematical Journal

Schubert varieties and cycle spaces

Alan T. Huckleberry and Joseph A. Wolf

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Let G0 be a real semisimple Lie group. It acts naturally on every complex flag manifold z=G/Q of its complexification. Given an Iwasawa decomposition G0=K0A0N0, a G0-orbit γ⊂Z, and the dual K-orbit κ⊂Z, Schubert varieties are studied and a theory of Schubert slices for arbitrary G0-orbits is developed. For this, certain geometric properties of dual pairs (γ,κ) are underlined. Canonical complex analytic slices contained in a given G0-orbit γ which are transversal to the dual K0-orbit $\gamma\cap \kappa$ 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 ΩW(D) is a Stein domain that contains the universally defined Iwasawa domain ΩI. This is one of the main ingredients in the proof that ΩW(D)=ΩAG for all but a few Hermitian exceptions. In the Hermitian case, ΩW(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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32E10: Stein spaces, Stein manifolds 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 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. http://projecteuclid.org/euclid.dmj/1082138583.

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