## Duke Mathematical Journal

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

### Schubert varieties and cycle spaces

Alan T. Huckleberry and Joseph A. Wolf

#### 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 $\gamma\cap \kappa$ 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 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

**Source**

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

**Dates**

First available in Project Euclid: 16 April 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.dmj/1082138583

**Digital Object Identifier**

doi:10.1215/S0012-7094-03-12021-9

**Mathematical Reviews number (MathSciNet)**

MR2019975

**Zentralblatt MATH identifier**

1048.32005

**Subjects**

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

#### Citation

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.