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Kähler Spaces, Nilpotent Orbits, and Singular Reduction
Johannes Huebschmann, Universite des Sciences et Technologies de Lille, Villeneuve, France
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Memoirs of the American Mathematical Society
2004; 96 pp; softcover
Volume: 172
ISBN-10: 0-8218-3572-6
ISBN-13: 978-0-8218-3572-2
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/172/814

For a stratified symplectic space, a suitable concept of stratified Kähler polarization encapsulates Kähler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kähler space which establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces: The closure of a holomorphic nilpotent orbit or, equivalently, the closure of the stratum of the associated pre-homogeneous space of parabolic type carries a (positive) normal Kähler structure. In the world of singular Poisson geometry, the closures of principal holomorphic nilpotent orbits, positive definite hermitian JTS's, and certain pre-homogeneous spaces appear as different incarnations of the same structure. The closure of the principal holomorphic nilpotent orbit arises from a semisimple holomorphic orbit by contraction. Symplectic reduction carries a positive Kähler manifold to a positive normal Kähler space in such a way that the sheaf of germs of polarized functions coincides with the ordinary sheaf of germs of holomorphic functions. Symplectic reduction establishes a close relationship between singular reduced spaces and nilpotent orbits of the dual groups. Projectivization of holomorphic nilpotent orbits yields exotic (positive) stratified Kähler structures on complex projective spaces and on certain complex projective varieties including complex projective quadrics. The space of (in general twisted) representations of the fundamental group of a closed surface in a compact Lie group or, equivalently, a moduli space of central Yang-Mills connections on a principal bundle over a surface, inherits a (positive) normal (stratified) Kähler structure. Physical examples are provided by certain reduced spaces arising from angular momentum zero.

Graduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology.

• Introduction
• Poisson algebras and Lie-Rinehart algebras
• Stratified polarized spaces
• The closure of a holomorphic nilpotent orbit
• Reduction and stratified Kähler spaces
• Associated representations and singular reduction
• Associated representations for the remaining classical case
• Hermitian Jordan triple systems and pre-homogeneous spaces
• The exceptional cases
• Contraction of semisimple holomorphic orbits
• Projectivization and exotic projective varieties
• Comparison with other notions of Kähler space with singularities
• References