**Memoirs of the American Mathematical Society**

2004;
96 pp;
Softcover

MSC: Primary 53;
Secondary 14; 17; 32; 81

Print ISBN: 978-0-8218-3572-2

Product Code: MEMO/172/814

List Price: $63.00

AMS Member Price: $37.80

MAA Member Price: $56.70

**Electronic ISBN: 978-1-4704-0415-4
Product Code: MEMO/172/814.E**

List Price: $63.00

AMS Member Price: $37.80

MAA Member Price: $56.70

# Kähler Spaces, Nilpotent Orbits, and Singular Reduction

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*Johannes Huebschmann*

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.

#### Readership

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

#### Table of Contents

# Table of Contents

## Kahler Spaces, Nilpotent Orbits, and Singular Reduction

- Table of contents v6 free
- Introduction 18 free
- 1. Poisson algebras and Lie-Rinehart algebras 1118 free
- 2. Stratified polarized spaces 1320
- 3. The closure of a holomorphic nilpotent orbit 1926
- 4. Reduction and stratified Kähler spaces 4754
- 5. Associated representations and singular reduction 5360
- 6. Associated representations for the remaining classical case 6976
- 7. Hermitian Jordan triple systems and pre-homogeneous spaces 7380
- 8. The exceptional cases 7784
- 9. Contraction of semisimple holomorphic orbits 8188
- 10. Projectivization and exotic projective varieties 8390
- 11. Comparison with other notions of Kähler space with singularities 8996
- References 93100