Memoirs of the American Mathematical Society 2004; 96 pp; softcover Volume: 172 ISBN10: 0821835726 ISBN13: 9780821835722 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 prehomogeneous spaces: The closure of a holomorphic nilpotent orbit or, equivalently, the closure of the stratum of the associated prehomogeneous 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 prehomogeneous 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 YangMills 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  Introduction
 Poisson algebras and LieRinehart 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 prehomogeneous spaces
 The exceptional cases
 Contraction of semisimple holomorphic orbits
 Projectivization and exotic projective varieties
 Comparison with other notions of Kähler space with singularities
 References
