Singular Poisson–Kähler geometry of Scorza varieties and their secant varieties

Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson–Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being af...

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Bibliographic Details
Published in:Differential geometry and its applications 2005-07, Vol.23 (1), p.79-93
Main Author: Huebschmann, Johannes
Format: Article
Language:English
Subjects:
Exotic projective variety
Holomorphic nilpotent orbit
Poisson algebra
Scorza variety
Stratified Kähler space
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Summary:Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson–Kähler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kähler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.
ISSN:0926-2245
1872-6984
DOI:10.1016/j.difgeo.2005.03.007