Severi varieties and holomorphic nilpotent orbits

Abstract

Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in terms of holomorphic nilpotent orbits. The projective space acquires an exotic K\"ahler structure with three strata, the cubic is an example of an exotic projective variety with two strata, and the corresponding Severi variety is the closed stratum in the exotic variety as well as in the exotic projective space. In the standard cases, these varieties arise also via K\"ahler reduction. An interpretation in terms of constrained mechanical systems is included.