Severi varieties and holomorphic nilpotent orbits

TL;DR

This paper explores the deep connections between Severi varieties, holomorphic nilpotent orbits, and exotic Kähler structures, revealing their geometric and algebraic properties through minimal orbits and stratifications.

Contribution

It establishes a novel link between Severi varieties and holomorphic nilpotent orbits in simple hermitian Lie algebras, and describes their geometric structures and stratifications.

Findings

Severi varieties correspond to minimal holomorphic nilpotent orbits.

The projective space has an exotic Kähler structure with three strata.

The cubic variety is an exotic projective variety with two strata.

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.