Compactifications of Adjoint Orbits and their Hodge Diamonds

TL;DR

This paper studies how compactifying adjoint orbits of semisimple Lie algebras affects their geometric and Hodge-theoretic properties, revealing variability in Hodge diamonds and singularities.

Contribution

It introduces a method to compactify adjoint orbits as projective varieties and analyzes how their Hodge structures change with different homogenizations.

Findings

Hodge diamonds vary significantly with homogenization choices

Compactifications often introduce degenerate singularities

Extensions of potentials to compactified varieties exhibit complex behavior

Abstract

A recent theorem of [GGSM1] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behaviour of their fibrewise compactifications. Expressing adjoint orbits and fibres as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenisation of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenisation, and that extensions of the potential to the compactification must acquire degenerate singularities.