Classical Symmetries of Complex Manifolds

TL;DR

This paper classifies complex manifolds admitting large symmetry groups from classical simple real Lie groups, showing they are essentially open subsets of complex flag manifolds associated with these groups.

Contribution

It provides a classification of complex manifolds with large classical symmetry groups, linking them to invariant open subsets of complex flag manifolds.

Findings

Manifolds are invariant open subsets of complex flag manifolds

Classification applies to large symmetry groups

Results connect group actions to geometric structures

Abstract

We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups.