Special Symplectic Connections

TL;DR

This paper classifies special symplectic manifolds, characterized by particular torsion-free connections, showing they are locally modeled by symplectic reductions of contact manifolds and establishing global properties including classification results.

Contribution

It demonstrates that all special symplectic manifolds are locally equivalent to symplectic reductions of contact manifolds and provides a classification in the compact simply connected case.

Findings

Special symplectic manifolds are locally symplectically equivalent to reductions of contact manifolds.

The paper classifies compact simply connected special symplectic manifolds.

Symplectic reduction provides a canonical construction for these manifolds.

Abstract

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.