Norm-squared of the momentum map in infinite dimensions with applications to K\"ahler geometry and symplectic connections

TL;DR

This paper explores the norm-squared of the momentum map in infinite-dimensional settings, providing new insights into symplectic geometry, with applications to almost complex structures and symplectic connections, revealing novel phenomena due to non-equivariance.

Contribution

It introduces a rigorous framework for the norm-squared of the momentum map in infinite dimensions and applies it to symplectic geometry, extending classical results to broader contexts.

Findings

Hessian at critical points is positive semi-definite along complexified orbits

Decomposition of the stabilizer under complexified action is determined

New central extensions of the symplectomorphism group encode geometric information

Abstract

We initiate the study of the norm-squared of the momentum map as a rigorous tool in infinite dimensions. In particular, we calculate the Hessian at a critical point, show that it is positive semi-definite along the complexified orbit, and determine a decomposition of the stabilizer under the complexified action. We apply these results to the action of the group of symplectomorphisms on the spaces of compatible almost complex structures and of symplectic connections. In the former case, we extend results of Calabi to not necessarily integrable almost complex structures that are extremal in a relative sense. In both cases, the momentum map is not equivariant, which gives rise to new phenomena and opens up new avenues for interesting applications. For example, using the prequantization construction, we obtain new central extensions of the group of symplectomorphisms that are encoding geometric information of the underlying finite-dimensional manifold.