Timothy Goldberg

Visiting assistant professor of mathematics at Lenoir-Rhyne University

Hamiltonian Actions in Integral Kähler and Generalized Complex Geometry

Reyer Sjamaar

Lie groups and symplectic geometry

Abstract: This thesis consists of two parts. The first concerns a specialization of the basic case of Hamiltonian actions on symplectic manifolds, and the second a generalization of the basic case. Brion proved a convexity result for the moment map image of an irreducible subvariety of a compact integral Kähler manifold preserved by the complexification of the Hamiltonian group action. Guillemin and Sjamaar generalized this result to irreducible subvarieties preserved only by a Borel subgroup. In another direction, O’Shea and Sjamaar proved a convexity result for the moment map image of the submanifold fixed by an anti-symplectic involution. Analogous to Guillemin and Sjamaar’s generalization of Brion’s theorem, in the first part of this thesis we generalize O’Shea and Sjamaar’s result, proving a convexity theorem for the moment map image of the involution fixed set of an irreducible subvariety preserved by a Borel subgroup. In the second part of this thesis, we develop the analogue of Sjamaar and Lerman’s singular reduction of Hamiltonian symplectic manifolds in the context of Hamiltonian generalized complex manifolds. Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized Kähler manifolds.

PhD

PhD