Sum of Hamiltonian manifolds

TL;DR

This paper investigates the Hamiltonian sum of two compact Hamiltonian G-manifolds sharing a common codimension 2 submanifold, establishing a relationship between their symplectic reductions and comparing their equivariant first Chern classes.

Contribution

It introduces a framework for the Hamiltonian sum of manifolds with a shared submanifold and relates their symplectic reductions and characteristic classes.

Findings

Symplectic reduction of the Hamiltonian sum matches the sum of reduced manifolds.

The equivariant first Chern class of the sum relates to those of the original manifolds.

Provides a method to compare characteristic classes in Hamiltonian sums.

Abstract

For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,\omega^+,\mu^+)$ and $(X^-,\omega^-,\mu^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite equivariant Euler classes of the normal bundles. We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of $X^\pm$.