Geometry of Toric Manifolds

Abstract: This project is an overview of Hamiltonian geometry on Kahler manifolds and of Kähler reduction. In the first section we define complex manifolds, give their basic properties and build some structures on them. We are mainly interested in Kähler manifolds which are a subset of symplectic manifolds. In the second section we discuss group actions on manifolds. We are only concerned with Hamiltonian actions for which we can compute their moment maps. From these we prove how to construct new manifolds using a process called Kähler reduction. Finally we define toric manifolds and review Delzant polytopes and their corresponding manifolds. In the third section we present detailed examples in order to clarify all previously given definitions.