Symplectic Toric Manifolds

Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties) from the symplectic viewpoint. The study of toric manifolds has many different entrances and has been scoring a wide spectrum of applications. For symplectic geometers, they provide examples of extremely symmetric and completely integrable hamiltonian spaces. In order to distinguish the algebraic from the symplectic approach, we call algebraic toric manifolds to the smooth toric varieties in algebraic geometry, and say symplectic toric manifolds when studying their symplectic properties. Native to algebraic geometry, the theory of toric varieties has been around for about thirty years. It was introduced by Demazure in [16] who used toric varieties for classifying some algebraic subgroups. Since 1970 many nice surveys of the theory of toric varieties have appeared (see, for instance, [14, 21, 28, 41]). Algebraic geometers and combinatorialists have found fruitful applications of toric varieties to the geometry of convex polytopes, resolutions of singularities, compactifications of locally symmetric spaces, critical points of analytic functions, etc. For the last ten years, toric geometry became an important tool in physics in connection with mirror symmetry [13] where research has been intensive. In this text we emphasize the geometry of the moment map whose image, the so-called moment polytope, determines the symplectic toric manifolds. The notion of a moment map associated to a group action generalizes that of a hamiltonian function associated to a vector field. Either of these notions formalizes the Noether principle, which states that to every symmetry (such as a group action) in a mechanical system, there corresponds a conserved quantity. The concept of a moment map was introduced by Souriau [45] under the french name application moment; besides the more standard english translation to moment map, the alternative momentum map is also used. Moment maps have been asserting themselves as a main tool to study problems in geometry and topology when there is a suitable symmetry , as illustrated in the book by Gelfand, Kapranov and Zelevinsky [22]. The material in some sections of the second part of these notes borrows largely from v vi that excellent text, where details …