Symplectic Toric Varieties

These are the notes of two lectures given at the Mini Spring School An introduction to the mathematics of string theory held at Adelaide University in November 2002. It is a leisurely introduction to the mathematics surrounding toric varieties. Lecture I. Aims of Lecture I. (i) To contrast topological, Riemannian, symplectic and complex structures; (ii) to set up various topological objects that will be given Riemannian, symplectic and complex structures in the next lecture. Toric varieties: a topological construction. Toric varieties are simple examples of Riemannian, symplectic and complex manifolds. There won’t be a lot of toric varieties in the first lecture, since I am more concerned with the basic mathematics that surrounds them. For that reason, I will begin with a short description of toric varieties. Consider the following two examples consisting of an interval and a triangle.