Integrable systems and special Kahler geometry
课程简介：Roughly speaking, an integrable system is a system of ordinary differential equations, which can be integrated by means of first integrals, i.e., functions remaining constant in the time-variable along any solution. I will, however, emphasize a more geometric approach to the problem of integrating ODEs stemming from the classical Hamiltonian mechanics. In this approach one is interested in a 2n-dimensional manifold M equipped with a nondegenerate (in a certain sense) 2-form ω, which is called a symplectic form. An integrable system can be described as a fibration π : M →B over an n-dimensional base B such that, roughly speaking, for any point b?B the fiber Mb := π-1(b) is Lagrangian, i.e., ?*bω= 0, where ?b: Mb→M is the canonical embedding. The main point of the first part of the lectures (roughly 3 lectures) is to explain this geometric approach in some details. While the material of the first part is well known, this will serve us as a model for what will come in the second part and is much less well understood. In the second part I will describe a complex version of integrable systems, which is essentially a complex symplectic (also known as hyperK?hler) manifold equipped with the structure of a holomorphic Lagrangian fibration. The situation becomes much more rigid in this case and it turns out that the base of a holomorphic Lagrangian fibration can be equipped with a so called special K?hler structure. The main point for the last part of the lectures will be to explain a relation between complex integrable systems and special K?hler geometry and to outline some questions for further research.