The degree of the logarithmic extension of the cotangent bundle to the moduli of pointed curves and Hitchin systems, spectral curves and KP equations.

摘要

This dissertation is divided into two parts. In the first part we prove a recursive relation for the Kahler-Einsein volume of the moduli space of genus g curves with n marked points in terms of the intersection of the Kahler-Einstein form with kappa classes. We will show that in the case where the genus is zero, the cohomology classes of the Kahler-Einstein form and kappa1 coincide. We will also show that if the genus is one, then the Kahler-Einstein volume can be computed in terms of kappa1 volume of moduli spaces of curves with either a smaller genus or less marked points. Using this we will show that the generating function for kappa1 volumes of genus one pointed curves satisfies a non-linear second order ordinary differential equation.;In the section part of this dissertation we examine the moduli space of stable Higgs bundles. We determine an effective family of spectral curves that appear in the Hitchin fibration. This effective family will provide an embedding into the Sato Grassmannian. The Hitchin integrable system will be shown to be the pull back of the KP-flows on the Sto Grassmannian. We will show that the formal adjoint on the space of pseudo-differential operators corresponds to the Serre duality operation on the moduli space of stable Higgs bundles. Using this correspondence, we can identify the moduli space of stable Sp2m-Higgs bundles as a reduction of the KP equations. The dual abelian fibration associated to the Sp2m-Higgs moduli space is constructed as the symplectic quotient of a Lie algebra action on the moduli space of GL-Higgs bundles.