Lectures on the moduli problems of vector bundles
December 6—December 7,
10:00am-12:00pm
December 9—December 10,2021
3:00pm-5:00pm
Xueqing Wen (Tsinghua University)
Lecture 1: An overview of vector bundles on curves and moduli problems
In this lecture, we will recall some basic notion of vector/Higgs bundles on curve, such as rank, degree, Riemann-Roch theorem, semistablity and so on. Moreover, we will formulate the moduli problem of semistable vector bundles and give some baby examples.
Lecture 2: GIT and the construction of moduli of Vector bundles and Higgs bundles
In this lecture, we will introduce the geometric invariant theory (GIT). We will start from some basic examples and end with the Hilbert-Mumford criterion. And we will briefly construct the moduli space of semistable vector/Higgs bundles using GIT.
Lecture 4: Deformation theory and geometry of the Hitchin map
In this lecture, we will state some basics about the deformation theory and compute the dimension of moduli spaces. And we will study the most important map on the moduli of semistable Higgs bundle—the Hitchin map. We will use the BNR correspondence the study its generic fibers.
Lecture 5: Parabolic vector bundle and parabolic Higgs bundle
Parabolic vector/Higgs bundles are a kind of generalization of vector/Higgs bundles. In this lecture, we will study the parabolic version of the Hitchin map and to reveal some connection with the representation theory. This is a joint work with Su Xiaoyu and Wang Bin.