Forschungsseminar Komplexe Analysis - Winter Term 2013/14

Representations of fundamental groups and Higgs bundles

Arakelov Geometry

Wednesdays 4-6pm - Arnimallee 6, SR 025/026. ( 16th October 2013 - 15th February 2014 )

During this semester we will continue our seminar on Higgs Bundles (see previous seminar and infohere). The main objective is to understand Hitchin's original paper:
[Hi87] N.Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.

Schedule 2013

♠ 16.10.2013 // Eva Martínez - Moduli spaces of stable rank 2 vector bundles on flag manifolds
Homogeneous vector bundles on rational homogeneous varieties carry a double nature between Algebraic Geometry and Lie Theory. I will show how this double nature allows us to describe a generically smooth component of the moduli space of stable rank 2 vector bundles on a complete flag manifold.

♠ 23.10.2013 // Arijit Dey - Bundles on toric variety
In the first part of my talk I will recall necessary facts on vector bundles over toric varieties and explain Klyachkos's classification theorem in terms of filtrations. In the second part of the talk I will explain my joint work with Mainak Poddar on equivariant principal G-bundles over nonsingular toric variety where G is an abelian linear algebraic group.

♠ 30.10.2013 // Ángel Muñoz - Hitchin’s self-duality equations
Description of the self-duality equations in terms of connections on principal bundles. A vanishing theorem for solutions of the self-duality equations. Relation with stability. (Section 2 of [Hi87])

♠ 06.11.2013 // Vincent Trageser - Review of classical Sobolev theory
We provide the main definitions and properties of Sobolev spaces of functions defined on a bounded domain in the n-dimensional euclidean space: Sobolev norm, Banach space, embedding theorem.

♠ 13.11.2013 // Eva Martínez - Sobolev spaces of sections in fiber bundles
We study the Sobolev spaces of sections of a fiber bundle over a compact Riemannian manifold. If time permits, we will give an introduction to elliptic operators.

♠ 20.11.2013 // Joana Cirici - Uhlenbeck’s compactness theorem
One of the most effective tools in gauge theory is Uhlenbeck's compactness. This result is essential in both Donaldson's proof of the Narasimhan-Seshadri correspondence and Hitchin's construction of ASD solutions on Riemann surfaces from stable pairs. In this talk I will give a proof of Uhlenbeck’s compactness for connections on SO(n)-bundles over a Riemann surface.

♠ 27.11.2013 // Ángel Muñoz - Stability of Hitchin pairs
In this talk we define the stability condition for Higgs bundles (or Hitchin pairs). We will characterize those rank two vector bundles that can appear in such pairs. This will be the last technical result that we need to prove the Hitchin Existence Theorem.

♠ 04.12.2013 // Joana Cirici - Hitchin's Existence Theorem
We shall see, using Uhlenbeck compactness, how to each each stable pair there corresponds a solution to the SO(3) self-duality equations, unique up to gauge equivalence (we will mainly follow Section 4 of [Hi87]).

♠ 11.12.2013 // Nikolai Beck - The gauge theoretic moduli space
We shall study the moduli space of all solutions of the self-dual Yang-Mills equations on a fixed principal bundle over a Riemann surface, modulo the group of gauge transformations.

♠ 18.12.2013 // Anna Wißdorf - Betti numbers of the moduli space of Higgs bundles of rank 2 and degree 1

Schedule 2014

♠ 08.01.2014 // Victoria Hoskins - Towards symplectic and hyperkähler quotients for non-reductive groups
The geometric invariant theory quotient of a complex affine space by a reductive group can be identified with a symplectic quotient by the associated maximal compact via the Kempf-Ness theorem. The reductive quotient also has a hyperkähler analogue constructed as a hyperkähler quotient. Both the symplectic and hyperkähler quotients are constructed using moment maps for the action. For non-reductive group actions, we ask if the there are also links to suitable symplectic and hyperkähler quotients. We focus our attention on the simplest non-reductive group, the additive group, as many examples of interest involve additive group actions. We start by reviewing the theory for reductive groups and see that for linear additive group actions the same results hold. Then we study non-linear additive group actions that look like a family of linear actions parametrised by a base. In both the symplectic and hyperkähler setting, we provide 'moment maps' in a modified sense. This is joint work in progress with Brent Doran.

♠ 15.01.2014 // Eva Martínez - Introduction to Arakelov Geometry I: Arakelov vector bundles over arithmetic curves
Arakelov Geometry gives techniques to study diophantic problems from a geometric point of view. Thus, on can understand this theory like a "translation" of Algebraic Geometry results to Number Theory and Complex Analysis ones. The goal of this talk will be to give a gentle introduction of Arakelov's philosophy by introducing what are so called Arakelov vector bundles over arithmetic curves.

♠ 22.01.2014 // Anna Wißdorf - Introduction to Arakelov Geometry II: Arakelov principal bundles
I will introduce the notion of Arakelov group scheme as defined by G. Harder and U. Stuhler. Using the analogy between vector bundles and GL(n)-principal bundles, this can be used to give a first approach to the definition of Arakelov principal bundles.

♠ 05.02.2014 // Nikolai Beck - Wonderful compactifications
In 1983 De Concini and Procesi constructed a compactification X of a semi-simple Lie-group G with certain "wonderful" properties. In this talk we will study the construction of X, its properties and the geometry of the orbits. If time permits, I will also explain how to obtain the linearized line bundles on X.

♠ 12.02.2014 // Anna Wißdorf - The theorem of Mehta-Ramanathan
I will explain the theorem of Mehta-Ramanathan. It shows, that a semistable sheaf on a smooth projective variety of dimension bigger than 2 remains semistable when restricting to a hypersurface of sufficiently high degree.