German-Spanish Workshop on Moduli Spaces of Vector Bundles 


Luis Álvarez-Cónsul

On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces


We study holomorphic (n+1)-chains En -> En-1 ->... -> E0 consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on n real parameters was introduced in joint work with O. Garcia-Prada and moduli spaces were constructed by A.H.W. Schmitt. In this talk we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples) in the work of Bradlow, Garcia-Prada and Gothen. Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.

This is joint work with Oscar Garcia-Prada and Alexander Schmitt: International Mathematics Research Papers, Volume 2006:10 (2006), Article ID 73597, 82 pages.

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Christopher Deninger

Vector bundles on p-adic curves and p-adic representations

This is a report on joint work with Annette Werner about a partial p-adic analogue of the classical Narasimhan-Seshadri correspondence.

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Hélène Esnault

k-linearization of the arithmetic fundamental group and applications

This is a report on joint work with Phùng Hô Hai on a characterization of sections of the absolute Galois group of k into the arithmetic fundamental group of X based in a geometric point, when X is smooth and absolutely connected and k has characteristc 0.

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Oscar García-Prada

Maximal surface group representations and Higgs bundles

We apply the theory of Higgs bundles over a compact Riemann surface to study representations of the fundamental group of the surface in a non-compact real Lie group, whose associated symmetric space is of Hermitian type.

Slides

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Marina Logares

Connected components of the moduli space of parabolic U(p,q)-Higgs bundles

Parabolic U(p,q)-Higgs bundles over a compact Riemann surface with a finite set of marked points are objects that correspond to representations of the fundamental group of the surface without the marked points, with fixed holonomy classes around them.

We count the number of connected components of the moduli space of parabolic U(p,q)-Higgs bundles. The main strategy is to use Bott-Morse theoretic techniques using the L2-norm of the Higgs field as Morse function. The connectedness properties of our moduli space reduce to the connectedness of certain moduli spaces of parabolic triples, so we study also the irreducibility of the moduli spaces of parabolic triples.


This is a joint work with Oscar García-Prada and Vicente Muñoz: math.AG/0603365.


Slides

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Stefan Schröer

The Brauer group

We shall discuss some aspects of Brauer groups and Azumaya algebras in the theory of moduli spaces.

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Andrei Teleman

Moduli spaces of holomorphic bundles over class VII surfaces and the GSS conjecture

We introduce the concept of stability for holomorphic bundles over compact manifolds in the general (non-Kählerian)  framework, and we explain how moduli spaces of poly-stable bundles over surfaces are identified with moduli spaces of instantons via the Kobayashi-Hitchin correspondence. We describe explicitly some moduli spaces of stable bundles over class VII surfaces and we show (using a combination of complex geometric and gauge-theoretical arguments) that any  class VII surface with b2=1 has curves. This implies the "GSS conjecture" and completes the classification of class VII surfaces in the case b2=1.