**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 *E _{n}*
->

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.

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Marina LogaresConnected 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.

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Stefan SchröerThe 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.