# Project A1: Algebraic vector bundles

## Principal investigator

## Participating scientists

## Summary

The classical Narasimhan-Seshadri correspondence relates stable vector bundles on a Riemann surface to unitary representations of the fundamental group. Recently a partial p-adic analogue of this correspondence has been developed. However basic problems are still open in the p-adic case and one of the main goals of the project is to solve them. We now give a more detailed summary. The project consists of the following parts which are loosely connected:- p-adic Narasimhan-Seshadri theory
- Duality of Néron-models

**p-adic Narasimhan-Seshadri theory.**
In the last years p-adic analogues
have been developed of the classical Narasimhan-Seshadri correspondence
and more generally of Simpson's Higgs bundle theory. The
correspondences relate certain semistable vector- resp. Higgs bundles
with p-adic representations of the geometric Grothendieck fundamental
group of the underlying variety.
The main open question concerns the class of vector bundles on
curves which are accessible to the p-adic theory. They can be characterized
in terms of strong semistability by their potential reduction behaviour.
The main aim of the project is to find a condition on the bundle
itself which guarantees this potential reduction behaviour. Hopefully
ordinary semisimplicity which is a necessary condition will suffice.
The difficulty of this problem comes from the poorly understood
notion of strong semistability.
Another interesting problem is to find a criterion for the question
when a p-adic representation comes from a vector bundle.
In the higher dimensional theory it is not even known if the class of
useful bundles can be characterized in terms of reduction properties let
alone by generic properties. This question should also be addressed.
If the preceeding problems could be solved it will be possible to
think about a p-adic analogue of Goldman's study of the dynamical
properties of the action of the mapping class group on moduli spaces
of vector bundles.

**Duality of Néron-models.**
Even the moduli spaces of line bundles still
present difficult open problems. The basic question in this direction
which should be addressed is this: Is it possible to understand the
Néron-model of the dual of an abelian variety A_{K} over a discretely
valued field in terms of line bundles on the Néron model of A_{K}? A
particular aspect of this problem concerns Grothendieck's pairing on
component groups which will be investigated in detail.