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 AK over a discretely valued field in terms of line bundles on the Néron model of AK? A particular aspect of this problem concerns Grothendieck's pairing on component groups which will be investigated in detail.

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