Moduli spaces classifying bundles over Azumaya algebras-the case of algebraic surfaces.

• Applicant: Professor Dr. Ulrich Stuhler
• Subject Area: Mathematics
• Term: from 2011 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 191121449

The aim of this project is the study of moduli spaces of vector bundles over algebraic varieties. The case of an algebraic surfaces is particular important here as the case of dimension one,that is algebraic curves, is rather well understood, at least in comparison to the general case.Therefore the case of dimension two, that is algebraic surfaces should be typical for the additional difficulties. The existence of the moduli schemes in question is guaranteed since a number of years under quite general circumstances. Unfortunately a more concrete and detailed understanding is very limited. Therefore it makes good sense to consider additional, but natural properties, which make the problems easier to handle. In our case we consider the additional action of an Azumaya algebra on the coherent sheaves resp. vector bundles. These are sheaves of associative algebras, which generically are central simple algebras over the function field of the base variety. The coherent sheaves considered are assumed to be simple modules over the generic stalk that is the central simple algebra. The first case satisfying these conditions is still strictly commutative. The sheaf of algebras is just the structure sheaf of the base variety, the coherent sheaves are invertible module sheaves over the variety. The emerging moduli problem is completely classical and leads to the Picard varieties. Their theory is much easier than the theory for general vector bundles. Therefore the cases which one wants to study in this project, should be considered in this context. The general theory of these moduli varieties in fact is more direct than the case of general bundles and was done by several people including N.Hoffmann and the author of this proposal. So for example it is not necessary to specify an ample divisor on the variety to apply geometric invariant theory. Also the moduli scheme are automatically proper resp. compact. Therefore there is some hope, that also the finer properties of these modular varieties beyond pure existence are better understandable. To have some success it is of course as always important to have a good number of well understood examples to be able to test more general conjectures. Also on the way to understanding the moduli problem some interesting deformation theoretic questions had to be studied under somewhat special assumptions. A more technical part of this project is to come here to a better understanding and to find similiar results in a more general setting.

• DFG Programme: Research Grants