Algebraic and Arithmetic Geometry Seminar

A functorial approach to the stability of vector bundles

by Dario Weissmann (Pisa)

Aula Seminari (Department of Mathematics)

Aula Seminari

Department of Mathematics


On a smooth projective curve the locus of stable bundles that remain stable on all etale Galois covers prime to the characteristic defines a big open in the moduli space of stable bundles. In particular, the bundles trivialized on some etale Galois cover of degree prime to the characteristic are not dense - in contrast to a theorem of Ducrohet and Mehta stating that all etale trivializable bundles are dense in positive characteristic. Etale trivializable bundles correspond to representations of the etale fundamental group and we obtain the upshot: the etale fundamental group behaves differently in positive vs. characteristic 0.

As an application we study the closure of the prime to p etale trivializable vector bundles. This closure is closely related to a stratification of the moduli space of stable vector bundles via their decomposition behaviour on Galois covers of degree prime to the characteristic. We obtain mostly sharp dimension estimates for the closure of the prime to p etale trivializable bundles as well as the decomposition strata.