Torsors, Vector Bundles and Fundamental Group Scheme – TOFIGROU

The methods used are not only the classical techniques well known in arithmetic geometry and in algebraic geometry in positive characteristic but also new methods «ad hoc« for each project. Let us just recall some of the most important: to study the theory of the fundamental group scheme it is important to use both the tannakian and the profinite construction. Tannakian methods are in general more suitable for geometric objects over fields while profinite constructions can fit better some arithmetic objects, like schemes defined over rings and not just fields. One of the main goal in this section is certainly the extension of the tannakian methods even to those arithmetic objects for which, so far, those methods don't exist. In order to obtain this one should deeply work Saavedra-Rivano's results on tannakian theory over rings, for many years forgotten by the scientific communities. This new method would be a wonderful tool to study further the problem of extending and lifting torsors.

The main results obtained so far are the following :

1) [In collaboration with Indranil Biswas, T.I.F.R. Mumbai] Let X be a normal rationally chain connected variety over an algebraically closed field of positive characteristic then we prove that the maximal local quotient of the fundamental group scheme is finite. This result is sharp. In particular the (global) fundamental group scheme of a normal Fano variety is finite. This generalize a result of Chambert-Loir and (independently) Kollar who proved that the étale fundamental group of a Fano variety is finite.

2) [In collaboration with Michel Emsalem (Université de Lille 1 and Carlo Gasbarri (Université de Stasbourg)]
Let S be a Dedekind scheme and X and a scheme faithfully flat and of finite type over S. The existence of an S-group scheme classifying all torsors above X under the action S-group schemes finite and flat was conjectured by Grothendieck in his famous SGA1 . When S is the spectrum of a field Madhav Nori showed the existence of this object in his PhD thesis and he called it the «fundamental group scheme.« We dealt with the case where S is a Dedekind scheme : in this case we were able not only to answer positively to Grothendick's expectations but we also found a larger object that classifies all torsors over X under the action of quasi-finite and flat group schemes.

We are planning to hire a PhD student for next year with (possibly) a scholarship given by the Université de Nice-Sophia Antipolis. In this case the student will work on problems related to the computaion of the fundamental group scheme, either over a field or over a Dedekind scheme (or both).

Antei M., Biswas I. On the fundamental group scheme of rational chain connected varieties
IMRN (to appear)

Antei M., Emsalem M., Gasbari C., Sur l'existence du schéma en groupes fondamental (submitted)

The project mixes two themes of Algebraic Geometry: the problem of extending and lifting torsors and the study of vector bundles as the main tool in the theory of the fundamental group scheme. Our two Research Packages have originally been inspired by each of them although the purpose of the project is precisely their interaction: the problems, the proposed approaches, and the participants have a background in either Arithmetic or Algebraic Geometry.

Research Package 1 «Torsors » aims at studying the problem of extending and lifting of torsors. We are given a scheme X defined over a discrete valuation ring R and we consider its generic fiber XK over K (where K is the field of fractions of R). Let G be a finite K-group scheme and Y a G-torsor over XK. Is it possible to find a finite and flat R-group scheme G’, model of G, and a G’-torsor Y’ over X model of the given torsor Y over XK? This problem originally comes from Grothendieck’s celebrated Théorie de la spécialisation du groupe fundamental where the author of SGA1 proves that, possibly after extending scalars, the problem has a solution when R is a complete discrete valuation ring with algebraically closed residue field of positive characteristic p, with X proper and smooth over R with geometrically integral fibres where p does not divide the order of G. In the last decades there have been many improvements. Simmetrically let us consider the special fiber Xs of X over k=R/m, the residue field of R. Let H be a finite k-group scheme and Y a H-torsor over Xs. Is it possible to find a finite and flat R-group scheme H’, which lifts H, and a H’-torsor Y’ over X which lifts the given torsor Y over Xs? On this direction Pop recently proved the famous Oort conjecture but much more can still be studied. We will push to its limits a new approach using the deformation of essentially finite vector bundles.

Research Package 2 « Vector Bundles and the Fundamental Group Scheme » concerns principally the study of the properties of the fundamental group scheme p(X,x) of a scheme X over a field k at a k-rational point x. Conjectured by Grothendieck and first defined by Nori it is, when X is reduced connected and proper over k, the k-group scheme naturally associated to the neutral tannakian category EF(X) whose objects are essentially finite vector bundles. Alternatively it can be constructed as the projective limit of finite and flat group schemes occurring as group of torsors over X. Thus in particular when k is just an algebraically closed field of characteristic zero it is nothing else but the étale fundamental group p^ét (X,x). Properties of the fundamental group schemes are in general difficult to study because for example it does not behaves well after base change; despite its complication it still shares some properties with the étale fundamental group. That is why it should hide inside itself many properties of the scheme and it is a natural object to study in the anabelian philosophy. The fundamental group scheme has been generalized in many different directions (by Gasbarri, Langer, Borne and Vistoli among others) and its properties have been widely studied for example by Mehta, Subramanian, Biswas, Esnault, Pauly and Antei. In particular let us recall that Gasbarri gave a construction for p(X,x) in the case of a scheme X over a Dedekind scheme S pointed at a S-valued point x. This makes clear how a deep knowledge of properties of the fundamental group scheme gives a usefool tool in answering to the main problem described in the section Research Package 1, and, of course, vice-versa.

The project is coordinated by Marco Antei.