Niels Borne

Parabolic sheaves on logarithmic schemes

Abstract We show how the natural context for the definition of parabolic sheaves on a scheme is that of logarithmic geometry. The key point is a reformulation of the concept of logarithmic structure in the language of symmetric monoidal categories, which might be of independent interest. Our main result states that parabolic sheaves can be interpreted as quasi-coherent sheaves on certain stacks of roots.

The Nori fundamental gerbe of a fibered category

We extend Nori’s theory of the fundamental group scheme to a theory of the fundamental gerbe, which applies to schemes, algebraic stacks, and more general fibered categories, even in the absence of a rational point. We give a Tannakian interpretation of the fundamental gerbe in terms of essentially finite bundles, extending Nori’s correspondence for complete varieties with a rational point. We also show how our formalism allows a natural formulation of Grothendieck’s Section Conjecture in arbitrary characteristic.

THE NORI FUNDAMENTAL GERBE OF TAME STACKS

Given an algebraic stack, we compare its Nori fundamental group with that of its coarse moduli space. We also study conditions under which the stack can be uniformized by an algebraic space.

Parabolic Sheaves and Logarithmic Geometry

The aim of this note is to give an introduction to the notion of parabolic sheaves on logarithmic schemes, as first defined in my joint work with Angelo Vistoli [BV12]. I will explain the examples we started from in order to, hopefully, enlighten the rather formal definitions given in loc. cit. I will conclude by a glimpse at subsequent developments.

Lifting Galois sections along torsors

The cuspidalization conjecture, which is a consequence of Grothendiecks section conjecture, asserts that for any smooth hyperbolic curve X over a finitely generated field k of characteristic 0 and any non empty Zariski open , every section of lifts to a section of . We consider in this article the problem of lifting Galois sections to the intermediate quotient introduced by Mochizuki [10]. We show that when and is an union of torsion sub-packets every Galois section actually lifts to . One of the main tools in the proof is the construction of torus torsors and over X and the geometric interpretation .

Extension of Galois Groups by Solvable Groups, and Application to Fundamental Groups of Curves

The issue of extending a given Galois group is conveniently expressed in terms of embedding problems. If the kernel is an abelian group, a natural method, due to Serre, reduces the problem to the computation of an etale cohomology group, that can in turn be carried out thanks to Grothendieck-Ogg-Shafarevich formula. After introducing these tools, we give two applications to fundamental groups of curves.