The fundamental group-scheme of an abelian variety

1983 ◽  
Vol 263 (3) ◽  
pp. 263-266 ◽  
Author(s):  
Madhav V. Nori
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Group Scheme

scholarly journals On the S-fundamental group scheme. II

2012 ◽  
Vol 11 (4) ◽  
pp. 835-854 ◽  
Author(s):  
Adrian Langer
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Group Scheme ◽  
Line Bundles ◽  
Fundamental Groups ◽  
Determinant Line Bundles

AbstractThe S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial étale fundamental group.

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

scholarly journals MODELS OF TORSORS AND THE FUNDAMENTAL GROUP SCHEME

2016 ◽  
Vol 230 ◽  
pp. 18-34 ◽  
Author(s):  
MARCO ANTEI ◽  
MICHEL EMSALEM
Keyword(s):  
Finite Number ◽  
Fundamental Group ◽  
Galois Group ◽  
Closed Subset ◽  
Valuation Ring ◽  
Group Scheme ◽  
Discrete Valuation Ring ◽  
Generic Fiber ◽  
Natural Morphism ◽  
Tannakian Category

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

The fundamental group-scheme

1982 ◽  
Vol 91 (2) ◽  
pp. 73-122 ◽  
Author(s):  
Madhav V Nori
Keyword(s):  
Fundamental Group ◽  
Group Scheme
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