scholarly journals On principal bundles over a projective variety defined over a finite field

2009 ◽  
Vol 4 (2) ◽  
pp. 209-221 ◽  
Author(s):  
Indranil Biswas
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Fundamental Group ◽  
Projective Variety ◽  
Rational Point ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Linear Algebraic Group ◽  
The Third ◽  
Extra Assumption

AbstractLet M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

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 Higgs bundles and fundamental group schemes

2019 ◽  
Vol 19 (3) ◽  
pp. 381-388
Author(s):  
Indranil Biswas ◽  
Ugo Bruzzo ◽  
Sudarshan Gurjar
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Higgs Bundles ◽  
Chern Classes ◽  
Group Schemes ◽  
Tannakian Category

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

SEMISTABILITY AND NUMERICALLY EFFECTIVENESS IN POSITIVE CHARACTERISTIC

2011 ◽  
Vol 22 (01) ◽  
pp. 25-46 ◽  
Author(s):  
INDRANIL BISWAS ◽  
YOGISH I. HOLLA
Keyword(s):  
Line Bundle ◽  
Parabolic Subgroup ◽  
Projective Variety ◽  
Positive Characteristic ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Smooth Projective Curve ◽  
Trivial Center

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and [Formula: see text], where each Gi is simple with trivial center. For i ∈ [1, m], let ρi : G → Gi be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρi(G) ⊂ Gi is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let EG → M be a principal G-bundle. We prove that the following six statements are equivalent: (1) The line bundle EG(χ) → EG/P associated to the principal P-bundle EG → EG/P for the character χ is numerically effective. (2) The sequence of principal G-bundles [Formula: see text] is bounded, where FM is the absolute Frobenius morphism of M. (3) The principal G-bundle EG is strongly semistable with respect to ξ, and c2( ad (EG)) is numerically equivalent to zero. (4) The principal G-bundle EG is strongly semistable with respect to ξ, and [c2( ad (EG))c1(ξ)d-2] = 0. (5) The adjoint vector bundle ad (EG) is numerically effective. (6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*EG → Y is semistable.

scholarly journals Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

2018 ◽  
Vol 2019 (19) ◽  
pp. 6089-6112
Author(s):  
Shu Kawaguchi ◽  
Kazuhiko Yamaki
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Discrete Valuation Ring ◽  
Complete Discrete Valuation Ring ◽  
System L ◽  
Semistable Model ◽  
Adjoint Bundle

Abstract Let R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

scholarly journals Second cohomology and nilpotency class 2

2007 ◽  
pp. 3-10
Author(s):  
Sinisa Crvenkovic ◽  
Vladimir Tasic
Keyword(s):  
Fundamental Group ◽  
Nilpotent Group ◽  
Projective Variety ◽  
Betti Numbers ◽  
Smooth Projective Variety ◽  
Nilpotency Class ◽  
Central Extensions

Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety.

scholarly journals Monodromy group for a strongly semistable principal bundle over a curve, II

2008 ◽  
Vol 1 (3) ◽  
pp. 583-607 ◽  
Author(s):  
Indranil Biswas ◽  
A. J. Parameswaran
Keyword(s):  
Vector Bundles ◽  
Principal Bundle ◽  
Monodromy Group ◽  
Rational Point ◽  
Group Scheme ◽  
Linear Algebraic Group ◽  
Tannakian Category ◽  
Adjoint Bundle ◽  
Semistable Vector Bundles ◽  
Nontrivial Character

AbstractLet X be a geometrically irreducible smooth projective curve defined over a field k. Assume that X has a k–rational point; fix a k–rational point x ε X. From these data we construct an affine group scheme X defined over the field k as well as a principal X–bundle over the curve X. The group scheme X is given by a ℚ–graded neutral Tannakian category built out of all strongly semistable vector bundles over X. The principal bundle is tautological. Let G be a linear algebraic group, defined over k, that does not admit any nontrivial character which is trivial on the connected component, containing the identity element, of the reduced center of G. Let EG be a strongly semistable principal G–bundle over X. We associate to EG a group scheme M defined over k, which we call the monodromy group scheme of EG, and a principal M–bundle EM over X, which we call the monodromy bundle of EG. The group scheme M is canonically a quotient of X, and EM is the extension of structure group of . The group scheme M is also canonically embedded in the fiber Ad(EG)x over x of the adjoint bundle.

scholarly journals Restrictions on the prime-to- fundamental group of a smooth projective variety

2015 ◽  
Vol 151 (6) ◽  
pp. 1083-1095
Author(s):  
Donu Arapura
Keyword(s):  
Fundamental Group ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Kähler Groups ◽  
One Relator Groups

The goal of this paper is to obtain restrictions on the prime-to-$p$ quotient of the étale fundamental group of a smooth projective variety in characteristic $p\geqslant 0$. The results are analogues of some theorems from the study of Kähler groups. Our first main result is that such groups are indecomposable under coproduct. The second result gives a classification of the pro-$\ell$ parts of one-relator groups in this class.

scholarly journals Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

2021 ◽  
pp. 2150086
Author(s):  
Donatella Iacono ◽  
Marco Manetti
Keyword(s):  
Line Bundle ◽  
Lie Algebra ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Canonical Bundle ◽  
Algebraically Closed Field

We investigate the deformations of pairs [Formula: see text], where [Formula: see text] is a line bundle on a smooth projective variety [Formula: see text], defined over an algebraically closed field [Formula: see text] of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair [Formula: see text] is homotopy abelian whenever [Formula: see text] has trivial canonical bundle, and so these deformations are unobstructed.

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