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.

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly 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 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.

On a criterion for hyperplane sections

1988 ◽  
Vol 103 (1) ◽  
pp. 59-67 ◽  
Author(s):  
Lucian Bǎdescu
Keyword(s):  
Line Bundle ◽  
Normal Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Cartier Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Hyperplane Sections

Throughout this paper we shall fix an algebraically closed field k. Consider the following:Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.

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.

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 Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

scholarly journals GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC

2018 ◽  
Vol 19 (2) ◽  
pp. 647-661 ◽  
Author(s):  
Kenta Sato ◽  
Shunsuke Takagi
Keyword(s):  
Projective Variety ◽  
Positive Characteristic ◽  
Hyperplane Section ◽  
Three Dimensional ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Canonical Singularities ◽  
Hyperplane Sections

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

scholarly journals Harder–Narasimhan reduction of a principal bundle

2004 ◽  
Vol 174 ◽  
pp. 201-223 ◽  
Author(s):  
Indranil Biswas ◽  
Yogish I. Holla
Keyword(s):  
Algebraic Group ◽  
Principal Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Mild Assumption

AbstractLet E be a principal G–bundle over a smooth projective curve over an algebraically closed field k, where G is a reductive linear algebraic group over k. We construct a canonical reduction of E. The uniqueness of canonical reduction is proved under the assumption that the characteristic of k is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of k is positive.

scholarly journals Rational connectivity and analytic contractibility

2019 ◽  
Vol 2019 (747) ◽  
pp. 45-62
Author(s):  
Morgan Brown ◽  
Tyler Foster
Keyword(s):  
Laurent Series ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Homotopy Equivalence ◽  
Formal Laurent Series ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Rationally Connected ◽  
Smooth Projective Varieties

Abstract Let {{k}} be an algebraically closed field of characteristic 0, and let {f:X\to Y} be a morphism of smooth projective varieties over the ring {k((t))} of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map {f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}} between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any {\mathbb{P}^{n}} -bundle over a smooth projective {k((t))} -variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over {k((t))} is contractible.

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