scholarly journals Griffiths Groups of Supersingular Abelian Varieties

2002 ◽  
Vol 45 (2) ◽  
pp. 213-219
Author(s):  
B. Brent Gordon ◽  
Kirti Joshi
Keyword(s):  
Projective Variety ◽  
Algebraic Closure ◽  
Torsion Group ◽  
Smooth Projective Variety ◽  
Algebraic Cycles ◽  
Closed Field ◽  
Ground Field ◽  
Tate Conjecture ◽  
Primary Torsion ◽  
Projective Surfaces

AbstractThe Griffiths group Grr(X) of a smooth projective variety X over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension r on X modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group Gr2() of a supersingular abelian variety over the algebraic closure of a finite field of characteristic p is at most a p-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of C. Schoen it is also shown that if the Tate conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field k of characteristic p > 2, then the Griffiths group of any ordinary abelian threefold over the algebraic closure of k is non-trivial; in fact, for all but a finite number of primes l ≈ p it is the case that Gr2() ⊗ ℤl ≈ 0.

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 SOME NEW EXAMPLES OF SMASH-NILPOTENT ALGEBRAIC CYCLES

2017 ◽  
Vol 59 (3) ◽  
pp. 623-634 ◽  
Author(s):  
ROBERT LATERVEER
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Algebraic Cycles ◽  
Numerical Equivalence

AbstractVoevodsky has conjectured that numerical equivalence and smash-equivalence coincide for algebraic cycles on any smooth projective variety. Building on work of Vial and Kahn–Sebastian, we give some new examples of varieties where Voevodsky's conjecture is verified.

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.

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 Remarks on Murre's conjecture on Chow groups

2013 ◽  
Vol 12 (1) ◽  
pp. 3-14 ◽  
Author(s):  
Kejian Xu ◽  
Ze Xu
Keyword(s):  
Abelian Variety ◽  
Function Field ◽  
Projective Variety ◽  
Product Variety ◽  
Smooth Projective Variety ◽  
Chow Groups ◽  
Closed Field ◽  
Irreducible Curve ◽  
Algebraically Closed Field

AbstractFor certain product varieties, Murre's conjecture on Chow groups is investigated. More precisely, let k be an algebraically closed field, X be a smooth projective variety over k and C be a smooth projective irreducible curve over k with function field K. Then we prove that if X (resp. XK) satisfies Murre's conjectures (A) and (B) for a set of Chow-Künneth projectors {, 0 ≤ i ≤ 2dim X} of X (resp. for {()K} of XK) and if for any j, , then the product variety X × C also satisfies Murre's conjectures (A) and (B). As consequences, it is proved that if C is a curve and X is an elliptic modular threefold over k (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture (B) is true for the fourfold X × C.

scholarly journals The Brauer group and indecomposable -cycles

2015 ◽  
Vol 152 (5) ◽  
pp. 1041-1051 ◽  
Author(s):  
Bruno Kahn
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Brauer Group ◽  
Algebraically Closed Field ◽  
Insight Into

We show that the torsion in the group of indecomposable $(2,1)$-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as $b_{2}-{\it\rho}>0$. We derive a new insight into Roǐtman’s theorem on torsion $0$-cycles over a surface.

scholarly journals Cycles de codimension 2 et H3 non ramifié pour les variétés sur les corps finis

2013 ◽  
Vol 11 (1) ◽  
pp. 1-53 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Bruno Kahn
Keyword(s):  
Cohomology Group ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Chow Group ◽  
Projective Surface ◽  
Tate Conjecture ◽  
Generic Fibre ◽  
Unramified Cohomology ◽  
Open Question ◽  
Global Principle

AbstractLet X be a smooth projective variety over a finite field $\mathbb{F}$. We discuss the unramified cohomology group H3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds, H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefold X equipped with a fibration onto a curve C, the generic fibre of which is a smooth projective surface V over the global field $\mathbb{F}$(C), the vanishing of H3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors on X implies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles on V. This sheds new light on work started thirty years ago.

EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER–LEFSCHETZ THEOREMS

2013 ◽  
Vol 15 (05) ◽  
pp. 1350003 ◽  
Author(s):  
G. V. RAVINDRA ◽  
AMIT TRIPATHI
Keyword(s):  
Characteristic Zero ◽  
Deformation Theory ◽  
Vector Bundles ◽  
Projective Variety ◽  
Hyperplane Section ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Open Set ◽  
Higher Rank ◽  
Lefschetz Theorems

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle [Formula: see text] on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

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 On Deformations of Pairs (Manifold, Coherent Sheaf)

2019 ◽  
Vol 71 (5) ◽  
pp. 1209-1241 ◽  
Author(s):  
Donatella Iacono ◽  
Marco Manetti
Keyword(s):  
Lie Algebra ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Coherent Sheaf ◽  
Closed Field ◽  
Trace Map ◽  
Deformation Problem ◽  
Graded Lie Algebra ◽  
Infinitesimal Deformations ◽  
Differential Graded Lie Algebra

AbstractWe analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.

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