scholarly journals ON VARIETIES WITH TRIVIAL TANGENT BUNDLE IN CHARACTERISTIC 0$" height="12pt">

2019 ◽  
pp. 1-17
Author(s):  
KIRTI JOSHI
Keyword(s):  
Abelian Variety ◽  
Tangent Bundle ◽  
Abelian Varieties ◽  
Projective Varieties ◽  
Crystalline Cohomology ◽  
Smooth Projective Varieties ◽  
Tangent Bundles ◽  
Torsion Free ◽  
Projective Surfaces

In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$ . Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.

scholarly journals Abelian Varieties as Automorphism Groups of Smooth Projective Varieties

2018 ◽  
Vol 2020 (7) ◽  
pp. 1942-1956
Author(s):  
Davide Lombardo ◽  
Andrea Maffei
Keyword(s):  
Automorphism Group ◽  
Projective Variety ◽  
Automorphism Groups ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Smooth Projective Varieties

Abstract We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.

scholarly journals Abelian varieties isogenous to a power of an elliptic curve

2018 ◽  
Vol 154 (5) ◽  
pp. 934-959 ◽  
Author(s):  
Bruce W. Jordan ◽  
Allan G. Keeton ◽  
Bjorn Poonen ◽  
Eric M. Rains ◽  
Nicholas Shepherd-Barron ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Opposite Direction ◽  
Abelian Varieties ◽  
Sufficient Conditions ◽  
Higher Dimensional ◽  
Free Left ◽  
Necessary And Sufficient ◽  
Finitely Presented ◽  
Torsion Free

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

scholarly journals Derived equivalent threefolds, algebraic representatives, and the coniveau filtration

2018 ◽  
Vol 167 (01) ◽  
pp. 123-131 ◽  
Author(s):  
JEFFREY D. ACHTER ◽  
SEBASTIAN CASALAINA-MARTIN ◽  
CHARLES VIAL
Keyword(s):  
Abelian Varieties ◽  
Projective Varieties ◽  
Chow Motives ◽  
Smooth Projective Varieties ◽  
Coniveau Filtration

AbstractA conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus on threefolds over perfect fields, and unconditionally secure results, which are implied by Orlov's conjecture, concerning the geometric coniveau filtration, and abelian varieties attached to smooth projective varieties.

scholarly journals Rational points on abelian varieties over function fields and Prym varieties

2020 ◽  
Author(s):  
Abolfazl Mohajer
Keyword(s):  
Structure Theorem ◽  
Function Fields ◽  
Abelian Varieties ◽  
High Rank ◽  
Prym Variety ◽  
Projective Varieties ◽  
Prym Varieties ◽  
Weil Group ◽  
Galois Covers ◽  
Smooth Projective Varieties

AbstractIn this paper, using a generalization of the notion of Prym variety for covers of projective varieties, we prove a structure theorem for the Mordell–Weil group of abelian varieties over function fields that are twists of abelian varieties by Galois covers of smooth projective varieties. In particular, the results we obtain contribute to the construction of Jacobians of high rank.

scholarly journals SEMISTABILITY OF RESTRICTED TANGENT BUNDLES AND A QUESTION OF I. BISWAS

2013 ◽  
Vol 24 (01) ◽  
pp. 1250122 ◽  
Author(s):  
PRISKA JAHNKE ◽  
IVO RADLOFF
Keyword(s):  
Riemann Surface ◽  
Abelian Variety ◽  
Tangent Bundle ◽  
Compact Riemann Surface ◽  
Projective Manifold ◽  
Holomorphic Map ◽  
Complex Projective ◽  
Tangent Bundles

Let M be a complex projective manifold with the property that for any compact Riemann surface C and holomorphic map f : C → M the pullback of the tangent bundle of M is semistable. We prove that in this case M is a curve or a finite étale quotient of an abelian variety answering a conjecture of Biswas.

scholarly journals On descending cohomology geometrically

2017 ◽  
Vol 153 (7) ◽  
pp. 1446-1478 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Affirmative Answer ◽  
Cohomology Groups ◽  
Projective Varieties ◽  
Smooth Projective Varieties

In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur’s question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel–Jacobi map admits a distinguished model over the rationals.

scholarly journals A Characterization of Ordinary Abelian Varieties by the Frobenius Push-Forward of the Structure Sheaf II

2018 ◽  
Vol 2019 (19) ◽  
pp. 5975-5988
Author(s):  
Sho Ejiri ◽  
Akiyoshi Sannai
Keyword(s):  
Abelian Variety ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Line Bundles ◽  
Characteristic P ◽  
Structure Sheaf ◽  
Direct Sum ◽  
Push Forward

Abstract In this paper, we prove that a smooth projective variety X of characteristic p > 0 is an ordinary abelian variety if and only if KX is pseudo-effective and $F_{*}^{e}{\mathcal {O}}_{X}$ splits into a direct sum of line bundles for an integer e with pe > 2.

scholarly journals GALOIS THEORY OF THE CANONICAL THETA STRUCTURE

2011 ◽  
Vol 07 (01) ◽  
pp. 173-202
Author(s):  
ROBERT CARLS
Keyword(s):  
Galois Theory ◽  
Theoretical Foundation ◽  
Abelian Varieties ◽  
Geometric Mean ◽  
Null Point ◽  
Point Counting ◽  
Characteristic 2 ◽  
Theoretic Characterization ◽  
Generalized Arithmetic

In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

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