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Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and…

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Overview

History and motivation

Analytic theory

Algebraic definition

Structure of the group of points

Polarisation and dual abelian variety

Abelian scheme

Semiabelian variety

Sources

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Abelian variety

Nodes140
Edges139
Triples64
Avg. degree1.99
Density0.014286
Components1

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Abelian variety

Top relations

related to Definition · 9
Abelian variety → An, By, Chow's, In, It, Kodaira, Riemann, Since, When
related to Polarisations · 8
Abelian variety → End, Jacobian, Jacobians, Not, Poincaré, Polarised, Rosati, Schottky
related to Riemann conditions · 8
Abelian variety → Choosing, Let, Riemann, Such, The, Then, There, V/L
related to history · 7
Abelian variety → Carl Jacobi, In, Jacobian, Niels Abel, The, This, When
related to The Jacobian of an algebraic curve · 6
Abelian variety → As, Every, From, Jacobian, More, The
related to Important theorems · 5
Abelian variety → It, Jacobian, Matsusaka's, One, This
related to Structure of the group of points · 5
Abelian variety → By, For, Hence, Its, Lefschetz
is a · 4
Abelian variety → group variety, isogeny from an abelian variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the as…, same as that of elliptic curve, smooth projective algebraic variety that is also an algebraic group
related to Abelian functions · 3
Abelian variety → An, For, This
related to Polarisations over the complex numbers · 3
Abelian variety → Over, Riemann, Two Riemann

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Important terminology

abelian variety varieties displaystyle group complex field algebraic dimension number defined riemann finite theory scheme torus curve projective mathbb curves

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Abelian varietyis asmooth projective algebraic variety that is also an algebraic group0.90text
Abelian varietyis asame as that of elliptic curve0.90text
Abelian varietyis agroup variety0.90text
Abelian varietyis aisogeny from an abelian variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the as…0.90text
reduction mod p of abelian varietiesinstance ofThis allows for a uniform treatment of phenomena0.80text
Abelian varietyrelated to Abelian functionsAn0.60section
Abelian varietyrelated to Abelian functionsFor0.60section
Abelian varietyrelated to Abelian functionsThis0.60section
Abelian varietyrelated to Algebraic definitionTwo0.60section
Abelian varietyrelated to DefinitionIt0.60section
Abelian varietyrelated to DefinitionBy0.60section
Abelian varietyrelated to DefinitionKodaira0.60section

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