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An [[Abelian variety|Abelian variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112301.png" /> canonically attached to an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112302.png" />, which is the solution of the following universal problem: There exists a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112303.png" /> such that any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112304.png" /> into an Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112305.png" /> factors into a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112307.png" /> (so named in honour of G. Albanese). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112308.png" /> is a complete non-singular variety over the field of complex numbers, the Albanese variety can be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a0112309.png" /> be the space of everywhere-regular differential forms of degree 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123010.png" />. Each one-dimensional cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123011.png" /> of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123012.png" /> determines a linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123014.png" />. The image of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123015.png" /> thus obtained is a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123017.png" />, and the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123018.png" /> coincides with the Albanese variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123019.png" />. From the algebraic point of view, an Albanese variety may be considered as a method of defining an algebraic structure on some quotient group of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123020.png" /> of zero-dimensional cycles of degree 0 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123022.png" /> is a non-singular complete algebraic curve, both its Picard variety and its Albanese variety are called its [[Jacobi variety|Jacobi variety]]. If the ground field has characteristic zero, then the equalities
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An
 
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[[Abelian variety|Abelian variety]] ${\rm Alb}(X)$ canonically attached to an
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123023.png" /></td> </tr></table>
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algebraic variety $X$, which is the solution of the following
 
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universal problem: There exists a morphism $\phi:X\to{\rm Alb}(X)$ such that any morphism
are valid. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123024.png" /> is called the irregularity of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123025.png" />. If the field has finite characteristic, the inequalities
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$f:X\to A$ into an Abelian variety $A$ factors into a product $f={\tilde f}\phi$, where ${\tilde f}:A\to{\rm Alb}(X)$
 
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(so named in honour of G. Albanese). If $X$ is a complete non-singular
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123026.png" /></td> </tr></table>
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variety over the field of complex numbers, the Albanese variety can be
 
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described as follows. Let $\Omega^1$ be the space of everywhere-regular
hold. If the ground field has positive characteristics it can happen that
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differential forms of degree 1 on $X$. Each one-dimensional cycle $\gamma$
 
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of the topological space $X$ determines a linear function $\omega\mapsto \int_\gamma\omega$ on
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011230/a01123027.png" /></td> </tr></table>
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$\Omega^1$. The image of the mapping $H_1(X,{\mathbb Z}) \to (\Omega^1)^*$ thus obtained is a lattice $\Gamma$ in
 
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$(\Omega^1)^*$, and the quotient space $(\Omega^1)^*/\Gamma$ coincides with the Albanese variety of
The Albanese variety is dual to the [[Picard variety|Picard variety]].
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$X$. From the algebraic point of view, an Albanese variety may be
 +
considered as a method of defining an algebraic structure on some
 +
quotient group of the group ${\mathbb Z}$ of zero-dimensional cycles of degree 0
 +
on $X$. If $X$ is a non-singular complete algebraic curve, both its
 +
Picard variety and its Albanese variety are called its
 +
[[Jacobi variety|Jacobi variety]]. If the ground field has
 +
characteristic zero, then the equalities  
 +
$${\rm dim}\;{\rm Alb}(X) = {\rm dim}_k\; H^0(X,\Omega_X^1) = {\rm dim}_k\; H^1(X,{\mathcal O}_X) $$
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are valid. The number
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${\rm dim}\;{\rm Alb}(X)$ is called the irregularity
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${\rm irr}(X)$ of the variety $X$. If the field has
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finite characteristic, the inequalities  
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$${\rm irr}\;(X) \le {\rm dim}\; H^0(X,\Omega_X^1) \text{ and } {\rm irr}\;(X)\le {\rm dim}\; H^1(X,{\mathcal O}_X) $$
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hold. If the ground
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field has positive characteristics it can happen that  
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$${\rm dim}\; H^0(X,\Omega_X^1) \ne  {\rm dim}\; H^1(X,{\mathcal O}_X) $$
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The
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Albanese variety is dual to the
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[[Picard variety|Picard variety]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Baldassarri,   "Algebraic varieties" , Springer (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang,   "Abelian varieties" , Springer (1983)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> M. Baldassarri, "Algebraic varieties" , Springer
 +
(1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
 +
S. Lang, "Abelian varieties" , Springer (1983)</TD></TR></table>

Latest revision as of 18:09, 12 September 2011

An Abelian variety ${\rm Alb}(X)$ canonically attached to an algebraic variety $X$, which is the solution of the following universal problem: There exists a morphism $\phi:X\to{\rm Alb}(X)$ such that any morphism $f:X\to A$ into an Abelian variety $A$ factors into a product $f={\tilde f}\phi$, where ${\tilde f}:A\to{\rm Alb}(X)$ (so named in honour of G. Albanese). If $X$ is a complete non-singular variety over the field of complex numbers, the Albanese variety can be described as follows. Let $\Omega^1$ be the space of everywhere-regular differential forms of degree 1 on $X$. Each one-dimensional cycle $\gamma$ of the topological space $X$ determines a linear function $\omega\mapsto \int_\gamma\omega$ on $\Omega^1$. The image of the mapping $H_1(X,{\mathbb Z}) \to (\Omega^1)^*$ thus obtained is a lattice $\Gamma$ in $(\Omega^1)^*$, and the quotient space $(\Omega^1)^*/\Gamma$ coincides with the Albanese variety of $X$. From the algebraic point of view, an Albanese variety may be considered as a method of defining an algebraic structure on some quotient group of the group ${\mathbb Z}$ of zero-dimensional cycles of degree 0 on $X$. If $X$ is a non-singular complete algebraic curve, both its Picard variety and its Albanese variety are called its Jacobi variety. If the ground field has characteristic zero, then the equalities $${\rm dim}\;{\rm Alb}(X) = {\rm dim}_k\; H^0(X,\Omega_X^1) = {\rm dim}_k\; H^1(X,{\mathcal O}_X) $$ are valid. The number ${\rm dim}\;{\rm Alb}(X)$ is called the irregularity ${\rm irr}(X)$ of the variety $X$. If the field has finite characteristic, the inequalities $${\rm irr}\;(X) \le {\rm dim}\; H^0(X,\Omega_X^1) \text{ and } {\rm irr}\;(X)\le {\rm dim}\; H^1(X,{\mathcal O}_X) $$ hold. If the ground field has positive characteristics it can happen that $${\rm dim}\; H^0(X,\Omega_X^1) \ne {\rm dim}\; H^1(X,{\mathcal O}_X) $$ The Albanese variety is dual to the Picard variety.

References

[1] M. Baldassarri, "Algebraic varieties" , Springer (1956)
[2] S. Lang, "Abelian varieties" , Springer (1983)
How to Cite This Entry:
Albanese variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Albanese_variety&oldid=18712
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article