# Difference between revisions of "Albanese variety"

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− | An [[Abelian variety|Abelian variety]] | + | An |

− | + | [[Abelian variety|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 | |

− | are valid. The number | + | $f:X\to A$ 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 | |

− | hold. If the ground field has positive characteristics it can happen that | + | 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|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|Picard variety]]. | ||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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=19583