nLab Albanese variety

Idea

The Albanese variety $Alb(X)$ of a projective algebraic variety $X$ with a chosen basepoint is the universal way of turning this pointed variety into an abelian variety. Moreover, the Albanese variety of the Albanese variety is the Albanese variety. Thus, taking the Albanese variety defines an idempotent monad on the category of pointed projective algebraic varieties.

Definition

By ‘variety’ let us mean a connected complete algebraic variety over an algebraically closed field. Given any variety $X$ with a chosen basepoint there is an abelian variety called the Albanese variety $Alb(X)$. This is defined by the following universal property: there is a map of pointed varieties called the Albanese map

$$i_X \colon X \to A(X)$$

such that any map of pointed varieties $f: X \to A$ where $A$ is abelian factors uniquely as $i_X$ followed by a map of abelian varieties (in particular, a group homomorphism):

$$\overline{f} \colon Alb(X) \to A.$$

That is:

$$f = \overline{f} \circ i_X$$

This process defines a functor

$$Alb: Var_* \to AbVar$$

from pointed varieties to abelian varieties which has a right adjoint

$$U: AbVar \to Var_*$$

sending any abelian variety to its underlying pointed variety. The right adjoint $U$ is faithful, but more remarkably it is also full: any basepoint-preserving map of varieties between abelian varieties is automatically a group homomorphism. (A proof of this fact is outlined in the article abelian variety.) Moreover, $U$ is monadic. As a consequence the composite functor

$$T = U \circ Alb$$

is an idempotent monad on $Var_*$, and its algebras are the abelian varieties.

It follows that the Albanese map $i_X \colon X \to A(X)$ is the unit of the monad $T$, and $Alb(Alb(X)) \cong Alb(X)$. For more details, see the nCafé discussion Two miracles in algebraic geometry.

Properties

The Albanese variety of $X$ is dual, as an abelian variety, to its Picard variety.

For $X$ a suitably well behaved (smooth complex, projective) algebraic variety of dimension $dim(X)$, its Albanese variety is the intermediate Jacobian in degree $2 dim(X)-1$:

$$Alb(X) \coloneqq J^{2 dim(X)-1}(X) \,.$$

Related concepts

• Picard scheme, Jacobian variety

References

• Is forming the Albanese variety a monad?, MathOverflow.

• Patrick Walls, Intermediate Jacobians and Abel-Jacobi maps, 2012 (pdf)

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<http://mathoverflow.net/questions/2548/albanese-schemes-when-does-an-initial-abelian-scheme-exist-under-a-given-sch>

nLab page on Albanese variety