nLab
Albanese variety

Context

Complex geometry

Differential cohomology

Contents

Idea

The Albanese variety Alb(X)Alb(X) of a projective algebraic variety XX 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 XX with a chosen basepoint there is an abelian variety called the Albanese variety Alb(X)Alb(X). This is defined by the following universal property: there is a map of pointed varieties called the Albanese map

i X:XA(X)i_X \colon X \to A(X)

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

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

That is:

f=f¯i X f = \overline{f} \circ i_X

This process defines a functor

Alb:Var *AbVar Alb: Var_* \to AbVar

from pointed varieties to abelian varieties which has a right adjoint

U:AbVarVar * U: AbVar \to Var_*

sending any abelian variety to its underlying pointed variety. The right adjoint UU 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, UU is monadic. As a consequence the composite functor

T=UAlb T = U \circ Alb

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

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

Properties

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

For XX a suitably well behaved (smooth complex, projective) algebraic variety of dimension dim(X)dim(X), its Albanese variety is the intermediate Jacobian in degree 2dim(X)12 dim(X)-1:

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

References


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

nLab page on Albanese variety

Last revised on August 19, 2016 at 01:13:36. See the history of this page for a list of all contributions to it.