MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary scheme? When do we know there exists an "Albanese" scheme Alb(X)? That is,

Under what conditions on a scheme X does there exist a morphism X → Alb(X) which factors uniquely into any map from from X to an abelian scheme?

share|cite|improve this question
Unhelpful stupid comment: "abelian scheme" is a relative notion. So really X should be a scheme over a base S (like in the answers). Otherwise someone could argue that X itself was an abelian scheme, of relative dimension 0, over the base X... – Kevin Buzzard Nov 4 '09 at 19:35
up vote 12 down vote accepted

The construction of an Albanese scheme and an Albanese map for proper and geometrically irreducible schemes over a perfect field goes back to the work of Chevalley, to this talk of Serre, and to Grothendieck, e.g. Theorem 3.3. in FGA Exp.VI . The idea is always to look at the dual of an appropriate component of the Picard scheme. One just has to pile enough conditions on the setup to ensure that the Picard functor is representable.

One thing that should be said here is that the Albanese scheme is not in general an abelian scheme but only a torsor over a semi-abelian scheme. Also if we drop the properness condition or consider our original scheme to be defined over a base scheme things become more interesting. In full generality it is possible to define an Albanese 1-motive over the base scheme (it is a complex of sheaves of abelian groups of small amplitude with typically representable cohomology) which has the desired universal property. There are various techniques for construction of this derived version of the Albanese scheme. Some use characteristic zero, resolution of singularities, and Nagata compactifications, and some use simplicial scheme resolutions. There are many cool works in this direction, e.g. the paper of Barbieri-Viale and Srinivas , and the more recent papers of Niranjan Ramachandran and Ayoub and Barbieri-Viale . The appendix of Mochizuki's paper mentioned in Lars' post is also excellent.

share|cite|improve this answer
Excellent indeed, thank-you! – Andrew Critch Oct 28 '09 at 4:31

While not having read it closely myself, I remember that there is an appendix in this paper by Shinichi Mochizuki claiming to develop the theory of Albanese varieties in modern language. Maybe you find it helpful. Skimming it, it seems that his main result is, that any separated geometrically integral scheme of finite type over a field admits an Albanese, see Corollary A.11 and the following remarks.

share|cite|improve this answer

You can construct it in arbitrary characteristic (over an algebraically closed field) using the (reduced, `iterated') Picard variety; check out theorem 3.4 of these notes.

share|cite|improve this answer
The link in the above answer is dead. – JSE Aug 14 '10 at 3:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.