Newsgroups: sci.math.research
From: gks1@can.pmms.cam.ac.uk (Gregory Sankaran)
Subject: Re: Abelian varieties
Date: Wed, 29 Mar 1995 13:26:14 GMT

In article <3l6h15$a1d@aries.ingress.com> jjk@ingress.com (Jerry Kovacic) writes:
>I am looking for examples of Abelian varieties.  I have been away from
>mathematics research for over 10 years, and was never an algebraic
>geometer in the first place, so I need something pretty elementary.  
>
>I am aware of the elliptic curve in projective 2-space defined 
>by the homogeneous equation   y^2 z = 4 x^3 - g_2 x z^2 - g_3 z^3 .  
>I am looking for something equally explicit.  I would like to know the 
>multiplication formula, and to be able to compute the Lie algebra.
>
>I learned algebraic geometry the old-fashioned way back in the 60's.
>So I am not comfortable with "spec", "sheaves", etc.  Thus I think
>Mumford's book would probably be over my head.
>
>Lang's book might be OK but there is a dearth of examples.
>
>Is there anything else out there that might be suitable?
>
>	Thanks,
>	Jerry Kovacic

The best handbook is Lange (not Lang!) and Birkenhake, Complex Abelian
Varieties (Springer): this takes a complex-analytic approach and
doesn't assume lots of scheme theory. It's quite dense, though. Other
books to investigate are by Kempf (also Springer) and Swinnerton-Dyer
(CUP, out of print but watch this space). The first part of Mumford's
book is also complex-analytic in its approach: the scheme theory comes
later. I do not really recommend Lang, though others may have a
different opinion.

Abelian varieties, even abelian surfaces, are more complicated than
elliptic curves and there isn't anything general like the Weierstrass
cubic equation. For instance, not all abelian surfaces can be embedded
in P^4, and if you want to know which ones can, and how, you will have
to learn about the Horrocks-Mumford bundle. There are results about
equations defining abelian varieties but they are not simple to state.

Gregory Sankaran