The cohomology of the moduli space of Abelian varieties

[Book reviewed by Fernando Q. Gouvêa, on 08/14/2013]

Algebraic geometers have been thinking about moduli spaces for a very long time, and the topic has a central role in that discipline and in various other branches of mathematics. Basically, a moduli space is some sort of geometric object whose points classify (equivalence classes of) other geometric objects in a way that makes sense of "continuously varying" the objects being classified.

David Ben-Zvi has a very nice article about moduli spaces in the Princeton Companion to Mathematics. Here, let’s only consider two easy examples. Suppose, first, that the objects we want to classify are lines on the plane. We remember that any such line is given by an equation of the form ax+by=c, and so we can represent the line by the triple (a,b,c). In order to have an honest line, we need a and b not to be both zero. And we need to identify two triples that differ by scaling, since (ka,kb,kc) defines the same line as (a,b,c) for any k≠0.

The set of all triples (a,b,c)≠(0,0,0), taken up to scaling, is the real projective plane P2(R). The only point excluded by our requirement that a and b can’t both be zero is (0,0,1), so the "space of lines in R2" is P2(R) minus one point. It is, of course, very tempting to add that extra line somehow, thereby compactifying the space of all lines. And, since P2(R) has a natural topology, we can talk about a continuous (or algebraic) family of lines: it is just a continuous (or algebraic) curve in P2(R).

In most cases, we don’t actually want to consider all objects in a class, but rather want to consider them up to some kind of equivalence. Here is a famous example. Suppose we start by considering four points in the extended complex plane C∪∞. Classifying all such sets of four points is easy (and dumb): the resulting space is just the Cartesian product of four copies of the extended plane. What is interesting, on the other hand is to bring in the standard Möbius transformations
z↦az+bcz+d
and to say two quadruples are equivalent if there exists a Möbius transformation taking one to the other. It turns out that any three points can always be transformed to the points (0,1,∞), so here’s what we do: given any quadruple (z1,z2,z3,z4), find a Möbius transformation that sends the first three to 0 ,1, and ∞. The quadruple will become (0,1,∞,w), and the number w turns out to classify all quadruples up to this equivalence. It can be any complex number except for 0, 1, and ∞, so once again we have a moduli space that is a "punctured" compact space.

Those examples are easy, but the things people actually think about are very, very hard. So hard, in fact, that the subject runs the risk of splitting up into small groups, each of which work on a particular kind of moduli problem. This book is an attempt to avoid this problem. Various authors were asked to provide an account of recent developments, techniques, and key examples in their area that would be accessible to others working in nearby fields. The editors asked, in particular, that the authors provide "contributions that illustrated ‘secret handshakes’, yogas and heuristics that experts use privately to guide intuition or simplify calculation but that are replaced by more formal arguments, or simply do not appear, in articles aimed at other specialists."

The book is addressed specifically to "producers of algebraic geometry", with the hope that "some consumers from cognate areas" might also profit. For that audience, these three volumes are likely to be of great value.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.
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