(Pic(X)) vs. (CP^infty)
There are likely inaccuracies in this post, as I wrote it quickly and am just beginning to learn the basics of algebraic geometry. Constructive criticism is strongly encouraged.
As we saw in a Precursor to Characteristic Classes, (CP^\infty) is the classifying space of complex line bundles over (X).
(CP^\infty) is, in some sense, the moduli space of line bundles over a point. There’s only one isomorphism class of line bundles over a point — but then this one line bundle has automorphism group (C^\times) (which is homotopy equivalent to (U(1))).
Allow me to introduce you to something that looks a LOT like (CP^\infty).
What is this map, (p \times C \to Pic(C) \times C), you might ask. Choose a point on our curve (C) and this defines a line bundle over (S) corresponding to a choice of the class of line bundles in (Pic(C)). In other words, we take a point on a (not sure if I require smoothness here) algebraic curve and turn it into a line bundle on that curve.
Warning: I’ve been told that there is a difference between topological line bundles and algebraic line bundles, unfortunately, I don’t know why or what it is! I mention this, for (Pic(X)) usually corresponds to *algebraic* line bundles over (X).
Why is the multiplicative formal group getting involved? Let’s briefly review what the multiplicative formal group law is (as a group scheme).
Thank you to Edward Frenkel for kindly explaining the difference between (CP^\infty) and (Pic(X)) (both classifying spaces of line bundles), and to Qiaochu Yuan for explaining why on earth (CP^\infty) is the moduli space of line bundles over a point. Any errors are mine, not theirs.