\(Pic(X)\) 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.