$Pic(X)$ vs. $CP^\infty$

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.

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.

11004347_10205043264553929_659910229_nWarning: 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).


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