I am now halfway through teaching a graduate course on “Topological methods in algebraic geometry”. The idea is to give a quick, informal introduction to various topics which are important, but a little too advanced to appear in most first-year graduate courses. I hope to communicate a homotopy-theoretic point of view, which can be applied in many different ways.
The topics we have covered so far are:
- Classifying spaces in topology, principal G-bundles
- Spectral sequences
- Rational homotopy theory
- Topology of Lie groups
- Faithfully flat descent
To summarize some of the early parts of the course: every geometer will learn about the Chern classes of a complex vector bundle. My idea for the course was to put this construction in a broader context, by considering principal G-bundles for any group G instead of just the general linear group (or the unitary group). This brings up some of the main ideas of homotopy theory, since the classifying space for principal G-bundles is related to G by taking the loop space. Discussing spectral sequences and rational homotopy theory shows that we can compute a lot in homotopy theory. Using these general methods, we can get a rather strong understanding of the topology of all Lie groups. At this point, the connection with algebraic geometry becomes serious, since many properties of a compact Lie group are best understood by considering its complexification, which is a complex reductive group.
Grothendieck’s theory of faithfully flat descent is a beautiful topic of commutative algebra, which appears in surprisingly few books. It generalizes the idea of constructing sheaves or vector bundles by gluing together sheaves or vector bundles on open subsets. The theory should be useful for understanding principal bundles in algebraic geometry, equivariant Chow groups, and algebraic stacks in the rest of the course. And oh yes, I also want to discuss derived categories. We’ll see how it goes.
Here are the books I have suggested for the course so far.
- R. Bott and L. Tu. Differential forms in algebraic topology.
- J. Milnor and J. Stasheff. Characteristic classes.
- J. McCleary. A user’s guide to spectral sequences.
- Y. Félix, S. Halperin, and J.-C. Thomas. Rational homotopy theory.
- Y. Félix, J. Oprea, D. Tanré. Algebraic models in geometry.
- W. Waterhouse. Introduction to affine group schemes.
And anyone who’s been paying attention knows that important ingredients in the secret sauce can be found here:
- J.-P. Serre. Oeuvres: collected papers, 4 vols.