One of the nice things about living in Hyde Park is the proximity to the University of Chicago. Consequently, over the summer I came in to the department from time to time to work in my office, where I have all my math books, fast internet connection, etc. One day in early September (note: the Chicago quarter doesn’t start until October, so technically this was still “summer”) I happened to run in to Volodya Drinfeld in the hall, and he asked me what I knew about fundamental groups of (complex) projective varieties. I answered that I knew very little, but that what I did know (by hearsay) was that the most significant known restrictions on fundamental groups of projective varieties arise simply from the fact that such manifolds admit a Kähler structure, and that as far as anyone knows, the class of fundamental groups of projective varieties, and of Kähler manifolds, is the same.
Based on this brief interaction, Volodya asked me to give a talk on the subject in the Geometric Langlands seminar. On the face of it, this was a ridiculous request, in a department that contains Kevin Corlette and Madhav Nori, both of whom are world experts on the subject of fundamental groups of Kähler manifolds. But I agreed to the request, on the basis that I (at least!) would get a lot out of preparing for the talk, even if nobody else did.
Anyway, I ended up giving two talks for a total of about 5 hours in the seminar in successive weeks, and in the course of preparing for these talks I learned a lot about fundamental groups of Kähler manifolds. Most of the standard accounts of this material are aimed at people whose background is quite far from mine; so I thought it would be useful to describe, in a leisurely fashion, and in terms that I find more comfortable, some elements of this theory over the course of a few blog posts, starting with this one.