8
$\begingroup$

I am interested in learning algebraic geometry and I talked to one of my professors today (who is, in part, an algebraic geometer) and he recommended I understand the analytic analogue of the ideas in algebraic geometry so it's not just abstract nonsense when I first see it. Some of specific words he mentioned were vector bundles and line bundles but he could not give any recommendations on the spot and recommended that I ask here.

There are only two sources that I know of which cover these subjects in a way that I think coincide with my relatively modest understanding of mathematics (which I will cover a bit later) are the first part of Hatcher's book on K-theory and Spivak's Comprehensive Introduction to Differential Geometry although the latter will, admittedly, cover much more than I need or could handle at the moment. If there are any more differential geometry concepts of which I should be aware as well, please feel free to include that as well. I am also aware that I will need to know some commutative algebra and complex analysis and I have gotten some solid recommendations on those topics (some from here, in fact).

These are the courses I've taken which I think are relevant to recommendations (all courses are undergraduate): algebra, analysis 1/advanced calculus, differential geometry, proof-based linear algebra, and I am familiar with some topology (in that I know what a topological space is and what a fundamental group is. I will definitely study that more over the summer), I did a reading course on algebraic curves covering the first three chapters of Fulton plus the proof of Bézout's theorem, and I have done an almost reading course in geometry/topology so I am aware of what manifolds are and some of the relevant topology.

  • 2
    Claire Voisin has a very nice two part book about complex algebraic geometry, I only have the first part but it is great.2012-05-09
  • 3
    Dear Brendan, I think that the first two volumes of Spivak are well worth reading if you don't know differential geometry/topology (and related topics) already. On the other hand, Miles Reid's *Undergraduate algebraic geometry* is an excellent entry point to algebraic geometry, which doesn't need very much background, and is not at all abstract. Regards,2012-05-09

4 Answers 4