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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.

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Claire Voisin has a very nice two part book about complex algebraic geometry, I only have the first part but it is great. – Olivier Bégassat May 9 '12 at 1:56
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, – Matt E May 9 '12 at 4:29

Bott and Tu's Differential Forms in Algebraic Topology. (This has a lot of other stuff too, but it's all stuff that you'll need analogues of in algebraic geometry.)

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+1. One of the all time great mathematics textbooks on any subject. – Mathemagician1234 May 9 '12 at 1:59
I've also heard that this is a fantastic book. However, I'm afraid my topology may not quite be up to scratch since I assume it's quite a topology heavy book so I don't know if I can just dive right into it. How much topology am I going to have to know to be able to read it? – Brendan Good May 9 '12 at 2:07
If I remember right, you really just need point-set topology and the definition of a smooth manifold. – user29743 May 9 '12 at 14:33

John Lee's Introduction to Smooth Manifolds has a good chapter on vector bundles. You may also want to check out Morita's Geometry of Differential Forms to see characteristic classes in the de Rham framework (which will probably be easier given your background than the more abstract, algebraic topological definitions of characteristic classes).

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Milnor and Stasheff's book "Characteristic Classes" has a lot of foundational material on vector bundles.

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Based on your background, I think you would enjoy reading Principles of Algebraic geometry by Griffiths and Harris.

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Maybe for a second reference but I think this is a very difficult and terse text to self learn from without a lot of graduate level work. – Eric O. Korman May 9 '12 at 3:20

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