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As someone who is

  • familiar with algebraic topology, say, at the level of Hatcher's book, and
  • familiar with homological algebra and categories and applications in topology
  • but has no idea what a variety is

what is a good place to start learning algebraic geometry?

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    $\begingroup$ Your background might tempt you to try to learn algebraic geometry as if it were a variant of algebraic topology, in which case you'd miss a lot of the motivation and flavor of the subject. So I'm inclined to recommend starting with something like Mumford's red book, which on the one hand gives a vivid and concrete presentation of the fundamentals from a very geometric, as opposed to homological or topological, point of view, but on the other hand gets you quickly to the point where you can start learning homological methods and seeing where they fit in. $\endgroup$ – Steven Landsburg Jul 22 '16 at 0:59
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    $\begingroup$ You should also have a good commutative algebra book close at hand --- say Atiyah-Macdonald. $\endgroup$ – Steven Landsburg Jul 22 '16 at 0:59
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    $\begingroup$ You could say the motivation you have to learn alg. geom. What are trying to achieve? Just a general overview or something more specific? $\endgroup$ – user40276 Jul 22 '16 at 2:22
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    $\begingroup$ As a topologist who has occassionally tried to learn some algebraic geometry, the two books I have found most useful are Shafarevich's "Basic Algebraic Geometry" (which touches on the relationship between complex vairities in the Zariski and the analytic topologies) and Eisenbud and Harris's "The Geometry of Schemes", which introduces schemes in a fairly topological/categorical language. $\endgroup$ – Mark Grant Jul 22 '16 at 6:13
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    $\begingroup$ @johnmangual: It's true that a scheme is (in part) a topological space, but not of the kind algebraic topologists usually think about (in particular wildly non-hausdorff). The more substantial connection is via the analytic topology on complex varieties. $\endgroup$ – potentially dense Jul 22 '16 at 12:49
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(I guess my opinion is no more worthy of being an answer than the opinions in the comments, but it's verbose, so let me put it in the answer box anyway.)

As a beginning PhD student I knew a reasonable amount of algebraic topology, similar to what you describe in the question. But I don't think it really gave me any extra or better choices in how to start learning algebraic geometry. As Steven Landsburg's comment suggests, the kind of objects one studies and the methods one uses in algebraic geometry are so much more specialised than arbitrary (even nice) topological spaces that you really need to start from scratch, with something like Shafarevich (as suggested by Mark Grant).

That isn't to say that having a good knowledge of algebraic topology won't be useful to you in learning algebraic geometry --- quite the opposite, in fact. (Example: characteristic classes.) And, as you progress in algebraic geometry, you will likely run into more and more topics where your topology knowledge gives you a great head-start on understanding. But it won't really help with those first steps.

On the other hand, if I had to nominate a beginning algebraic geometry textbook that is oriented towards the topological point of view, I guess it would be Principles of Algebraic Geometry by Griffiths and Harris. But, great resource though it is, I don't think I can recommend that book to anyone in good conscience as a first introduction to algebraic geometry.

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    $\begingroup$ Huybrechts "Complex Geometry: An Introduction" is similar to Griffiths and Harris, but perhaps closer to algebraic topology, as it discusses Atiyah classes and holonomy groups (if I remember correctly). $\endgroup$ – Ben McKay Jul 23 '16 at 9:49
  • $\begingroup$ @BenMcKay: thanks for the extra information. $\endgroup$ – potentially dense Jul 23 '16 at 18:47
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    $\begingroup$ I recommend deep study of basic topics with examples, rather than abstract study of general topics, so I suggest Rick Miranda's book Algebraic Curves and Riemann Surfaces, since this discusses the topic from which modern algebraic geometry began. on the opposite side, with your background maybe serre's annals paper FAC. you won't learn anything about actual algebraic varieties but you will know cohomology. $\endgroup$ – roy smith Jul 25 '16 at 2:33
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    $\begingroup$ As a topologist, I also found Huybrechts to be very readable (and Differential Analysis on Complex Manifolds by Raymond O. Wells is good background for Huybrechts). On the other hand, I found it to be only one area of algebraic geometry. It didn't, for example, teach me about schemes. So, as others have observed, it depends a bit what yuo want to learn. Algebraic geometry is a large field. $\endgroup$ – Greg Friedman Jul 26 '16 at 18:13
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A similar question was asked while ago, which was made a community wiki later on. You may find it useful, if not exactly what you would have expected. Here is

A learning roadmap for algebraic geometry

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