I want to start self-studying Algebraic Geometry at some point in the near future. There are plenty of posts discussing prerequisites, but one thing I couldn't find a discussion of: How much Differential Geometry is needed to appreciate Algebraic Geometry?
I know very little Differential Geometry. I read comments saying Differential Geometry is needed to motivate the equivalent of curvature in Algebraic Geometry. This is what's making me concerned.
If Differential Geometry is needed, do you think Bishop's book "Tensor Analysis on Manifolds" sufficient? Is there a better (but still concise) book out there?
I don't think you need to learn a lot of differential geometry before beginning to learn algebraic geometry. It will definitely help with some things, but most of it you can learn along the way. When you encounter a new concept that you have a hard time understanding, or whose definition you have a hard time motivating, you can take a step aside and look at the analogous constructions on the differential geometry side. You don't need to master it all to begin with.
Having a decent background in commutative algebra is probably much more important at the beginning. The spaces you'll encounter at first are quite simple from the topological point of view, and you don't need to be a differential geometer to think about them. However, to understand their subtleties, you'll need to have a good understanding of basic commutative algebra.
Having a good background in complex analysis is also quite useful. Assuming you already know basic complex analysis, you might want to pick up a good book on compact Riemann surfaces (such as Rick Miranda's), because studying compact Riemann surfaces is as close as you'll come to doing algebraic geometry without even being aware of it.
Nevertheless, a useful book on the "differential" side, in my opinion, is Bott & Tu's Differential Forms in Algebraic Topology. It's a fantastic introduction to the basic ideas of cohomology, and it requires relatively little background to read, which is definitely a plus.