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One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a differential geometry course. Since there are a lot of Differential geometry text books with different intention, he asked me what he should learn in differential geometry so that it helps him to improve his knowledge in algebraic geometry

I can not answer this question so I decided to post it here. Please help him.


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Isn't just studying differential geometry good enough for every purpose? Just a thought. – Bruno Stonek May 14 '14 at 10:04
No, may be you misunderstood my question. It is like asking a book on differential geometry for algebraic geometer. To be more precise, there are a lot of informations in different Differential Geometry books, with different intentions, my friend want to find what is the right intentions, or what are the most important informations that he should know for better understanding and learning algebraic geometry. – Arsenaler May 14 '14 at 11:56
I would say (and perhaps this is just a rephrasing of what Bruno Stonek was saying) that many of the topics in a first course on differential geometry (things like vector bundles, connections, de Rham cohomology, characteristic classes) are extremely relevant and helpful when studying algebraic geometry. Having said that, one idea is the following: look in Griffiths--Harris' book Principles of Algebraic Geometry. Find all the terms from differential geometry that you don't understand. Then choose the book or books that explain these things in the way you like best. – user64687 May 14 '14 at 13:21

To start with the basics, the strangest thing in differential geometry for somebody coming from an algebraic-geometry background is the idea of partition of unity. So this would have to be internalized as an introduction to a DG point of view as opposed to AG. Next, I would suggest performing the following thought experiment: take $\mathbb{CP}^2$ blown up an $k$ points. Now reverse the orientation of this beautiful algebraic variety. Does the resulting manifold make any sense?

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