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I'm interested in algebraic geometry (I'm working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and I have seen some basic definitions from differential geometry, e.g. vector bundles, (co)tangent spaces, differential forms. However, I don't have great intuition for such objects and as a result I feel a bit hindered as far as developing good geometric intuition in AG.

Are there any suggestions as to resources I can look at for efficiently gaining a solidly intuitive, but not necessarily deep, understanding of differential geometry specifically for the purpose of motivating related ideas in algebraic geometry? Or, perhaps, is it essential that I learn differential geometry as thoroughly as I can before trying to study algebraic geometry seriously?

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    $\begingroup$ I don't think it's essential to learn DG thoroughly before AG. Vakil himself has said something like "Sure, it would be great if the student has a background in differential geometry, varieties, complex analysis, algebraic number theory, homological algebra... before beginning this course, but this is rarely the case and certainly not necessary." But your point still stands - it is helpful to have some intuition from DG. Whenever Vakil mentions something from DG, I quickly skim what Lee's "Smooth Manifolds" has to say about it, coming back for a deeper read if I can't do Vakil's exercises. $\endgroup$ – Ragib Zaman May 31 '14 at 6:09
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    $\begingroup$ Learn everything simultaneously. It hasn't steered me wrong so far. $\endgroup$ – Qiaochu Yuan May 31 '14 at 6:56
  • $\begingroup$ No I don't think so. Differential geometry is from calculus & analysis. Algebraic geometry is from algebra. $\endgroup$ – mathreadler Apr 13 '18 at 21:39
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...Are there any suggestions as to resources I can look at for efficiently gaining a solidly intuitive, but not necessarily deep, understanding of differential geometry specifically for the purpose of motivating related ideas in algebraic geometry?...

If one woulds like to develop the intuition in differential geometry, I suggest:

  • do Carmo M. P. - Differential Geometry of Curves and Surfaces;
  • Spivak M. - A comprehensive introduction to differential geometry; Volume 1; Volume 2, chapters 1, 2 and 3; Volume 3, chapters 2 and 3;
  • Wells R. O. Jr. and Garcia-Prada O. - Differential Analysis on Complex Manifolds, chapters 1, 2, 3 and 5.

If one woulds like to develop the intuition in algebraic geometry, I suggest:

  • Eisenbud D., Harris J. - The Geometry of Schemes, the chapter 2;
  • Fischer G. - Plane Algebraic Curves;
  • Hulek K. - Elementary Algebraic Geometry;
  • Mumford D. - The Red Book of Varieties and Schemes.

And, as recap in the complex differential and algebraic frameworks, I suggest:

  • Griffiths P. and Harris J. - Principles of Algebraic Geometry, chapters 0, 1 and 2;
  • Neeman A. - Algebraic and Analytic Geometry;
  • Voisin C. - Hodge Theory and Complex Algebraic Geometry, the first parts of volumes I and II.
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