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I heard a few rumours that knowledge of differential geometry is crucial to really understand anything in algebraic geometry. I want to ask You, people I don't know IRL to give Your opinion. If You know also about Lie's groups and algebras, You can write also.

I know that commutative algebra and sth my faculty calls "algebraic methods of geometry and topology" are necessary, but what about DG? Is it "must have" or is it only making learning algebraic geometry easier? And why it is like that?

I don't know anything about both of these geometries at that, moment, I'm asking in case to plan 2020/2021 season of my studies.

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    $\begingroup$ I personally don't see how differential geometry is necessary to understand varieties, schemes, sheaves and cohomology. Going just from the title, that algebraic methods course sounds directly relevant, though. $\endgroup$
    – Arthur
    Commented Nov 26, 2019 at 20:17
  • $\begingroup$ Not so much differential geometry in general as complex geometry in particular. en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry $\endgroup$ Commented Nov 26, 2019 at 20:20
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    $\begingroup$ IMO they are not relevant, but it's good to have some understanding of ideas in differential geometry to see how notions in AG are their analogues. An example of that is the notion of a Zariski (co)tangent space. $\endgroup$
    – Wojowu
    Commented Nov 26, 2019 at 20:24
  • $\begingroup$ @Wojowu is this historical reason that algebraic geometry inherited concepts of differential one bcs was created later? But these are only concepts? $\endgroup$
    – robin3210
    Commented Nov 26, 2019 at 20:30
  • $\begingroup$ I just noticed typo in my previous comment, I meant not required, they are certainly relevant! Anyway, I think you are right that the reason may be at least in part historical. But I also think that differential world is in general "easier", so concepts are easier to formalize than those in algebraic geometry. $\endgroup$
    – Wojowu
    Commented Nov 26, 2019 at 20:33

2 Answers 2

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knowledge of differential geometry is crucial to really understand anything in algebraic geometry

This might be true, depending on what you mean by "really". I think there are roughly two ways that differential geometry influences algebraic geometry:

First of all, the definition of a scheme (a space which locally looks like an affine scheme) basically mimics the definition of a manifold (a space which locally looks like a Euclidean space). As a result, arguments from differential geometry that let you translate local results to global results are also seen (in a modified form) in algebraic geometry. While it's not necessary to know any differential geometry to understand these arguments, it certainly can't hurt to be familiar with the ideas.

Secondly, many constructions on schemes are designed to replicate constructions from differential geometry in the algebraic setting, and while it's possible to learn the relevant definitions by rote, it's extremely helpful to have an intuition for these concepts from differential geometry. For example, if you've never seen the proof that the concrete tangent spaces of a manifold embedded in Euclidean space agree with the abstract tangent spaces $(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, it will probably seem quite strange to define tangent spaces in this way in algebraic geometry. The same is true for differential forms, vector bundles, etc. While you could learn all of this from scratch in algebraic geometry, it will probably feel rather unmotivated (especially because most texts assume you have some intuition about the corresponding concepts in differential geometry).

With that said, not all of algebraic geometry uses these constructions. There's quite a lot you can do with just algebra! You should just be aware that geometric concepts are quite pervasive in algebraic geometry (hence the name of the field), so you'll probably encounter them in most directions you look.

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    $\begingroup$ Really nice answer. While it's implicitly addressed in your "differential forms, vector bundles, etc.", I'll mention that I think it's enormously helpful to have seen cohomology in some form before you learn about sheaf cohomology in algebraic geometry. What kind of cohomology one picks is up to taste, but I think De Rham cohomology is certainly a good choice, especially since it is very geometric in nature. To some, I'm sure sheaf cohomology is quite geometric too, but for me, having seen De Rham cohomology first gave me some flavor of what kind of theorems I should expect to be true in... $\endgroup$ Commented Feb 20, 2020 at 9:05
  • $\begingroup$ ...the algebro-geometric setting, even if the algebraic analogue did not have an obvious interpretation. (I am thinking of things like Poincare duality, Serre duality, etc.) $\endgroup$ Commented Feb 20, 2020 at 9:06
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Knowing differential geometry can help you to, by analogy, understand parts of algebraic geometry, but I would say that the actual material in differential geometry is almost completely inapplicable directly to algebraic geometry.

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