http://mathoverflow.net/questions/49024/mirror-symmetry-with-algebraic-geometry-closed

I don't think this is a bad question. I think it should be re-opened. In the comments I have given the answer that I would give to this question if it were not closed.

How about re-writing the question to make it appear less like a fishing expedition and to make it more focused?

A running problem I have with these kinds of questions is they either look like we're rewarding fishing expeditions or we're turning them into podiums to give little lectures on the questions we hope for. There's a certain amount of effort required of the poster, moreover re-writing the question to make it more focused should really be something the OP does, not the question answerer. The lack of feedback from the OP leads me to think this is a fishing expedition.

I agree with the two Kevins. In fact, I would suggest (with the OP's consent) that we put Kevin Walker's version of the question, with Kevin Lin's comments as one of the answers. I have always been interested in why so much of the geometry of mirror symmetry is heavy on the algebraic geometry rather than differential / complex analytic geometry.

My interpretation is that:

[1]: algebraic geometry methods are easier to apply and much more well-developed. I don't mean algebraic geometry is easy, I just mean that the tools, by their nature, give more concrete results (for example, toric varieties), as opposed to geometric analysis methods, which by their nature often yield non-constructive or non-explicit results.
[2]: the algebraic geometers got into the Mirror Symmetry game much earlier and made more rapid progress than the differential geometers. (And they wrote many of the books.)

And I've reopened.

The comment thread at this moment:

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This question has been closed. It is way too broad for MO, which is for specific, focused questions. – Pete L. Clark 8 hours ago

This is not a bad question! Aaron Bergman's answer to this question mathoverflow.net/questions/30629/… for example might be a good answer to OP's question. – Kevin Lin 7 hours ago

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Essentially: algebraic geometry sometimes enters the picture in string theory and physics because, while we start with a compact Kähler manifold, for some reason or another we maybe get an integral Kähler class, and thus our manifold is projective by the Kodaira embedding theorem, and thus it is algebraic by Chow's theorem. Conversely, we may be actually interested in possibly non-algebraic compact Kähler manifolds in the physics or string theory, but the algebraic manifolds will provide at least a pretty big class of nice examples to play with. – Kevin Lin 7 hours ago

2

And at least for smooth projective algebraic varieties, GAGA theorems tell us that many things (like e.g. sheaf cohomology) are the same whether we consider our space as an algebraic variety or as an analytic thing. – Kevin Lin 7 hours ago

1

I agree with Kevin Lin that the question (if phrased more clearly) is actually interesting. – Spiro Karigiannis 4 hours ago

I think the question should not have been closed. – Kevin Walker 2 hours ago

Further meta discussion should go in the meta thread Kevin Lin created: meta.mathoverflow.net/discussion/826/… – Ryan Budney 1 hour ago

Excellent. It's a good addition to MO now.