tea.mathoverflow.net - Discussion Feed (Mirror symmetry with algebraic geometry)

Spiro Karigiannis comments on "Mirror symmetry with algebraic geometry" (11679)

2010-12-11T10:26:10-08:00

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

Kevin Lin comments on "Mirror symmetry with algebraic geometry" (11676)

2010-12-11T10:05:36-08:00

I have deleted my comments and added an answer. Thanks for re-opening this question!

Scott Morrison comments on "Mirror symmetry with algebraic geometry" (11672)

2010-12-11T09:06:51-08:00

The comment thread at this moment:

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

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

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

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

Scott Morrison comments on "Mirror symmetry with algebraic geometry" (11671)

2010-12-11T09:06:08-08:00

I propose to copy and paste the comment thread over to here, delete it and replace it with a link to this thread.

Scott Morrison comments on "Mirror symmetry with algebraic geometry" (11670)

2010-12-11T09:05:13-08:00

And I've reopened.

Ryan Budney comments on "Mirror symmetry with algebraic geometry" (11669)

2010-12-11T08:23:01-08:00

Okay, I've made the edits.

Spiro Karigiannis comments on "Mirror symmetry with algebraic geometry" (11668)

2010-12-11T08:13:17-08:00

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.)

Kevin Walker comments on "Mirror symmetry with algebraic geometry" (11666)

2010-12-11T07:43:09-08:00

Is the issue lack of focus, or just poor English skills? It seems to me that a fairly direct paraphrase of the question is:

"Why is it that mirror symmetry has many relations with algebraic geometry, rather than with complex geometry or differential geometry? (In other words, how is it that solutions to polynomials become relevant, given that these do not appear in the physics which motivates mirror symmetry?) I would especially appreciate nontechnical answers."

It's not clear to me how that could be made more focused by someone who does not already know a fair amount about mirror symmetry. I'm not saying that the question is ideal and could not improved in any way. I'm just saying that it's not so far from ideal that it should be closed.

Ryan Budney comments on "Mirror symmetry with algebraic geometry" (11665)

2010-12-11T07:09:14-08:00

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.

Kevin Walker comments on "Mirror symmetry with algebraic geometry" (11664)

2010-12-11T06:42:00-08:00

I agree -- I think it's a reasonable question.

Kevin Lin comments on "Mirror symmetry with algebraic geometry" (11650)

2010-12-11T01:34:58-08:00

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.