Calabi-Yau surfaces and algebraic geometry for mathematicans

[bham10246]

Hi, I posted this under General Physics but I thought I should post this under the math section too in case there are mathematicians who do not read posts under General Physics.

I read some papers online about Calabi-Yau surfaces but I have some basic questions about them if you can answer them for me.

1. Who proved that Calabi-Yau mfds fit into string theory? Why does C-Y surface fit into string theory, instead of any other surfaces?
2. I googled C-Y surfaces and saw many beautiful pictures of them. But why do they look like that? Do you have explanation other than the fact that they have a trivial canonical bundle?
3. What are different kinds of singularity types that can occur on a C-Y surface (just an intuition of them will be fine)? Why is the orbifold singularity considered to be the simplest singularity on a C-Y surface?
4. Why do singularities form on C-Y surfaces and what physical meaning do they have?
5. Why should we resolute singularities?
6. An orbifold is locally C^n/G where G is a finite subgroup of SL(n,C) and C = complex numbers. So where are the singularities? How do you see that?
7. What is a physics meaning (the physical perspective) of a canonical bundle?
8. How much of algebraic geometry do mathematicians and physicists use to study Calabi-Yau manifolds and string theory? I'm sure they do use PDE/ODE and differential geometry, correct?

Thank you in advance for your time!

 

[n_bourbaki]

I can only answer some of your points.

Some would say we study resolutions of singularities purely because of a classical hang up that smooth things are 'nicer'.

A singularity, from a quotient by a group action, is a fixed point in the orginal space: generically the quotien X --> X/G is |G| to 1, but at some places it is not |G| to 1 - these give singularities. A simple example is R/{+/-1} (not in SL, but that's not the issue). The map R --> R/{+/-1} is genericall 2 to 1 except at the origin where it is one to one. The quotient space "is" the positive real axis union zero, hence not a smooth 1-d manifold.

There are many equivalent definitions of C-Y, and I think the point why physicists are interested is that the manifold admits a Kahler metric. But that is only a guess. You can also do integration on them, as they have a no-where vanishing volume form.

You certainly won't use differential geometry to study them (they not even differential manifolds, necessarily). You want to study algebraic geometry, a lot of it. Though you might get into that by classical differential geometry.

One also has nice results in derived categories for C-Y surfaces, etc, that if I recall correctly are thought to link into some kind of 'duality' for string theories.

 

[bham10246]

Thank you n_bourbaki! Your explanation makes sense.

I found a paper from arXiv and it seemed interesting. The hardest part was that since it was a random paper, I had to look up definitions and basic concepts for every other word. I had to read through the paper a few times and think about it for a long time in order for it to begin to make some sense to me!

 

[bham10246]

For your example above:

"A singularity, from a quotient by a group action, is a fixed point in the orginal space: generically the quotien X --> X/G is |G| to 1, but at some places it is not |G| to 1 - these give singularities. A simple example is R/{+/-1} (not in SL, but that's not the issue). The map R --> R/{+/-1} is genericall 2 to 1 except at the origin where it is one to one. The quotient space "is" the positive real axis union zero, hence not a smooth 1-d manifold."

How do we resolute and remove the singularity, i.e., the origin?

 

[bham10246]

Do you have an example for this: "Some would say we study resolutions of singularities purely because of a classical hang up that smooth things are 'nicer' " ?

Do you mean in terms of covering it locally with Euclidean open sets or when defining curves or geodesics on the surface?

 

[n_bourbaki]

The simplest example of resolving a singularity is for the singular curve xy=0 in affine two space. I.e. the union of the x and y axis, which is singular at the origin. A desingularisation is easy to imagine, e.g. consider the algebraic set sitting in A^3 (and think of the copy of A^2 given by the first two coords)

{ (0,y,0) : y in R} U { (x,0,1) : x in R} with the 'obvious' projection maps.

You can picture this as removing the origin from the singular curve, then pulling the two lines apart, now filling in the holes.

 

[n_bourbaki]

Do you have an example for this: "Some would say we study resolutions of singularities purely because of a classical hang up that smooth things are 'nicer' " ?

Do you mean in terms of covering it locally with Euclidean open sets or when defining curves or geodesics on the surface?

No, since I'm talking about algebraic geometry, and hence the zariski topology. Smooth things are just 'nicer' - they have a well defined notion of dimension in terms of algebra and derived categories or cohomology, for instance.

 

[Hurkyl]

No, since I'm talking about algebraic geometry, and hence the zariski topology. Smooth things are just 'nicer' - they have a well defined notion of dimension in terms of algebra and derived categories or cohomology, for instance.

This confuses me. At the very least, dimension (in the sense of algebraic geometry) is well-defined for all Noetherian schemes, (and all Noetherian topological spaces!) not just smooth ones. And sheaf cohomology is well-defined for any topological space whatsoever. (And even for sites!)

 

[n_bourbaki]

An elision, I apologise. The point I was trying to make was that for a smooth variety, one can read off its dimension from the vanishing of cohomology (or Homs in the derived category), i.e. the global dimension (which equals the Krull dimension you allude to for regular Noetharian stuff, or something - I'm no expert in this). For non-smooth things it's not so clear what one ought to use, at least to some of us playing around with derived equivalences. At the very least 'dimension' ought to be some kind of invariant of the derived category.

If you attempt to work out the usual definitions of dimension, you (may?) get infinity, irrespective of what you started with. If you take some large open smooth subvariety, or a resolution, you'll get an answer independent of the type of singularity - and algebraically speaking there are different types of singularity - log, ADE, Gorenstein, whatever. Neither is a particularly satisfying state of affairs.

 

[Hurkyl]

Ah, I see how I misread what you wrote.

 

[n_bourbaki]

Probably because I wasn't particularly precise. I've added to my previous post as you were replying.