### Cliff Harveyowner

Mathematics - **Diving for Calabi-Yau manifolds**. Initially, i had written the following paragraphs in reverse order, more like being washed up ashore, saved from the deep. But i figured you might prefer to peacefully approach

*K3 surfaces*, the secret topic of this post. Assuming you know what

*polynomials*are, we'll start nice and slow:

What are

**Rational functions**? You get those by dividing one polynomial by another N(x)/D(x). Their degree is handily defined to be the maximum power appearing in any of the two polynomials. Since the denominator mustn't be zero, we'll get funny "singularities" wherever the D(x) polynomial has a zero. Just as polynomials themselves, rational functions are quite an easy subject, so don't be shy.

Some facts: They're representative for

*meromorphic functions*. Just like holomorphic functions but singularities are allowed to appear at a finite set of points. You can "Taylor" them to your required precision, too.

https://en.wikipedia.org/wiki/Rational_function

The

**Elliptic integral**appeared when people set out to calculate arc lengths of ellipses. It's an integral like this:

x

**∫ R(t, √P(t)) dt**

c

That's a path integral, and you have to pick a starting point

**c**for your path.

The result of the integration cannot, in general, be stated using elementary functions. But when we add the three

*Legendre canonical forms*, also known as the

*elliptic integrals*of 1st, 2nd, and 3rd kind, we can write the solutions as integral over rationals combined with these forms. The

*reduction formula*does this.

https://en.wikipedia.org/wiki/Elliptic_integral

I just hate those notations, but luckily, there's also the

*Carlson symmetric form*:

https://en.wikipedia.org/wiki/Carlson_symmetric_form

The

**Algebraic variety**is the object of a

*projective*geometric theory, in contrast to the

*linear algebraic groups*, giving the better known

*affine*theory. That's to be expected, because there's a division happening in rational functions, and you get projective spaces by "dividing" lines out, so you can look at the remainder.

One can define

*algebraic varieties*as "the set of solutions of a system of polynomial equations". They're just like a regular manifolds, but may have singular points.

https://en.wikipedia.org/wiki/Algebraic_variety

A special case is the

**Abelian variety**, defined as: A

*projective algebraic variety*that is also an

*algebraic group*. Such a group is given by

*regular functions*, which are just another name for polynomials in this context. It means we call a curve

*Abelian variety*when one can find a set of polynomial functions that generate a group.

Historically, the

*abelian variety*played the role of generalizing the elliptic integral from involving polynomials of 3rd and 4th degree to ones involving 5th or higher degree. Originally studied over the complex numbers, over which they turned out to be exactly the

*complex tori*that can be embedded in a

*complex projective space*.

**Solomon Lefschetz**worked that out, and also was the first to use the term

*Abelian variety*.

Example, an

*elliptic curve*is an

*Abelian variety*of dimension 1.

https://en.wikipedia.org/wiki/Abelian_variety

*Abelian varieties*appear naturally as

**Jacobian variety**of algebraic curves. We define:

**J(C)**of a nonsingular algebraic curve

**C**of genus

**g**is the

*moduli space*of degree 0 line bundles. You fill your surface with nonoverlapping lines, forget that a line has many points, and look at how lines are related.

As a group, the

*Jacobian*of a curve is isomorphic to [...] divisors of rational functions. Our lines correspond to multiples of rational functions, just like any fraction can be given as a multiple of itself:

a/b = (x·a)/(x·b)

Note that plotting f(x) = x·a gives a line going through the origin.

https://en.wikipedia.org/wiki/Jacobian_variety

*Abelian varieties*also appear as

**Albanese variety**of a curve

**C**, a very natural generalization of the

*Jacobian*to higher dimensions. The construction of the

*Jacobian variety*as a quotient of the space of holomorphic 1-forms (our "lines") generalizes straight forward to give the

*Albanese variety*. Btw, when we're working over complex numbers, holomorphic means polynomial, 1-forms are the linear ones.

https://en.wikipedia.org/wiki/Albanese_variety

**Enriques-Kodaira Classification**of compact complex surfaces into 10 classes. For 9 of them we have a fairly complete understanding, but the remaining "general" type has too much in it. (Okay, our understanding for type VII depends on the

*global spherical shell conjecture*.)

If you look at the diagram now you'll notice that the 9 well-understood classes are at most 1-d portions, lines of dots. The "general" type is the area left, i'm not surprised it's full of stuff.

https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification

One class are the

**K3 surfaces**, named after

**Ernst Kummer**,

**Erich Kähler**and

**Kunihiko Kodaira**. Actually there are four classes of

*K3 surfaces*, but

**Andre Weil**ran out of people to name fourth one. He chose the mountain K2 instead, which, as you can see, didn't really make it into modern language....

They're the complete smooth surfaces with trivial canonical bundle. That's exactly those 2-dimensional

*Calabi-Yau manifolds*that aren't

*complex tori*. The latter are simple

*Abelian varieties*and much

Another way to define

*Calabi-Yau surfaces*is as

*complete smooth surfaces*with

*trivial canonical bundle*. The canonical bundle is one passing through zero, it's called

*trivial*when it's made of "lines". Maybe a better way to understand them is to state that all of them have

**Kodaira dimension**0...

https://en.wikipedia.org/wiki/K3_surface

The

**Kodaira dimension**k is one less than the number of algebraically independent generators. It's the rate of growth k such that P(d)/d^k is bounded. All Abelian varieties have zero Kodaira dimension! While k > 0 corresponds to hyperbolic, k = 0 to flat, sometimes, k is undefined or negative. We then say k = -oo and those cases correspond to surfaces with

*positive curvature*.

So our surfaces are the "flat" ones in the above sense.

Initially, i had written this post in reverse, trying to figure out what this thread by +David Roberts mentioning K3 surfaces, Chern forms, and the magic number 24 was all about:

https://plus.google.com/+DavidRoberts/posts/CoBq5Y5Haqp

**More links**

Since the

*Jacobian variety*is the connected component of the identity in the

*Picard group*of our curve, i should have told you about

*Picard varieties*and their associated

*Picard groups*as well. I guess i'll have to do that next time, and maybe then we can also talk about how to blow something up:

https://en.wikipedia.org/wiki/Blowing_up

See also the

*Abelian surface*, which are just

*Abelian varieties*in 2d.

https://en.wikipedia.org/wiki/Abelian_surface

*Hodge diamonds*give another

*invariant*, just like the

*Kodaira dimension*is:

https://en.wikipedia.org/wiki/Homological_mirror_symmetry#Hodge_Diamond

The diagram is by R.e.b.:

https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification#/media/File:Geography_of_surfaces.jpg

The portraits i found here:

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Kodaira.html

https://upload.wikimedia.org/wikipedia/commons/5/52/Federigo_Enriques.jpg

#ScienceSunday - #algebraic #geometry , #elliptic integrals, #jacobian #variety

5

1

2 comments

There's a nice Scholarpedia article by Yau: http://www.scholarpedia.org/article/Calabi-Yau_manifold

Add a comment...