Calabi - Yau Manifolds - MathOverflow

Calabi - Yau Manifolds

2010-10-18T22:59:15Z

I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one from other. But my question is the following :

"What is the most strict definition of Calabi-Yau Manifolds"

By that I mean the definition from which all the others follow.

Answer by Laie for Calabi - Yau Manifolds

2010-10-18T23:15:37Z

There are different views about how Calabi-Yau varieties should be defined. A characterization that is most appropriate for many applications of these spaces is to define them as compact Kaehler varieties with vanishing first Chern class. Sometimes stricter definitions are adopted, but these lead to the exclusion of certain degenerate cases, such as the product of a K3 surface with an elliptic curve or the triple products of elliptic curves, that really should not be excluded.

Answer by Richard Borcherds for Calabi - Yau Manifolds

2010-10-18T23:35:52Z

In http://en.wikipedia.org/wiki/Calabi_Yau_manifold there is a discussion of some of the many definitions of CY manifold and the relations between them. Yau defines them in http://www.scholarpedia.org/article/Calabi-Yau_manifold as "compact, complex Kähler manifolds that have trivial first Chern classes (over R). In most cases, we assume that they have finite fundamental groups." The strictest definition would also require vanishing integral Chern class.

Answer by Spiro Karigiannis for Calabi - Yau Manifolds

2010-10-21T15:54:10Z

There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can be found in Joyce's book:

Compact Manifolds with Special Holonomy

The other answers you have gotten so far seem to be from the algebraic geometry side of things, and are fine in that context. From a Riemannian geometry point of view, the most natural definition of a Calabi-Yau manifold (whether compact or non-compact) is a $2n$-dimensional Riemannian manifold for which the holonomy of the Levi-Civita connection is exactly $SU(n)$. Allowing the holonomy to be a proper subgroup of $SU(n)$ is also common. In that case, hyperKahler (which is holonomy $Sp(n/2)$ in dimension $4n$) can also be considered as being Calabi-Yau, for example.

This Riemannian geometry definition is equivalent to the existence of the "Calabi-Yau package": a Riemannian metric $g$, an integrable complex structure $J$ (orthogonal with respect to $g$) together which induce the associated Kahler form $\omega$ by $\omega(X,Y) = g(JX, Y)$, and a holomorphic volume form $\Omega$, which is a holomorphic $(n,0)$-form on $M$. These tensors must satisfy:

(1) $\nabla \omega = 0$ (equivalent to $\nabla J = 0$, the Kahler condition) This is also equivalent to $d\omega = 0$ because we are assuming $J$ to be integrable.

(2) $\nabla \Omega = 0$

(3) $\frac{\omega^n}{n!} = c_n \, \Omega \wedge \bar \Omega$ for some universal constant $c_n$ depending only on the dimension which I can never remember.

These conditions imply, in particular that $g$ is Ricci-flat and $c_1(M) = 0$. Also, if the holonomy is exactly $SU(n)$ rather than a proper subgroup, then it also follows that $h^{p,0} = h^{0,p} = 0$ for all $1 \leq p \leq n-1$.