Calabi - Yau Manifolds

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.

• There are also varieties with vanishing first Chern class in de Rham (or rational, or real coeff.) cohomology but nonvanishing $c_1$ in integer cohomology, i.e. $c_1$ is a torsion class. Do they deserve the name of Calabi-Yau manifolds? Yau's theorem doesn't say anything about the existence of Ricci-flat metrics in this case, if I don't misremember. Commented Oct 18, 2010 at 23:22
• @unknown: Yau says there exists a Ricci-flat metric as soon as X is compact, Kahler and has real first Chern class zero. The manifolds with vanishing real c_1 can be obtained as finite quotients of manifolds with vanishing integer c_1. Beauville shows this and classifies these manifolds in Variétés Kähleriennes dont la première classe de Chern est nulle. Commented Oct 19, 2010 at 5:43

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 $1 \leq p \leq n-1$.

• ya even I was reading through the Joyce's book and noticed the same.but I was just wondering if there is more strict definition than the holonomy.Thanks for pointing out. Commented Oct 21, 2010 at 23:35
• In string theoretic point of view it must not change the supersymmetry, the holonomy group must be $G ⊂ SU(n)$ but not contained in any $SU(n-1)$ subgroup of $SU(n)$.
– user21574
Commented Jul 18, 2017 at 16:36

For singular varieties , Siu introduced Calabi-Yau varieties by using numerical Kodaira dimension

Let $X_0$ be a projective variety with canonical line bundle $K\to X_0$ of Kodaira dimension $$\kappa(X_0)=\limsup\frac{\log \dim H^0(X_0,K^{\otimes \ell})}{\log\ell}$$ This can be shown to coincide with the maximal complex dimension of the image of $X_0$ under pluri-canonical maps to complex projective space, so that $\kappa(X_0)\in\{-\infty,0,1,...,m\}$. Also since in general we work on Singular K\"ahler variety we need to notion of numerical Kodaira dimension instead of Kodaira dimension.

$$\kappa_{num}(X)=\sup_{k\geq 1}\left[\limsup_{m\to \infty}\frac{\log\dim_{\mathbb C}H^0(X,mK_X+kL)}{\log m}\right]$$

where $L$ is an ample line bundle on $X$.Note that the definition of $\kappa_{num}(X)$ is independent of the choice of the ample line bundle $L$ on $X$. Siu formulated that the abundance conjecture is equivalent as $$\kappa_{kod}(X) =\kappa_{num}(X)$$

The Minimal Model and abundance conjectures would imply that every variety of Kodaira dimension $\kappa (X)=0$ is birational to a Calabi-Yau variety with terminal singularities.

Note that when $K_X$ is psudoeffective vanishing numerical Kodaira dimension is equivalent that $X$ is Calabi-Yau variety

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.

• Wow. My already reasonably high regard for scholarpedia has increased even more. Commented Oct 19, 2010 at 9:44

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.