Calabi-Yau

Let $M$ be a Kaehler manifold. Then $M$ is Calabi-Yau if $c_{1}(K_{M})=0$. If $M$ is a variety, we say that $M$ is a Calabi-Yau variety if it has vanishing canonical class.

Examples

All elliptic curves are Calabi-Yau, since they are parallelizeable.

A degree $n+1$ hypersurface $M$ in $\mathbb {P} ^{n}$ is Calabi-Yau. For from the exact sequence

$0\to TM\to T\mathbb {P} ^{n}\to {\mathcal {O}}(n+1)|_{M}\to 0$

we get that $c(T\mathbb {P} ^{n})=c(TM)(1+(n+1)\omega )$. But this implies that

$c(TM)=1+c_{1}(TM)+\ldots ={\frac {(1+\omega )^{n+1}}{1+(n+1)\omega }}$

from which one easily concludes that $c_{1}(TM)=c_{1}(K_{M})=0$.

Calabi Conjecture

The first Chern class of a Kaehler manifold is homologous to the Ricci curvature. For this reason, Calabi conjectured that when $c_{1}=0$, there existed a metric on the manifold whose Ricci form identically vanished. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.