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.

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

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