Calabi-Yau threefold
= Calabi-Yau threefold
{c}
Here a Calabi-Yau threefold is a compact complex three-dimensional <Kähler manifold> with trivial <canonical bundle> and $H^1(M,\mathbb C)=0$. The <Hodge decomposition theorem for compact Kähler manifolds> gives $H^1(M,\mathcal O_M)=0$, and <Serre duality for compact complex manifolds> then gives $H^2(M,\mathcal O_M)=0$. This supplies a positive integral Kähler class and hence a projective embedding.