Calabi–Yau manifold

Remarks

Several stricter definitions are in use. Many authors also ask that $H^0(X, \Omega^i_X) = 0$ for $0 < i < n$, or that $X$ be simply connected, or that the holonomy group of a Ricci-flat Kähler metric be exactly $SU(n)$. Example: a smooth quintic hypersurface $X \subset \mathbb{P}^4$ has $K_X = \mathcal{O}_X(5 - 5) = \mathcal{O}_X$ by the adjunction formula, so it is Calabi–Yau. Example: an elliptic curve $\C/\Lambda$ has the nowhere vanishing form $dz$. Non-example: $\mathbb{P}^1$ has $K = \mathcal{O}(-2)$, which has no nonzero sections, so it is not Calabi–Yau. A complex torus of dimension $2$ has trivial canonical bundle, but $H^0(X, \Omega^1) \cong \C^2$, so it fails the stricter definition.