Canonical bundle

Summary

The canonical bundle $\omega_X$ of a smooth variety $X$ of dimension $n$ is the line bundle of differential forms of top degree $n$. It is the $n$-th exterior power of the sheaf of differentials. A divisor $K_X$ with $\mathcal{O}_X(K_X) \cong \omega_X$ is a canonical divisor. The canonical bundle appears in Serre duality, in the adjunction formula and in the Kodaira dimension. Its sign governs the broad classification of varieties: Fano varieties have anti-ample $\omega_X$, Calabi–Yau varieties trivial $\omega_X$, and varieties of general type big $\omega_X$.