Canonical bundle

In mathematics, the canonical bundle of a non-singular algebraic variety ${\displaystyle V}$ of dimension ${\displaystyle n}$ over a field is the line bundle ${\displaystyle \,\!\Omega ^{n}=\omega }$, which is the nth exterior power of the cotangent bundle ${\displaystyle \Omega }$ on ${\displaystyle V}$.

Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle ${\displaystyle T^{*}V}$. Equivalently, it is the line bundle of holomorphic n-forms on ${\displaystyle V}$. This is the dualising object for Serre duality on ${\displaystyle V}$. It may equally well be considered as an invertible sheaf.

The canonical class is the divisor class of a Cartier divisor ${\displaystyle K}$ on ${\displaystyle V}$ giving rise to the canonical bundle — it is an equivalence class for linear equivalence on ${\displaystyle V}$, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −${\displaystyle K}$ with ${\displaystyle K}$ canonical.

The anticanonical bundle is the corresponding inverse bundle ${\displaystyle \omega ^{-1}}$. When the anticanonical bundle of ${\displaystyle V}$ is ample, ${\displaystyle V}$ is called a Fano variety.