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 ${\displaystyle n}$th 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 ${\displaystyle n}$-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.

Suppose that ${\displaystyle X}$ is a smooth variety and that ${\displaystyle D}$ is a smooth divisor on ${\displaystyle X}$. The adjunction formula relates the canonical bundles of ${\displaystyle X}$ and ${\displaystyle D}$. It is a natural isomorphism

In terms of canonical classes, it is

This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of ${\displaystyle X}$ from the singularities of ${\displaystyle D}$.

Let ${\displaystyle X}$ be a normal surface. A genus ${\displaystyle g}$ fibration ${\displaystyle f:X\to B}$ of ${\displaystyle X}$ is a proper flat morphism ${\displaystyle f}$ to a smooth curve such that ${\displaystyle f_{*}{\mathcal {O}}_{X}\cong {\mathcal {O}}_{B}}$ and all fibers of ${\displaystyle f}$ have arithmetic genus ${\displaystyle g}$. If ${\displaystyle X}$ is a smooth projective surface and the fibers of ${\displaystyle f}$ do not contain rational curves of self-intersection ${\displaystyle -1}$, then the fibration is called minimal. For example, if ${\displaystyle X}$ admits a (minimal) genus 0 fibration, then is ${\displaystyle X}$ is birationally ruled, that is, birational to ${\displaystyle \mathbb {P} ^{1}\times B}$.