Editing Canonical bundle (section)

{{Short description|Concept in algebraic geometry}}
In [[mathematics]], the '''canonical bundle''' of a [[non-singular variety|non-singular]] [[algebraic variety]] <math>V</math> of dimension <math>n</math> over a field is the [[line bundle]] <!-- Using \,\! to force PNG rendering, else formula won't show up (used again below) --> <math>\,\!\Omega^n = \omega</math>, which is the <math>n</math>th [[exterior power]] of the [[cotangent bundle]] <math>\Omega</math> on <math>V</math>.

Over the [[complex number]]s, it is the [[determinant bundle]] of the holomorphic [[cotangent bundle]] <math>T^*V</math>. Equivalently, it is the line bundle of holomorphic <math>n</math>-forms on <math>V</math>.
This is the [[Duality (mathematics)|dualising object]] for [[Serre duality]] on <math>V</math>. It may equally well be considered as an [[invertible sheaf]].

The '''canonical class''' is the [[divisor class]] of a [[Cartier divisor]] <math>K</math> on <math>V</math> giving rise to the canonical bundle &mdash; it is an [[equivalence class]] for [[linear equivalence]] on <math>V</math>, and any divisor in it may be called a '''canonical divisor'''. An '''anticanonical''' divisor is any divisor &minus;<math>K</math> with <math>K</math> canonical.

The '''anticanonical bundle''' is the corresponding inverse bundle <math>\omega^{-1}</math>. When the anticanonical bundle of <math>V</math> is [[ample line bundle|ample]], <math>V</math> is called a [[Fano variety]].