Canonical bundle

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In mathematics, the canonical bundle of a non-singular algebraic variety <math>V</math> of dimension <math>n</math> is the line bundle

<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>. That is, it is the bundle of holomorphic <math>n</math>-forms on <math>V</math>, if <math>V</math> is defined over the complex number field. This is the dualising object for Serre duality on <math>V</math>. It may equally be considered 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 — 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 <math>-\!K</math> with <math>\!\,K</math> canonical. The anticanonical bundle is the corresponding inverse bundle <math>\,\!\omega^{-1}</math>.