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>.