Canonical bundle

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In mathematics, the canonical bundle of a non-singular algebraic variety [V] of dimension [n] is the line bundle

[\,\!\Omega^n = \omega]

which is the [n^] exterior power of the cotangent bundle [\Omega] on [V]. That is, it is the bundle of holomorphic [n]-forms on [V], if [V] is defined over the complex number field. This is the dualising object for Serre duality on [V]. It may equally be considered an invertible sheaf.

The canonical class is the divisor class of a Cartier divisor [K] on [V] giving rise to the canonical bundle — it is an equivalence class for linear equivalence on [V], and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor [-\!K] with [\!\,K] canonical. The anticanonical bundle is the corresponding inverse bundle [\,\!\omega^].

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