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In mathematics, the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle \,\!\Omega ^{n}=\omega } , which is the nth exterior power of the cotangent bundle Ω on V.
Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising object for Serre duality on V. It may equally well be considered as 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 ω. When the anticanonical bundle of V is ample, V is called a Fano variety.
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