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# Canonical bundle

(Redirected from Canonical class)

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 nth exterior power of the cotangent bundle Ω 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 $\,\!\omega^{-1}$.

03-10-2013 05:06:04