Canonical bundle
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In mathematics, the canonical bundle of a non-singular algebraic variety of dimension
over a field is the line bundle
, which is the nth exterior power of the cotangent bundle
on
.
Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle . Equivalently, it is the line bundle of holomorphic n-forms on
.
This is the dualising object for Serre duality on
. It may equally well be considered as an invertible sheaf.
The canonical class is the divisor class of a Cartier divisor on
giving rise to the canonical bundle — it is an equivalence class for linear equivalence on
, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −
with
canonical.
The anticanonical bundle is the corresponding inverse bundle . When the anticanonical bundle of
is ample,
is called a Fano variety.