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
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 
with 
canonical. The anticanonical bundle is the corresponding inverse bundle 
On a singular variety X, there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique Weil divisor class on X. It is this class, denoted by KX that is referred to as the canonical divisor on X.
Alternately, again on a normal variety X, one can consider 
, the − d'th cohomology of the normalized dualizing complex of X. This sheaf corresponds to a Weil divisor class, which is equal to the divisor class KX defined above. In the absence of the normality hypothesis, the same result holds if X is S2 and Gorenstein in dimension one.
