Singular Case
On a singular variety, 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 . It is this class, denoted by that is referred to as the canonical divisor on
Alternately, again on a normal variety, one can consider, the 'th cohomology of the normalized dualizing complex of . This sheaf corresponds to a Weil divisor class, which is equal to the divisor class defined above. In the absence of the normality hypothesis, the same result holds if is and Gorenstein in dimension one.
Read more about this topic: Canonical Bundle
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