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Concept# Canonical bundle

Summary

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.
Adjunction formula
Suppose that X is a smooth variety and that D is a smooth divisor on X. The adjunction formula relates the canonical bundles of X and D. It is a natural isomorphism
In terms of canonical classes, it is
This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of X from the singularities of D.
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 S2 and Gorenstein in dimension one.
If the canonical class is effective, then it determines a rational map from V into projective space.

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