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Canonical class

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The class of divisors, with respect to linear equivalence on an algebraic variety , which are divisors of differential forms of maximal degree. If is a non-singular algebraic variety and , then in local coordinates a form can be written as

The divisor of is locally equal to the divisor of this rational function . This construction does not depend on the choice of local coordinates and gives the divisor of on all of . Since for any other form of the same degree as , , it follows that , and corresponding divisors are equivalent. The canonical class thus constructed is the first Chern class of the sheaf of regular differential forms of degree . Its numerical characteristics (degree, index, self-intersections, etc.) are effectively calculable invariants of the algebraic variety.

If is a non-singular projective curve of genus , then . For elliptic curves and, more generally, for Abelian varieties, . If is a non-singular hypersurface of degree in projective space , then , where is a hyperplane section of it.

See also Canonical imbedding.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001


Comments

References

[a1] S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) Zbl 0491.14006
How to Cite This Entry:
Canonical class. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Canonical_class&oldid=34201
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article