Jump to content

Arithmetic genus

From Wikipedia, the free encyclopedia

In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.

Projective varieties

[edit]

Let X be a projective scheme of dimension r over a field k, the arithmetic genus of X is defined asHere is the Euler characteristic of the structure sheaf .[1]

Complex projective manifolds

[edit]

The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely

When n=1, the formula becomes . According to the Hodge theorem, . Consequently , where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

When X is a compact Kähler manifold, applying hp,q = hq,p recovers the earlier definition for projective varieties.

Kähler manifolds

[edit]

By using hp,q = hq,p for compact Kähler manifolds this can be reformulated as the Euler characteristic in coherent cohomology for the structure sheaf :

This definition therefore can be applied to some other locally ringed spaces.

See also

[edit]

References

[edit]
  • P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library (2nd ed.). Wiley Interscience. p. 494. ISBN 0-471-05059-8. Zbl 0836.14001.
  • Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3
  1. ^ Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. New York, NY: Springer New York. p. 230. doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8. S2CID 197660097.

Further reading

[edit]