# Arithmetic genus

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.

## Complex projective manifoldsEdit

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

*p*_{a}=*h*^{n,0}−*h*^{n − 1, 0}+ ... + (−1)^{n − 1}*h*^{1, 0}.

When *n* = 1 we have^{[clarification needed]} χ = 1 − *g* where *g* is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

## Kähler manifoldsEdit

By using *h*^{p,q} = *h*^{q,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 alsoEdit

## ReferencesEdit

- 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

## Further readingEdit

- Hirzebruch, Friedrich (1995) [1978].
*Topological methods in algebraic geometry*. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-58663-6. Zbl 0843.14009.