KNOWPIA
WELCOME TO KNOWPIA

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.

Let *X* be a projective scheme of dimension *r* over a field *k*, the *arithmetic genus* of *X* is defined as Here is the Euler characteristic of the structure sheaf .^{[1]}

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 *h*^{p,q} = *h*^{q,p} recovers the earlier definition for projective varieties.

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.

- 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

**^**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.

- 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.