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.

Projective varieties

Let X be a projective scheme of dimension r over a field k, the arithmetic genus ${\displaystyle p_{a}}$  of X is defined as

$${\displaystyle p_{a}(X)=(-1)^{r}(\chi ({\mathcal {O}}_{X})-1).}$$

 

Here ${\displaystyle \chi ({\mathcal {O}}_{X})}$  is the Euler characteristic of the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$ .^[1]

Complex projective manifolds

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

${\displaystyle p_{a}=\sum _{j=0}^{n-1}(-1)^{j}h^{n-j,0}.}$ 

When n=1, the formula becomes ${\displaystyle p_{a}=h^{1,0}}$ . According to the Hodge theorem, ${\displaystyle h^{0,1}=h^{1,0}}$ . Consequently ${\displaystyle h^{0,1}=h^{1}(X)/2=g}$ , 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.

Kähler manifolds

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 ${\displaystyle {\mathcal {O}}_{M}}$ :

${\displaystyle p_{a}=(-1)^{n}(\chi ({\mathcal {O}}_{M})-1).\,}$ 

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

See also

• Genus (mathematics)
• Geometric genus

References

• 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

• 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.