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 $p_{a}$ of X is defined as $p_{a} (X) = (- 1)^{r} (χ (𝒪_{X}) - 1) .$ Here $χ (𝒪_{X})$ is the Euler characteristic of the structure sheaf $𝒪_{X}$ .^[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

p_{a} = \sum_{j = 0}^{n - 1} (- 1)^{j} h^{n - j, 0} .

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

p_{a} = (- 1)^{n} (χ (𝒪_{M}) - 1) .

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

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

↑ 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.

Arithmetic genus

Projective varieties

Complex projective manifolds

Kähler manifolds

See also

References

Further reading