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}(\chi ({\mathcal {O}}_{X})-1).

Here

\chi ({\mathcal {O}}_{X})

is the Euler characteristic of the structure sheaf

{\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

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

p_{a}=(-1)^{n}(\chi ({\mathcal {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.

Arithmetic genus

Contents

Projective varieties[]

Complex projective manifolds[]

Kähler manifolds[]

See also[]

References[]

Further reading[]