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 manifolds

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

pa = hn,0 hn 1, 0 + ... + (1)n 1h1, 0.

When n = 1 we have χ = 1 g where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

Kähler manifolds

By using hp,q = hq,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.