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.

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

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.

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.