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 $$\mathcal{O}_X$$.

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 hp,q = hq,p recovers the earlier definition for projective varieties.

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 $$\mathcal{O}_M$$:


 * $$ p_a=(-1)^n(\chi(\mathcal{O}_M)-1).\,$$

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