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 <math>p_a</math> of X is defined as<math display="block">p_a(X)=(-1)^r (\chi(\mathcal{O}_X)-1).</math>Here <math>\chi(\mathcal{O}_X)</math> is the Euler characteristic of the structure sheaf <math>\mathcal{O}_X</math>.
Complex projective manifolds
The arithmetic genus of a complex projective manifold
of dimension n can be defined as a combination of Hodge numbers, namely
:<math>p_a=\sum_{j=0}^{n-1} (-1)^j h^{n-j,0}.</math>
When n=1, the formula becomes <math>p_a=h^{1,0}</math>. According to the Hodge theorem, <math>h^{0,1}=h^{1,0}</math>. Consequently <math>h^{0,1}=h^1(X)/2=g</math>, 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<sup>p,q</sup> = h<sup>q,p</sup> recovers the earlier definition for projective varieties.
Kähler manifolds
By using h<sup>p,q</sup> = h<sup>q,p</sup> for compact Kähler manifolds this can be
reformulated as the Euler characteristic in coherent cohomology for the structure sheaf <math>\mathcal{O}_M</math>:
: <math> p_a=(-1)^n(\chi(\mathcal{O}_M)-1).\,</math>
This definition therefore can be applied to some other
locally ringed spaces.
See also
- Genus (mathematics)
- Geometric genus