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 of X is defined asHere is the Euler characteristic of the structure sheaf .
The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely
When n=1, the formula becomes . According to the Hodge theorem, . Consequently , 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.
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 :
This definition therefore can be applied to some other locally ringed spaces.