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
When n = 1 we have χ = 1 − g where g is the usual 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 :
This definition therefore can be applied to some other
locally ringed spaces.