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 (topological) meaning of genus of a surface, so the definitions are compatible.
Kähler manifolds.
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 formula_1:
This definition therefore can be applied to some other 
locally ringed spaces.