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 of X is defined asHere is the Euler characteristic of the structure sheaf.

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, the formula becomes. According to the Hodge theorem,. Consequently, where g is the usual meaning of genus of a surface, so the definitions are compatible.
When X is a compact Kähler manifold, applying hp,''q = h''q,''p'' recovers the earlier definition for projective varieties.

Kähler manifolds

By using hp,''q = h''q,''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.