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. 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 hp,q = hq,p recovers the earlier definition for projective varieties. 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.