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. 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. 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. 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 O M {displaystyle {mathcal {O}}_{M}} : This definition therefore can be applied to some other locally ringed spaces.