Arithmetic genus

In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface.

The arithmetic genus of a projective complex manifold of dimension n can be defined as a combination of Hodge numbers, namely

p_a = h^n,0 − h^{n − 1, 0} + ... + (−1)^{n − 1}h^{1, 0}.

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 h^p,q = h^q,p for compact Kähler manifolds this can be reformulated as Euler characteristic in coherent cohomology for the structure sheaf $\mathcal{O}_M$:

$p_a=(-1)^n(\chi(\mathcal{O}_M)-1).\,$

This definition therefore can be applied to some other locally ringed spaces.

See also

• Geometric genus

References

• P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 494. ISBN 0-471-05059-8.