Arithmetic genus

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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 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)ⁿh^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 this can also be manipulated to a formula that is an Euler characteristic in coherent cohomology for the structure sheaf :

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This definition therefore can be applied to any locally ringed space.

See also: geometric genus