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.
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Complex projective manifolds
The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely
pa = hn,0 − hn − 1, 0 + ... + (−1)n − 1h1, 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.
Kähler manifolds
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.