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 projective manifold of dimension n can be defined as a combination of Hodge numbers, namely

pa = hn,0hn − 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.

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 \mathcal{O}_M:

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

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

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