Arithmetic genus

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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.

Complex projective manifolds[edit]

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.

Kähler manifolds[edit]

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:


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

See also[edit]


Further reading[edit]