Arithmetic genus

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.

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. Let X be a projective scheme of dimension r over a field k, the arithmetic genus p_a of X is defined as p_a(X)=(-1)^r (\chi(\mathcal{O}_X)-1). Here \chi(\mathcal{O}_X) is the Euler characteristic of the structure sheaf \mathcal{O}_X .

Complex projective manifolds

The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely p_a=\sum_{j=0}^{n-1} (-1)^j h^{n-j,0}. When n=1, the formula becomes p_a=h^{1,0} . According to the Hodge theorem, h^{0,1}=h^{1,0} . Consequently h^{0,1}=h^1(X)/2=g , where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When X is a compact Kähler manifold, applying h p,q = h q,p recovers the earlier definition for projective varieties.

Kähler manifolds

By using h p,q = h q,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.

See also

Genus (mathematics) Geometric genus

References

Further reading

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