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.

Projective varieties

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

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 ${𝒪}_M$:

$p_a=(-1{)}^n(\chi ({𝒪}_M)-1).$

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

See also

• Genus (mathematics)
• Geometric genus

References

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Further reading

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