Arithmetic genus
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In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalisations of the genus of an algebraic curve.
The arithmetic genus of a non-singular variety of dimension n over the complex numbers can be defined as a combination of Hodge numbers, namely
- pa = hn,0 − hn − 1 ,0 + ... + (−1)nh1,0.
By using hp,q = hq,p this can also be manipulated to a formula that is an Euler characteristic in coherent cohomology for the structure sheaf OM:
- pa = (−1)n(χ( OM) − 1).
This definition therefore can be applied to varieties over any field. When n = 1 we have χ = 1 − g where g is the usual meaning of genus, so the definitions are compatible.
There is a second meaning of arithmetic genus, applied to singular curves.
See also: geometric genus
