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

: p a = h n ,0 − h n − 1 ,0 + ... + (−1) n h 1 ,0.

By using h p , q = h q , p this can also be manipulated to a formula that is an Euler characteristic in coherent cohomology for the structure sheaf O M :

: p a = (−1) n (χ( O M ) − 1).

This definition therefore can be applied to varieties over any field (mathematics). 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.