In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the arithmetic genus of an algebraic variety

Algebraic variety

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

 is one of some possible generalizations of the genus of an algebraic curve or Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

.

The arithmetic genus of a projective complex manifold
of dimension n can be defined as a combination of Hodge numbers, namely

p_a = h^n,0 − h^{n − 1, 0} + ... + (−1)^{n − 1}h^{1, 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.

By using h^p,q = h^q,p for compact Kähler manifolds this can be
reformulated as Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 in coherent cohomology for the structure sheaf :

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