# Genus of a surface

A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field $k$. There are two different genera — the arithmetic genus and the geometric genus. The geometric genus $p _ {g}$ of a complete smooth algebraic surface $X$ is equal to

$$p _ {g} = \mathop{\rm dim} _ {g} H ^ {0} ( X , \Omega _ {X} ^ {2} ) ,$$

i.e. to the dimension of the space of regular differential $2$- forms (cf. Differential form) on $X$. The arithmetic genus $p _ {a}$ of a complete smooth algebraic surface $X$ is equal to

$$p _ {a} = \chi ( X , {\mathcal O} _ {X} ) - 1 = \ \mathop{\rm dim} _ {k} H ^ {2} ( X , {\mathcal O} _ {X} ) - \mathop{\rm dim} _ {k} H ^ {1} ( X , {\mathcal O} _ {X} ) .$$

The geometric and arithmetic genera of a complete smooth algebraic surface $X$ are related by the formula $p _ {g} - p _ {a} = q$, where $q$ is the irregularity of $X$, which is equal to the dimension of the space of regular differential $1$- forms on $X$.

How to Cite This Entry:
Genus of a surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_surface&oldid=53543
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article