# 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 $X$.

#### References

 [1] I.R., et al. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001