# 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 |

#### Comments

#### References

[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |

[a2] | A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023 |

[a3] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001 |

**How to Cite This Entry:**

Genus of a surface.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Genus_of_a_surface&oldid=47081