# Genus of a surface

From Encyclopedia of Mathematics

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

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

The geometric and arithmetic genera of a complete smooth algebraic surface are related by the formula , where is the irregularity of , which is equal to the dimension of the space of regular differential -forms on .

#### References

[1] | I.R., et al. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) |

#### Comments

#### References

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

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

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

**How to Cite This Entry:**

Genus of a surface.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Genus_of_a_surface&oldid=16127

This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article