# Genus of a surface

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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)