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Inmathematics, thearithmetic genus of analgebraic variety is one of a few possible generalizations of thegenus of an algebraic curve orRiemann surface.

LetX be aprojective scheme of dimensionr over a fieldk, thearithmetic genus${\displaystyle p_a}$ ofX is defined as$${\displaystyle p_a(X)=(-1{)}^r(\chi ({\mathcal{O}}_X)-1).}$$Here${\displaystyle \chi ({\mathcal{O}}_X)}$ is theEuler characteristic of the structure sheaf${\displaystyle {\mathcal{O}}_X}$.^[1]

The arithmetic genus of acomplex projective manifold of dimensionn can be defined as a combination ofHodge numbers, namely

${\displaystyle p_a=\sum _{j=0}^{n-1}(-1{)}^j{h}^{n-j,0}.}$

Whenn=1, the formula becomes${\displaystyle p_a=h^{1,0}}$. According to theHodge theorem,${\displaystyle h^{0,1}=h^{1,0}}$. Consequently${\displaystyle h^{0,1}=h^1(X)/2=g}$, whereg is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

WhenX is a compactKähler manifold, applyingh^p,q =h^q,p recovers the earlier definition for projective varieties.

By usingh^p,q =h^q,p for compact Kähler manifolds this can be reformulated as theEuler characteristic incoherent cohomology for thestructure sheaf${\displaystyle {\mathcal{O}}_M}$:

${\displaystyle p_a=(-1{)}^n(\chi ({\mathcal{O}}_M)-1).}$

This definition therefore can be applied to some otherlocally ringed spaces.

• Genus (mathematics)
• Geometric genus

• P. Griffiths;J. Harris (1994).Principles of Algebraic Geometry. Wiley Classics Library (2nd ed.). Wiley Interscience. p. 494.ISBN 0-471-05059-8.Zbl 0836.14001.
• Rubei, Elena (2014),Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter,ISBN 978-3-11-031622-3

• Hirzebruch, Friedrich (1995) [1978].Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin:Springer-Verlag.ISBN 3-540-58663-6.Zbl 0843.14009.

• Topological methods of algebraic geometry

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