{{Short description|Property of an algebraic variety}} {{Technical|date=August 2023}}
In mathematics, the '''arithmetic genus''' of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
== Projective varieties == Let ''X'' be a projective scheme of dimension ''r'' over a field ''k'', the ''arithmetic genus'' <math>p_a</math> of ''X'' is defined as<math display="block">p_a(X)=(-1)^r (\chi(\mathcal{O}_X)-1).</math>Here <math>\chi(\mathcal{O}_X)</math> is the Euler characteristic of the structure sheaf <math>\mathcal{O}_X</math>.<ref>{{Cite book |last=Hartshorne |first=Robin |author-link=Robin Hartshorne |url=http://link.springer.com/10.1007/978-1-4757-3849-0 |title=Algebraic Geometry |date=1977 |publisher=Springer New York |isbn=978-1-4419-2807-8 |series=Graduate Texts in Mathematics |volume=52 |location=New York, NY |pages=230 |doi=10.1007/978-1-4757-3849-0|s2cid=197660097 }}</ref>
==Complex projective manifolds== The arithmetic genus of a complex projective manifold of dimension ''n'' can be defined as a combination of Hodge numbers, namely
:<math>p_a=\sum_{j=0}^{n-1} (-1)^j h^{n-j,0}.</math>
When ''n=1'', the formula becomes <math>p_a=h^{1,0}</math>. According to the Hodge theorem, <math>h^{0,1}=h^{1,0}</math>. Consequently <math>h^{0,1}=h^1(X)/2=g</math>, where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible.
When ''X'' is a compact Kähler manifold, applying ''h''<sup>''p'',''q''</sup> = ''h''<sup>''q'',''p''</sup> recovers the earlier definition for projective varieties.
==Kähler manifolds== By using ''h''<sup>''p'',''q''</sup> = ''h''<sup>''q'',''p''</sup> for compact Kähler manifolds this can be reformulated as the Euler characteristic in coherent cohomology for the structure sheaf <math>\mathcal{O}_M</math>:
: <math> p_a=(-1)^n(\chi(\mathcal{O}_M)-1).\,</math>
This definition therefore can be applied to some other locally ringed spaces.
==See also== *Genus (mathematics) * Geometric genus
==References== * {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths |author2=J. Harris |authorlink2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | edition=2nd | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | zbl=0836.14001 | page=494 }} * {{Citation | last1=Rubei | first1=Elena | title=Algebraic Geometry, a concise dictionary | publisher=Walter De Gruyter | location=Berlin/Boston | isbn=978-3-11-031622-3 | year=2014}} <references /> ==Further reading== * {{cite book | last=Hirzebruch | first=Friedrich | authorlink=Friedrich Hirzebruch | title=Topological methods in algebraic geometry | others=Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel | edition=Reprint of the 2nd, corr. print. of the 3rd | orig-date=1978 | series=Classics in Mathematics | location=Berlin | publisher=Springer-Verlag | year=1995 | isbn=3-540-58663-6 | zbl=0843.14009 }}
Category:Topological methods of algebraic geometry