arithmetic genus

Let X be a projective scheme of dimension r over a field k, the arithmetic genus $p\_a$ of X is defined as$$p\_a(X)=(-1)^r\; (\backslash chi(\backslash mathcal\{O\}\_X)-1).$$Here $\backslash chi(\backslash mathcal\{O\}\_X)$ is the Euler characteristic of the structure sheaf $\backslash mathcal\{O\}\_X$.{{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 }}

The arithmetic genus of a complex projective manifold

of dimension n can be defined as a combination of Hodge numbers, namely

:$p\_a=\backslash sum\_\{j=0\}^\{n-1\}\; (-1)^j\; h^\{n-j,0\}.$

When n=1, the formula becomes $p\_a=h^\{1,0\}$. According to the Hodge theorem, $h^\{0,1\}=h^\{1,0\}$. Consequently $h^\{0,1\}=h^1(X)/2=g$, 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^p,q = h^q,p recovers the earlier definition for projective varieties.

By using h^p,q = h^q,p for compact Kähler manifolds this can be

reformulated as the Euler characteristic in coherent cohomology for the structure sheaf $\backslash mathcal\{O\}\_M$:

: $p\_a=(-1)^n(\backslash chi(\backslash mathcal\{O\}\_M)-1).\backslash ,$

This definition therefore can be applied to some other

locally ringed spaces.

• Genus (mathematics)
• Geometric genus

• {{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}}

• {{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