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 p_a of X is defined as
p_a(X)=(-1)^r (\chi(\mathcal{O}_X)-1).
Here \chi(\mathcal{O}_X) is the Euler characteristic of the structure sheaf \mathcal{O}_X.[1]
Complex projective manifolds
The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely
p_a=\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 hp,q = hq,p recovers the earlier definition for projective varieties.
Kähler manifolds
By using hp,q = hq,p for compact Kähler manifolds this can be
reformulated as the Euler characteristic in coherent cohomology for the structure sheaf \mathcal{O}_M:
p_a=(-1)^n(\chi(\mathcal{O}_M)-1).\,
This definition therefore can be applied to some other locally ringed spaces.
See also
References
- 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
- ^ Hartshorne, Robin (1977). Algebraic Geometry. Vol. 52. Graduate Texts in Mathematics. New York, NY: Springer New York. p. 230. doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8. S2CID 197660097
Further reading
- Hirzebruch, Friedrich (1995 [1978]). Topological methods in algebraic geometry. Classics in Mathematics. Reprint of the 2nd, corr. print. of the 3rd ed. Berlin: Springer-Verlag. ISBN 3-540-58663-6. Zbl 0843.14009.