In mathematics, an abelian surface is 2-dimensional abelian variety.
One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are called abelian surfaces and are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century.
Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4.
Hodge diamond:
| 1 | ||||
|---|---|---|---|---|
| 2 | 2 | |||
| 1 | 4 | 1 | ||
| 2 | 2 | |||
| 1 |
Examples: A product of two elliptic curves. The Jacobian variety of a genus 2 curve.
Abelian surfaces with a given number of points
References
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR2030225
Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3; 978-0-521-49842-5, MR1406314
Birkenhake, Ch. (2001), "a/a110040", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
