Abelian surface
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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.
[edit] 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, MR2030225, ISBN 978-3-540-00832-3
- Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, 34 (2nd ed.), Cambridge University Press, MR1406314, ISBN 978-0-521-49510-3; 978-0-521-49842-5
- Birkenhake, Ch. (2001), "Abelian surface", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
