# Abelian surface

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 *S*^{1}×*S*^{1}×*S*^{1}×*S*^{1} so the fundamental group is **Z**^{4}.

1 | ||||

2 | 2 | |||

1 | 4 | 1 | ||

2 | 2 | |||

1 |

**Examples:** A product of two elliptic curves. The Jacobian variety of a genus 2 curve.

## See also

## 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, MR 2030225 - Beauville, Arnaud (1996),
*Complex algebraic surfaces*, London Mathematical Society Student Texts,**34**(2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314 - Birkenhake, Ch. (2001), "a/a110040", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4