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 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.

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), "Abelian surface", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=a/a110040