In mathematics, an abelian surface is 2-dimensional abelian variety

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

.

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

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

.
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

Isogeny

In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....

) 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

Jacobian variety

In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

of a genus 2 curve.