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:

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