mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an abelian surface is a 2-dimensional

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

. One-dimensional complex tori are just

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

s and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations. Essentially, these are conditions on the parameter space of period matrices for complex tori which define an algebraic subvariety. This subvariety contains all of the points whose period matrices correspond to a period matrix of an abelian variety. The algebraic ones are called abelian surfaces and are exactly the 2-dimensional

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

. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century. Invariants: The plurigenera are all 1. The surface is

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Defini ...

to ''S''¹×''S''¹×''S''¹×''S''¹ so the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

is Z⁴. Hodge diamond: Examples: A product of two elliptic curves. The

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...

of a genus 2 curve.

References

* * * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub