mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most

complex tori In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...

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. 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 to ''S''¹×''S''¹×''S''¹×''S''¹ so the fundamental group is Z⁴. Hodge diamond: Examples: A product of two elliptic curves. The Jacobian variety of a genus 2 curve.

References

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

See also

References