# Abelian surface

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In algebraic geometry an Abelian surface over a field ${\displaystyle K}$ is a two dimensional Abelian variety. Every abelian surface is a finite quotient of a Jacobian variety of a smooth hyperelliptic curve of genus two or a product of two elliptic curves. Abelian surfaces are one of the two types of algebraic surfaces with trivial canonical class, the other type being algebraic K3 surfaces.

### Polarization

Abelian surfaces have a trivial canonical class. Therefore they are usually considered together with a choice of some non-trivial effective divisor on them. This divisor is called the polarization on the Abelian surface; A pair ${\displaystyle (A,C)}$ of an Abelian surface and a polarization is call a polarized Abelian surface. Given a polarized Abelian variety ${\displaystyle (A,D)}$ we define the polarization map ${\displaystyle A\to Pic^{0}(A)}$ by sending a point ${\displaystyle a}$ to the divisor class ${\displaystyle [\tau _{-a}C-C]}$. This map is a group morphism. The kernel of the map is a finite Abelian group with at most four generators. The ismorphism type of the kernel is called the type of the polarization. If the kernel is trivial then polarization is called principal; in this case the arithmetic genus of ${\displaystyle C}$ is ${\displaystyle 2}$. Most of the classical theory of Abelian surfaces deal with the case where ${\displaystyle C}$ is a smooth curve of genus 2.

### The Kummer surface

The quotient of an Abelian variety by the involution ${\displaystyle x\mapsto -x}$ is called the Kummer variety of the Abelian variety. The Kummer varieties of Jacobians of hyperelliptic curves of genus 2 exhibit many beautiful properties - see the article Kummer surfaces.