Abelian variety

Definition

An abelian variety is a complete algebraic variety that has a group structure, allowing for both addition and subtraction of its points. It serves as a generalization of elliptic curves, which are specific types of abelian varieties defined over the complex numbers or finite fields. Abelian varieties play a crucial role in various areas of mathematics, connecting algebraic geometry with number theory and complex analysis.

5 Must Know Facts For Your Next Test

1. Abelian varieties can be defined over any algebraically closed field, not just the complex numbers, and they maintain their group structure in this broader context.
2. The dimension of an abelian variety indicates how many independent directions there are for addition within the variety; for instance, elliptic curves are 1-dimensional abelian varieties.
3. Abelian varieties can be classified by their endomorphisms, which are homomorphisms from the variety to itself that preserve the group structure.
4. The study of abelian varieties is deeply connected to the theory of modular forms, as they can be represented by points in certain moduli spaces.
5. Riemann-Roch theorem applies to abelian varieties, allowing for powerful results regarding the relationships between line bundles and divisor classes on these varieties.

Review Questions

• Compare and contrast abelian varieties and elliptic curves, focusing on their properties and significance in algebraic geometry.
  • Abelian varieties generalize elliptic curves, with the latter being specific cases of abelian varieties that are 1-dimensional. While all elliptic curves have a group structure that allows for point addition, abelian varieties can exist in higher dimensions and retain similar properties. Both play significant roles in algebraic geometry; however, abelian varieties encompass a broader class with more complex structures and applications in number theory.
• Discuss how complex tori serve as examples of abelian varieties and their relevance in understanding these mathematical structures.
  • Complex tori illustrate the concept of abelian varieties by providing concrete examples built from complex vector spaces. When you take a complex vector space and quotient it by a lattice, you get a complex torus which has a natural group structure. This example helps to understand the geometry and properties of higher-dimensional abelian varieties, bridging the gap between abstract definitions and tangible geometric objects.
• Evaluate the importance of Jacobian varieties in relation to algebraic curves and how they enhance our understanding of abelian varieties.
  • Jacobian varieties are crucial as they link algebraic curves to the world of abelian varieties, offering insights into the properties and characteristics of the curves themselves. By associating each curve with its Jacobian variety, mathematicians can explore features such as divisors and linear systems through the group structure present in the Jacobian. This connection enriches the study of both algebraic curves and abelian varieties, revealing deeper connections in algebraic geometry and number theory.

Related terms

Elliptic Curve: A smooth, projective algebraic curve of genus one with a specified point, which can be given a group structure that allows for addition of its points.

Complex Tori: A complex torus is an example of an abelian variety, constructed by taking a complex vector space and quotienting it by a lattice.

Jacobian Variety: The Jacobian variety is an abelian variety associated with an algebraic curve, providing a way to study the curve's properties through its points and group structure.