Dual abelian variety

Definition

A dual abelian variety is a concept in algebraic geometry that refers to a specific construction associated with an abelian variety, essentially serving as its counterpart or 'dual'. It captures important properties of the original abelian variety, particularly relating to its morphisms and isogenies, and provides insights into the structure of its points and the relationships between them. The dual abelian variety allows for a deeper understanding of the isogenies of abelian varieties by linking them to their dual counterparts.

5 Must Know Facts For Your Next Test

1. The dual abelian variety can be represented using the homomorphism of the original abelian variety into its dual via the evaluation map.
2. There exists a natural isomorphism between an abelian variety and its double dual, which highlights the duality in this context.
3. Isogenies can be understood through their action on dual abelian varieties, leading to significant results in the study of endomorphisms.
4. The notion of dual abelian varieties extends to higher dimensions and plays a crucial role in the theory of complex tori and algebraic cycles.
5. The relationship between an abelian variety and its dual is essential in understanding the Picard group associated with it, influencing line bundles and divisor classes.

Review Questions

• How does the concept of dual abelian varieties enhance our understanding of morphisms between abelian varieties?
  • The concept of dual abelian varieties deepens our understanding of morphisms by illustrating how these varieties interact through their respective structures. Morphisms between abelian varieties can be viewed as induced maps on their duals, revealing how geometric properties translate across this duality. By examining these relationships, we gain insights into the behavior of endomorphisms and how they reflect back on the original variety.
• In what ways do isogenies relate to dual abelian varieties, and why is this relationship significant?
  • Isogenies are crucial when discussing dual abelian varieties because they provide a framework for understanding how these varieties can be transformed into one another while preserving certain algebraic structures. The existence of an isogeny from an abelian variety to its dual indicates a deep connection between their geometric properties. This relationship is significant as it allows mathematicians to explore symmetries and invariants within the context of both varieties, enriching the study of their characteristics.
• Evaluate the implications of having a polarization on an abelian variety with respect to its dual and related structures.
  • Having a polarization on an abelian variety significantly influences its relationship with its dual by providing an ample isogeny that reveals additional geometric features. This polarization facilitates a clearer understanding of the interactions between divisors on the original variety and line bundles on its dual. Furthermore, polarizations play a vital role in establishing criteria for embeddings into projective spaces and impact various applications in arithmetic geometry, making it essential for analyzing both the original and dual structures comprehensively.

Related terms

abelian variety: An abelian variety is a complete algebraic variety that has a group structure, meaning it allows for a notion of addition and possesses a zero element.

isogeny: An isogeny is a morphism between two abelian varieties that is also a group homomorphism, preserving the structure of the varieties.

polarization: Polarization refers to an isogeny from an abelian variety to its dual that is also ample, providing important geometric properties.