Toric Varieties

Definition

Toric varieties are a special class of algebraic varieties that are defined by combinatorial data associated with polyhedral cones and their fans. They allow the study of geometric properties through combinatorial techniques, making them a bridge between algebraic geometry and convex geometry. The structure of toric varieties provides a way to understand how these geometric objects can be constructed from simpler pieces, particularly in the context of algebraic groups and their actions.

5 Must Know Facts For Your Next Test

1. Toric varieties can be constructed from combinatorial data given by fans, where each cone corresponds to a piece of the variety.
2. These varieties often come equipped with a natural action of algebraic groups, allowing for deep connections with group theory and representation theory.
3. Every toric variety can be embedded in a projective space, making them particularly useful in projective geometry.
4. The study of toric varieties is linked to several important topics, such as mirror symmetry and string theory, due to their rich combinatorial structure.
5. Toric varieties provide a geometric realization of combinatorial objects, leading to applications in areas like mirror symmetry and the computation of cohomological properties.

Review Questions

• How do polyhedral cones and fans contribute to the construction and understanding of toric varieties?
  • Polyhedral cones serve as the fundamental building blocks for toric varieties, where each cone represents a specific piece of the variety's structure. Fans, composed of these cones, provide the combinatorial framework that captures how these pieces fit together. By analyzing the interactions between these cones and fans, one can derive significant geometric properties of toric varieties, revealing their intricate structure.
• Discuss the relationship between toric varieties and algebraic groups, focusing on their actions and implications for the geometry of these varieties.
  • Toric varieties exhibit a natural action by algebraic groups, which means that there are symmetries and transformations that preserve their structure. This relationship allows for exploring deeper geometrical and topological properties through group theory. For instance, understanding how these group actions manifest can lead to insights about the classification of toric varieties and their moduli spaces.
• Evaluate the significance of toric varieties in modern algebraic geometry and their role in broader mathematical contexts such as mirror symmetry.
  • Toric varieties have emerged as vital objects within modern algebraic geometry due to their rich combinatorial nature and ability to connect disparate areas like convex geometry and representation theory. Their applications extend into mirror symmetry, where they serve as examples for duality phenomena between certain types of varieties. This interplay highlights the relevance of toric varieties not just as standalone entities but as pivotal elements that bridge various mathematical theories and concepts.

Related terms

Polyhedral Cone: A polyhedral cone is a subset of a vector space formed by the non-negative combinations of a finite set of vectors, which can be used to define toric varieties.

Fan: A fan is a collection of polyhedral cones that satisfy certain compatibility conditions, providing a combinatorial structure that encodes the geometry of a toric variety.

Algebraic Group: An algebraic group is a group that is also an algebraic variety, allowing for the study of group actions on varieties, which is essential in understanding the geometric properties of toric varieties.