Toric Varieties

Definition

Toric varieties are a special class of algebraic varieties that are defined combinatorially from fans, which are collections of cones in a lattice. These varieties provide a bridge between algebraic geometry and combinatorial geometry, allowing for the study of geometric objects using combinatorial data. They often arise in the study of torus actions and play a crucial role in various areas of mathematics, including intersection theory and mirror symmetry.

5 Must Know Facts For Your Next Test

1. Toric varieties can be constructed from any fan, making them highly flexible and adaptable to different combinatorial settings.
2. They have a rich structure that allows for effective computation of invariants, such as intersection numbers and cohomology groups.
3. One significant property of toric varieties is that they can often be embedded into projective space, enabling the application of projective techniques in their study.
4. Toric varieties are particularly important in mirror symmetry, where they correspond to dual pairs of Calabi-Yau manifolds.
5. The Grothendieck group of the toric variety captures important information about its geometry and can be used to compute other invariants.

Review Questions

• How do fans relate to the construction of toric varieties, and why are they significant?
  • Fans serve as the foundational combinatorial structure for constructing toric varieties. Each fan consists of cones that correspond to various coordinate charts within the toric variety, allowing mathematicians to assemble these charts into a coherent geometric object. The significance lies in how fans encapsulate all the necessary combinatorial data needed for defining toric varieties, thus bridging the gap between algebraic and combinatorial geometry.
• Discuss the role of torus actions in understanding the geometric properties of toric varieties.
  • Torus actions play a pivotal role in defining the geometric properties of toric varieties by introducing symmetries that help explore their structure. These actions allow mathematicians to study how the variety behaves under transformations and how it relates to other geometric constructs. Moreover, they facilitate computations related to invariants and contribute to understanding more complex phenomena like moduli spaces and deformation theory.
• Evaluate the impact of toric varieties on modern algebraic geometry and their connections to other areas like mirror symmetry.
  • Toric varieties have significantly influenced modern algebraic geometry by providing a clear framework that links combinatorial data with geometric structures. Their utility extends beyond pure geometry; they have applications in mirror symmetry, where they reveal dualities between pairs of Calabi-Yau manifolds. This connection has led to deeper insights into both algebraic topology and string theory, showcasing how ideas from different mathematical realms can interact and enrich one another.

Related terms

Fans: A fan is a collection of cones that describes how to piece together affine varieties to construct a toric variety, providing the combinatorial data needed for its definition.

Torus Action: A torus action is a group action of a torus on a variety, which helps define the geometric structure of toric varieties and allows for the study of their symmetries.

Polyhedral Geometry: Polyhedral geometry is the study of polyhedra and their properties, which is closely related to toric varieties as these varieties can be associated with polytopes in this field.