Toric varieties

Definition

Toric varieties are a special class of algebraic varieties that can be defined by combinatorial data from convex geometry, specifically polyhedral cones and fans. They provide a bridge between algebraic geometry and combinatorial geometry, with important connections to both Kähler geometry and complex manifolds, especially in understanding the geometric structures and properties of these spaces.

5 Must Know Facts For Your Next Test

1. Toric varieties can be constructed from fans, which are collections of cones that describe how local affine patches fit together.
2. These varieties are particularly useful because they can be explicitly described using combinatorial data, making them easier to work with in certain contexts.
3. One important property of toric varieties is that they can possess Kähler metrics, linking them to concepts in Kähler geometry.
4. They have applications in areas such as mirror symmetry, where they can help relate different types of geometrical spaces.
5. Toric varieties often exhibit a rich structure that allows for effective computation of various geometric invariants, such as intersection numbers.

Review Questions

• How do the combinatorial aspects of toric varieties facilitate their study in relation to Kähler geometry?
  • The combinatorial nature of toric varieties allows for a clear understanding of their geometric properties through the use of fans and polyhedral cones. This connection provides tools to study Kähler metrics on these varieties. By analyzing the combinatorial data, one can derive results about the complex structure and curvature properties that are essential in Kähler geometry.
• Discuss the significance of fans in the definition and construction of toric varieties.
  • Fans play a crucial role in the construction and understanding of toric varieties. They organize the polyhedral cones that define local charts of the variety and ensure that these charts fit together consistently. By providing a combinatorial structure, fans enable mathematicians to study complex geometrical objects through algebraic methods, facilitating deeper insights into both algebraic and complex geometry.
• Evaluate the impact of toric varieties on modern algebraic geometry and their connections to other mathematical areas.
  • Toric varieties have significantly influenced modern algebraic geometry by providing a framework where algebraic and combinatorial techniques converge. Their explicit construction using fans allows for rigorous computations and has facilitated developments in mirror symmetry and deformation theory. The interplay between toric varieties and other areas like symplectic geometry and string theory illustrates their versatility and importance in current mathematical research.

Related terms

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

Fan: A collection of polyhedral cones that is used to describe the combinatorial structure of a toric variety, helping to organize the relationships between different cones.

Kähler manifold:

A type of complex manifold that has a Kähler metric, which allows for the integration of symplectic and Riemannian geometry, often studied in relation to toric varieties.