Toric varieties

Definition

Toric varieties are a special class of algebraic varieties that are defined by combinatorial data, particularly through fans or polyhedral cones. They arise from the theory of torus actions on varieties, allowing for a geometric representation of combinatorial structures and providing powerful tools for calculations in K-theory.

5 Must Know Facts For Your Next Test

1. Toric varieties can be constructed from combinatorial data such as fans, making them a bridge between geometry and combinatorics.
2. They exhibit nice properties such as being smooth or singular, depending on the underlying fan structure and its dimensionality.
3. The K-theory of toric varieties can often be computed using tools like the localization theorem and the Mayer-Vietoris sequence.
4. Toric varieties provide a way to understand various algebraic structures through their torus action, leading to simplifications in complex calculations.
5. They have applications not just in pure mathematics but also in areas like mirror symmetry and algebraic geometry, demonstrating their broad utility.

Review Questions

• How do the combinatorial structures associated with toric varieties influence their geometric properties?
  • The combinatorial structures, such as fans and polyhedral cones, directly dictate the geometric properties of toric varieties. Each cone in a fan corresponds to an open set in the variety, determining how the overall shape behaves. The intersections of these cones also influence whether the variety is smooth or has singularities, thus connecting the geometry of the variety with its combinatorial data.
• Discuss how the Mayer-Vietoris sequence applies to computations involving toric varieties in K-theory.
  • The Mayer-Vietoris sequence is essential for computing K-groups of spaces that can be decomposed into simpler components. For toric varieties, which can often be expressed as unions of simpler varieties corresponding to their cones, this sequence allows mathematicians to derive K-theory information by analyzing these components. This approach simplifies computations by breaking down complex varieties into more manageable pieces.
• Evaluate the role of toric varieties in understanding localization sequences within K-theory and their broader implications.
  • Toric varieties play a pivotal role in localization sequences in K-theory by serving as examples where both topological and algebraic methods converge. Their rich combinatorial structures allow for effective application of localization techniques, leading to insights into the K-groups associated with other varieties. This connection not only enhances our understanding of K-theory but also opens pathways to explore further implications in areas like mirror symmetry and representation theory, highlighting the versatility and importance of toric varieties in modern mathematics.

Related terms

Fans: A fan is a collection of cones that cover a polyhedral space, used to describe the combinatorial structure underlying toric varieties.

Polyhedral cones: These are the building blocks of toric varieties, representing convex sets that can be used to define the geometric properties of the variety.

K-theory: A branch of algebraic topology and algebraic geometry that studies vector bundles and their generalizations, often applied in conjunction with toric varieties for computing K-groups.