Toric Varieties

Definition

Toric varieties are a special class of algebraic varieties that can be described using combinatorial data associated with fans, which are collections of cones in a lattice. They provide a bridge between algebraic geometry and combinatorics, allowing for the study of geometric objects through their combinatorial structures, often arising in the context of torus actions and symplectic geometry.

5 Must Know Facts For Your Next Test

1. Toric varieties can be explicitly constructed from fans, making them easier to study and understand compared to general varieties.
2. They often have rich combinatorial and geometric structures, allowing for applications in areas like mirror symmetry and string theory.
3. The set of closed orbits of a torus action on a toric variety corresponds to the maximal cones in its fan, illustrating how geometry relates to combinatorics.
4. Toric varieties can be realized as projective varieties by embedding them in projective space, linking them to projective geometry.
5. Many important properties of algebraic varieties, such as dimension and singularity type, can be easily determined using the associated combinatorial data from their fans.

Review Questions

• How do fans contribute to the definition and understanding of toric varieties?
  • Fans are essential for defining toric varieties as they provide the combinatorial data necessary for constructing these varieties. Each fan consists of cones that dictate how the corresponding algebraic variety is structured. By analyzing the properties of the cones within a fan, one can derive important geometric features of the toric variety, such as its dimension and singularities, making fans a key tool in understanding the underlying geometry.
• Discuss the significance of torus actions in relation to toric varieties and their geometric interpretations.
  • Torus actions are significant because they allow us to study the symmetries of toric varieties. The closed orbits of these actions correspond to maximal cones in the associated fan, establishing a direct link between algebraic geometry and combinatorial structures. This relationship helps us visualize and analyze complex geometric properties, such as how different varieties interact under various transformations induced by the torus action.
• Evaluate how the process of symplectic reduction is related to the construction and study of toric varieties.
  • Symplectic reduction plays a crucial role in connecting toric varieties with symplectic geometry. By examining the quotient of a symplectic manifold under a Hamiltonian group action (often involving a torus), we can construct new manifolds that may exhibit toric structures. This relationship highlights how symplectic techniques can yield insights into algebraic varieties like toric varieties, emphasizing the interplay between geometry, algebra, and combinatorics in modern mathematical research.

Related terms

Fans: A fan is a collection of cones in a vector space that satisfy certain intersection properties, serving as the combinatorial data used to define toric varieties.

Torus Action: A torus action is a continuous action of an algebraic torus on an algebraic variety, providing insights into the structure and symmetry of the variety.

Symplectic Reduction:

Symplectic reduction is a process that produces new symplectic manifolds by quotienting out by a group action, which can be related to constructing toric varieties.