9.2 Polytopes and their duality

Polytopes are key players in toric geometry, bridging the gap between geometry and combinatorics. They're like the building blocks of toric varieties, helping us understand their structure and properties. Think of them as multidimensional shapes that hold secrets about complex geometric spaces.

Dual polytopes and fans are two sides of the same coin in toric geometry. By flipping between these dual perspectives, we can unlock insights about toric varieties, their cohomology, and even classify them. It's like having a secret decoder ring for toric geometry!

Polytopes and their duals

Defining polytopes in toric geometry

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• A polytope generalizes polygons and polyhedra to higher dimensions, defined as the convex hull of a finite set of points in a real vector space
• In toric geometry, a polytope P ⊂ N_R is the convex hull of lattice points in the real vector space N_R associated with the lattice N
• The faces of a polytope P correspond to the vertices of its dual polytope P^∨, with complementary dimensions (edges of P correspond to codimension-1 faces of P^∨)

Dual polytopes and their properties

• The dual polytope P^∨ of a polytope P is the set of points y in the dual vector space M_R satisfying ⟨x, y⟩ ≤ 1 for all x ∈ P, where M is the dual lattice of N
• The polar dual P° is obtained by taking the dual polytope P^∨ and translating it so the origin becomes an interior point
• The normal fan of a polytope P is the fan formed by the cones over the faces of the dual polytope P^∨, encoding the combinatorial structure of P (face lattice and incidence relations)

Constructing polytopes from toric data

Constructing polytopes from toric varieties

• A toric variety X_Σ is associated with a fan Σ in the lattice N, and the fan Σ can be obtained from a polytope P ⊂ M_R in the dual lattice M
• To construct a polytope P from a toric variety X_Σ, take the dual fan Σ^∨ in the lattice M and define P as the convex hull of the primitive generators of the rays in Σ^∨
• The vertices of the polytope P correspond to the torus-fixed points of the toric variety X_Σ, and the edges of P correspond to the torus-invariant curves connecting the fixed points

Constructing toric varieties from polytopes

• To construct a toric variety X_Σ from a polytope P ⊂ M_R, take the normal fan Σ of P in the lattice N and define X_Σ as the toric variety associated with the fan Σ
• The faces of the polytope P correspond to the torus-invariant subvarieties of the toric variety X_Σ, with matching dimensions and codimensions
• Examples of toric varieties constructed from polytopes include projective spaces (simplex), Hirzebruch surfaces (trapezoid), and weighted projective spaces (simplex with integer labels)

Duality between polytopes and fans

Proving the duality

• The duality between polytopes and fans can be proven using the definitions of dual polytopes and normal fans
• Given a polytope P ⊂ M_R, its dual polytope P^∨ ⊂ N_R is defined as the set of points y such that ⟨x, y⟩ ≤ 1 for all x ∈ P, and the normal fan Σ of P is the fan formed by the cones over the faces of P^∨
• Conversely, given a fan Σ in N_R, the dual fan Σ^∨ in M_R is defined as the set of cones σ^∨ = {y ∈ M_R | ⟨x, y⟩ ≥ 0 for all x ∈ σ}, and the polytope P associated with Σ is the convex hull of the primitive generators of the rays in Σ^∨

Properties of the duality

• To prove the duality, show that (P^∨)^∨ = P and (Σ^∨)^∨ = Σ using the definitions and properties of dual cones and dual lattices
• The duality establishes a bijective correspondence between the faces of P and the cones of Σ, preserving incidence relations and dimensions
• The duality allows translating between the combinatorial properties of polytopes (face lattice, f-vector) and the geometric properties of fans (cone structure, ray generators)

Polytope duality for toric geometry problems

Computing invariants using polytope duality

• The cohomology ring of a smooth projective toric variety X_Σ can be computed using the Stanley-Reisner ring of the dual polytope P^∨, isomorphic to the quotient of a polynomial ring by the ideal generated by the non-faces of P^∨
• The Ehrhart polynomial of a lattice polytope P counts the lattice points in dilations of P and is related to the Hilbert function of the toric variety X_Σ associated with the normal fan of P
• The Minkowski sum of polytopes corresponds to the fiber product of toric varieties, allowing the study of toric varieties through polytope combinatorics

Classifying toric varieties using polytopes

• Polytope duality can classify toric varieties with certain properties, such as Fano varieties (associated with reflexive polytopes) and Gorenstein varieties (associated with lattice polytopes whose dual polytopes are also lattice polytopes)
• The combinatorial structure of a polytope (face lattice, f-vector) provides information about the singularities and resolution of the associated toric variety
• Examples of classifications include smooth Fano threefolds (18 reflexive polytopes), toric del Pezzo surfaces (16 reflexive polygons), and toric Calabi-Yau varieties (4319 reflexive 4-polytopes)