Toric GeometryEdit

Toric geometry is the study of spaces that are as symmetric as they are treatable: algebraic varieties endowed with an action of an algebraic torus that has a dense orbit. The central idea is to encode the geometry of these spaces in combinatorial data, most notably lattices, cones, and fans. This translation between geometry and polyhedral combinatorics makes a wide range of questions—ranging from singularities and morphisms to cohomology and mirror symmetry—both explicit and tractable.

At its core, toric geometry reveals how simple symmetry (the torus action) can organize and constrain complex geometric behavior. The resulting theory sits at the crossroads of algebraic geometry, polyhedral geometry, and symplectic geometry, and it provides a transparent laboratory where abstract ideas can be tested with concrete, computable objects. Throughout its development, toric geometry has served as a bridge between pure mathematics and theoretical physics, notably in string theory and mirror symmetry.

Foundations of toric geometry

Algebraic torus and torus actions

An algebraic torus is a group variety isomorphic to a product of copies of the multiplicative group, roughly speaking a higher-dimensional analogue of the familiar circle. A toric variety is a normal algebraic variety X equipped with an action of this torus T ≅ (C*)^n that contains a dense orbit. The geometry of X reflects the combinatorics of how T acts, and many properties of X are visible by studying this action.

For reference, see torus and toric variety.

Lattices, cones, and fans

The combinatorial backbone of toric geometry rests on two free abelian lattices: N ≅ Z^n and its dual M = Hom(N, Z). One studies strongly convex rational polyhedral cones in the real vector space N_R = N ⊗_Z R and builds objects called fans, which are collections of cones closed under taking faces and intersecting along faces. These fans encode how local affine pieces glue together to form a global toric variety.

The dual cone σ^∨ lives in M_R and determines the semigroup of lattice points that define a corresponding affine piece.
A fan Δ in N_R yields a toric variety X_Δ by gluing the affine varieties X_σ for σ ∈ Δ.

Key terms and ideas are developed in connection with lattices, polyhedrons, and fan (toric geometry).

Affine and projective toric varieties

For each cone σ, the associated affine toric variety X_σ is constructed as Spec C[S], where S = σ^∨ ∩ M is a finitely generated semigroup. Gluing these affine pieces according to the combinatorics of a fan Δ produces the global toric variety X_Δ.

Projective toric varieties arise from lattice polytopes, and their geometry is closely tied to the polytopes’ combinatorics. If P ⊂ M_R is a lattice polytope, the projective toric variety X_P is determined by P, and sections of ample line bundles correspond to lattice points in P. This deep link between polytopes and projective toric varieties is a central theme in the subject.

For related concepts, see toric variety and lattice polytope.

Smoothness, singularities, and resolutions

Not all toric varieties are smooth. The smoothness of a toric variety corresponds to a simple combinatorial condition on its fan: each cone σ must be generated by part of a basis of the lattice N. When this fails, toric varieties have controlled singularities, and one can often obtain a smooth variety by subdividing the cones in the fan (a process that corresponds to a resolution of singularities in the geometric category).

The Delzant criterion provides a particularly transparent criterion in the symplectic setting: a smooth, compact toric manifold corresponds to a fan whose cones are generated by lattice vectors forming part of a basis, and the moment map image is a Delzant polytope.

See Delzant polytope and reflexive polytope for related constructions.

Morphisms and functoriality

Toric morphisms correspond to maps of fans: a lattice morphism f: N' → N induces a map of fans in the opposite direction, and hence a morphism of the associated toric varieties. This contravariant relationship is a hallmark of toric geometry and helps translate geometric questions into combinatorial ones.

Cohomology, intersection theory, and the Chow ring

For smooth, complete toric varieties, the cohomology and intersection theory are governed by the combinatorics of the fan. The Chow ring (or cohomology ring) can be described in terms of divisor classes associated with the rays of the fan, subject to linear relations and Stanley–Reisner-type relations that encode combinatorial incidences of cones. This translates many geometric calculations into linear-algebra or combinatorial counts.

Mirror symmetry and reflexive polytopes

Toric geometry plays a prominent role in mirror symmetry. Batyrev's construction uses reflexive polytopes—a special class of lattice polytopes—to produce pairs of Calabi–Yau hypersurfaces inside dual toric varieties, yielding a powerful combinatorial route to constructing mirror pairs. This connection has spurred extensive cross-pollination with polyhedral geometry and string theory.

Computational aspects

The explicitly combinatorial nature of toric geometry makes it amenable to computation. Tasks such as constructing a toric variety from a fan, studying singularities, performing resolutions by fan refinements, and computing cohomology rings can be aided by polyhedral software and computer algebra systems. Tools and topics include polymake and Macaulay2, among others, which support polyhedral computations and toric variety constructions.

Examples and classical families

Projective space, projective space, as a toric variety, arises from a simple fan whose rays correspond to the coordinate hyperplanes. This foundational example connects algebraic geometry to combinatorics in a direct way.
Hirzebruch surfaces, such as the surface F_a, provide smooth complete toric surfaces with a controlled, parameterized family of fans.
The product P^1 × P^1 is another basic smooth toric surface, realized by a straightforward fan in two dimensions.
Weighted projective spaces appear as toric varieties with quotient-type singularities; their fans reflect the weighted structure and give a natural testing ground for questions about singularities and resolutions.
More generally, toric surfaces and higher-dimensional toric varieties form a broad class that includes many familiar spaces as special cases or deformations.

See also sections and related topics under Hirzebruch surface, weighted projective space, and toric variety.

Connections and influences

The combinatorial core of toric geometry links directly to lattice polytope theory and polyhedral geometry, enabling explicit calculations of degrees, volumes, and intersection numbers.
Its explicit nature makes toric geometry a valuable framework for testing conjectures in broader algebraic geometry, including questions about singularities and birational geometry.
In symplectic geometry, the moment-map picture provides a parallel, very concrete interpretation of toric varieties as symplectic manifolds with torus actions, with the moment polytope encoding symplectic data.
In mathematical physics, toric methods underpin constructions in mirror symmetry and string theory, where combinatorial data leads to physically meaningful geometric families.