In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.

Toric varieties from tori

The original motivation to study toric varieties was to study torus embeddings. Given the algebraic torus T, the group of characters Hom(T,C^x) forms a lattice. Given a collection of points A, a subset of this lattice, each point determines a map to C and thus the collection determines a map to C^|A|. By taking the Zariski closure of the image of such a map, one obtains an affine variety. If the collection of lattice points A generates the character lattice, this variety is a torus embedding. In similar fashion one may produce a parametrized projective toric variety, by taking the projective closure of the above map, viewing it as a map into an affine patch of projective space.

Given a projective toric variety, observe that we may probe its geometry by one-parameter subgroups. Each one parameter subgroup, determined by a point in the lattice, dual to the character lattice, is a punctured curve inside the projective toric variety. Since the variety is compact, this punctured curve has a unique limit point. Thus, by partitioning the one-parameter subgroup lattice by the limit points of punctured curves, we obtain a lattice fan, a collection of polyhedral rational cones. The cones of highest dimension correspond precisely to the torus fixed points, the limits of these punctured curves.

The toric variety of a fan

Suppose that N is a finite-rank free abelian group. A strongly convex rational polyhedral cone in N is a convex cone (of the real vector space of N) with apex at the origin, generated by a finite number of vectors of N, that contains no line through the origin. These will be called "cones" for short.

For each cone σ its affine toric variety U_σ is the spectrum of the semigroup algebra of the dual cone.

A fan is a collection of cones closed under taking intersections and faces.

The toric variety of a fan is given by taking the affine toric varieties of its cones and gluing them together by identifying U_σ with an open subvariety of U_τ whenever σ is a face of τ. Conversely, every fan of strongly convex rational cones has an associated toric variety.

The fan associated with a toric variety condenses some important data about the variety. For example, a variety is smooth if every cone in its fan can be generated by a subset of a basis for the free abelian group N.

Morphisms of toric varieties

Suppose that Δ₁ and Δ₂ are fans in lattices N₁ and N₂. If f is a linear map from N₁ to N₂ such that the image of every cone of Δ₁ is contained in a cone of Δ₂, then f induces a morphism f_* between the corresponding toric varieties. This map f_* is proper if and only if the preimage of |Δ₂| under the map f is |Δ₁|, where |Δ| is the underlying space of a fan Δ given by the union of its cones.

Resolution of singularities

A toric variety is nonsingular if its cones of maximal dimension are generated by a basis of the lattice. This implies that every toric variety has a resolution of singularities given by another toric variety, which can be constructed by subdividing the maximal cones into cones of nonsingular toric varieties.

The toric variety of a convex polytope

The fan of a rational convex polytope in N consists of the cones over its proper faces. The toric variety of the polytope is the toric variety of its fan. A variation of this construction is to take a rational polytope in the dual of N and take the toric variety of its polar set in N.

The toric variety has a map to the polytope in the dual of N whose fibers are topological tori. For example, the complex projective plane CP² may be represented by three complex coordinates satisfying

${\displaystyle |z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}=1,\,\!}$

where the sum has been chosen to account for the real rescaling part of the projective map, and the coordinates must be moreover identified by the following U(1) action:

${\displaystyle (z_{1},z_{2},z_{3})\approx e^{i\phi }(z_{1},z_{2},z_{3}).\,\!}$

The approach of toric geometry is to write

${\displaystyle (x,y,z)=(|z_{1}|^{2},|z_{2}|^{2},|z_{3}|^{2}).\,\!}$

The coordinates ${\displaystyle x,y,z}$ are non-negative, and they parameterize a triangle because

${\displaystyle x+y+z=1;\,\!}$

that is,

${\displaystyle \quad z=1-x-y.\,\!}$

The triangle is the toric base of the complex projective plane. The generic fiber is a two-torus parameterized by the phases of ${\displaystyle z_{1},z_{2))$; the phase of ${\displaystyle z_{3))$ can be chosen real and positive by the ${\displaystyle U(1)}$ symmetry.

However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at ${\displaystyle x=0}$ or ${\displaystyle y=0}$ or ${\displaystyle z=0}$ because the phase of ${\displaystyle z_{1},z_{2},z_{3))$ becomes inconsequential, respectively.

The precise orientation of the circles within the torus is usually depicted by the slope of the line intervals (the sides of the triangle, in this case).

Relation to mirror symmetry

The idea of toric varieties is useful for mirror symmetry because an interpretation of certain data of a fan as data of a polytope leads to a combinatorial construction of mirror manifolds.