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.
The original motivation to study toric varieties was to study torus embeddings. Given the algebraic torus
T
\hom(T,C*)
l{A}
C*
\left(C*\right)|l{A|}
l{A}
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.
Suppose that
N
Zn
N
N
N
v1,...,vk
cone(v1,\ldots,vk)=\left\{\sum
| ka | |
| i |
vi\colonai\inR\geq\right\}
\sigma
U\sigma
\sigma
A fan is a collection of cones closed under taking intersections and faces. The underlying space of a fan
\Sigma
|\Sigma|
The toric variety of a fan of strongly convex rational cones is given by taking the affine toric varieties of its cones and gluing them together by identifying
U\sigma
U\tau
\sigma
\tau
The fan associated with a toric variety condenses some important data about the variety. For example, the Cartier divisors are associated to the rays of the fan. Moreover, a toric variety is smooth, or nonsingular, if every cone in its fan can be generated by a subset of a basis for the free abelian group
N
Suppose that
\Sigma1
\Sigma2
N1
N2
f
N1
N2
\Sigma1
\Sigma2
f
f*
f*
|\Sigma2|
f
|\Sigma1|
A toric variety is projective if it can be embedded in some complex projective space.
Let
P
v
P
P
v
v
P
P
It is well known that projective toric varieties are the ones coming from the normal fans of rational polytopes.
CP2
2
| 2 | |
| |z | |
| 3| |
=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:
(z1,z2,z3) ≈ ei\phi(z1,z2,z3).
The approach of toric geometry is to write
(x,y,z)=
| 2) | |
| (|z | |
| 3| |
.
The coordinates
x,y,z
x+y+z=1;
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
z1,z2
z3
U(1)
However, the two-torus degenerates into three different circles on the boundary of the triangle i.e. at
x=0
y=0
z=0
z1,z2,z3
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).
Note that this construction is related to symplectic geometry as the map
\begin{cases}CP2&\toR\geq\\(z1,z2,z3)&\mapsto|z1|+|z2|+|z3|\end{cases}
U(1)
CP2
From the fundamental theorem for toric geometry, the classification of smooth compact toric varieties of complex dimension
n
m
n
m
The Picard number of a fan
\Sigma
n
m
m-n
\Sigma
n
1
CPn
e1,e2,\ldots,en
| n | |
| f=-\sum | |
| i=1 |
ei
e1,e2,\ldots,en
N
cone(e1,\ldots,en)
cone(e1,\ldots,ei-1,f,ei+1,\ldots,en)
i=1,\ldots,n
n
2
3
The classification for Picard number greater than
3
Smooth toric surfaces are easily characterized, they all are projective and come from the normal fan of polygons such that at each vertex, the two incident edges are spanned by two vectors that form a basis of
Z2
Every toric variety has a resolution of singularities given by another toric variety, which can be constructed by subdividing the maximal cones of its associated fan into cones of smooth toric varieties.
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.