Toric Varieties vs. Horofunction Compactifications of Polyhedral Norms
We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension and horofunction compactifications of with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geometry, normed spaces in convex analysis, and horofunction compactifications in metric geometry are directly and explicitly related.
- 1 Introduction
- 2 Toric varieties
- 3 Polyhedral metrics
- 4 Horofunction compactification of metric spaces
- 5 Horofunction compactification of polyhedral normed spaces
In this paper we give a correspondence between the three seemingly different concepts toric varieties, horofunction compactifications and polyhedral norms. Toric varieties provide a basic class of algebraic varieties which are relatively simple. The nonnegative part and the moment map of toric varieties are essential ingredients of the rich structure of toric varieties. The horofunction compactification of metric spaces is a general method to construct compactifications of metric spaces introduced by Gromov [Gr1, §1.2] in 1981 (see §4 below). Finally, polyhedral norms on give a special class of normed linear spaces (or Minkowski spaces [Th]) and metric spaces (see §3 below).
In this paper we establish an explicit geometric connection between projective toric varieties of dimension and horofunction compactifications of with respect to rational polyhedral norms.
In every dimension , there exists a bijective correspondence between projective toric varieties of dimension and rational polyhedral norms on up to scaling such that:
The nonnegative part of a projective toric variety is homeomorphic to the horofunction compactification of with respect to the distance induced by the corresponding polyhedral norm .
Equivalently, the image of the moment map of the toric variety is homeomorphic to the horofunction compactification of with respect to the distance induced by the corresponding polyhedral norm .
This correspondence is canonical and given as follows: The unit ball of a rational polyhedral norm is a rational convex polytope in which contains the origin as an interior point, which in turn gives a fan in by taking cones over the faces of , and hence gives a toric variety . Note that the fan does not change when the polytope is scaled, and hence the correspondence is up to scaling on the polyhedral norms .
This result adds another perspective on the close relations between integral convex polytopes and toric projective varieties, for a detailed description see [Od1, Chap 2].
Theorem 1.1 implies the following
Let be a rational polyhedral norm on , and its unit ball. Let be the polar set
This gives a bounded realization of the horofunction compactification of . The same result holds for the horofunction compactification of any polyhederal norm on whether it is rational or not and is proven in [JS, Theorem 1.2].
It is well-known that algebro-geometric and cohomology properties of toric varieties are determined by combinatorial and convex properties of their fans (see [Fu] [CLS] [Od1]). Consequently, the existence of a correspondence between toric varieties and polyhedral norms is not surprising. But it is probably not obvious that there exists such a direct connection between horofunction compactifications of in metric geometry and important parts of toric varieties : the nonnegative part and the image of the moment map of .
The correspondence in Theorem 1.1 can give numerical invariants of toric varieties. In the statement of Theorem 1.1, we have fixed the standard integral structure of when we discuss toric varieties and the rationality of polyhedral norms. Consequently, by requiring the standard basis of to be unit vectors, we can also fix the standard Euclidean metric on . Though scaling of the polyhedral norm does change its unit ball, i.e., the polytope, it does not change the fan induced from . On the other hand, we can use the following canonical normalization of polyhedral norms: Every vertex of the unit ball of is integral, and one of them is primitive.
With this normalization, by Theorem 1.1, each projective toric variety gives a unique polyhedral norm and its unit ball , which is a polytope in . Besides computing the volume of with respect to the standard Euclidean metric on , we can also compute the volume of with respect to a suitable notion of volume induced from the norm . According to [AT], there are four commonly used definitions of volumes on normed vector spaces: Busemann volume, Holmes-Thompson volume, Gromov volume, and Benson volume. Consequently, we obtain the following corollary:
Choose one of the four volumes mentioned above, for each projective toric variety , there is a canonical number given by the volume of the unit ball of the normalized polyhedral norm corresponding to the toric variety .
One natural question is the meaning of such volumes for toric varieties. Note that if we use the standard Euclidean metric on , then the volume of the convex polytope is related to the implicit degree of the projective toric variety . See [So, §5].
The correspondence in Theorem 1.1 also raises the question of how to understand toric varieties by using metric geometry.
Horofunction compactification and noncommutative geometry
Before we explain some detailed definitions of toric varieties and horofunction compactifications in later sections, we point out a connection between horofunction compactifications of normed vector spaces and reduced -algebras of discrete groups and consequently the noncommutative geometry.
After the horofunction compactification of a proper metric space was introduced by Gromov [Gr1, §1.2] in 1981, the horofunction compactification of a complete simply connected nonpositively curved manifold was identified with the geodesic compactification in [BGS, §3]. This gives a direct connection between the geometry of geodesics and analysis, or rather a class of special functions, on the manifold. Among nonpositively curved simply connected Riemannian manifolds, horofunctions are difficult to compute except for symmetric spaces of noncompact type (see [Ha] [GJT]). For noncompact locally symmetric spaces of nonpositive curvature, the horofunction compactification was identified in [JM] and [DFS]. It will be seen below that with polyhedral norms provide another class of spaces for which all horofunctions can be computed [Wa1].
It turned out that the horofunction compactifications of with respect to norms are unexpectedly related to the noncommutative geometry developed by Alain Connes (see [Co], [Ri1] and [Ri2]). This brought another perspective to horofunction compactifications and motivated the work in this paper.
Let be a countable discrete group such as and . Let be the convolution -algebra of complex valued functions of finite support, i.e., of compact support, on . Let be the usual -representation of on . Then the norm-completion of in the space of operators of is the reduced -algebra of , denoted by .
Let be a length function of , i.e., a function such that (1) , (2) for all , , (3) for all , .
For example, the word length on with respect to a set of generators gives rise to such a length function.
Let be the multiplication operator on defined by the length function , which is usually unbounded. Then serves as a Dirac operator in the noncommutative geometry of . The following fact is true: For every , the commutator is a bounded operator on . This allows one to define a semi-norm on :
We recall that a state on a -algebra is a positive linear functional of norm 1. The set of all states of a -algebra is denoted by , and is a convex subset of the space of linear functionals of . Extreme points of are called pure states of . When , the space of continuous functions on a compact topological space , then states on correspond to probability measures on , and pure states correspond to evaluations on .
In [Ri1, p. 606], Rieffel called a semi-norm on a Lip-norm if the topology on induced from coincides with the weak -topology, and he called a unital -algebra equipped with a Lip-norm a compact quantum metric space.
In [Ri1], Rieffel asked the question: Given a discrete group , is the seminorm on coming from a length function on a Lip-norm?
He could only handle the case and prove the following result:
Proposition 1.4 ([Ri1], Thm 0.1).
Let be a length function on which is either the word length for some finite generating set or the restriction to of some norm on . Then the induced seminorm is a Lip-norm on , and hence is a compact quantum metric space.
In proving this result, Rieffel made crucial use of horofunction compactifications of with respect to norms. In this paper, he also raised the following question ([Ri1, Question 6.5]): Is it true that, for every finite-dimensional vector space and every norm on it, every horofunction (i.e., a boundary point of the horofunction compactification of ) is a Busemann function, i.e., the limit of an almost-geodesic ray?
2. Toric varieties
In this section, we give a summary of several results on toric varieties which are needed to understand and prove Theorem 1.1. The basic references for this section are [Fu], [CLS], [Od1], [Od2], [AM], [Cox], and [So].
A toric variety over is an irreducible variety over such that
the complex torus is a Zariski dense subvariety of and
the action of on itself by multiplication extends to an action of to .
We fix the standard lattice in , which gives an integral structure, and also a -structure, .
Recall that a rational polyhedral cone is a cone generated by finitely many elements of , or equivalently of :
Usually, is assumed to be strongly convex: , i.e., does not contain any line through the origin. A face of a cone is the intersection of with the 0-level set of a linear functional which is nonnegative on . The relative interior and relative boundary of a cone are the interior respectively boundary of in the linear subspace spannend by .
For each strongly convex rational polyhedral cone , define its dual cone by
Then is also a convex rational polyhedral cone, though it is not strongly convex anymore unless .
A fan in is a collection of strongly convex rational polyhedral cones such that
if , then every face of also belongs to ;
if , then their intersection is a common face of both of them, and hence belongs to .
In this paper, we only deal with fans which consist of finitely many polyhedral cones.
It is known that there is a strong correspondence between fans and toric varieties, namely:
For every fan of , there is an associated toric variety , which is a normal algebraic variety.
If a toric variety is a normal variety, then is of the form for some fan in .
Because of this correspondence, toric varieties are often required to be normal, for example in [Fu]. In this paper, we follow this convention and require toric varieties to be normal.
The construction of a toric variety from a fan and a description of its topology in terms of is crucial to the proof of Theorem 1.1. Therefore we give a short description here:
Given a fan in , its associated toric variety is constructed as follows:
Each cone gives rise to an affine toric variety . Specifically, is a finitely generated semigroup. Let be a set of generators of this semigroup, i.e., every element of is of the form , with being non-negative integers. Then the Zariski closure of the image of in under the embedding
is the affine toric variety . Note that we use Laurent monomials for the notation: for all where denotes the -th component of .
For any two cones in , if is a face of , then is a Zariski dense subvariety of .
The toric variety is obtained by gluing these affine toric varieties together:
where the relation is given by the inclusion relation in (2): Note that for any two cones , the intersection , if nonempty, is a common face of both and , and hence can be identified with a subvariety of both and .
Many properties of can be expressed in terms of the combinatorial properties of the fan . We state one about the orbits of in , details can be found for example in [CLS, p. 119], [Cox, §9] or [Fu, §3.1].
For every toric variety , there is a bijective correspondence between orbits of the torus in and cones in the fan . Denote the orbit in corresponding to by . Then is a complex torus isomorphic to . In particular, the open and dense orbit corresponds to the trivial cone .
2.1. A topological model of toric varieties
In order to better understand the toric variety as a compactification of , we want to give a topological description of which exhibits its dependence on clearly and also describes explicitly sequences in which converge to points in the complement .
To do so, note that in terms of the standard integral structure , we can realize by:
and when Re, it holds . Then the exponential map gives an identification
Conversely, using the logarithmic function , we get an identification
In the following, we denote the complex torus by .
Given any fan in , we will define a bordification of and show in Proposition 2.10 below that is homeomorphic to the toric variety as -topological spaces.
For each cone , define a boundary component
Note that this is a complex torus of dimension equal to . When , then . Later we will identify with .
Define a topological bordification by
with the following topology: A sequence , where and , converges to a point for some if and only if the following conditions hold:
The real part can be written as such that when it holds
the first part is contained in the relative interior of the cone and its distance to the relative boundary of goes to infinity,
the second part is bounded.
the image of in under the projection
converges to the point .
Note that the imaginary part of lies in the compact torus , and the second condition controls both the imaginary part and the bounded component of the real part .
The above definition of and the identification of with in Proposition 2.10 follows the construction and discussion in [AM, pp. 1-6]. We note that there is one difference with the convention there: On page 2 in [AM], the complex torus is identified with , and the real part is the compact torus , and the imaginary part is , which can be identified with .
The toric variety is compact if and only if the support of is equal to , i.e., gives a rational polyhedral decomposition of . Similarly, it is clear from the definition that the bordification is a compactification of if and only if the support of is equal to .
To obtain a continuous action of on , we note that or acts on and every boundary component by translation. These translations are compatible in the following sense.
For any sequence , if is convergent in , then for any vector , or rather its image in , the shifted sequence is also convergent. Furthermore,
This implies the following result.
The action of on itself by multiplication extends to a continuous action on , and the decomposition in Equation 2.3 into gives the orbit decomposition of with respect to the action of .
We note that the multiplication of the torus on itself and the boundary components corresponds to translation in and . Then the proposition follows from Lemma 2.8. ∎
One key result we need for the proof of Theorem 1.1 is the following description of the toric variety as a topological -space. Since this proposition and its proof are not explicitly written down in literature, we give an outline of the proof on page 2.1 for the convenience of the reader.
The identity map on extends to a homeomorphism , which is equivariant with respect to the action of , and the -orbits in the toric variety are mapped homeomorphically to the boundary components .
The identification between and allows one to see that when a sequence of points in the real part of the complex torus goes to infinity along the directions contained in a cone of the fan , the sequence will converge to a point of a complex torus of smaller dimension. Hence the compact torus , which is the fiber over in the toric variety, will collapse to a torus of smaller real dimension .
This result is well-known and can be found for example in [AM, pp. 1-6] [Od2, §10] [Cox, p. 211] or [Fu, p. 54]. Such a picture of toric varieties including also the compact part of the torus is often described in connection with the moment map of toric varieties (for a reference see [Fu, p. 79] or [Mi]). We will come back to this map later. But first we need a description of it as a bordification of instead of a bounded realization via the image of the moment map. A bordification of the noncompact part of the torus is described in this way in [AM, p. 6] (see also [Od2, §10]) together with a map from the toric variety to this bordification.
First, we recall some properties of orbits of in a toric variety . The 1-1 correspondence between -orbits in and cones in mentioned in Proposition 2.3 above can be described more explicitly (see [Fu, p. 28], [CLS, p. 118] and [Cox, p. 212]):
For every cone , there is a distinguished point in the affine toric variety . It is contained in the orbit of in corresponding to (see Proposition 2.3), and hence the orbit is equal to the orbit .
The distinguished point can be described as follows. The smallest cone of the fan corresponds to the affine toric variety , and the distinguished point is in this case. The -orbit through this point gives .
In general, we note that every one-parameter subgroup is of the form
We need to identify this distinguished point with a corresponding distinguished point in the bordification .
Under the identification of with in Proposition 2.10, this distinguished point in corresponds to the image of the origin of in under the projection . When the orbit is identified with , where , then corresponds to .
As mentioned before, for the trivial cone of the fan , the distinguished point is . Under the identification in Equation 2.2 on page 2.2, the distinguished point corresponds to the image of the origin of under the projection .
Therefore, corresponds to in . For any nontrivial cone , the distinguished point in the orbit is equal to the limit in , where is an integral vector contained in the relative interior of the cone .
We need to determine the limit in the bordification . When we identify with as above in Equation 2.2, the complex curve () in is the image of a complex line in with slope given by , and hence its real part is a straight line in through the origin with slope , i.e., , , and is contained in .
By the definition of the topology of above, converges to the distinguished point in , i.e., to the image of the origin of in . This proves Lemma 2.13. ∎
For any cone , a sequence in converges to the distinguished point in the toric variety if and only if it converges to the distinguished point in the topological model
We note that for the open subset , under the embedding of in Equation 2.1 on page 2.1, the coordinates of the distinguished point are either 0 or 1 depending on whether the element in is zero or positive on . This implies that a sequence converges to the distinguished point if and only if the following conditions are satisfied:
for any with it holds as ,
for any with it holds as .
Note that the vectors in with span the dual cone , i.e., linear combinations of these vectors with nonpositive coefficients give . In terms of the identification , write with as in the definition of the topology of , then the above condition on is equivalent to the following conditions:
The real part can be written as such that when ,
the first part is contained in the interior of the cone and its distance to the relative boundary of goes to infinity,
the second part is bounded.
The image of in under the projection
converges to the image in of the zero vector in .
By the definition of , this is exactly the conditions for the sequence to converge to the distinguished point in . This proves Lemma 2.14. ∎
Proof of Proposition 2.10.
The idea of the proof is to use the continuous actions of on and to extend the equivalence of convergence of interior sequences to the distinguished point in Lemma 2.14 to other boundary points.
Under the action of , the orbit in gives . As pointed out in Proposition 2.3 on page 2.3 and Proposition 2.12, the orbit in gives the orbit corresponding to . It can be seen that the stabilizer of the distinguished point in is equal to the subgroup (see [CLS, Lemma 3.2.5]). By the definition of , the stabilizer of the point is also equal to . Therefore, there is a canonical identification between and .
By Lemma 2.14, for any sequence in , in if and only if in . Take any such sequence with . Let be any converging sequence with . For both the toric variety and the bordification , the continuous actions of on and in Proposition 2.9 imply that the sequence converges to in , and to in respectively. This implies that a sequence of interior points in converges to a boundary point in the orbit if and only if it converges to a corresponding point in . Since is an arbitrary cone in and is an arbitrary convergent sequence in , this proves the topological description of toric varieties in Proposition 2.10.
2.2. Fans coming from convex polytopes
One way to construct fans in is to start with a rational convex polytope which contains the origin as an interior point. As a more detailed reference for this construction see [Fu, Section 1.5].
Recall that a convex polytope in is the convex hull of finitely many points of . If the vertices of are contained in , then is called a rational convex polytope
Assume that is a rational convex polytope in and contains the origin as an interior point. Then each face of spans a rational polyhedral cone
i.e., the face is a section of the cone , and these cones form a fan in , denoted by . See for example Figure 3 below. Denote the toric variety defined by the fan by . Since the support of the fan is equal to , is compact. Note that for any integer , the scaled polytope is also a rational polytope and gives the same fan, .
It is known that not every fan in comes from such a rational convex polytope , as the following example shows.
[Fu, p.25] Take the fan generated by the eight halflines through the origin and one of the following eight points:
Then it is not possible to find eight points, one on each of the halflines, such that for each of the six cones the four corresponding generating points lie on one affine hyperplane.
The toric varieties defined by fans which do come from rational convex polytopes as above have a simple characterization:
A toric variety is a projective variety if and only if the fan is equal to the fan induced from a rational convex polytope containing the origin as an interior point as above.
The polar set of a convex polytope is defined by
When is a rational convex polytope containing the origin as an interior point, then is also a rational convex polytope containing the origin as an interior point.
When is symmetric with respect to the origin, then is equivalent to another common definition of polar set:
There is a duality between and which is given by an one-to-one correspondence between the set of faces of and the set of faces of which reverses the inclusion relation.
The correspondence is as follows: Let be a face of . Then there is exactly one face of , called the dual face of , which satisfies the following two conditions:
for any and it holds: ,
Proof of Proposition 2.16.
The fan associated with the dual polytope above is called the normal fan of the polytope and is defined for example in [Cox, pp. 217-218] or [Fu, Proposition, p. 26]. The toric varieties there are defined by the normal fans of and not by the fans as in this paper. But since the polar set of is equal to , , the above statement in Proposition 2.16 is equivalent to that in [Cox, Theorem 12.2] which is stated in terms of normal fans of rational polytopes. ∎
2.3. Real and nonnegative part of toric varieties and the moment map
In , the real part is , and in the complex torus , the real part is , which has -connected components. The positive part of is , and the positive part of is .
[So, Definition 6.2] For any toric variety , the closure of the positive part is called the nonnegative part of , denoted by .
Under the identification in Proposition 2.10, can be described as follows:
For any fan of , the nonnegative part is homeomorphic to the space
with the following topology: An unbounded sequence converges to a boundary point in for a cone if and only if one can write such that the following conditions are satisfied:
when , is contained in the cone and its distance to the relative boundary of goes to infinity,
the image of in under the projection converges to .
If we denote by , then for any two cones , it holds that is contained in the closure of if and only if is a face of . Therefore, the nonnegative part can be rewritten as
Consequently, we have
The translation action of on extends to a continuous action on , and the decomposition of in Equation 2.5 is the decomposition into -orbits. This decomposition of the nonnegative part of the toric variety is a cell complex dual to the fan . If for a rational convex polytope containing the origin as an interior point, then this cell complex structure is isomorphic to the cell structure of the polar set .
Using the moment map for projective toric varieties, we can realize this cell complex of and hence the compactification by a bounded convex polytope:
Let be a rational convex polytope containing the origin as an interior point, and the associated projective variety. By definition, each cone of corresponds to a unique face of , which gives by Proposition 2.19 a dual face of the polar set .
The moment map induces a homeomorphism
such that for every cone , the positive part of the orbit as a complex torus, or equivalently the orbit in Proposition 2.21, is mapped homeomorphically to the relative interior of the face corresponding to the cone .
3. Polyhedral metrics
In this section, we recall the definition of polyhedral norms on . They are rather special in view of the Minkowski geometry of normed real vector spaces and the Hilbert geometry of bounded convex subsets of real vector spaces.
Let be an asymmetric norm on , i.e., a function satisfying:
For any , if , then .
For any and ,
For any two vectors , .
In particular, and may not be equal to each other. If the second condition is replaced by the stronger condition: for all , then is symmetric and is a usual norm on .
Normed vector spaces have been extensively studied. They are also called Minkowski geometry in [Th]. Asymmetric norms on vector spaces have also been studied systematically, see [Cob]. In terms of their connection with convex domains below, they are natural.
Given an asymmetric norm on , the unit ball of ,
is a closed convex subset of which contains the origin as an interior point. Conversely, given any convex closed subset of which contains the origin as an interior point, we can define the Minkowski functional on by
It can be checked easily that defines an asymmetric norm on . If is symmetric with respect to the origin, i.e., , then is a norm on .
It is also easy to see that the unit ball of is equal to . Since any asymmetric norm on is uniquely determined by its unit ball, it is of the form for some closed convex domain in containing the origin in its interior.
When is a polytope, the asymmetric norm is called a polyhedral norm. If is a rational polytope with respect to the integral structure , the norm is also called a rational polyhedral norm.
Remark 3.2 (Connections to the Minkowski and Hilbert geometry).
This interplay between convex subsets of and norms on plays a foundational role in the convex analysis of Minkowski geometry, see for example [Gru] and [Th]. If the lattice is taken into account, connections with number theory and counting of lattices points are established and the structure becomes richer. The geometry of numbers relies crucially on these connections, see also [GrL] and [Ba].
There is another metric space associated with a convex domain of . It is the domain itself equipped with the Hilbert metric defined on it. When is the unit ball of , this is the Klein’s model of the hyperbolic plane. In general, the Hilbert metric is a complete metric on defined through the cross-ratio. See [deL] for details. Since is diffeomorphic to , the Hilbert metric induces a metric on .
When is the interior of a convex polytope , the Hilbert metric on is quasi-isometric to a polyhedral norm [Be] [Ve]. The polyhedral Hilbert metric associated with a polytope is isometric to a normed vector space if and only if the polytope is the simplex [FK, Theorem 2]. Furthermore, polyhedral Hilbert metrics have also special isometry groups [LW]. See also [LN] for other special properties of these Hilbert metrics.
4. Horofunction compactification of metric spaces
In this section we recall briefly the horofunction compactification of metric spaces, which was first introduced in [Gr1, §1.2].
Let be a proper metric space, for example, a locally compact metric space. The metric can be asymmetric, i.e., it satisfies all conditions of a usual metric except for the symmetry: possible. Such metrics arise naturally in view of polyhedral norms on as we saw in the previous subsection.
Let be the space of continuous functions on with the compact-open topology. Let be the quotient space . Denote the image of a function in by .
Define a map
If we fix a basepoint , then we can consider functions normalized to take value 0 at , and get a map