Journal of Mathematics

Volume 2017 (2017), Article ID 2542796, 15 pages

https://doi.org/10.1155/2017/2542796

## Toric Geometry of the Regular Convex Polyhedra

Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

Correspondence should be addressed to Elisa Prato

Received 30 July 2016; Accepted 31 January 2017; Published 23 March 2017

Academic Editor: Frédéric Mynard

Copyright © 2017 Fiammetta Battaglia and Elisa Prato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry.

#### 1. Introduction

Among the five regular convex polyhedra, the regular tetrahedron and the cube are examples of simple rational convex polytopes. To these, the standard smooth toric geometry applies, in both the symplectic and complex category. From the symplectic viewpoint, in fact, the regular tetrahedron and the cube satisfy the hypotheses of the Delzant theorem [1], and it is easily seen that they correspond, respectively, to and . From the complex viewpoint, on the other hand, the toric variety associated with the regular tetrahedron is while corresponds to the cube (see, e.g., [2, Section 1.5]); these can also be obtained as quotients by a complex version of the Delzant procedure described by Audin in [3, Chapter VI].

The regular octahedron is still rational, but it is no longer simple. The toric variety associated with the regular octahedron is however well known and is described, for example, by Fulton in [2, Section 1.5]; it can also be obtained as a complex quotient by applying the Cox construction [4, Theorem 2.1].

The regular dodecahedron, on the other hand, is simple but it is the first of the five regular convex polyhedra that is not rational. It is shown by Prato in [5] that, by applying her extension of the Delzant procedure to the case of general simple convex polytopes [6], one can associate with the regular dodecahedron a* symplectic toric quasifold*. Quasifolds are a generalization of manifolds and orbifolds: they are not necessarily Hausdorff and they are locally modeled by the quotient of a manifold modulo the action of a discrete group.

In this article, we recall all of the above and we complete the picture, first of all, by associating with the regular dodecahedron a* complex toric quasifold*. We do so by applying a generalization, given by the authors in [7], of the procedure described by Audin in [3, Chapter VI] to the case of general simple convex polytopes. As in the smooth case, the symplectic and complex quotients can be identified [7, Theorem 3.2], endowing the corresponding toric quasifold with a Kähler structure.

We go on to address the case of the regular icosahedron. From the toric viewpoint, this is certainly the most complicated of the five regular convex polyhedra, since it is neither simple nor rational. However, we can apply generalizations by Battaglia and Prato [8] and Battaglia [9, 10] of toric quasifolds [6, 7] and of the Cox construction [4] to arbitrary convex polytopes; this allows us to associate with the regular icosahedron, in both the symplectic and complex category, a space that is stratified by quasifolds. As for all -dimensional polytopes, here there are only zero-dimensional singular strata and an open dense regular stratum. Moreover, by [10, Theorem 3.3], the symplectic and complex quotients can be identified, endowing the regular stratum with the structure of a Kähler quasifold.

Notice, finally, that we are still missing a symplectic toric space corresponding to the regular octahedron; this too can be found by applying Battaglia’s work on arbitrary convex polytopes. What we get here is a space that is stratified by symplectic manifolds (see [9, Remark 6.6]); moreover, by [10, Theorem 3.3], this symplectic quotient can be identified with the complex quotient, and the regular stratum is Kähler.

The article is structured as follows: in Section 2 we recall a few necessary facts on convex polytopes; in Section 3 we recall from [6, 7] how to construct symplectic and complex toric quasifolds from simple convex polytopes; in Section 4 we recall from [9, 10] the construction of the symplectic and complex toric spaces corresponding to arbitrary convex polytopes; finally, in Sections 5, 6, 7, and 8 we describe the symplectic and complex toric spaces corresponding to the five regular convex polyhedra.

#### 2. Facts on Convex Polytopes

Consider a dimension convex polytope .

*Definition 1 (simple polytope). * is said to be* simple* if each of its vertices is contained in exactly facets.

Assume now that has facets. Then there exist elements in and real numbers such thatLet us consider the open faces of . They can be described as follows. For each such face there exists a possibly empty subset such thatA partial order on the set of all open faces of is defined by setting (we say contained in ) if . Notice that if and only if . The polytope is the disjoint union of its open faces. Let ; we have the following definitions.

*Definition 2. *A -dimensional open face of the polytope is said to be* singular* if .

*Definition 3. *A -dimensional open face of the polytope is said to be* regular* if .

*Remark 4. *Let be a -dimensional singular face in , then . Therefore any polytope in is simple and the singular faces of a nonsimple polytope in are vertices.

We refer the reader to Ziegler’s book [11] for additional basic facts on convex polytopes. We now go on to recall what is meant by quasilattice and quasirational polytope.

*Definition 5 (quasilattice). *A* quasilattice* in is the -span of a set of -spanning vectors, , of .

Notice that is a lattice if and only if it is generated by a basis of .

*Definition 6 (quasirational polytope). *Let be a quasilattice in . A convex polytope is said to be* quasirational* with respect to the quasilattice if the vectors in (1) can be chosen in .

Remark that each polytope in is quasirational with respect to the quasilattice that is generated by the elements in (1). We note that if can be chosen inside a lattice, then the polytope is rational.

#### 3. The Simple Case

Let be an -dimensional simple convex polytope. We are now ready to recall from [6] and [7] the construction of the symplectic and complex toric quasifolds associated with . For the definition and main properties of symplectic and complex quasifolds we refer the reader to [6, 12] and [7]. For the purposes of this article, we will restrict our attention to the special case . We begin by remarking that both constructions rely on the notion of quasitorus, which we recall.

*Definition 7 (quasitorus). *Let be a quasilattice in . We call* quasitorus* of dimension the group and quasifold .

Notice that, if the quasilattice is a lattice, we obtain the classical notion of torus. The quasilattice also acts naturally on : Therefore, in the complex category we have the following.

*Definition 8 (complex quasitorus). *Let be a quasilattice in . We call* complex quasitorus* of dimension the group and complex quasifold .

In analogy with the smooth case, we will say that is the* complexification* of . Assume now that our polytope is quasirational with respect to a quasilattice and writefor some elements and some real numbers ; again, here is the number of facets of . Let denote the standard basis of and . Consider the surjective linear mapping and its complexification Consider the quasitori and . The mappings and each induce group epimorphismsWe define to be the kernel of the mapping and to be the kernel of the mapping . Notice that neither nor are honest tori unless is a honest lattice. The Lie algebras of and are, respectively, and . The mappings and induce isomorphisms Let us begin with the symplectic construction. Consider the space , endowed with the symplectic formand the action of the torus :This action is effective and Hamiltonian, with moment mapping given by Choose now , with as in (4). Denote by the Lie algebra inclusion and notice that is a moment mapping for the induced action of on . Consider now the orbit space . Then we have, from [6, Theorem 3.3], the following.

Theorem 9 (generalized Delzant construction). *Let be a quasilattice in and let be a 3-dimensional simple convex polytope that is quasirational with respect to . Assume that is the number of facets of and consider vectors in that satisfy (4). For each , the orbit space is a compact, connected 6-dimensional symplectic quasifold endowed with an effective Hamiltonian action of the quasitorus such that, if is the corresponding moment mapping, then .*

We say that the quasifold with the effective Hamiltonian action of is the* symplectic toric quasifold* associated with .

Let us now pass to the complex construction. Following the notation of the previous section, consider, for any open face of , the -orbit Consider the open subset of given byNotice thatwhere ranges over all the vertices of the polytope . Moreover, since the polytope is simple, we have that In fact, in this case, . The group acts on the space . Consider the space of orbits . We then have, from [7, Theorem 2.2], the following.

Theorem 10. *Let be a quasilattice in and let be a 3-dimensional simple convex polytope that is quasirational with respect to . Assume that is the number of facets of and consider vectors in that satisfy (4). For each , the corresponding quotient is a complex quasifold of dimension 3, endowed with a holomorphic action of the complex quasitorus having a dense open orbit.*

We say that the quasifold with the holomorphic action of is the* complex toric quasifold* associated with .

Finally, we conclude this section by recalling that the natural embeddinginduces a mapping that sends each -orbit to the corresponding -orbit. This mapping is equivariant with respect to the actions of the quasitori and . Then, under the same assumptions of Theorems 9 and 10, we have, from [7, Theorem 3.2], the following.

Theorem 11. *The mapping : is an equivariant diffeomorphism of quasifolds. Moreover, the induced symplectic form on the complex quasifold is Kähler.*

For the smooth case see Audin [3, Proposition 3.1.1] but also Guillemin [13, Appendix 1, Theorem 1.4].

#### 4. The Nonsimple Case

Consider now a nonsimple convex polytope and assume that is quasirational with respect to a quasilattice . The idea here is to repeat the constructions of the previous section. If we do so, we again find the groups and and the quasitori and , isomorphic to and , respectively. However, the symplectic construction produces spaces that are* stratified by symplectic quasifolds*, while the complex construction yields spaces that are* stratified by complex quasifolds*. For the exact definitions of these notions we refer the reader to [9, Section 2] and [10, Definition 1.5]. We remark that in the smooth case these definitions yield the classical definition of Goresky and MacPherson [14]. Many of the important features of these stratified structures will be clarified when addressing the relevant examples (see Sections 6 and 8).

Let us consider the symplectic case first. The main difference with respect to the case of simple polytopes is that here there are points in the level set that have isotropy groups of positive dimension; therefore is no longer a smooth manifold. From Proposition 3.3 and Theorems 5.3, 5.10, 5.11, 6.4 in [9] we have the following.

Theorem 12 (generalized Delzant construction: nonsimple case). *Let be a quasilattice in and let be a 3-dimensional convex polytope that is quasirational with respect to . Assume that is the number of facets of and consider vectors in that satisfy (4). For each , the quotient is a compact, connected 6-dimensional space stratified by symplectic quasifolds, endowed with an effective continuous action of the quasitorus . Moreover, there exists a continuous mapping : such that . Finally, the restriction of the -action to each stratum is smooth and Hamiltonian, with moment mapping given by the restriction of .*

When the polytope is rational, these quotients are examples of the symplectic stratified spaces described by Sjamaar and Lerman in [15]; in particular, the strata are either manifolds or orbifolds [9, Remark 6.6]. We remark that the nonsimple rational case was addressed also by Burns et al. in [16, 17]; they gave an in-depth treatment and a classification theorem in the case of isolated singularities.

Let us now examine the complex case. Here, one still considers the open subset as defined in (13) but while in the simple case the orbits of on were closed, here there are nonclosed orbits. We recall first that , where ; this actually happens also in the simple case. Then, from [10, Theorem 2.1], we have the following.

Theorem 13 (closed orbits). *Let . Then , the -orbit through , is closed if and only if there exists a face such that is in . Moreover, if is nonclosed, then its closure contains one, and only one, closed -orbit.*

Therefore, in order to define a notion of quotient, one defines the following equivalence relation: two points and in are equivalent with respect to the action of the group if and only if where the closure is meant in . The space is then defined to be the quotient with respect to this equivalence relation. Notice that if the polytope is simple, ; thus, by Theorem 13, -orbits through points in are always closed and the quotient is just the orbit space endowed with the quotient topology. From [10, Proposition 3.1,Theorem 3.2] we have the following.

Theorem 14. *Let be a quasilattice in and let be a 3-dimensional convex polytope that is quasirational with respect to . Assume that is the number of facets of and consider vectors in that satisfy (4). For each , the corresponding quotient is endowed with a stratification by complex quasifolds of dimension 3. The complex quasitorus acts continuously on , with a dense open orbit. Moreover, the restriction of the -action to each stratum is holomorphic.*

We remark that when is a lattice and the vectors are primitive in , the quotient coincides with the Cox presentation [4] of the classical toric variety that corresponds to the fan normal to the polytope . As for classical toric varieties, there is a one-to-one correspondence between -dimensional orbits of the quasitorus and -dimensional faces of the polytope. In particular, the dense open orbit corresponds to the interior of the polytope and the singular strata correspond to singular faces.

We remark that, like in the simple case, the natural embedding induces an identification between symplectic and complex quotients. From [10, Theorem 3.3] we have the following.

Theorem 15. *The mapping : is a homeomorphism which is equivariant with respect to the actions of and , respectively. Moreover, the restriction of to each stratum is a diffeomorphism of quasifolds. Finally, the induced symplectic form on each stratum is compatible with its complex structure, so that each stratum is Kähler.*

We conclude by pointing out that has two different kinds of singularities, namely, the stratification and the quasifold structure of the strata. The nonsimplicity of the polytope yields the decomposition in strata of the corresponding topological space, while its nonrationality produces the quasifold structure of the strata and also intervenes in the way the strata are glued to each other. This last feature can be observed only in spaces with strata of positive dimension; this led to a definition of stratification that naturally extends the usual one [9, Section 2].

#### 5. Simple and Rational: The Regular Tetrahedron and the Cube

The regular tetrahedron and the cube are both simple and rational. Let us recall the construction of the corresponding symplectic and complex toric manifolds. We follow the notation of Section 3, which also applies to the smooth case.

Let us begin with the regular tetrahedron (see Figure 1) having vertices Consider the sublattice of that is generated by the corresponding four vectors Notice that ; therefore any three of these four vectors form a basis of . Moreover, where . Thus the regular tetrahedron satisfies the hypotheses of Delzant’s theorem [1] with respect to . From the symplectic viewpoint, it is readily verified here that and therefore that is given by where denotes the -sphere of radius . From the complex viewpoint, it is easy to see that , that , and therefore that Consider now the cube having vertices (see Figure 2). Notice that where , , , , , and . We can again apply the Delzant procedure, this time relatively to the lattice , and we get that the 3-dimensional group is given by and, therefore, the symplectic toric manifold is given by where the ’s have all radius . The corresponding complex toric manifold , on the other hand, is given by This provides an elementary example of a general fact: the symplectic toric manifold depends on the polytope, while the complex toric manifold only depends on the fan that is normal to the polytope. For instance, let us consider the cube having vertices , with a positive real number, with the same vectors as above. Then the corresponding symplectic toric manifold is the product of three spheres of radius . In conclusion, the symplectic structure varies, while the complex toric manifold remains the same.