LVMB manifolds and simplicial spheres
LVM and LVMB manifolds are a large family of examples of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied (for example) by Buchstaber and Panov. The aim of this paper is to generalize the polytopes associated to LVM manifolds to the LVMB case and study its properties. In particular, we show that it belongs to a very large class of simplicial spheres. Moreover, we show that for every sphere belonging to this class, we can construct a LMVB manifold whose associated sphere is the given sphere. We use this latter result to show that many moment-angle complexes can be endowed with a complex structure (up to product with circles).
It is not easy to construct non kähler compact complex manifolds. The simplest example is the well know Hopf’s manifold ([H], 1948), which gives a complex structure on the product of spheres as a quotient of by the action of a discrete group. The Hopf’s manifold has many generalizations: firstly, by Calabi and Eckmann [CE] who give a structure of complex manifold on any product of spheres (of odd dimension). Then by Santiago Lopez de Medrano, Alberto Verjovsky ([LdM] and [LdMV]) and Laurent Meersseman [M]. In these last generalizations, the authors obtain complex structures on products of spheres, and on connected sums of products of spheres, also constructed as a quotient of an open subset in but by the action of a nondiscrete quotient this time. These manifolds are known as LVM manifolds.
The construction in [M] has (at least) two interesting features: on the one hand, the LVM manifolds are endowed with an action of the torus whose quotient is a simple convex polytope and the combinatorial type of this polytope characterizes the topology of the manifold. On the other hand, for every simple polytope P, it is possible to construct a LVM manifold whose quotient is .
In [B], Frédéric Bosio generalize the construction of [M] emphasing on the combinatorial aspects of LVM manifolds. This aim of this paper is to study the LVMB manifolds (i.e. manifolds constructed as in [B]) from the topological and combinatorial viewpoints. In particular, we will generalize the associated polytope of a LVM manifold to our case and prove that this generalization belongs to a large class of simplicial spheres (named here rationally starshaped spheres).
In the first part, we briefly recall the construction of the LVMB manifolds as a quotient of an open set in by an holomorphic action of . In the second part, we study fundamental sets, the combinatorial data describing a LVMB manifold and its connection to pure simplicial complexes111In this paper, the expression "simplicial complex" (or even "complex") stands for "abstract simplicial complex" and a "geometric simplical complex" is a geometric complex whose cells are (straight) simplices in some Euclidean space .. Mainly, we show that an important property appearing in [B] (the property) is related to a well-known class of simplicial complexes: the pseudo-manifolds. In the same part, we also introduce the simplicial complex associated to a LVMB manifold and we show that this complex generalizes the associated polytope of a LVM manifold. In the third and forth parts, we are mainly interested in the properties of this complex and we show that the complex is indeed a simplicial sphere. To do that, we have to study another action whose quotient is a toric variety closely related to our LVMB manifold. This action was already studied in [MV] and [CFZ] but we need a more thorough study. Finally, in the fifth part, we make the inverse construction: starting with a rationally starshaped sphere, we construct a LVMB manifold whose associated complex is the given sphere. Using this construction, we show an important property for moment-angle complexes: up to a product of circles, every moment-angle complex arising from a starshaped sphere can be endowed with a complex structure of LVMB manifold.
To sum up, we prove the following theorems:
Let be a LVMB manifold. Then its associated complex is a rationally starshaped sphere. Moreover, if is indeed a LVM manifold, can be identified with (the dual of) its associated polytope.
Every rationally starshaped sphere can be realized as the associated complex for some LVMB manifold.
Up to a product of circles, every moment-angle complex arising from a starshaped sphere can be endowed with a complex structure of LVMB manifold.
In this short section, we fix several notations which will be used throughout the text:
We put for every in .
is the closed unit disk in and its boundary.
Moreover, will be the map defined by (where is the usual exponential map of ).
The characters of will be noted : .
And the one-parameter subgroups of will be noted : .
is the usual non hermitian inner product on
We will identify , as a -vector space, to via the morphism
We set and , so
In , Conv(A) (resp. pos(A)) is the convex hull (resp. the positive hull) of a subset A.
For every in , we set .
1 Construction of the LVMB manifolds
In this section, we briefly recall the construction of LVMB manifolds, following the notations of [B]. Let and be two positive integers such that . A fundamental set is a nonempty subset of whose elements are of cardinal . Elements of are called fundamental subsets.
For practical reasons, we sometimes consider fundamental sets whose elements does not belong to but to another finite set (usually ).
Let be a subset of . We say that is acceptable if contains a fundamental subset. We set as the set of all acceptable subsets. Finally, an element of will be called indispensable if it belongs to every fundamental subset of . We say that is of type (or if we want to emphasize the number of indispensable elements).
For instance, is a fundamental set of type .
Two combinatorial properties (named and 222for Substitute’s Existence and Substitute’s Existence and Uniqueness respectively, cf. [B]) will be very important in the sequel:
Moreover, a fundamental set is minimal for the property if it verifies this property and has no proper subset such that verifies the property. Finally, we associate to two open set in and in defined as follow:
The open is the complement of an arrangement of coordinate subspaces in . We also remark that, for all , we have so the following definition is consistant (where is the complex projective space of dimension ):
In example 1, verifies the property and the set is
Now, we suppose that is odd and we fix a family of elements of . We can define an action (called acceptable holomorphic action) of on defined by:
If , the previous action is just the classical one defining .
In the sequel, we focus only on families such that for every , spans (seen as a -affine space). In this case, we said that is a acceptable system. If , and , we have , so is invariant for the acceptable holomorphic action. Moreover, the restriction of this action to is free (cf. [B], p.1264).
As a consequence, we note the orbit space for the action restricted to . Notice that we can also consider an action of on whose quotient is again . We call a good system if can be endowed with a complex structure such that the natural projection is holomorphic. Such a manifold is known as a LVMB manifold. Since a quotient of a holomorphic manifold by a free and proper action (cf. [Hu], p.60) can be endowed with a complex structure, we only have to check the last property. Here, according to , we define a proper action of a Lie Group on a topological space as a continuous action such that the map defined by is proper. Notice that in our case, the group is not discrete. If is a discrete group which acts freely and properly on a space , then the action is properly discontinuous.
According to [B], we make the following definition:
Let be a acceptable system. We say that it verifies the imbrication condition if for every in , the convex hulls and does not have disjoint interiors.
In [B], the following fundamental theorem is proved:
Theorem 3 ([B], p.1268):
A acceptable system is a good one if and only if verifies the property and the imbrication condition.
In [B], it is also proved that a good system is minimal for the property.
Finally, a LVM manifold is a manifold constructed as in [LdMV] or [M]. We don’t explain here the whole construction of the LVM manifolds. The only thing we need here is that is a special case of LVMB. Indeed, we have the following theorem (and we use this theorem as a definition of LVM manifolds):
Theorem 4 ([B], p.1265):
Every LVM manifold is a LVMB manifold. To be more precise, let be the set of points of which are not in the convex hull of any subset of with cardinal . Then, a good system is the good system of a LVM manifold if and only if there exists an unbounded component in such that is exactly the set of subsets of with for cardinal such that is contained in .
We come back to example 1. If we set , and , then the imbrication condition is fulfilled and is a good system. As a consequence, theorem 3 and theorem 4 imply that can be endowed with a structure of a LVM manifold. In [LdMV], the LVM manifolds constructed from a good system of type are classified up to diffeomorphism. Here, has type and we have . Notice that we can also use the theory of moment-angle complex (cf.the last section of this text) to do this calculation.
To conclude this section, we recall how is constructed the polytope associated to a LVM manifold . It is clear that the natural action of on preserves and that this action commutes with the holomorphic action. So, we have an induced action of on . Up to a translation of the (which does not change the quotient ), theorem 4 allow us to assume that belongs to (this condition is known as Siegel’s condition) and in this case, the quotient of the action of on can be identified with:
This set is clearly a polytope and it can be shown that it is simple (cf. [BM], lemma ). The polytope is called the polytope associated to the LVM manifold .
For the previous example, we have to perform a translation on the vectors with the view to respect the Siegel’s condition. For example, is also the LVM manifold associated with the good system where , and . A calculation shows that the polytope is the square
2 Fundamental sets and associated complex
In this section, we will briefly study the fundamental sets introduced in [B] and defined above and construct a simplicial complex whose combinatorial properties reflect the geometry of a LVMB manifold. Here, is a fundamental set of type (). The integer is not supposed to be odd. For the moment, our aim is to relate the above properties to some classical ones of simplicial complexes.
Let us begin with some vocabulary. Faces, or simplices are subsets of a simplicial complex. If a complex is pure-dimensional (or simply pure), the simplices of maximal dimension are named facets and the faces of dimension 0 (resp. ) are the vertices (resp. the ridges) of the complex.
The first important definition in this paper is the following:
Let be a fundamental set of type . The associated complex of is the set of subsets of whose complement (in ) is acceptable.
As we will see later, this complex is the best choice for a combinatorial generalization of the associated polytope of a LVM manifold.
First properties of are the following:
Let be a fundamental set of type . Then, its associated complex is a simplicial complex on . Moreover, is pure-dimensional of dimension and has vertices. Its vertices are precisely the non-indispensable elements of for and its facets are exactly the complements of the subsets of .
is obviously a simplicial complex. Moreover, the maximal simplices of are exactly the complements of minimal subsets of , that are fundamental subsets. The latters have the same number of elements, so every maximal simplex of has elements. Finally, an element is a vertex of if and only if contains a fundamental subset, that is is not indispensable.
The complex associated to the fundamental set of example 1 is the complex with facets . So, is the boundary of a square.
Conversely, every pure complex can be realized as the associated complex of a fundamental set: let be a pure-dimensional simplicial complex on the set with dimension and vertices. Then, for every integer , there exists two integers and a fundamental set of type whose associated complex is . If is fixed, this fundamental set is unique.
Moreover, the property can be expressed as a combinatorial property of :
verify the property if and only if
where is the set of facets of .
First, we assume that verifies the property. Let be a facet of and . Then belongs to . The property implies that there is in (so ) such that belongs to . As it was claimed, is also a facet of . Moreover, we obviously have . Finally, if is a facet of with and , then, we would have in , which contradicts the property. The proof of the converse is analogous.
Let be a fundamental set. Then its associated complex verify the property if and only if every ridge of is contained in exactly two facets of .
To begin, we assume that verifies the property. Let be a ridge of . By definition, is included in a facet of . We put . By proposition 2.2, there exists (and so ) such that is a facet of . We have so is contained in at least two facets of . Let assume that is contained in a third facet . However, in this case, we would have , which contradicts proposition 2.2.
Conversely, let be a facet of and . If , then is a ridge of and by hypothesis, is contained in exactly two facets and . One of them, say , is . The other is and we have . Then we have (on the contrary, we would have ) and . Moreover, if is another facet of with , so contains and by hypothesis, (i.e ). If , we remark that the element such that is a facet of is . Indeed, if , then is a facet of . And if , then and, as a consequence, is not in .
Let be a fundamental set of type . We define the (unoriented) graph stating that its vertices are fundamental subsets of and two vertices and are related by an edge if and only if there exist such that . Equivalently, we relate two subsets of if and only if they differ exactly by one element. is called replacement graph of .
Let be a fundamental set of type which verifies the property. Then, there exists an integer , and fundamental sets of type which are minimal for the property and such that is the disjoint union .
We proceed by induction on the cardinal of . If is minimal for the property, then there is nothing to do. Let assume that it is not the case: there exists a proper subset of which is minimal for the property. We put for its complement . It is obvious that is a fundamental set (of type ). We claim that verifies the property. Let be an element of and . If , then, putting , we have that is an element of . It is the only choice (for ) since is in and verifies the property. Let assume now that is not in . Since is an element of , there exists exactly one such that is an element of , too. We claim that cannot be in . Indeed, if it was the case, since is minimal for the property, there would exist exactly one such that . But is in and . So, and . As a consequence, is a fundamental set of type which verifies the property with cardinal strictly smaller than . Applying the induction hypothesis on , we have the decomposition of we were looking for.
The decomposition of the previous proposition induces a decomposition of the vertex set of . In the proof, we show that an element of is related only to others elements in . Consequently, each set is the vertex set of a connected component of . This also implies that this decomposition is unique up to the order. We call connected components of the sets .
Let be a fundamental set of type and its replacement graph. We assume that verifies the property. Then, the following assertions are equivalent:
is minimal for the property.
has only one connected component.
Let be a simplicial complex. is a pseudo-manifold if these two properties are fulfilled:
every ridge of is contained in exactly two facets.
for all facets of , there exists a chains of facets of such that is a ridge of for every .
For instance, every simplicial sphere is a pseudo-manifold. More generally, a triangulation of a manifold (that is a simplicial complex whose realization is homeomorphic to a topological manifold) is also a pseudo-manifold. Now, the proposition below shows that the notion of pseudo-manifold is exactly the combinatorial property of which characterizes the fact that is minimal for the property:
Let be a fundamental set with . Then, is a pseudo-manifold if and only if is minimal for the property.
On the one hand, let assume that is minimal for the property. This implies that every ridge of is contained in exactly two facets (cf. corollary 1). Moreover, let be two distinct facets of . So, and are two fundamental subsets. By minimality for the property, is connected (cf. corollary 2). Consequently, there exists a sequence of fundamental subsets such that and differ by exactly one element. We note the acceptable subset with elements. Its complement is thus a face of with elements. If we put , we have so . Consequently, is a pseudo-manifold.
On the other hand, we assume that is a pseudo-manifold. Then, thanks to corollary 1, verifies the property. Moreover, will be minimal for this property if and only if is connected (cf. corollary 2). Let be two distinct elements of . Then and are facets of . Since is a pseudo-manifold, there exists a sequence , of facets of such that for every , and share a ridge of . This means that and are fundamental subsets which differ only by an element, and consequently, and are related in . This implies that is connected and is minimal for the property.
The case where corresponds to . It is not a pseudo-manifold since the only facet is and it doesn’t contains any simplex with dimension strictly smaller.
Finally, we prove the following proposition which is the motivation for the study of the associated complex:
Let be a good system associated to a LVM manifold and its associated polytope. Then the associated complex of is combinatorially equivalent to the border of the dual of .
Since is a good system associated to a LVM manifold, there exists a bounded component in such that is exactly the set of subsets of with for cardinal such that is included in the convex hull of . Up to a translation, (whose effect on the action is just to introduce an automorphism of and so does not change the action, cf. [B]), we can assume that is exactly the set of subsets of with for cardinal such that the convex hull of contains . In this setting, according to the page 65 of [BM], the border of is combinatorially characterized as the set of subsets of verifying
So, is a subset of if and only if is acceptable, i.e. .
As a consequence, from a viewpoint of set theory, and are the same set. We claim that the order on these sets and are reversed. On the one hand, the order on is the usual inclusion (as for every simplicial complex). On the other other, we recall the order on the face poset of given in [BM]:
every face of is represented by a tuple. So, facets of are represented by a singleton and vertices by a tuple. A face represented by is contained in another face represented by if and only if . So, combinatorially speaking, the poset of is , which prove the claim. Finally, the poset for the dual is , and the proof is over.
Consequently, is exactly the object we are looking for to generalize the associated polytope of a LVM manifold to the case of LVMB manifolds. What we have to do now is to study its properties. Our main goal for the moment is to prove the following theorem:
Let be a good system of type . Then is a simplicial sphere.
The theorem is trivial in the LVM case since the associated complex is a polytope.
To prove the previous theorem, we have to focus on good systems which verify an additional condition, called condition : there exists a real affine automorphism of such that has coordinates in for every . For instance, if all coordinates of are rationals, then verify condition (K). Note that the imbrication condition is an open condition. As a consequence, it is sufficient to prove the previous theorem for good system verifying the condition . Indeed, since is dense in , a good system which does not verify the condition can be replaced by a good system which verifies the condition, with the same associated complex .
The main interest of condition stands in the fact that we can associate to our holomorphic acceptable action an algebraic action (called algebraic acceptable action) of on (or an action of on ):
Let be a fundamental set of type verifying condition . We set for every and . We can define an action of on setting:
Using the notation for the character of defined by , we can resume the formula describing the acceptable algebraic action by:
It is clear that is invariant by this action. So we can define as the topological orbit space of by the algebraic action. As for the holomorphic acceptable action, we can define an action of on whose quotient is also . In [CFZ], proposition 2.3, it is shown that the holomorphic acceptable action of on can be seen as the restriction of the algebraic acceptable action to a closed cocompact subgroup of . As a consequence, we can define an action of on whose quotient can be homeomorphically identified with .
The principal consequence of this result is the following:
is Hausdorff and compact.
Let be the canonical surjection. is a compact Lie group so is a closed map (cf. [Br], p.38). Consequently, is Hausdorff. Finally, since is continuous and is compact, we can conclude that is compact.
Another important consequence for the sequel of the article is that the algebraic action on (or ) is closed. Moreover, since every complex compact commutative Lie group is a complex compact torus (i.e. a complex Lie group whose underlying topological space is , cf. [L], Theorem 1.19), we can say that is a complex compact torus.
Using an argument of [BBS], we show that:
is in the stabilizer of if and only if , we have
Let us fix some linear order on . Up to a permutation of the homogenous coordinates of , we can assume that for every (notice that such a permutation is an equivariant automorphism of ). We set the smallest index of nonzero coordinates of . Then, for every which stabilizes , we have
So . In particular, we have for every . If , then .
In [BBS], it is shown that is a fixed point for the algebraic acceptable action if and only if , we have .
Consequently, we have:
Every element of have a finite stabilizer for the algebraic acceptable action.
First, we recall that an element of has at least nonzero coordinates and amongs this coordinates, there are coordinates such that spans as a real affine space. Up to a permutation, we can assume that is contained in and in . In this case, we have for every and every in the stabilizer of . We put . Writing , we get that and verify the following systems:
where is the matrix defined by and . The system being acceptable, spans as a real affine space, which means exactly that the real matrix M is inversible.
Consequently, (i.e pour tout j) and . So, can take only a finite number of values. As a conclusion, the stabilizer of is finite, as it was claimed.
An analogous proof shows that the stabilizer of is finite, too.
4 Connection with toric varieties
In this section, our main task is to recall that is a toric variety and to compute its fan. When the fan is simplicial, we can constructed a simplicial complex as follow: we note for the set of rays of and order its elements by . Then, the complex is the simplicial complex on defined by:
This complex is the underlying complex of . We recall a theorem which will be very important in the sequel:
Let X be a normal separated toric variety and its fan. We suppose that is simplicial. Then, the three following assertions are equivalent:
is complete in .
’s underlying simplicial complex is a -sphere.
4.1 Toric varieties
To begin with, it is clear that and are separated normal toric varieties. As it is explained in [CLS], we can associate to a separated normal toric variety with group of one-parameter subgroups a fan in the real vector space whose cones are rational with respect to the lattice 333a cone is said to be rational for if there a family of elements of such that . Moreover, if is free in , we say that is simplicial. So, we can compute the fan associated to :
Let be the canonical basis for . Then, the fan describing in is
To compute the fan of a toric variety, one has to calculate limits for its one-parameter subgroups. The embedding of in is the inclusion and the one-parameter subgroups of have the form with .
So, the limit of when t tends to exists in if and only if for every . In this case, the limit is with (Kronecker’s symbol).
Of course, the limit has to be in , which implies for to be acceptable. But so the condition means exactly that belongs to .
For the fundamental set of example 1, we have . As a consequence, the fan is the fan in whose facets are the dimensional cones and (where is the canonical basis of ).
We can also easily compute orbits of for the action of . For , we set . Then if , its orbit is .
So, we can describe as the partition:
Moreover, in the orbit-cone correspondance between and (cf. [CLS] ch.3), corresponds to .
In quite the same way, we can show that the fan of is in , with defined as the canonical basis of and .
In [CLS], it is proven that is an irreducible separated compact normal toric variety. In the next section, we will detail the construction with the view to identify its group of one-paramater subgroup and the structure of its fan.
For the good system with and , and , the algebraic action is
Using the automorphism of defined by , we can see that the quotient of the algebraic action is also the quotient of by the action defined by
so is the product and the projection comes from the usual definition of as a quotient of
To conclude this section, let be the map defined by
The algebraic acceptable action on is just the restriction to of the natural action of the torus on .
Let be in . Then for every . We set and and we have for every . That means that we have where is the real matrix defined by and is the vector . Since affinely spans , we can extract an inversible matrix of order from and we can deduce that . This means that for every . Moreover, if , we get that is an element of . Consequently, is an element of , where . Finally, we can deduce that is a th root of unity for every . Particularly, there are only a finite number of elements in the kernel of .
Generally, is not injective. For instance, if we consider