Small covers over wedges of polygons
Suyoung Choi, Hanchul Park

TL;DR
This paper classifies small covers and real toric manifolds with orbit spaces derived from polygons through wedging, using combinatorial puzzles to systematically analyze their structure.
Contribution
It introduces a systematic combinatorial method to classify small covers and real toric manifolds over wedge polygons, expanding understanding of their topological and combinatorial properties.
Findings
Classification of small covers over wedge polygons
Development of a combinatorial puzzle method
Identification of new classes of real toric manifolds
Abstract
A small cover is a closed smooth manifold of dimension having a locally standard -action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or, simply, a real toric manifold), that is, a real locus of projective toric manifold. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method finding toric spaces called puzzles.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Commutative Algebra and Its Applications · Computational Geometry and Mesh Generation
Small covers over wedges of polygons
Suyoung Choi
Department of Mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon, 443-749, Republic of Korea
and
Hanchul Park
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro Dongdaemun-gu, Seoul 130-722, Republic of Korea
Abstract.
A small cover is a closed smooth manifold of dimension having a locally standard -action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or, simply, a real toric manifold), that is, a real locus of projective toric manifold. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method finding toric spaces called puzzles.
Key words and phrases:
puzzle, real toric variety, simplicial wedge, small cover, real toric manifold
2010 Mathematics Subject Classification:
14M25, 57S25, 52B11, 13F55, 18A10
The first author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning(NRF-2016R1D1A1A09917654).
Contents
1. Introduction
One of basic objects in toric topology is the small cover which is an -dimensional closed smooth manifold with a locally standard -action whose orbit space is a simple convex polytope. It was firstly introduced in [9] as a generalization of real projective toric variety; when is a toric variety of complex dimension , there is a canonical involution on and its fixed points set forms a real subvariety of real dimension , called a real toric variety. It should be noted that admits a -action induced from the action of torus on , and its orbit space is equal to . A non-singular complete toric variety is called a toric manifold, and the corresponding real toric variety is called a real toric manifold. Hence, if a toric manifold is projective, then its orbit space is a convex polytope, and hence is a small cover. For an -dimensional small cover , the boundary complex of its orbit space is a polytopal simplicial complex of dimension . We simply say that is a small cover over or a small cover over .
Two small covers are said to be Davis-Januskiewicz equivalent (simply, D-J equivalent) if there is a weakly equivariant homeomorphism between them which covers the identity map of their orbit spaces. It is known by [9] that small covers are classified by a pair of a polytopal simplicial complex and a -characteristic map over . The definition of -characteristic map will be given in the next section. One fundamental problem in toric topology is to classify small covers as well as (real) toric manifolds up to D-J equivalence. See [10, 3, 4, 14, 8].
However, the class of small covers (up to D-J equivalence) is too huge to be classified. Hence, we have to restrict our attention to a smaller but interesting subclass of manifolds. We denote by the boundary complex of the regular -gon, and by the boundary complex of the -dimensional cube. Two remarkable classes of polytopal simplicial complexes are ones obtainable by a sequence of wedges (definition will be given in Section 2) from either (i) or (ii) . A simplicial complex in the class (ii) is called a wedged polygon, and it is denoted by for some positive integer -tuple . The reason why such classes are interesting came from the classical results for toric manifolds. Up to present, there are complete classifications of toric manifolds up to Picard number ([12, 2, 8]). Interestingly, if is a toric manifold of Picard number , then the boundary complex of its orbit space should be in the class (i) or (ii). In general, if the complex dimension of is and its Picard number is , then is dimensional star-shaped complex having vertices. Therefore, if , then must be the boundary complex of the product of two simplices [15] which can be obtained by a sequence of wedges from . Furthermore, it is shown in [11] that if and supports a toric manifold, then is obtainable by a sequence of wedges from either or . Therefore, it is reasonable to focus on toric spaces whose orbit space has the boudary complex belonging in the class (i) or (ii).
A study of toric manifolds and small covers over in the class (i) is well established. Note that a simplicial complex in the class (i) is the boundary complex of the product of simplices. A toric manifold over is known as a generalized Bott manifold, and its real part is called a generalized real Bott manifold. In [5], it has been shown that every small cover over is indeed a generalized real Bott manifold and they have been classified up to D-J equivalence in terms of block matrices.
Nevertheless, for the class (ii), a study of toric manifolds and small covers over has not been established for long times except for a wedged pentagon which is in the Picard number case. Kleinschmidt and Sturmfels [13] showed that every toric manifold over is projective, and, by using this fact, Batyrev [2] classified toric manifolds over .
Very recently, the authors in [8, 7] found a new way how to find all characteristic maps over a wedge of from information of , and, using this new technique, in [6], they have classified all toric manifolds over in the class (ii) and showed that all toric manifolds over are projective. This generalizes both the main results of [13] and [2]. Furthermore, they could count the number of the small covers (and real toric manifolds) over in [8]. Hence, as the next step, it is natural to classify small covers or real toric manifolds over , as stated in Question 6.3 of [7]. This is our main purpose.
This paper is organized as follows. In Section 2, we review some notions including simplicial wedges, characteristic maps, and diagrams and puzzles. In Section 3, we compute the diagram of , and, by using this, we give the classification of the small covers over in Section 4. In addition, we classify real toric manifolds over in Section 5. In the final section, we shall provide a summary of the paper. There, we enclose all combinatorial concepts required to understand the results, and present the main results and examples. This section would be helpful for the readers even if s/he did not read other parts of the paper carefully.
2. Backgrounds
The reader is recommended to refer [7] for many terms of this paper, even though we will present definitions of essential notions.
Let be a simplicial complex on the vertex set . We say that a subset of the vertex set is a minimal non-face of if itself is not a face of , but every proper subset of is a face of . Every simplicial complex is determined by describing its minimal non-faces.
Definition 2.1** ([1]).**
Let be an -tuple of positive integers. The simplicial complex is defined by the vertex set
[TABLE]
and the minimal non-faces
[TABLE]
for each minimal non-face of .
In particular, for distinct vertices of , denote by the -tuple such that the th entries are 2, for , and the other entries are 1, and, then we use the notation
[TABLE]
When , the operation is known as the (simplicial) wedge operation, and it is easy to show that can be obtained from a sequence of wedges. See Figure 1 for an example of the simplicial wedge.
Let us further assume that is a polytopal simplicial complex, that is, is a simplicial complex isomorphic to the boundary of some simplicial polytope. Note that if is polytopal, then every is also polytopal ([8, Proposition 2.2]).
Definition 2.2**.**
A -characteristic map over , or simply a characteristic map, is a map such that the following holds:
[TABLE]
The condition ( ‣ 2.2) is known as the non-singularity condition. Two characteristic maps over are called Davis-Januszkiewicz equivalent, or D-J equivalent, if there exists a linear isomorphism such that . The equivalence classes themselves are called Davis-Januszkiewicz classes or D-J classes.
When is a polytopal simplicial complex of dimension , it is a fundamental fact shown in [9] that -characteristic maps over up to D-J equivalence correspond one-to-one to small covers over up to weakly -equivariant homeomorphism fixing orbit spaces. A characteristic map is often represented by an -matrix called the characteristic matrix, whose th column vector corresponds to for . Two characteristic matrices and correspond to two D-J equivalent characteristic maps if and only if can be obtained from by a series of elementary row operations.
For a polytopal simplicial complex and a face of , its link is also polytopal. If supports a characteristic map , then the link also supports a natural characteristic map over a link of called the projection as follows.
Definition 2.3**.**
For a characteristic map and a face of , the projection of with respect to is defined as
[TABLE]
when is a vertex of . The projection is well defined up to D-J equivalence. When is a vertex, one can simply write .
Remark 2.4**.**
Note that a link of corresponds to a face of its dual simple polytope. Furthermore, the projection is the characteristic map for a submanifold fixed by a subtorus. When is a vertex, then the submanifold is known as a characteristic submanifold.
Remark 2.5**.**
Throughout this paper, we will not necessarily distinguish the concept of characteristic maps and D-J classes unless otherwise mentioned. In actual calculation, we will mainly deal with characteristic maps while freely using elementary row operations. Also observe that the projection is actually an operation of D-J classes.
Let be the boundary complex of the regular -gon for . We call a wedged polygon. Our objective of the current paper is to figure out the D-J classes over . For this objective, let us review the definition of the diagram of a polytopal simplicial complex . Once is calculated, one can construct every characteristic map over for any due to [7].
Definition 2.6**.**
Let be a polytopal simplicial complex and define , and as follows.
- (1)
is the set of the D-J classes over . 2. (2)
is the set of the D-J classes over where is a vertex of . 3. (3)
is the set of the D-J classes over where and are distinct vertices of .
The triple is called the diagram of and is denoted by . The elements of , and are called the nodes, edges, and realizable squares of respectively.
To any edge of , one can correspond the set where are two D-J classes over which are the two projections of , namely, and . This can be regarded as an edge connecting and which is colored , which looks like
[TABLE]
In this case, we say that and are -adjacent. We can abuse the set for the edge by the uniqueness result of [8, Theorem 1.1]. That is, if both of the characteristic maps and over correspond to , then either they are D-J equivalent or they are the “twins” of each other (see Remark 2.7). An edge is said to be trivial if . The edge set contains every possible trivial edge. That is, for every D-J class and every vertex , .
Like edges, for any realizable square , we have the four projections , and onto , so that
[TABLE]
and the four edges
[TABLE]
We call a figure of the form (2.1) a square.
Remark 2.7**.**
For a D-J class over such that and , there surely exists a D-J class over such that and , because of symmetry of the simplicial wedge. That means, every edge of has its “twin”. The twins must be distinguished when counting D-J classes over , but in most other situations they need not be distinguished. A similar argument goes for four different realizable squares over corresponding to the same square up to symmetry.
Here, we introduce a general method how to find D-J classes over using the given diagram referring to [7]. Let be a polytopal simplicial complex. Let be an -tuple of positive integers. We consider an edge-colored graph with colors constructed as follows: is the graph determined by the -skeleton of the simple polytope , where is the -dimensional simplex. One remarks that each edge of can be uniquely written as
[TABLE]
where is a vertex of , , , and is an edge of . Then we color on the edge . Let us call a subgraph of a subsquare if it comes from a 2-face of which has edges. Two edges
[TABLE]
of are said to be parallel if . Every parallel edge of has the same color.
Observe that the diagram whose realizable squares are forgotten can be regarded as a pseudograph whose edges are colored.
Definition 2.8**.**
A realizable puzzle over is a pseudograph homomorphism such that
- (1)
preserves the coloring of the edges, and 2. (2)
each image of the subsquare of is a realizable square in .
Theorem 2.9** ([7], Theorem 5.4).**
There is a one-to-one correspondence
[TABLE]
We have one more useful statement as follows.
Proposition 2.10** ([7], Proposition 4.3).**
Let be any fixed node of and suppose that there are two realizable puzzles . Then if and only if and for every node of which is adjacent to .
The essential form of the above statement is for the realizable squares. That is, if a square of the form (2.1) is realizable, then is uniquely determined by , , and . If this holds, we say that the two edges and spans a realizable square.
According to Remark 5.7 of [7], for any edge and any vertex , the square of the form
[TABLE]
is always realizable. We say the realizable square is reducible. Realizable squares which are not reducible are called irreducible.
3. The diagram
According to Theorem 2.9, the diagram can be used to construct every D-J class over . Therefore, our objective here is to provide the diagram . Throughout the paper, we identify the vertex set of with in counterclockwise order.
3.1. Node of
We begin by calculating the node set of . Let us label the vectors by and , respectively. Then one can regard a characteristic map as a finite circular sequence consisting of and , no letter in which appears consecutively. Furthermore, any cardinality two subset of is a basis of and the other element is the sum of the two, and hence any permutation of and does not affect the D-J type of the characteristic map .
Proposition 3.1**.**
Let be the number of D-J classes over . Then,
[TABLE]
Proof.
We may assume that and . If , then the number of possible cases for is equal to and is either or . So total number is . If , then is determined uniquely, and the number of possible cases for is equal to . Therefore, , where and . ∎
It should be observed that a D-J class over can be regarded as a special kind of partitions of : for each (up to D-J equivalence), is divided into at most three disjoint subsets , , and . Using this observation, we can give an alternative definition of the D-J class over . A subset of is called non-consecutive if are not contained in for any .
Definition 3.2**.**
The set is a D-J class over if it satisfies the following:
- (1)
is a weak partition of , that is, it is a partition of and one of could be empty. 2. (2)
all of and are non-consecutive.
3.2. Edge of
The next step is for the edges of .
Definition 3.3**.**
For a characteristic map over and , the support of for , denoted by , is . Since whenever and are D-J equivalent, the support is well defined for a D-J class .
Example 3.4**.**
The characteristic map over represented by
[TABLE]
or simply written as , is D-J equivalent to , and they correspond to the D-J class or simply . The support of for is .
Lemma 3.5**.**
Two D-J classes and over are -adjacent if and only if .
Proof.
A proof is easily given by direct matrix calculation. Note that and are -adjacent if and only if there is a characteristic matrix over whose two projections are and . As a version of Lemma 4.2 of [7] for the -characteristic maps, it is known that for a characteristic map
[TABLE]
over a simplicial complex where denotes the th column vector of , the D-J class over one of whose projection is can be expressed by a matrix of the following form, called a standard form for an edge,
[TABLE]
for for . The numbers above the horizontal line are indicators for vertices of simplicial complexes.
In our case, let us assume that . For a suitable rearrangement of columns, can be written as
[TABLE]
whose first two columns correspond to and respectively. Check that the projection is
[TABLE]
and is
[TABLE]
One can see that in order for to satisfy non-singularity condition, which is equivalent to that . ∎
Recall that a characteristic map over can be regarded as a finite circular sequence consisting of and . If is a non-consecutive subset with , then is divided into finite sequences called pieces of determined by . If , then one obtains pieces consisting of and , each of which will be called a -piece. Likewise, -pieces and -pieces are defined.
For a -piece , the inversion of is obtained from by exchanging and . The following proposition completely describes the edges of .
Proposition 3.6**.**
Let and be two characteristic maps over and . We also assume that . Then the D-J classes with respect to and are -adjacent if and only if becomes after replacing a number of -pieces by their inversions.
Proof.
The proof is similar to Lemma 3.5. Pick a -piece of together ’s on its ends. For example, consider a part of
[TABLE]
which corresponds to . According to the proof of Lemma 3.5, should look like
[TABLE]
at the same place, for . Since satisfies non-singularity condition, we have . If , then the corresponding -piece is fixed. If , then the inversion is applied to the -piece. This works regardless of the shape of the -piece, completing the proof. ∎
Example 3.7**.**
Over , assume that and or . Then there are apparently three characteristic maps obtained by a number of inversions of -pieces: , , and . But since the whole exchanging of and does not change the D-J equivalence type, and . In general, there are D-J classes -adjacent to the D-J class with respect to .
Let be a characteristic map over and let , be two vertices of such that . Then and divide into two pieces and . Then we obtain a new characteristic map by replacing every -piece lying in by its inversion. Although this definition depends on the choice of , is well defined up to D-J equivalence provided . More generally, let be a non-consecutive subset of even cardinality and let be a characteristic map over such that for . Then we define
[TABLE]
One can easily see that the operation is well defined up to D-J equivalence and does not depend on the order of .
Remark 3.8**.**
Intuitively, can be understood as the following. One can regard as a set of points on the circle. Then one paints each piece of the circle black or white like the chessboard, such that neighboring pieces have different colors. Then one performs inversion for every -piece in the black region. There are exactly two ways of such colorings, but is independent of coloring up to D-J equivalence.
Definition 3.9**.**
For a D-J class over , an ordered pair is called an e-set (compatible with ) if it satisfies the following
- (1)
, 2. (2)
is of even cardinality, and 3. (3)
.
Note that is an e-set for arbitrary . We call an empty e-set. We have a natural map which maps an edge of to an e-set in the following way: is the color of , and is the unique set such that .
Example 3.10**.**
Let us consider the edge when ,
[TABLE]
Then and .
When is a node incident to in and , is compatible with . Conversely, for any node of and its compatible e-set , there is a unique edge incident to such that .
3.3. Square of
The last step is to find realizable squares of . Recall that, if two edges and span a realizable square, then the fourth vertex is unique. That is, if both of the two squares
[TABLE]
are realizable, then as D-J classes. In order to find a realizable square, we firstly consider over whose projections are given three characteristic maps , and , namely, , and . Then, if exists, then it must be uniquely determined by , , and , as well as .
We have the matrix (3.1) for an edge of the diagram. For realizable squares, every D-J class over for can be expressed by the following matrix called a standard form for a square,
[TABLE]
by Proposition 4.3 of [7]. Note that its two projections with respect to and have the form of (3.1) when and respectively. We can regard them as the two edges and . One can see that is a genuine characteristic matrix over if and only if satisfies the non-singularity condition as a characteristic map over . In that case is the fourth node of the realizable square.
By Lemma 3.5, we know that and . We have two cases depending on whether or not. Let us consider the first case when . We may assume that . Let be a standard form over . Among vertices of , we pick and any other point as a representative and we write . The submatrix of the matrix consisting of the columns corresponding to , and would look like the following:
[TABLE]
where . To compute and from this matrix, we add the third and fourth row to the first one to obtain
[TABLE]
Since the columns corresponding to and are coordinate vectors, one can obtain by deleting rows and columns containing the underlined entries:
[TABLE]
As consequence, we conclude that and similarly and . When , should be and one concludes and . When , or for any . In this case, observe that and indicate whether the -piece at the position of is inverted or not. Therefore we reach the following theorem.
Theorem 3.11**.**
Consider the characteristic map over of the form
[TABLE]
where and are -pieces for . Let and be distinct two vertices of such that . Then any two edges and span a realizable square with fourth node . Furthermore, one can assume that
[TABLE]
and
[TABLE]
where each of and is either the identity or the inversion, and the fourth node is computed as follows:
[TABLE]
∎
Example 3.12**.**
Let , , and be characteristic maps over represented by
[TABLE]
and pick distinct elements and in . Then the fourth node determined by the edges and is
[TABLE]
The remaining second case is when . In this case, we can assume that and for all . Just like before, let be a standard form for . We again pick , and like before and we write . The submatrix of the matrix consisting of the columns corresponding to , and would look like the following:
[TABLE]
where . Again like before, we obtain
[TABLE]
One observes that if is non-singular then because if both of and were 1, then one of or should be zero, violating the non-singularity condition.
Let and be two characteristic maps, not D-J classes, over . We denote by the set . For a subset , we denote by the set .
Lemma 3.13**.**
Let , and be characteristic maps over such that and for . Then the following are equivalent.
- (1)
Two edges and span a realizable square. 2. (2)
. 3. (3)
.
Proof.
Note that if and only if in (3.2), and if and only if in (3.2). In other words, indicates the position of the inverted -pieces and corresponds to the inverted -pieces. Since , one has .
Let us show the implication . Suppose that there is an element such that and . Because is adjacent to an inverted -piece, . Without loss of generality, one can assume that . Then and reminding that , we obtain that because . Then (3.2) implies that , which contradicts to the non-singularity condition.
To show the implication , consider parts of and where they could intersect. There are two possibilities of restricted on :
[TABLE]
Similarly, there are two possibilities of restricted on :
[TABLE]
In order to satisfy both and , the only possibility is
[TABLE]
where lies on and is on . But it contradicts to the condition .
For the remaining part , one observes that (3.2) guarantees is obtained from by performing two inversion maps given by two edges and . Since each inversion does not change the endpoints of and , is well defined and non-singular, proving that the square is realizable. ∎
The realizable square in Lemma 3.13 is called a realizable square of type 2 if it is irreducible. A realizable square of is called of type 1 if it is not of type 2.
Example 3.14**.**
Put , and let , , and be characteristic maps over represented by
[TABLE]
Then
[TABLE]
From now on, we study the realizable square in the language of e-sets, starting from the following definition. We fix a characteristic map over .
Definition 3.15**.**
When , two e-sets and compatible with are said to be type 1 or type 2 -related, if edges and span a realizable square of type 1 or type 2, respectively.
When , and compatible with are always said to be type 1 -related.
Now, let us find a combinatorial criterion for two given e-sets to be type 2 -related or not. Let be a non-empty e-set of and fix a vertex . Recall Remark 3.8 to observe that the set divides to two disjoint subsets and corresponding to “black” and “white” arcs respectively. We can assume that . Then we write .
Proposition 3.16**.**
Assume that we are given two non-empty e-sets and compatible with . Then and are type 2 -related if and only if and .
Proof.
To show the “only if” part, let us assume that and . We can further assume that and . Then observe that and . Then, it immediately follows by Lemma 3.13.
To show the “if” part, one picks any such that
[TABLE]
and the proof goes analogously. ∎
Remark 3.17**.**
We remark that an empty e-set is always type 1 -related with arbitrary e-set compatible with . One can see that two non-empty e-sets and compatible with are type 1 -related if and only if .
4. Classification of small covers over
We are now ready to describe the realizable puzzles over . Let us fix which is an -tuple of positive integers. We denote by the set of pairs such that is a node in and is a finite sequence of e-sets of compatible with ,
[TABLE]
such that the members of are pairwise -related.
Theorem 4.1**.**
There is a bijection
[TABLE]
Proof.
By Theorem 2.9, we will use the realizable puzzle instead of the D-J class over . We label nodes of in the following way. Let be the node set of . Recall that is the 1-skeleton of the polytope
[TABLE]
By labeling the vertices of by , we may identify where
[TABLE]
Let us use the notation . A node adjacent to can be written as
[TABLE]
where the th entry is , and the other entries are . In order to construct a pair from a realizable puzzle , we put and the sequence consisting of when and . Then, is indeed an element of . In summary, is determined by and for all adjacent to .
The converse construction is the essential part of the proof. Our aim here is to construct a realizable puzzle using the data of . The basic philosophy is Proposition 2.10. We assign an e-set to each edge of incident to analogously to above argument. This is extended to the whole edges of using the following rules.
- (1)
Every parallel edge is assigned the same e-set. 2. (2)
For a triangle and the corresponding e-sets , ,
[TABLE]
This assignment is indeed well defined and the verification is very easy. Recall that is an edge-colored graph. If is an edge of whose color is , then its assigned e-set is for some . Now we define a pseudograph homomorphism by the following rules.
- (1)
. 2. (2)
If and are adjacent nodes in and the assigned e-set to the edge is , then
[TABLE]
We must check that is well defined and is a realizable puzzle. The following diagram
[TABLE]
indicates some nodes and edges of , whose two nodes map to and by respectively. Here, and are edges and the ordered pairs are assigned e-sets. We additionally assume that the and are parallel and thus if , and the and are parallel and if .
In order to show both well-definedness and realizablity of , we have to show the transitivity property of e-sets: if , and are mutually -related as e-sets compatible with , then , and are mutually -related as e-sets compatible with .
STEP 1. Let us firstly show that , and are e-sets compatible with . Note that by the definition of . Since , we also have . Therefore, is compatible with .
Note that is either or . If , then
[TABLE]
Since , we have , and, thus
[TABLE]
If (and thus ), then and must be type 2 -related. Hence, . Note that since and , we have . Assume that and . Then, is the map exchanging and in . Therefore, . Therefore,
[TABLE]
In both cases, is compatible with . Similar arguments can be applied to .
STEP 2. Observe that and are -related since the three edges , , and determine a realizable square when , or a triangle when . Similarly, and are -related.
STEP 3. If , then , and . Therefore, since and are -related, and are -related. If , then or . In the second case, we must have and thus and we can use the result of STEP 2. Hence, in any case, and are -related. Similar argument can be applied to the case . Hence, the claim holds when one of and is an empty set.
From now on, let us assume that none of and is an empty set. We may use Proposition 3.16 and Remark 3.17 as criteria whether the e-sets are -related in the next Step.
STEP 4. Let us show that and are -related. We divide this case into a few smaller cases:
- •
: By (4.1), we have
[TABLE]
Therefore, and are type 1 -related.
- •
and : Note that , and observe that
[TABLE]
Therefore, , and hence, and are type 1 -related.
- •
and : Note that is a partition of . Also observe that
[TABLE]
and
[TABLE]
Thus and . Likewise , concluding . Since and , and are type 2 -related.
- •
: If , then and are type 1 -related. Now let us consider the case where . In this case, and are type 2 -related. Hence, . If and , then and . Therefore, and are type 2 -related. If , then , and because . Since and must be type 2 -related, we have . To show , we are enough to prove that
[TABLE]
To show (4.3), pick any element reminding that the right hand side contains . Then is the disjoint union of two consecutive sets and . Now is an element of if and only if is odd. Thus one of or must be odd, and . In conclusion, and are type 2 -related.
The above Steps 1–4 prove our claim.
Let us finish our goal using the above claim we just have shown. Pick a node of and consider the minimal path from to given by the sequence
[TABLE]
Then is given by a chain of maps
[TABLE]
for the e-sets corresponding to the above sequence. To check that this is well defined, we start from the node . The edge connecting and and that connecting and spans a realizable square since the e-sets and are -related. Therefore is well defined. Next, we move on the node . Then the claim shows that and are -related and thus is well defined. The inductive application of this procedure proves that is well defined for every node of . Similarly, our claim shows that every subsquare is realizable, completing the proof. ∎
For a D-J class over , put
[TABLE]
Corollary 4.2**.**
The number of small covers over is equal to the sum of for all D-J classes over .
Some examples and calculations for Corollary 4.2 will be given in Section 6.
5. Real toric manifolds over
The objective of this section is to specify every real toric manifold over the wedged polygon in terms of realizable puzzles. In [6], the authors have described all smooth complete toric varieties over wedged polygons up to D-J equivalence. By taking the mod 2 reduction, one can obtain the classification of real toric manifolds over wedged polygons up to D-J equivalence. Before that we need some preperation. We identify with the vertex set of as before. Let us start with the real toric manifolds over .
Lemma 5.1**.**
Let be a -characteristic map over . The small cover given by , written as , is also a real toric manifold if and only if
- (1)
* is D-J equivalent to , or* 2. (2)
the image of has exactly three elements.
In other words, is a real toric manifold unless is D-J equivalent to
[TABLE]
where .
Proof.
First of all, recall the classical fact that every complete smooth toric surface is either or a consecutive equivariant blow-up of a Hirzebruch surface. Then their mod 2 reductions cannot be D-J equivalent to for . More precisely, we can define a blow-up of a -characteristic map over to be
[TABLE]
for , or
[TABLE]
for , which is a -characteristic map over . Observe that is not a blow-up of other -characteristic map, and gives a real toric manifold if and only if .
Conversely, we are going to show that every with three values gives a real toric manifold. One can assume that and we use an induction on . Note that there is an inverse operation of the blow-up for , called a blow-down, if there is a vertex of such that , , and are all distinct, when , , and are considered modulo . When is viewed as a circular sequence, it should have a subsequence up to D-J equivalence, i,e, , , and . If , then the blow-down obtained by deleting in the subsequence also has three values. If is the only one with the value , then we have for and we have a subsequence . Therefore there is an blow-down obtained by deleting . Since , the blow-down still has three values. By induction, is obtained by consecutive blow-ups from a -characteristic map over and the proof is complete. ∎
The next step is for real toric manifolds over . The following is from [6].
Proposition 5.2**.**
[6, Proposition 3.1]** Let and be two complete non-singular fans with rays in and the matrix
[TABLE]
a characteristic matrix for . Suppose that the two fans are 1-adjacent in the diagram for toric manifolds. Then either of the following holds:
- (1)
two fans are the same. 2. (2)
two fans share a ray generated by .
In the second case, a characteristic matrix for can be written as the following:
[TABLE]
for some and . Conversely, for every , is 1-adjacent to whenever contains a ray generated by .
An edge in is said to be real toric if either or the following holds:
- (1)
there exists so that , and 2. (2)
each piece of determined by does not contain or contains all three values.
An edge satisfying above can be represented by the set , or a circular sequence consisting of , , and whose two points at and are marked with different marks and respectively. For example, consider the edge where
[TABLE]
It can be represented by or . It is real toric because does not contain and contains all three values. On the other hand, the edge is not real toric since contains and has only two values.
Lemma 5.3**.**
An edge in gives a real toric manifold over for some if and only if it is real toric.
Proof.
Let us denote by the mod 2 reduction of a characteristic map . Let us deal with “If” part first. That is, we are given an edge satisfying and . We start from the circular sequence or and apply consecutive blow-ups to finally obtain any edge satisfying and (2). Observe that every edge obtained in this way corresponds to a real toric manifold over by Proposition 5.2. Or, equivalently, one starts from a circular sequence with two different marks satisfying (1) and (2), and performs blow-downs conserving two marks to end with or . First, let us consider the case that each piece of determined by does not contain . Then each piece consists of and and for example it will look like
[TABLE]
together with the two marked ’s. Then one can repeat blow-downs until the piece has length 1, obtaining or . When a piece contains , watch for its two neighbors at and . If they are different to each other, then we can remove at by blow-down. If not, then without loss of generality . Then one searches for at and in the piece and at and and so on. Since the piece contains by condition (2), one can eventually find in the piece. For example, the following piece
[TABLE]
where the underlined is at the position , can be reduced to
[TABLE]
by consecutive blow-downs. Finally we can remove on the underline.
For “only if” part, the condition (1) is a direct consequence of Proposition 5.2. Suppose that the condition (2) does not hold. Then we have a non-singular fan in spanning the upper half-plane whose mod 2 reduction has the form up to D-J equivalence. Then by reflection of across the -axis, we obtain a complete non-singular fan whose mod 2 reduction has only two values and . This is a contradiction with Lemma 5.1 and the proof is done. ∎
Let be an -tuple of positive integer.
Theorem 5.4**.**
A realizable puzzle corresponds to a real toric manifold if and only if either every node of maps to the same node of , or there is a quadruple such that
- (1)
* and ,* 2. (2)
the edges and are real toric, and 3. (3)
every nontrivial edge of maps to either the edge or .
In the latter case, every irreducible realizable square has the form
[TABLE]
Proof.
The proof is an immediate consequence of Proposition 3.2 of [6] and Lemma 5.3. ∎
6. Summary
This self-contained section is the summary for this article. We describe our classification in terms of combinatorial language and provide several counting examples. We shall introduce again all notions required to understand the main results.
Let us fix a positive integer . For a finite set , a weak partition of is a partition of such that some of could be empty. A subset of is said to be non-consecutive if are not contained in for any .
We consider a weak partition of such that all of and are non-consecutive. We call a D-J class over . Two elements and of are said to be -equivalent if for some .
Remark 6.1**.**
It is convenient to describe a D-J class over using a word consisting of alphabets , , and . For example, the D-J class can be written as or ; the representation is unique up to permutation of , , and .
Definition 6.2**.**
For a D-J class over , an ordered pair is called an e-set (compatible with ) if it satisfies the following
- (1)
, 2. (2)
is of even cardinality, and 3. (3)
all elements of are mutually -equivalent, that is, for some .
In particular is always an e-set for any .
Let be an e-set of and fix a vertex . Note that divides to two disjoint subsets and corresponding to “black” and “white” arcs respectively. We can assume that . Then we write .
Definition 6.3**.**
Let and be two e-sets compatible with .
- (1)
Two e-sets and of are type 1 -related if and only if either one of and is empty, or and are -equivalent. 2. (2)
Two e-sets and of are type 2 -related if and only if and are nonempty, and are not -equivalent, and .
In addition, and are said to be -related if they are either type 1 -related or type 2 -related.
For a D-J class over and a positive integer -tuple , we denote by the set of finite sequences of e-sets of compatible with
[TABLE]
such that the members of are mutually -related.
Theorem 6.4**.**
The number of small covers over up to D-J equivalence is
[TABLE]
where runs through all D-J classes over .
For a given , it should be interesting to consider a subset of , which is the set of such that its e-sets are mutually type 1 -related.
Proposition 6.5**.**
Let be a D-J class over , and . Then,
[TABLE]
where w_{i}=\left\{\begin{array}[]{ll}0,&\hbox{ if \mu_{i}=\varnothing;}\\ \sum_{k\in\mu_{i}}(j_{k}-1),&\hbox{otherwise.}\end{array}\right..
Proof.
Note that the number of even-subsets of is . For each , we have to choose such that is an even-subset of . One remarks that if and with a non-empty set , then implies that either or . Therefore, there are cases. Together with the case that every is an empty set, the proposition is proved. ∎
Example 6.6** (Example 2.9 of [4]).**
Assume that and . There are only three D-J classes , , and . For each , -related and are always of type 1, that is, . By Proposition 6.5, we have
[TABLE]
Thus, the number of D-J classes over is
[TABLE]
Example 6.7** (Theorem 8.3 of [8]).**
Assume that and . Then, there are five D-J classes over . Put
[TABLE]
Similarly to Example 6.6, for each , one can see that , and we have where all indices are given by modulo . Therefore, the number of D-J classes over is
[TABLE]
Example 6.8**.**
Assume that and . There are D-J classes over . In order to compute the number of D-J classes over , since we can compute easily due to Proposition 6.5, it is enough to consider . We note that there are only a few possible pairs of which can be type 2 related:
[TABLE]
Here is the list of all type 2 -related pairs for each .
[TABLE]
Hence, one can see that the sum of for all D-J classes over is
[TABLE]
Real toric manifolds over are small covers and there is an analogue of for real toric manifolds which is a subset of . Let us fix a D-J class over and pick two distinct points which are -equivalent. Then we can assume that . The points and divide to two disjoint sets and in the way explained above. In this settings, we say and satisfies property RT with if or , and are all nonempty, for each .
Definition 6.9**.**
Let be a D-J class over for . We denote by the set of satisfying the following; for such a sequence , one can take two elements satisfying property RT with such that every entry of is , , or for . When is the unique D-J class over , we define to be the singleton whose unique element is so that every entry of is for .
Remark 6.10**.**
For , we have a family of D-J classes . When is a D-J class over , if and only if for . In fact, there is no pair and satisfying property RT with for .
Theorem 6.11**.**
The number of real toric manifolds over is
[TABLE]
where runs through all D-J classes over .
Remark 6.12**.**
For any D-J class over and , we have the inclusions
[TABLE]
When , the two inclusions are actually equalities (see Chapter 8 of [8]). But in general, almost all inclusions are strict. Indeed, they are strict if and .
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