Complex Hantzsche-Wendt manifolds

Complex Hantzsche-Wendt manifolds are flat Kähler manifolds with holonomy group Z2n-1⊂SU(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathbb {Z}}}}_2^{n-1}\subset SU(n)$$\end{document}. They are important examples of Calabi-Yau manifolds of abelian type. In this paper we describe them as quotients of a product of elliptic curves by a finite group G~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{G}$$\end{document}. This will allow us to classify all possible integral holonomy representations and give an algorithm classifying their diffeomorphism types.


Introduction
The class of complex Hantzsche-Wendt manifolds was defined in [2] as a complex analogue to the class of Hantzsche-Wendt manifolds. The latter is a class of flat orientable n-dimensional manifolds with holonomy group Z n−1 2 , a generalisation of the original Hantzsche-Wendt manifold which is the unique 3-dimensional manifold with holonomy group Z 2 2 . Any Hantzsche-Wendt manifold is odd-dimensional, thus it cannot be a complex manifold. However, we use the following definition: • The complex dimension of CHW manifold is an odd number. Moreover, it is not hard to construct an example of CHW manifold in any odd dimension (using results on the existence of Hantzsche-Wendt manifold, see [2,Lemma 4.2]). • The Hodge numbers of a CHW manifold M are given by: (for the calculation see [2,Theorem 4.3]). • Any CHW manifold is an algebraic variety (which follows from the equation h 2,0 = 0).
All three-dimensional CHW manifolds were classified (up to diffeomorphism) in [2]. However they were also found independently by different authors. In particular they were listed in [9] as examples of Calabi-Yau threefolds of Type A (more detailed description, which agree with our results, can be found in [4]). We point out that there are many inequivalent definitions of Calabi-Yau manifolds in literature and that the CHW manifolds do not satisfy all of them. However, it is interesting to observe, that the Hodge numbers of CHW manifolds satisfy equations h n,0 = h 0,0 = 1 and h p,0 = 0 for p = 1, 2, . . . , n − 1, which is typical to the Calabi-Yau manifolds.
In this paper we describe the structure of CHW manifolds of any complex dimension n as orbit spaces of a free action of a finite group on the product of n elliptic curves. Details of that description will be given in Theorem 1. In particular for n = 3 it agrees with [4, Proposition 6.2, pt 1.] and [6,Theorem 5.5]. Next we show in Corollary 1 that the integral holonomy representation decomposes as direct product of two subrepresentations, of which the first is diagonal and the second is isomorphic to the diagonal representation over Q but not necessarily over Z. We classify representations of the second type in Proposition 1 using equivalence classes of some subgroups of Z n 2 (using different terminology, isometry classes of binary linear codes of length ≥ 2). After that, for any CHW manifold M we define two matrices (M) and (M) that encode the objects from which M is build up (see Definition 3 and Definition 4). We examine which pairs of matrices , correspond to a CHW manifold (Theorem 3) and we define an equivalence relation (see Definition 5) such that pairs ( , ) and ( , ) are equivalent if and only if corresponding manifolds are diffeomorphic (Theorem 2). This way we obtain an algorithm of classification of fundamental groups of CHW manifolds, which is analogue to the algorithm of classification of Hantzsche-Wendt manifolds given in [8]. At the end of this paper we present results obtained using our algorithm in complex dimension five. There are exactly 8616 diffeomorphism classes of CHW fivefolds. We were not able to determine exact number of CHW manifolds in dimension seven due to hardware restrictions. However, by analysis of our method we find a lower bound (equal to 48321790784) on the number of CHW sevenfolds with diagonal holonomy representation. It is interesting to compare this results with the results on number of Hantzsche-Wendt manifolds: there are two five-dimensional and 62 seven-dimensional HW manifolds ( [7,8]).
Results of this paper were published in the author's PhD Thesis ( [3], in Polish language only).
Let us review some basic facts on flat manifolds, which will be needed further. A flat manifold M is a compact Riemannian manifold with the Riemann curvature tensor equal to zero. It is a quotient space R n / , where = π 1 (M) is a torsion-free, cocompact and discrete subgroup of the group E(n) = O(n) R n of isometries of R n . Those groups are called Bieberbach groups (torsion-free crystallographic groups). They are an essential tool in studying flat manifolds since they determine them up to diffeomorphism. The algebraic structure of crystallographic groups is well understood: by first theorem of Bieberbach any crystallographic subgroup of E(n) fits into short exact sequence: where ⊂ R n is a maximal abelian subgroup of and is a lattice in R n (isomorphic to Z n ) and H = / is a finite group called holonomy group of . If is Bieberbach group, then H is the holonomy of manifold R n / . The short exact sequence above allows to define faithful integral representation of the group H (which will be reffered to as the integral holonomy representation). The second Bieberbach theorem asserts that two crystallographic groups 1 , 2 are isomorphic if and only if they are conjugated in the group A(n) = G L n (R) R n . It follows, that if 1 and 2 are isomorphic, then the images of integral holonomy representations of 1 and 2 are conjugated in G L n (Z) (though representations are not necessarily isomorphic).
Flat Kähler manifolds are characterised by a number of equivalent conditions ([5, Theorem 3.1 and Proposition 3.2]). In particular, is fundamental group of a flat Kähler manifold of complex dimension n if and only if it is discrete, cocompact, torsion-free subgroup of U (n) C n .

Structure theorem
We are going to describe the structure of complex Hantzsche-Wendt manifolds. Assume that M is such manifold. Recall that M is a quotient of the form M = C n / , where is a Bieberbach group. We have short exact sequence: where ⊂ is the maximal abelian subgroup of isomorphic to Z 2n . Since is a lattice in C n , then T = C n / is a complex torus and M is a quotient M = T / Z n−1 2 (where the group Z n−1 2 acts biholomorphically). We shall show that T is isogenous to the product of elliptic curves. Consequently M is a quotient M =T/Ĝ whereT = E 1 × E 2 × . . . × E n is a product of some elliptic curves andĜ is an extension of Z n−1 2 . Let us begin with some notation. Let : Z n−1 2 → SL(V ) be a faithful representation, where dim C V = n. The image of (up to conjugation) is the subgroup of diagonal integer matrices of SU (n) (see [2,proof of Lemma 4.2]). Thus we have decomposition V = V 1 ⊕V 2 ⊕ . . . ⊕ V n . Moreover we can find generators g 1 , g 2 . . . , g n of Z n−1 Lemma 1 Let ⊂ V be a lattice (of rank 2n) and i = V i ∩ . Then E i = V i / i are elliptic curves for i = 1, 2, . . . , n.
Proof Let us consider maps 1 + g i : V → V where 1 is the identity map and 1 ≤ i ≤ n. It is easy to see that for all i we have Im(1 (1 + g i )(λ l ). This means that elements (1 + g i )(λ l ) for l = 1, 2, . . . , 2n spans V i , in other words (1 + g i )( ) is a lattice in V i . Since (1 + g i )( ) ⊂ i , we get our result.

Lemma 2 Let T = V / and E i = V i / i as in previous lemma. Let us define a map
Then μ is an isogeny. Moreover elements (t 1 , t 2 , . . . , t n ) ∈ ker μ are 2-torsion points which cannot have exactly one non-zero entry t i .
To show that any element (t 1 , . . . , t n ) of the kernel of μ is a 2-torsion point, let us observe that for any i = 1 . . . , n we have Now we are ready to prove our structure theorem for CHW manifolds.

Theorem 1 Let n be an odd number. A manifold M is a n-dimensional CHW manifold if and only if there exist:
• elliptic curves E 1 , . . . , E n ,

is a non-zero 2-torsion point and
x i i = t i , which satisfy the following conditions:

. , n} such that m is odd and for any
for which M is of the form: Proof First observe thatĜ is finite. Since W consists of 2-torsion points, then any translation t w by an element of W is an involution commuting with mapsf i for i = 1, . . . , n. Moreover, since x i i is a 2-torsion point,f i also is an involution. Condition (iii) ensures thatf n = f 1f2 . . .f n−1 and from condition (i) any commutator

andĜ is a central extension of this group by a group
It is enough to show that elements ofĜ have not fixed points. Letĝ = t wfi 1f i 2 . . .f i k be an arbitrary element of G. Using relations inĜ we can assume that i 1 < i 2 < . . . < i k and that k is odd. Denote the set {i 1 , . . . , i k } by I . Let also y i denote the coordinates of w, i.e. w = (y 1 , y 2 , . . . , y n ). For any z = (z 1 , . . . , z n ) we have: Ifĝ had a fixed point z, then for all i ∈ I we would have x i I + y i = 0. However, we assumed the contrary in the condition (ii). From the definition of M it is clear now, that M is a complex flat manifold. It is also clear that the holonomy group of M is isomorphic to the group generated by linear parts of the mapsf i (that is to the group Z n−1 2 ) and that this group consists of matrices of determinant one. Thus M is a CHW manifold.
. We also know from Lemma 2 that the kernel of the isogeny μ : n i=1 E i → T given by μ(z 1 , z 2 , . . . , z n ) = n i=1 z i consists of 2-torsion points. This kernel will coincide with the group W . However, at this moment we only see that it is a subgroup of group of 2-torsion points which is isomorphic to Z 2n 2 . We can choose n − 1 elements (g 1 , x 1 ), . . . , (g n−1 , x n−1 ) of the group where the linear parts g i are defined by diagonal matrices (where the only one 1 entry is on i-th place) in a basis corresponding to the decomposition of V . Then we define (x 1 i , . . . , x n i ) as an arbitrarily chosen element of μ −1 (x i + ). In case i = n we use condition (iii) to define elements x j n , then μ(x 1 n , . . . , For any i, map g i yields an automorphismg i of the torus T . Similarly we have biholomorphic maps f i : Observe that from Lemma 2 this equality holds also in E i . In similar way from relation Hence x i j ∈ ker μ, where x i j is an element defined as in condition (i). Next, since is torsion-free, then the biholomorphic maps f i have no fixed points (if any of them had a fixed point, then it would be easy to find an element of which does not act freely). A map f : T → T given by equation f (z) = g(z) + x (where g is an automorphism of T ) has no fixed points if and only if x / ∈ im(id − g). Observe that any element of the group generated by automorphismsg i is of the formg . This is equivalent to the condition (ii) from the definition of elements x j I (because μ(x 1 I , . . . , x n I ) =x I ). Applying above condition to the special case I = {i} we get, that there is no element (y 1 , . . . , y n ) ∈ ker μ such that y i = x i i . Thus x i i = 0. It follows also that for any i and for any two elements (y 1 , . . . , y n ) and (y 1 , . . . , y n ) of ker μ if both i-th entries y i and y i are non-zero, then they are equal. This allows us to define elements t i ∈ E i and shows that W = ker μ is a subgroup of a group isomorphic to Z n−1 2 . To finish the proof, observe that directly from the definition ofĜ we get M = V/ = T/ = T 1 × · · · × T n /Ĝ.

Integral holonomy representation
It is known that Hantzsche-Wendt manifolds have diagonal integral holonomy ([10, Theorem 3.1]). This cannot be true for complex Hantzsche-Wendt manifolds. Indeed, let n be an odd number and diag be a faithful representation of Z n−1 2 in the SL n (Z) such that its image consists of diagonal matrices. We take any representation of Z n−1 2 such that it is isomorphic to diag over Q but not over Z. Then we are able to construct Hantzsche-Wendt manifold with integral holonomy representation ⊕ diag . We shall show that the structure theorem implies that any integral holonomy representation of CHW manifold is of that form. We will denote by L diag a module corresponding to the representation diag . Proof Let ⊂ be a sublattice defined as = n i=1 i , where i are defined as in Lemma 1. For any i, lattice i has two generators which we denote by λ 1 i and λ 2 i . This gives us following decomposition: where both summands are diagonal modules. Now observe that the group of 2-torsion points in T i = V i / i is generated by 1 2 λ 1 i + i and 1 2 λ 2 i + i . Without loss of generality we assume that Since is generated by elements of and by elements of W = ker μ, then we have decomposition: The first component is the diagonal module. The second component is diagonal over Q because the set of elements {λ 2 i : 1 ≤ i ≤ n} is its basis.
First step in order to classify fundamental groups of CHW manifolds is to find all integral representations of odd degree n (in fact, their images in G L(n, Z)). It is possible to do it in terms of the subgroup W . Proof Let L be a Z n−1 2 -module corresponding to the integral representation : Z n−1 Fix a set g 1 , . . . , g n−1 of generators of Z n−1 2 in a such way, that for any g i its eigenspace (over Q) of 1 is one-dimensional. Denote also g n = g 1 g 2 . . . g n . Define: such that • f σ is isomorphic to . Thus L / and L/ represent the same class in Sub(n). Conversely, if W and W are representatives of the same class [W ] = [W ] ∈ Sub(n), then there exists σ such that σ W = W . Let and be a representations associated to W and W . Changing the order of generators according to σ we get an isomorphism of these representations, thus their images are conjugate.

Remark 1
We can think of subgroups of Z n−1 2 as subspaces of the vector space F k 2 . Such subspaces are also known as binary linear codes. Recall that the Hamming distance of two elements of such a code is the number of positions where these elements differ. The distance of a linear code is the minimal Hamming distance between two different elements of this code. A linear isometry of codes is a linear map between codes that preserves the Hamming distance. Using this language, the set Sub(n) is the set of linear isometry classes of (n, * , ≥ 2) binary codes (where * can be any number between 0 and n − 1).
There is a notion of dual code (subspace of vectors orthogonal to a given code with respect to the standard bilinear form). It is easy to see that a code dual to code of distance ≥ 2 is a code that has no column of zeros in its generator matrix. Such codes were studied in [11].
In particular, using notation from that paper, |Sub(n)| = n−1 k=1 S nk . Further informations on enumeration and generation of such codes can be found in [1].
For each of this classes there is only one manifold with corresponding holonomy representation.

Fundamental groups
Now we are in position to classify CHW manifolds up to diffeomorphism, in other words, to classify their fundamental groups. We will construct an algorithm which can be used to list all such Bieberbach groups. It can be also used to check if two CHW manifolds are diffeomorphic. The algorithm will be similar to the algorithm for HW manifolds obtained in [8].
Assume that M is a CHW manifold. According to Theorem 1, we can associate to M a collection of elements x j i , t i for i, j = 1, 2, . . . , n and a subgroup W of group generated by elements t i . Recall that for all i = j an element x i j (defined in Theorem 1) is an element of W . Thus for every k = 1, 2, . . . n we have This allows us to introduce the following definition.

Definition 3 Let M be a CHW manifold and let {x
j i , t i | 1 ≤ i, j ≤ n} be a set of elements associated to M as in Theorem 1 (we use also notation x k i j from that Theorem). Then we define (M) as a binary n × n matrix with coefficients ψ i j given by:

Proposition 2 Let M be a CHW manifold of complex dimension n together with a fixed set
of elements x j i and t j , for 1 ≤ i, j ≤ n and a subgroup W ⊂ t 1 , . . . , t n , defined as in the proof of Theorem 1. Then the matrix (M) has following properties: 1. ψ ii = 0 for i = 1, 2, . . . , n, 2. ψ 21 = ψ 12 = ψ 13 = . . . = ψ 1n = 0, 3. for all i = j there exists an element w = (w 1 , . . . , w n ) ∈ W such that w i = w j = 0 and for any k / ∈ {i, j} we have w k = t k ⇐⇒ ψ ik = ψ jk , 4. sum of the coefficients in every column is 0 (mod 2).
Proof The first statement follows from Theorem 1, as x i i is a 2-torsion point. The second statement follows directly from the definition of the matrix (M) and the definition of x k i j . The remaining statements are easily implied by conditions (i) (because we can take w = x i j ) and (iii) of Theorem 1.

Corollary 2 If CHW manifold M is a threefold, then (M) is a zero matrix. If M is any CHW manifold such that the associated subgroup W has at most two elements (in other words, binary linear code corresponding to the holonomy representation is at most one-dimensional) then (M) is a zero matrix.
Proof From the two first statements of the Proposition 2, if n = 3 then (M) has only three coefficients: ψ 23 , ψ 31 and ψ 32 which could be non-zero. However as there is only one such an element in every column of (M), from the last statement of Proposition 2 we conclude that (M) = 0.
Assume now that W has at most 2 elements. If it has only 1 element (that is W is trivial) then obviously (M) = 0. So assume that |W | = 2 and let w = (w 1 , . . . , w n ) be the nonzero element of W . Let i be an index such that w i = 0. Then if we take any j = i and use third statement from the Proposition 2, then we get that for any k / ∈ {i, j} we have ψ ik = ψ jk (as the only one element of W with 0 on the i-th coordinate is the trivial element). This shows that for any k = i, in k-th column of matrix (M) all elements but the i-th and the k-th are the same. The k-th element in the k-th column is always 0 (this is the first statement in Proposition 2). Since n is an odd number and all elements in every column of sum up to 0, then all elements in every column (but the i-th) which are not on the diagonal are equal. From the second statement of Proposition 2 we see that these columns consist only of zeros. Since we could find another number l = i such that w l = 0, then the i-th column has the same property and is a zero matrix. Now we shall define second matrix associated to a CHW manifold. We will denote by Z 2 [τ ] an additive group on elements {0, 1, τ, 1 + τ } isomorphic to the Klein 4-group. If M is a CHW manifold and E 1 , . . . , E n are elliptic curves defined as in Lemma 1, then for each i = 1, 2, . . . , n we define isomorphisms κ i from the group E i (2) of 2-torsion points of E i to the group Z 2 [τ ] by the values on generators:

Definition 4
Let M be a CHW manifold of complex dimension n with associated collection of objects {x j i , t i | 1 ≤ i, j ≤ n} as in Theorem 1. Let (M) be the matrix defined in the Definition 3. Then (M) is a n × n matrix with coefficients φ i j ∈ Z 2 [τ ] defined as follows: where k ∈ {1, 2, . . . , n} is the smallest number such that k = j and ψ k j = ψ i j .
Observe, that since ψ k j = ψ i j , then 2(x where l = 1 for j > 1 and l = 2 for j = 1. Thus matrix (M) is well defined.

Proposition 3 Let M be a CHW manifold of complex dimension n together with a fixed set
of elements x j i and t j , for 1 ≤ i, j ≤ n. Then the matrix (M) has following properties: 1. φ 21 = φ 12 = φ 13 = . . . = φ 1n = 0, 2. if k j is the smallest index such that ψ k j j = 1 then φ k j j = 0, 3. for all j ∈ {1, 2, . . . , n} sum of the elements of the j-th column of (M) is given by the equation: Proof The first two properties follows directly from the definition of (M). The last property is implied by the condition (iii) of the Theorem 1. Indeed, we have: where l = 2 if j = 1 and l = 1 otherwise. Let k be the minimal index such that ψ k j = 1.
But from definition of (M), equality ψ k j = 1 means that 2(x Because κ j is an isomorphism, we get our result.

Theorem 2 Let M, M be a CHW manifolds of complex dimension n with isomorphic integral holonomy representations corresponding to W ⊂ Z n 2 . Assume that (M), (M), (M) and (M) are matrices associated to M and M respectively. Then M is diffeomorphic to M if and only if the pairs ( (M), (M)) and ( (M ), (M )) are equivalent (in the sense of Definition 5).
Proof At the beginning of the proof we shall show that a pair of matrices ( (M), (M)) determines a diffeomorphism type of M. Assume that M and M are CHW manifolds of complex dimension n (with the same holonomy representation corresponding to W ) to which we associated the same matrices, (M) = (M ) and (M) = (M ). These matrices were defined using some objects, which we will denote x j i , t j ∈ E j (for manifold M) and (x are the maximal lattices. Moreover it follows that for any i = 1, 2, . . . , n we have E i = (E i ) . Up to diffeomorphism we may also assume, that x i i = (x i i ) and t i = (t i ) , for all i = 1, 2, . . . , n. Let l j = 1 if j > 1, l 1 = 2. We will show that without any loss of generality we may assume that x j l j = (x j l j ) . Indeed, if it is not the case, take a ∈ V such that the coset of a in T = V / is equal to Consider the matrices: Lattice j has a basis a, b such that 1 2 a + = x j j and 1 2 b + = t j . Using this basis we can see that each of the above defined matrices induces a diffeomorphism of the elliptic curve E j to itself. These diffeomorphisms fix all 2-torsion points. Moreover for any two points x, y from the set { 1 2 t j , − 1 2 t j , x j j + 1 2 t j , x j j + 1 2 t j } we can find i such that diffeomorphism induced by A i maps x to y. Thus up to diffeomorphism we can assume that since W = ker μ. This means that under this operation the fundamental group is unchanged. Here it may be necessary to use all but the first normalization. We will need the second normalization for the same reason as above. Third normalization must be applied if we add to the i-th row of (M) a vector τ w such that w i = 0. In this case the element x i i would be changed to x i i + t i , but then we would rename the 2-torsion points of E i (see the proof of Theorem 1). This renaming is precisely the third normalization. Observe also that adding a vector to a single row of (M) corresponds to a different choice of elements x j i , and this implies also that elements x j n would be chosen differently (which follows from the condition (iii) of the Theorem 1). 3. Now we will consider the remaining operation. Let k be a number such that for any w ∈ W we have w k = 0. Then there are two 2-torsion points which could be named t k equally well and the third operation corresponds to the different choice of this element. Clearly here we do not need any normalization.
The coefficients of (M) and (M) are determined by the points x j i (and implicitly by t j ). These points are not uniquely determined though. In the next step we shall show that different choice of points x j i gives rise to equivalent pair of matrices. There are two cases. First assume, that k is a number such that there exists w ∈ W with w k = 0. This determines the choice of an element t k . However, since the n-tuple (x 1 k , x 2 k , . . . , x n k ) is determined up to elements of ker μ, then there are two such n-tuples which differ on the k-th place by t k . This means that there are two possible choices of x k k . Transition between them can be done precisely by the second operation (adding a vector to a row of ) together with the fourth normalization. In the second case, for all w ∈ W we have w k = 0. Then the point x k k is uniquely determined, but t k could be equally well substituted by t k + x k k . This corresponds to the third operation from Definition 5. • If a = 0, then it follows from direct calculation that = A . Moreover since the holonomy representations are isomorphic, then A induce an isomorphism φ A of elliptic curves E i and E i . Thus we can assume that elements (x The theorem above can be used to check whether two CHW manifolds are diffeomorphic in terms of matrices (M) and (M). However, in order to classify all diffeomorphism types of n-dimensional CHW manifolds for a given odd number n, we need to know which pairs of matrices from the set M (defined in the Definiton 5) correspond to a CHW manifold. For this reason we introduce following condition.

Definition 6
Let n be an odd number and ( , ) ∈ M (where the set M is defined as in the Definition 5). Let φ i j and ψ i j denote the coefficients of matrices and respectively. We say that ( , ) satisfies the torsion-free condition, if for any I = {i 1 , . . . , i m } ⊂ {1, 2, . . . , n} of odd size m there exists j = i p ∈ I such that: where ψ I j ∈ Z is a number defined by equation: (by direct calculation similar to the one used in the proof of Proposition 3). If ψ I j is even, then we get and thus the torsion-free condition is satisfied. Now we assume that all equivalent pairs to ( , ) satisfy the torsion-free condition. To construct a CHW manifold M we use matrices and . Precisely, we take any elliptic curves E 1 , E 2 , . . . , E n , and for any i ∈ {1, 2, . . . , n} we choose two different 2-torsion points x i i , t i ∈ E i . Then by Theorem 1 we need also a group W which is the subgroup to which the matrices ( , ) are associated, and points x j i which we obtain in following way: x j i = κ j (φ i j ) + ψ i j y j , where y j ∈ E j is a point such that 2y j = t j (which we may choose arbitrarily but only once). Conditions (i) and (iii) of Theorem 1 are then easily satisfied from the properties of matrices and . The only thing we need to prove is that the condition (ii) is also satisfied. Assume the contrary, that there exists w = (w 1 , w 2 , . . . , w n ) and a subset I ⊂ {1, 2, . . . n} of odd size such that for every j ∈ I we have x  1, 1, 1, 0) 0 1290 (1, 1, 1, 1, 1) 0 420