ELLIPTIC GENERA, TORUS MANIFOLDS AND MULTI-FANS

AKIO HATTORI AND MIKIYA MASUDA

1. Introduction

The rigidity theorem of Witten-Bott-Taubes-Hirzebruch [W, BT, H] tells us that, if the circle group acts on a closed almost complex (or more generally unitary) manifold whose first Chern class is divisible by a positive integer N greater than 1, then its equivariant elliptic genus of level N is rigid. Applying this to a non-singular compact toric variety we see that its elliptic genus of level N is rigid if its first Chern class is divisible by N. But, using a vanishing theorem of Hirzebruch [H], we can show moreover that the genus actually vanishes. In this note we shall extend this result to torus manifolds [M, HM]. A torus manifold is an oriented closed manifold of even dimension which admits an action of a torus of half the dimenstion of the manifold with some orientation data concerning codimension two fixed point set components of circle subgroups. Though there is not an almost complex nor unitary structure on a torus manifold in general, one can still define elliptic genus of level N on torus manifolds. In the almost complex or unitary case the elliptic genus is defined as the index of a twisted Dirac operator, so that the equivariant index is a virtual character of the torus acting on the manifold. For general torus manifolds the equivariant elliptic genus of level N is defined in terms of the multifan associated with the torus manifold. Thus the genus is in fact defined for multi-fans (precisely speaking for complete simplicial multi-fans). The fact that it is a character of a torus is proved by using the multiplicity formula for Duistermaat-Heckman function for multi-polytopes given in [HM]. The Chern class is also defined for multi-fans, and rigidity and vanishing of elliptic genus of level N can be formulated and proved. One of the main results is Corollary 4.4 which states that, if the first Chern class of a complete non-singular multi-fan is divisible by N, then the elliptic genus of level N vanishes. The proof of rigidity and vanishing of level N elliptic genus follows the idea of the proof given in [H]. The formula for Duistermaat-Heckman function is also used to modify topological terms into combinatorial terms so as to be applicable to multi-fans. When N = 2, the torus manifold is a spin manifold. The corresponding multi-fan might be called a spin multi-fan. As a corollary we see that its signature vanishes in this case.

The T_{y}-genus can be considered as a special value of equivariant elliptic genus. Thus, if the first Chern class is divisible by N, then T_{y}-genus vanishes for (−y)^{N} = 1. One can derive some applications from this fact. For example if Δ is a complete non-singular multifan of dimension n with first Chern class c_{1}(Δ) divisible by N and with non-vanishing Todd genus, then N must be equal or less than n + 1 (Proposition 5.2). In the extremal case N = n + 1, if Δ is assumed to be a complete non-singular ordinary fan, then Δ must be isomorphic to the fan of projective space ℙ^{n}. Hence a complete non-singular toric variety M of dimension n with c_{1}(M) divisible by n + 1 must be isomorphic to ℙ^{n} as toric varity (Corollary 5.4). We show furthermore that, in case c_{1}(M) is divisible by n, M is isomorphic to a certain projective space bundle over ℙ^{1} (Corollary 5.8).

The paper is organizd as follows. In Section 2 we recall some basic facts about multifans from [HM]. In Section 3 we define equivariant first Chern class of multi-fans and discuss properties concerning its divisibility. Section 4 is the main part of the paper. The equivariant elliptic genus of complete non-singular multi-fans is introduced and the main results are stated and proved here. The main results are Theorem 4.3 and Corollary 4.4. Section 5 is devoted to applications. Finally in Section 6 we discuss phenomena which arises when the multi-fan is not assumed to be non-singular.

2. Multi-fans

We refer to [HM] for notions and notations concerning multi-fans and torus manifolds. We shall summarize some of them. Let Δ = (Σ, C, w^{±}) be an n-dimensional complete non-singular multi-fan. Here Σ is an augmented simplicial set which means that Σ is a simplicial set with empty set ∗ = ∅ added as (−1)-dimenional simplex. Σ^{(k)} denotes the k − 1 skeleton of Σ so that ∗ ∈ Σ^{(0)}. We assume that Σ = ∑_{k=0}^{n} Σ^{(k)}. There is associated with Δ an n-dimensional lattice L (the notation N was used in [HM]). C is a map from Σ^{(k)} into the set of k-dimensional strongly convex rational polyhedral cones in the vector space L_{ℝ} = L ⊗ ℝ for each k such that, if J is a face of I, then C(J) is a face of C(I). w^{±} are maps Σ^{(n)} → ℤ which, when Σ is complete, satisfy certain compatibility conditions. We set w(I) = w^{+}(I) − w^{−}(I). T will denote an n-dimensinal torus. We identify Hom(S^{1}, T) = H_{2}(BT) with the lattice L. Thus there is a unique primitive vector v_{i} ∈ H_{2}(BT) which generates the cone C(i) for each i ∈ Σ^{(1)}. Non-singularity of Σ means that {v_{i} | i ∈ I} is a basis of the lattice L = H_{2}(BT) for each I ∈ Σ^{(n)}. Let {u_{i}^{I}} be the dual basis of {v_{i} | i ∈ I} in the dual lattice L^{∗} = H^{2}(BT). H^{2}(BT) is canonically identified with Hom(T, S^{1}). The latter is embedded in the character ring R(T). In fact R(T) can be considered as the group ring of the group Hom(T, S^{1}). It is convenient to write the element in R(T) corresponding to u ∈ H^{2}(BT) by t^{u}. The homomorphism v^{∗} : R(T) → R(S^{1}) induced by an element v ∈ H_{2}(BT) = Hom(S^{1}, T) can be written in the form

v^{∗}(t^{u}) = t^{⟨u,v⟩},

where ⟨u, v⟩ is the dual pairing and t^{m} ∈ Hom(S^{1}, S^{1}) ⊂ R(S^{1}), m ∈ ℤ, is defined by t^{m}(λ) = λ^{m}.

We define the equivariant cohomology H_{T}^{∗}(Δ) of a complete non-singular multi-fan Δ as the face ring of the simplicial complex Σ. Namely let {x_{i}} be indeterminates indexed by Σ^{(1)}, and let R be the the polynomial ring over the integers generated by {x_{i}}. We denote by I the ideal in R generated by monomials ∏_{i∈J} x_{i} such that J ∉ Σ. H_{T}^{∗}(Δ) is by definition the quotient R/I. We regard H^{2}(BT) as a submodule of H_{T}^{2}(Δ) by the formula

u = ∑_{i∈Σ^{(1)}}⟨u, v_{i}⟩x_{i}. (1)

This determines an H^{∗}(BT)-module structure of H_{T}^{∗}(Δ). For each I ∈ Σ^{(n)} we define the restriction homomorphism i_{I}^{∗} : H_{T}^{∗}(Δ) → H^{∗}(BT) by

i_{I}^{∗}(x_{i}) = \begin{cases}u_{i}^{I} & for i ∈ I \\ 0 & for i ∉ I. \end{cases} (2)

It follows from (1) that i_{I}^{∗}|H^{2}(BT) is the identity map for any I, and ∑_{I∈Σ^{(n)}} i_{I}^{∗} is injective.

The following lemma is a consequence of Corollary 7.4 in [HM].

Lemma 2.1. For any x = ∑_{i∈Σ^{(1)}} c_{i}x_{i} ∈ H_{T}^{2}(Δ), c_{i} ∈ ℤ, the element

∑_{I∈Σ^{(n)}} w(I)\frac{t^{i_{I}^{∗}(x)}}{∏_{i∈I}(1 − t^{−u_{i}^{I}})}

in the quotient ring of R(T) actually belongs to R(T).

We also use an extended version of Corollary 7.4 in [HM]. Given K ∈ Σ^{(k)} an (n − k)- dimensional multi-fan Δ_{K} = (Σ_{K}, C_{K}, w_{K}^{±}), which we called projected multi-fan, was defined in [HM]. Σ_{K} consists of such I ∈ Σ that K ⊂ I. If I is in Σ^{(l)} then I is considered as lying in Σ_{K}^{(l−k)}. An element of Σ_{K}^{(1)} is of the form K ∪ {i} which we identify with i. When K = ∅ then Δ_{K} = Δ. The lattice L_{K} is the quotient of L by the susbmodule generated by {v_{i} | i ∈ K}, and C_{K}(I) ∈ L_{K} ⊗ ℝ is the projection image of C(I) ∈ L_{ℝ} = L ⊗ ℝ. The torus T_{K} corresponding to L_{K} is a quotient of T. We consider the polynomial ring R_{K} generated by {x_{i} | i ∈ K ∪ Σ_{K}^{(1)}} and the ideal I_{K} generated by monomials ∏_{i∈J} x_{i} such that J ∉ Σ_{K}. We define the equivariant cohomology H_{T}^{2}(Δ_{K}) of Δ_{K} with respect to the torus T as the quotient ring R_{K}/I_{K}. H^{2}(BT) is regarded as a submodule of H_{T}^{2}(Δ_{K}) by a formula similar to (1). The projection H_{T}^{2}(Δ) → H_{T}^{2}(Δ_{K}) is defined by putting x_{i} = 0 for i ∉ K ∪ Σ_{K}^{(1)}, which is an H^{2}(BT)-module homomorphism. The restriction homomorphism i_{I}^{∗} : H_{T}^{2}(Δ_{K}) → H^{2}(BT) is also defined for I ∈ Σ_{K}^{(n−k)}. Given x = ∑_{i} c_{i}x_{i} ∈ H_{T}^{2}(Δ_{K}; ℝ), c_{i} ∈ ℝ, let A^{∗} be the affine subspace in the dual space L_{ℝ}^{∗} defined by ⟨u, v_{i}⟩ = c_{i} for i ∈ K. Then we introduce a collection F_{K} = {F_{i} | i ∈ Σ_{K}^{(1)}} of affine hyperplanes in A^{∗} by setting

F_{i} = {u | u ∈ A^{∗}, ⟨u, v_{i}⟩ = c_{i}}.

The pair P_{K} = (Δ_{K}, F_{K}) will be called a multi-polytope associated with x; see [HM] for the case K = ∅. For I ∈ Σ_{K}^{(n−k)}, i.e. I ∈ Σ^{(n)} with I ⊃ K, we put u_{I} = ∩_{i∈I}F_{i} ∈ A^{∗}. Note that u_{I} is equal to i_{I}^{∗}(x). The dual vector space L_{K}^{∗} ⊗ ℝ of L_{K} ⊗ ℝ is canonically idenitified with the subspace {u | ⟨u, v_{i}⟩ = 0, i ∈ K} of L^{∗} ⊗ ℝ = H^{2}(BT; ℝ). It is parallel to A^{∗}, and u_{i}^{I} lies in L_{K}^{∗} ⊗ ℝ for I ∈ Σ_{K}^{(n−k)} and i ∈ I \ K. A vector v ∈ L_{K} ⊗ ℝ is called generic if ⟨u_{i}^{I}, v⟩ ≠ 0 for any I ∈ Σ_{K}^{(n−k)} and i ∈ I \ K. We take a generic vector v ∈ L_{K} ⊗ ℝ, and define, for I ∈ Σ_{K}^{(n−k)} and i ∈ I \ K,

(−1)^{I} := (−1)^{#\{j∈I\K|⟨u_{j}^{I},v⟩>0\}} and (u_{i}^{I})^{+} := \begin{cases}u_{i}^{I} & if ⟨u_{i}^{I}, v⟩ > 0 \\ −u_{i}^{I} & if ⟨u_{i}^{I}, v⟩ < 0. \end{cases}

We denote by C_{K}^{∗}(I)^{+} the cone in A^{∗} spanned by the (u_{i}^{I})^{+}, i ∈ I \ K, with apex at u_{I}, and by ϕ_{I} its characteristic function. With these understood, we define a function DH_{P_{K}} on A^{∗} \ ∪_{i}F_{i} by

DH_{P_{K}} := ∑_{I∈Σ_{K}^{(n−k)}}(−1)^{I}w(I)ϕ_{I}.

As in [HM] we call this function the Duistermaat-Heckman function associated with P_{K}.

Lemma 2.2. The support of the function DH_{P_{K}} is bounded, and the function is independent of the choice of generic vector v.

The proof is similar to that of Lemma 5.4 in [HM]. We shall denote by P_{K+} the multi-polytope associated with x_{+} = ∑_{i∈K} c_{i}x_{i} + ∑_{i∈Σ_{K}^{(1)}}(c_{i} + ϵ)x_{i} where 0 < ϵ < 1. The following theorem is a generalization of Corollary 7.4 in [HM].

Theorem 2.3. Let x and x_{+} be as above with all c_{i} integers. Then

∑_{u∈A^{∗}∩L^{∗}} DH_{P_{K+}}(u)t^{u} = ∑_{I∈Σ_{K}^{(n−k)}} w(I)\frac{t^{i_{I}^{∗}(x)}}{∏_{i∈I\K}(1 − t^{−u_{i}^{I}})}.

In particular the right hand side belongs to R(T).

The proof is similar to that of Corollary 7.4 in [HM].

Take I ∈ Σ_{K}^{(n−k)} and let G_{I} be the subgroup of the permutation group of I consisting of those elements which are identity on K. Let L_{I} be the set of all linear forms ∑_{i∈I} m_{i}u_{i}^{I} with integer coefficients m_{i}. The group G_{I} acts on L_{I}. Let O_{I} denote the set of orbits of that action. If I′ is also in Σ_{K}^{(n−k)}, take a bijection f : I → I′ which is the identity on K. It induces a bijection f_{∗} : O_{I} → O_{I′}. It is easy to see that f_{∗} does not depend on the choice of particular f. Thus we can write them simply O. For α ∈ O we define t_{I}^{α} by

t_{I}^{α} = ∑_{l∈α} t^{l},

where α is regarded as contained in O_{I} in the sum at the right hand side.

Corollary 2.4. For any α ∈ O the expression

∑_{I∈Σ_{K}^{(n−k)}} w(I)\frac{t_{I}^{α}}{∏_{i∈I\K}(1 − t^{−u_{i}^{I}})}

belongs to R(T).

Proof. Let Ḡ denote the subgroup of permutation group of K ∪ Σ_{K}^{(1)} consisting of those elements which are identity on K. It acts on the set L̄ of linear forms l = ∑_{i∈K∪Σ_{K}^{(1)}} m_{i}x_{i}. Let Ō denote the set of orbits of that action on L̄. There is an obvious map of O in Ō. We denote the image of α ∈ O in Ō by ᾱ. We define the length of a linear form l as the maximal number of i’s with i ∈ Σ_{K}^{(1)} and m_{i} ≠ 0. The length is invariant under the action of Ḡ, so that the length |ᾱ| of ᾱ ∈ Ō is defined as that of a linear form contained in ᾱ. The length is also defined for α ∈ O independently of I and we have |α| = |ᾱ|. The proof will proceed by induction on the length of α. If |α| = 0, then it is clear that α consists of a single linear form l = ∑_{i∈K} m_{i}x_{i}. Applying Theorem 2.3 to x = l we see that the statement of Corollary is true in this case. Suppose that |α| > 0. Then it is not difficult to see that

∑_{l∈ᾱ} t^{i_{I}^{∗}(l)} = t_{I}^{α} + ∑_{|β|<|α|} a_{β}t_{I}^{β}

where a_{β} ∈ ℤ does not depend on I. We apply Theorem 2.3 and use induction assumption to conclude that

∑_{I∈Σ_{K}^{(n−k)}} w(I)\frac{t_{I}^{α}}{∏_{i∈I\K}(1 − t^{−u_{i}^{I}})}

belongs to R(T).

If we take K to be the empty set, the G_{I} is the permutation group of I, and t_{I}^{α} is a symmetric character with respect to G_{I}. We obtain

Corollary 2.5. For any α ∈ O the expression

∑_{I∈Σ^{(n)}} w(I)\frac{t_{I}^{α}}{∏_{i∈I}(1 − t^{−u_{i}^{I}})}

belongs to R(T).

3. Equivariant first Chern class

Let Δ be a complete non-singular multi-fan as in the previous section. The class

∑_{i∈Σ^{(1)}} x_{i}

will be called equivariant first Chern class of Δ, and will be denoted by c_{1}^{T} (Δ). Its image c_{1}(Δ) in H^{2}(Δ) = H_{T}^{2}(Δ)/H^{2}(BT) is called first Chern of Δ. Let N > 1 be an integer. The first Chern class c_{1}(Δ) is divisible by N if and only if c_{1}^{T} (Δ) is of the form

c_{1}^{T} (Δ) = Nx + u, x ∈ H_{T}^{2}(Δ), u ∈ H^{2}(BT).

We set u^{I} = i_{I}^{∗}(c_{1}^{T} (Δ)) = ∑_{i∈I} u_{i}^{I} ∈ H^{2}(BT).

Lemma 3.1. The following three conditions are equivalent:

(1) the first Chern class c_{1}(Δ) is divisible by N,

(2) u^{I} mod N is independent of I ∈ Σ^{(n)},

(3) there is an element u ∈ H^{2}(BT) such that ⟨u, v_{i}⟩ = 1 mod N for all i ∈ Σ^{(1)}.

Proof. Suppose c_{1}^{T} (Δ) is of the form

c_{1}^{T} (Δ) = Nx + u, x ∈ H_{T}^{2}(Δ), u ∈ H^{2}(BT).

Then u^{I} = i_{I}^{∗}(c_{1}^{T} (Δ)) mod N is equal to i_{I}^{∗}(u) = u, and hence is independent of I. Thus (1) implies (2).

Suppose that u mod N is equal to u^{I} mod N for any I ∈ Σ^{(n)}. Then ⟨u, v_{i}⟩ = ⟨u^{I}, v_{i}⟩ mod N, and hence ⟨u, v_{i}⟩ = ∑_{j∈I}⟨u_{j}^{I}, v_{i}⟩ = 1 mod N for i ∈ I. Thus (2) implies (3).

Suppose ⟨u, v_{i}⟩ = 1 mod N for any i ∈ Σ^{(1)}. Then, by (1),

c_{1}^{T} (Δ) − u = ∑_{i∈Σ^{(1)}}(1 − ⟨u, v_{i}⟩)x_{i} = 0 mod N.

Hence c_{1}^{T} (Δ) is of the form c_{1}^{T} (Δ) = Nx + u because H_{T}^{2}(Δ) is a free module. Thus (3) implies (1).

Remark. Let Ĥ_{T}^{2}(M) = H_{T}^{2}(M)/H^{∗}(BT)-torsion. In [M] it was shown that, if M is a torus manifold and Δ(M) is its associated multi-fan, then there is a canonical identification of H_{T}^{2}(Δ(M)) with Ĥ_{T}^{2}(M), and, in case M is a unitary torus manifold, c_{1}^{T} (M) ∈ H_{T}^{2}(M) descends to c_{1}^{T} (Δ). It follows that, if M is a unitary torus manifold and c_{1}(M) is divisible by N, then c_{1}(Δ(M)) is also divisible by N.

Let v ∈ H^{2}(BT) be a generic integral vector. If we fix I and write

v = ∑_{i∈I} m_{i}v_{i},

then m_{i} = ⟨u_{i}^{I}, v⟩. Since u^{I} = ∑_{i∈I} u_{i}^{I}, we have ⟨u^{I}, v⟩ = ∑_{i} m_{i}. Let m ≥ 1 be an integer. Fixing a generic integral vector v and m, we put

I_{(m)} := {i ∈ I | m does not divide m_{i}}

for I ∈ Σ^{(n)}. It will be called modm face of I.

Lemma 3.2. Let v, m and I be as above, and let K = I_{(m)} be the mod m face of I. If I′ ∈ Σ^{(n)} contains K, then K is also the modm face of I′. Moreover, ⟨u_{i}^{I′}, v⟩ = ⟨u_{i}^{I}, v⟩ mod m for i ∈ K.

Proof. We put m_{i} = ⟨u_{i}^{I}, v⟩ and m′_{i′} = ⟨u_{i′}^{I′}, v⟩. Then we have

v = ∑_{i∈I} m_{i}v_{i} = ∑_{i′∈I′} m′_{i′}v_{i′}.

By assumption m_{i} = 0 mod m for i ∉ K. Hence

∑_{i∈K} m_{i}v_{i} = ∑_{i′∈I′} m′_{i′}v_{i′} mod m.

or

∑_{i∈K}(m′_{i} − m_{i})v_{i} + ∑_{i′∈I′, i′∉K} m′_{i′}v_{i′} = 0 mod m.

Since {v_{i′} | i′ ∈ I′} is a basis of the free module H_{2}(BT), we see that m′_{i′} = 0 mod m for i′ ∈ I′, i′ ∉ K and m′_{i} = m_{i} mod m for i ∈ K.

We shall say that I and I′ are (v, m)-equivalent and write I ∼ I′ if I_{(m)} = I′_{(m)}. This defines an equivalence relation ∼ in Σ^{(n)}. Lemma 3.2 implies that its equivalence class X forms a projected multi-fan Δ_{K} = (Σ_{K}, C_{K}, w_{K}^{±}) where K is the common mod m face of the members of X. We shall call this K the core of the equivalence class X.

Lemma 3.3. Let X be an equivalence class of (v, m)-equivalence relation. For x ∈ H_{T}^{2}(Δ) the value ⟨i_{I}^{∗}(x), v⟩ mod m does not depend on the choice of I in X.

Proof. Write x = ∑_{i∈Σ^{(1)}} a_{i}x_{i}. Let K denote the common mod m face of I ∈ X. Then

⟨i_{I}^{∗}(x), v⟩ = ∑_{i∈K} a_{i}⟨u_{i}^{I}, v⟩ + ∑_{i∈I,∉K} a_{i}⟨u_{i}^{I}, v⟩.

Since ⟨u_{i}^{I}, v⟩ = 0 mod m for i ∉ K and ⟨u_{i}^{I}, v⟩ mod m does not depend on I in X by Lemma 3.2, ⟨i_{I}^{∗}(x), v⟩ mod m does not depend on the choice of I in X.

Corollary 3.4. Assume that c_{1}(Δ) is divisible by N, and write c_{1}^{T} (Δ) = Nx + u, u ∈ H^{2}(BT). Let v and m be as above. If we write ⟨u_{i}^{I}, v⟩ in the form

⟨u_{i}^{I}, v⟩ = mh_{i} + r_{i} with 0 ≤ r_{i} < m,

then the sum ∑_{i∈I} h_{i} mod N depends only on the (v, m)-equivalence class X of I.

Proof. We put h_{I} = ∑_{i∈I} h_{i} and r_{I} = ∑_{i∈I} r_{i}. Then ⟨u^{I}, v⟩ = mh_{I} + r_{I}. By Lemma 3.3 ⟨i_{I}^{∗}(x), v⟩ is of the form

⟨i_{I}^{∗}(x), v⟩ = mh′_{I} + r′,

for I ∈ X, where r′ is independent of I. Therefore, if we write ⟨u, v⟩ = r′′, then

⟨u^{I}, v⟩ = Nmh′_{I} + Nr′ + r′′.

If we compare this with

⟨u^{I}, v⟩ = mh_{I} + r_{I},

we see that Nr′ + r′′ is of the form Nr′ + r′′ = mh′ + r_{I} and h_{I} = Nh′_{I} + h′. This shows that h_{I} mod N depends only on X.

Under the situation of Corollary 3.4, the mod N value of ∑_{i∈I} h_{i} will be called (v, m)- type of X and will be denoted by h(v, m, X). Similarly the mod N value of ⟨u^{I}, v⟩ (which is independent of I ∈ Σ^{(n)} by Lemma 3.1) will be called v-type and will be denoted by h(v).

Lemma 3.5. Assume c_{1}(Δ) is divisible by N. Any non-zero b ∈ ℤ/N can occur as v-type when v varies over generic vectors in H_{2}(BT).

Proof. This follows readily from the fact that {u_{i}^{I} | i ∈ I} is a basis of H^{2}(BT).

4. Elliptic genus of level N

We define the equivariant elliptic genus of a multi-fan. For that purpose we first consider the theta function

θ_{y}(λ) = (1 + yλ) ∏_{n=1}^{∞}(1 + yq^{n}λ)(1 + y^{−1}q^{n}λ^{−1}).

It is defined for y ∈ ℂ^{∗} and q ∈ ℂ with |q| < 1. For q ≠ 0 we have

θ_{y}(qλ) = y^{−1}λ^{−1}θ_{y}(λ). (3)

We define

ϕ_{y}(λ) = \frac{θ_{y}(λ)}{θ_{−1}(λ)}.

It transforms by

ϕ_{y}(qλ) = −y^{−1}ϕ_{y}(λ) or ϕ_{y}(q^{−1}λ) = −yϕ_{y}(λ), (4)

by virtue of (3). We exclude the case y = −1. It follows that, if (−y)^{N} = 1, then ϕ_{y} is the pull-back of a rational function on the torus ℂ^{∗}/q^{N}. In this case we shall regard ϕ_{y} as the function on that torus.

Let Δ = (Σ, C, w^{±}) be a complete non-singular multi-fan. We set formally

φ_{y}(t) = ∑_{I∈Σ^{(n)}} w(I) ∏_{i∈I} ϕ_{y}(t^{−u_{i}^{I}}),

and call it the equivariant elliptic genus of the multi-fan Δ.

Lemma 4.1. Let φ_{y}(t) = ∑_{n=0}^{∞} φ_{y,n}(t)q^{n} be the expansion into power series, then φ_{y,n}(t) belongs to R(T) ⊗ ℤ[y, y^{−1}] where ℤ[y, y^{−1}] denotes the ring of Laurent polynomials over ℤ.

Proof. φ_{y,n}(t) is a linear combination with coefficients in ℤ[y, y^{−1}] of the expression of the following form:

∑_{I∈Σ^{(n)}} w(I)\frac{t_{I}^{α}}{∏_{i∈I}(1 − t^{−u_{i}^{I}})},

where α ∈ O as was introduced in Section 2. Therefore it belongs to R(T) ⊗ ℤ[y, y^{−1}] by Corollary 2.5.

Let v ∈ H_{2}(BT) be a generic vector. We set

φ_{y}^{v}(t) = ∑_{I∈Σ^{(n)}} w(I) ∏_{i∈I} ϕ_{y}(t^{−⟨u_{i}^{I},v⟩}).

If (−y)^{N} = 1, then φ_{y}^{v}(t) is an elliptic function on ℂ^{∗}/q^{N}, because ϕ_{y}(t) is such a function. In this case φ_{y}^{v}(t) is called elliptic genus of level N. We have furthermore

Lemma 4.2. Suppose that c_{1}(Δ) is divisible by N. Let v ∈ H_{2}(BT) be a generic vector and h(v) the v-type. Then the elliptic genus φ_{y}^{v}(t) of level N transfoms by

φ_{y}^{v}(qt) = ζ^{h(v)}φ_{y}^{v}(t),

where ζ = −y.

Proof.

∏_{i∈I} ϕ_{y}((qt)^{−⟨u_{i}^{I},v⟩}) = ζ^{∑_{i∈I}⟨u_{i}^{I},v⟩} ∏_{i∈I} ϕ_{y}(t^{−⟨u_{i}^{I},v⟩})

by (4). But ∑_{i∈I}⟨u_{i}^{I}, v⟩ = h(v) mod N which is independent of I. Hence we obtain

φ_{y}^{v}(qt) = ζ^{h(v)}φ_{y}^{v}(t).

The following theorem and corollary are versions of rigidity theorem and vanishing theorem for multi-fans.

Theorem 4.3. Let Δ be a non-singular complete multi-fan and v ∈ H_{2}(BT) a generic vector. Assume that c_{1}(Δ) is divisible by an integer N > 1. Then the equivariant elliptic genus φ_{y}^{v}(t) ((−y)^{N} = 1) of level N is rigid, i.e. φ_{y}^{v}(t) is constant.

Corollary 4.4. Under the same situation as in Theorem 4.3 the equivariant elliptic genus φ_{y}(t) ((−y)^{N} = 1) of level N constantly vanishes.

Proof. We postpone the proof of Theorem 4.3. As to Corollary 4.4, we take a generic vector v such that ζ^{h(v)} ≠ 1, which is possible by Lemma 3.5. Since φ_{y}^{v}(t) is constant by Theorem 4.3 for any v, φ_{y}(t) is also constant, which we denote by φ_{y}. Since φ_{y} = ζ^{h(v)}φ_{y} by Lemma 4.2, φ_{y} must be equal to 0.

The degree 0 term φ_{y,0}(t) in the expansion in Lemma 4.1 reduces to the T_{y}-genus T_{y}[Δ] (cf. [HM]). We obtain

Corollary 4.5. If c_{1}(Δ) is divisible by N, then the T_{y}-genus T_{y}[Δ] vanishes for (−y)^{N} = 1.

For N = 2 and y = 1 the T_{y}-genus equals the signature Sign(Δ). Hence

Corollary 4.6. The signature Sign(Δ) of a spin multi-fan vanishes.

Remark. If M is a torus manifold, we define its elliptic genus to be that of its associated multi-fan Δ(M). If c_{1}(M) is divisible by N, c_{1}(Δ(M)) is also divisible by N as was remarked in Section 3, and hence its equivariant elliptic genus of level N vanishes for (−y)^{N} = 1. In case N = 2 we have the following conclusion. The equivariant Stiefel-Whitney class w_{2}^{T} (M) ∈ H_{T}^{2}(M; ℤ/2) is defined and descends to c_{1}^{T} (Δ(M)) mod 2. If M is a spin torus manifold, then w_{2}^{T} (M) lies in H^{2}(BT; ℤ/2). Therfore c_{1}^{T} (Δ(M)) is divisible by 2. It follows from Corollary 4.6 that the signature Sign(M) of M vanishes. This can be also deduced from Corollary in 1.5 of [HS].

The rest of this section is devoted to the proof of Theorem 4.3. We assume throughout that c_{1}(Δ) is divisible by N. It suffices to show that φ_{y}^{v}(t) has no poles since it is an elliptic function. It is clear that possible poles λ satisfy λ^{m}q^{−s} = 1 for some integers m ≥ 1 and s. Hence it suffices to show that φ_{y}^{v}(tq^{\frac{s}{m}}) has no poles λ with λ^{m} = 1. Let

Σ^{(n)} = ⊔_{ν}X_{ν}

be the decomposition into (v, m)-equivalence classes. Let K_{ν} denote the core of X_{ν}. We fix a class X_{ν} and write

⟨u_{i}^{I}, v⟩ = m_{i} = mh_{i} + r_{i}, 0 ≤ r_{i} < m,

for I ∈ X_{ν}. From the definition of core it follows that

r_{i} = 0 ⇔ i ∉ K_{ν}.

We decompose K_{ν} into a disjoint union K_{ν} = ⊔_{r=1}^{m−1}K_{ν}^{r} where

K_{ν}^{r} = {i ∈ K_{ν} | r_{i} = r}.

Note that ∑_{i∈I} h_{i} = h(v, m, X_{ν}) mod N. With these understandings a straightforworard calculation yields

Lemma 4.7.

φ_{y}^{v}(tq^{\frac{s}{m}}) = ∑_{ν} ζ^{sh(v,m,X_{ν})}φ_{y}^{v}(tq^{\frac{s}{m}})_{ν},

where

φ_{y}^{v}(tq^{\frac{s}{m}})_{ν} = ∑_{I∈X_{ν}} \frac{w(I)}{∏_{i∈I\K_{ν}}(1 − t^{−⟨u_{i}^{I},v⟩})}

· ∏_{i∈I\K_{ν}} ((1 + yt^{−⟨u_{i}^{I},v⟩}) ∏_{n=1}^{∞} \frac{(1 + yq^{n}t^{−⟨u_{i}^{I},v⟩})(1 + y^{−1}q^{n}t^{⟨u_{i}^{I},v⟩})}{(1 − q^{n}t^{−⟨u_{i}^{I},v⟩})(1 − q^{n}t^{⟨u_{i}^{I},v⟩})})

· ∏_{r=1}^{m−1} ∏_{i∈K_{ν}^{r}} (\frac{1 + yq^{− \frac{s}{m} r}t^{−⟨u_{i}^{I},v⟩}}{1 − q^{− \frac{s}{m} r}t^{−⟨u_{i}^{I},v⟩}} ∏_{n=1}^{∞} \frac{(1 + yq^{n}q^{− \frac{s}{m} r}t^{−⟨u_{i}^{I},v⟩})(1 + y^{−1}q^{n}q^{\frac{s}{m} r}t^{⟨u_{i}^{I},v⟩})}{(1 − q^{n}q^{− \frac{s}{m} r}t^{−⟨u_{i}^{I},v⟩})(1 − q^{n}q^{\frac{s}{m} r}t^{⟨u_{i}^{I},v⟩})}.)

Lemma 4.8. Fix ν. Let

φ_{y}^{v}(tq^{\frac{s}{m}})_{ν} = ∑_{n=0}^{∞} φ̄_{n}(t)q^{\frac{n}{m}}

be the expansion of φ_{y}^{v}(tq^{\frac{s}{m}})_{ν} into power series in q^{\frac{1}{m}}. Then each φ̄_{n}(t) is in R(S^{1}) ⊗ ℂ.

Proof. The proof is similar to that of Lemma 4.1. φ̄_{n}(t) is a linear combination with complex coefficients of the expression of the form:

∑_{I∈Σ_{Kν}^{(n−kν)}} w(I)\frac{t_{I}^{⟨α,v⟩}}{∏_{i∈I\K_{ν}}(1 − t^{−⟨u_{i}^{I},v⟩})}

where k_{ν} is the dimension of C(K_{ν}), α ∈ O is as in Section 2, and t_{I}^{⟨α,v⟩} = ∑_{l∈α} t^{⟨l,v⟩}. Thus φ̄_{n}(t) is in R(S^{1}) ⊗ ℂ by Corollary 2.4.

We shall show that each φ_{y}^{v}(tq^{\frac{s}{m}})_{ν} has no poles at λ^{m} = 1. Let λ be a possible pole with λ^{m} = 1. Let φ̄_{n,I}(t) be the contribution from I in φ̄_{n}(t). There is an open set U containing λ such that φ̄_{n,I}(t) is holomorphic in U \ \{λ\} for each I. ∑_{n=0}^{∞} φ̄_{n,I}(t)q^{\frac{n}{m}} converges uniformly in any compact set in U − {λ}. Note that ∑_{I} φ̄_{n,I}(t) = φ̄_{n}(t) is holomorphic in U because it is a finite Laurent series by Lemma 4.8. We now quote a lemma from [H].

Lemma 4.9 (Hirzebruch). Let b_{n,j} be meromorphic functions on U with j running over some finite set J. Suppose that they satisfy the following properties:

(1) b_{n,j} is holomorphic in U \ {λ},

(2) b_{n} = ∑_{j} b_{n,j} is holomorphic in U,

(3) ∑_{n=0}^{∞} b_{n,j} converges uniformly in any compact set in U \ {λ}. Then ∑_{n=0}^{∞} b_{n} converges uniformly in any compact set in U and is a holomophic extension of ∑_{j∈J} ∑_{n=0}^{∞} b_{n,j} | U \ {λ}.

We apply this Lemma to φ̄_{n,I}(t)q^{\frac{n}{m}} and φ̄_{n}(t)q^{\frac{n}{m}}. It follows that φ_{y}^{v}(tq^{\frac{s}{m}})_{ν} and hence φ_{y}^{v}(tq^{\frac{s}{m}}) has no pole at t = λ. Hence φ_{y}^{v}(t) has no poles and it is a constant. This proves Theorem 4.3.

Remark. In case s = 0, we can use Lemma 4.1 instead of Lemma 4.8,

5. Applications

Let Δ be a complete simplicial multi-fan of dimension n. If I is in Σ^{(n)} and v is a generic vector, one can write v in the form v = ∑_{i∈I} a_{i}v_{i} with non zero real numbers a_{i}. Fixing a generic vector v, we set

μ(I) = #{i ∈ I | a_{i} > 0, }

and define

h_{q}(Δ) := ∑_{I∈Σ^{(n)}, μ(I)=q} w(I),

for each integer q with 0 ≤ q ≤ n. We also define

e_{k} := ∑_{K∈Σ^{(k)}} T_{0}[Δ_{K}],

where Δ_{K} is the projected multi-fan associated with K. If the Todd genus T_{0}[Δ] equals 1 and w(I) = 1 for all I ∈ Σ^{(n)}, then

h_{q}(Δ) = #{I ∈ Σ^{(n)} | μ(I) = q}, (5) and

e_{k} = #Σ^{(k)}. (6)

Remark. Note that these conditions are always satisfied for complete simplicial ordinary fans. In particular, if M is a complete toric variety, the fan Δ(M) associated with M satisfies these conditions. It is known that T_{−t}[Δ(M)] is equal to the Poincaré polynomial P (t) of M if M is a non-singular projective toric variety, see e.g. [F].

The following Lemma was proved in [HM].

Lemma 5.1.

T_{y}[Δ] = ∑_{q=0}^{n} h_{q}(Δ)(−y)^{q}

= ∑_{m=0}^{n} e_{n−m}(Δ)(−1 − y)^{m}.

Moreover h_{0} = h_{n} and they are the Todd genus of the multi-fan Δ.

It was also proved there that the equivariant T_{y}-genus of a torus manifold M is always rigid, and hence reduces to the ordinary T_{y}-genus T_{y}[M]. The proof given there can also be applicable in the case of complete non-singular multi-fan.

Proposition 5.2. Let Δ be a complete non-singular multi-fan with Todd genus T_{0}[Δ] ≠ 0. If c_{1}(Δ) is divisible by a positive integer N, then N is equal to or less than n + 1. In the cases N = n + 1 and N = n the T_{y}- genus must be of the following forms

T_{y}[Δ] = T_{0}[Δ] ∑_{q=0}^{n}(−y)^{q} (N = n + 1), (7)

and

T_{y}[Δ] = T_{0}[Δ](1 − y) ∑_{q=0}^{n−1}(−y)^{q} (N = n). (8)

Proof. Suppose that c_{1}(Δ) is divisible by N. Then, by Corollary 4.5 T_{y}[Δ] considered as a polynomial in −y has roots at all N-th roots of unity other than 1. Hence it must be divisible by ∑_{q=0}^{N−1}(−y)^{q}. On the other hand it is a polynomial of degree n with constant term T_{0}[Δ] by Lemma 5.1 since T_{0}[Δ] ≠ 0. Therefore we must have N + 1 ≤ n.

Suppose that N = n + 1. Then the same reasoning as above proves (7). If N = n, then T_{y}[Δ] is divisible by ∑_{q=0}^{n−1}(−y)^{q}. Since the constant term and the coefficient (as a polynomial of −y) of the highest term do not vanish by assumption and Lemma 5.1, T_{y}[Δ] must be of the form (8).

Lemma 5.3. Let Δ be a complete non-singular multi-fan with T_{0}[Δ] = 1 and w(I) = 1 for all I ∈ Σ^{(n)}. If T_{y}[Δ] is of the form (7), then

#Σ^{(1)} = n + 1 and #Σ^{(n)} = n + 1.

If T_{y}[Δ] is of the form (8), then

#Σ^{(1)} = n + 2 and #Σ^{(n)} = 2n.

Moreover, in case n ≥ 3, #Σ^{(2)} = \frac{1}{2}n(n + 3).

Proof. The equality (7) with T_{0}[Δ] = 1 means that h_{q} = 1 for all q with 0 ≤ q ≤ n. This implies that #Σ^{(n)} = n + 1 by (5). Putting the above values of h_{q} in Lemma 5.1 and using (6) we see that #Σ^{(1)} = e_{1} = n + 1.

Similarly the equality (8) with T_{0}[Δ] implies that h_{q} = 1 for q = 0, n and h_{q} = 2 for 1 ≤ q ≤ n−1. This implies that #Σ^{(n)} = 2n by (5), and yields, together with Lemma 5.1 and (6), the equalities #Σ^{(1)} = e_{1} = n + 2 and #Σ^{(2)} = e_{2} = \frac{1}{2}n(n + 3) in case n ≥ 3.

Corollary 5.4. Let M be a complete toric variety of dimension n. If c_{1}(M) is divisible by n + 1, then M is isomorphic to the projective space ℙ^{n} as a toric variety.

Proof. By Proposition 5.2 and Lemma 5.3 the fan associated with M has n + 1 1dimensional cones and n + 1 n-dimensional cones. Such a fan is unique (up to automorphisms of the lattice L) and coincides with the fan associated with ℙ^{n}. Since a toric variety is determined by its fan, M must be ℙ^{n}.

In order to handle the case N = n we investigate the fan associated with a projective space bundle over a projective space. Let ξ denote the hyperplane bundle (dual of the tautological line bundle) over ℙ^{r}. Let 1 ≤ r < n and set η = (∑_{i=r+1}^{n} ξ^{k_{i}}) ⊕ 1 where k_{i} are integers and 1 denotes the trivial line bundle. The associated projective space bundle of η will be denoted by M. It is a complex manifold. A point of M is expressed in homogeneous coordinate

[z_{0}, z_{1}, …, z_{r}, w_{r+1}, …, w_{n}, w_{n+1}] (9)

where z_{i}, w_{j} ∈ ℂ, (z_{0}, z_{1}, …, z_{r}) ≠ (0, 0, …, 0), (w_{r+1}, …, w_{n}, w_{n+1}) ≠ (0, …, 0, 0), and if, α ∈ ℂ^{∗}, then

[αz_{0}, αz_{1}, …, αz_{r}, α^{k_{r+1}}w_{r+1}, …, α^{k_{n}}w_{n}, w_{n+1}]

and

[z_{0}, z_{1}, …, z_{r}, αw_{r+1}, …, αw_{n}, αw_{n+1}]

are identified with (9).

Let (n + 1)-dimensional torus T^{n+1} = S^{1} × ⋯ × S^{1} act on M by

\begin{matrix}(t_{0}, t_{1}, …, t_{n}) & [z_{0}, z_{1}, …, z_{r}, w_{r+1}, …, w_{n}, w_{n+1}] \\ = [t_{0}z_{0}, t_{1}z_{1}, …, t_{r}z_{r}, t_{r+1}w_{r+1}, …, t_{n}w_{n}, w_{n+1}] \end{matrix}

The action is a holomorphic action. The subgroup D′ = {(t, …, t, t^{k_{r+1}}, …, t^{k_{n}}) | t ∈ S^{1}} of T^{n+1} acts trivially on M. Hence the quotient T = T^{n+1}/D′ acts on M. Put

M_{i} = \{ \begin{matrix}\{[z_{0}, z_{1}, …, z_{r}, w_{r+1}, …, w_{n}, w_{n+1}] | z_{i} = 0\} & for 0 ≤ i ≤ r \\ \{[z_{0}, z_{1}, …, z_{r}, w_{r+1}, …, w_{n}, w_{n+1}] | w_{i} = 0\} & for r + 1 ≤ i ≤ n + 1. \end{matrix}

We also put

S_{i} = {(1, …, 1, t_{i}, 1, …, 1) ∈ T^{n+1}} for 0 ≤ i ≤ n

and

S_{n+1} = {(1, 1, …, 1, t_{r+1}, …, t_{n}) ∈ T^{n+1} | t_{r+1} = ⋯ = t_{n}}.

We shall denote by the same letter the image of S_{i} in T. It is easy to see that S_{i} pointwise fixes M_{i}, and there are no other circle subgroups of T which have (complex) codimension 1 fixed point set components.

Let ṽ_{i} ∈ Hom(S^{1}, T^{n+1}) denote the inclusion homomorphism of S^{1} into the i-th factor of T^{n+1} for 0 ≤ i ≤ n. Put v_{i} = π_{∗}(ṽ_{i}) ∈ Hom(S^{1}, T) where π_{∗} is the homomophism induced by the projection π : T^{n+1} → T. ¿From the definition it follows that there is a relation

v_{0} + v_{1} + ⋯ + v_{r} + k_{r+1}v_{r+1} + ⋯ + k_{n}v_{n} = 0. (10)

We also put

v_{n+1} = −(v_{r+1} + ⋯ + v_{n}) ∈ Hom(S^{1}, T). (11)

Then S_{i} is the image of v_{i} : S^{1} → T. Moreover S^{1} acts via v_{i} on each fiber of the normal bundle of M_{i} in M by standard complex multiplication. Thus {v_{i} | i = 0, …, n, n + 1} coincides with the set of primitive edge vectors of 1-dimensional cones of the multi-fan Δ(M) = (Σ(M), C(M), w(M)^{±}) associated with the torus manifold M, and we have Σ(M)^{(1)} = {0, 1, …, n, n + 1}, cf. [HM].

To determine the whole augmented simplicial set Σ(M), we need to look at the fixed point set M^{T}. For i ∈ {0, 1, … r}, put I_{i} = {0, 1, … r}\{i}, and for j ∈ {r +1, … n+1}, put J_{j} = {r + 1, … n + 1} \ {j}. It is not difficult to see that M^{T} consists of points

M_{I_{i}} ∩ M_{J_{j}}, i ∈ {0, 1, … r}, j ∈ {r + 1, … n + 1},

where M_{I_{i}} = ∩_{k∈I_{i}}M_{k} and M_{J_{j}} = ∩_{l∈J_{j}}M_{l}. This implies that

Σ(M)^{(n)} = {I_{i} ∪ J_{j} | i ∈ {0, 1, … r}, j ∈ {r + 1, … n + 1}}.

In particular

#Σ(M)^{(n)} = (r + 1)(n − r + 1).

It follows that

#Σ(M)^{(n)} ≥ 2n and #Σ(M)^{(n)} = 2n if and only if r = 1 or n − 1.

Let τ be the tautological line bundle over the projective space bundle M. Its dual τ^{∗} restricts to the hyperplane bundle on each fiber of π : M → ℙ^{r}. Let ω ∈ H^{2}(M) be the first Chern class of τ^{∗}. Then, by the Leray-Hirsch theorem, H^{∗}(M) is a free H^{∗}(ℙ^{r})- module over generators 1, ω, ω^{2}, …, ω^{n−r}. In particular, H^{2}(M) is a free module over ω, ω′, where ω′ is the image of the canonical generator of H^{2}(ℙ^{r}) by π^{∗}. We have

Lemma 5.5.

c_{1}(M) = (n − r + 1)ω + ( ∑_{i=r+1}^{n} k_{i} + r + 1)ω′.

Proof. The tautological line bundle τ is a subbundle of π^{∗}η, and the tangent bundle along the fibers T_{f}M of π : M → ℙ^{r} is isomorphic to Hom(τ, π^{∗}η/τ) = τ^{∗} ⊗ (π^{∗}η/τ). Hence

c_{1}(T_{f}M) = (n − r)c_{1}(τ^{∗}) + c_{1}(π^{∗}η/τ)

= (n − r)ω + c_{1}(π^{∗}η) − c_{1}(τ)

= (n − r)ω + (∑_{i} k_{i})ω′ + ω

= (n − r + 1)ω + (∑_{i} k_{i})ω′.

Since the tangent bundle T M is isomorphic to π^{∗}T ℙ^{r} ⊕ T_{f}M, and c_{1}(π^{∗}T ℙ^{r}) = (r + 1)ω′, we have

c_{1}(M) = (n − r + 1)ω + (∑_{i} k_{i} + r + 1)ω′.

As an immediate consequence of Lemma 5.5 we obtain

Corollary 5.6. Let M = ℙ(η) be as above. Then c_{1}(M) is divisible by n if and only if r = 1 and ∑_{i=r+1}^{n} k_{i} + 2 is divisible by n.

We now consider complete non-singular multi-fans having first Chern class divisible by n.

Lemma 5.7. Let Δ = (Σ, C, w^{±}) be a complete non-singular multi-fan of dimension n such that

T_{0}(Δ) = 1, w(I) = 1 for all I ∈ Σ^{(n)},
#Σ^{(1)} = n + 2, #Σ^{(n)} = 2n, and #Σ^{(2)} = \frac{1}{2}n(n + 3) in case n ≥ 3.

Then it is equivalent to the multi-fan of a ℙ^{n−1} bundle over ℙ^{1} or a ℙ^{1}-bundle over ℙ^{n−1}.

Proof. Let {v_{i}}_{i=0}^{n+1} be the primitive edge vectors of the 1-dimensional cones. In view of (10) and (11) it suffices to show that, under a suitable numbering, they satisfy the relations

v_{1} + v_{2} + ⋯ + v_{n} = 0, v_{0} + v_{n+1} + ∑_{i=2}^{n} k_{i}v_{i} = 0, (12)

or

v_{1} + v_{2} + ⋯ + v_{n} + kv_{n+1} = 0, v_{0} + v_{n+1} = 0. (13)

We first deal with the case n ≥ 3. From the completeness we see that each 1-dimensional cone is a face of at least n 2-dimensional cones, and it is a face of at most n + 1 2dimensional cones because the number of 1-dimensional cones are n+2. Since the number of 2-dimensional cones is \frac{1}{2}n(n+3), we conclude that there are two edge vectors, say v_{0} and v_{n+1} such that v_{0} (v_{n+1} respectively) spans 2-dimensional cones with each of remaining vectors v_{1}, v_{2}, …, v_{n}, and each v_{i}, 1 ≤ i ≤ n, spans 2-dimensional cones with v_{j}, j ≠ i. Thus the projected multi-fan Δ_{{0}} has exactly n 1-dimensional cones. It is complete and non-singular as a projected multi-fan of a complete non-singular multi-fan Δ. It follows that Δ_{{0}} is equivalent to the fan of ℙ^{n−1}, and the projected edge vectors v̄_{i}, 1 ≤ i ≤ n, satisfy the relation

v̄_{1} + v̄_{2} + ⋯ + v̄_{n} = 0.

This implies the relation

v_{1} + v_{2} + ⋯ + v_{n} = kv_{0}.

Similarly we have

v_{1} + v_{2} + ⋯ + v_{n} = k′v_{n+1}.

If k = 0 then k′ = 0 since v_{n+1} ≠ 0, and v_{1}, v_{2}, …, v_{n} lie on a hyperplane v_{1}+v_{2}+⋯+v_{n} = 0. Since the multi-fan Δ is complete and non-singular, the primitive vectors v_{0} and v_{n+1} lie on the different sides of that hyperplane and must satisfy a relation as (12).

If k ≠ 0 then k′ ≠ 0 and v_{0} and v_{n+1} are linearly dependent primitive vectors. Therefore we must have v_{0} + v_{n+1} = 0. Thus (13) holds. This proves Lemma 5.7 in the case n ≥ 3.

The case n = 2 is similar and easier. We see that there are four primitive edge vectors v_{0}, v_{1}, v_{2}, v_{3} in 2-dimensional vector space V = L ⊗ ℝ such that

v_{1} + v_{2} = kv_{0} = k′v_{3}.

By the same reasoning as above we derive

v_{1} + v_{2} = 0, v_{0} + v_{3} + kv_{2} = 0 or v_{1} + v_{2} + kv_{3} = 0, v_{0} + v_{3} = 0.

Corollary 5.8. Let M be a complete non-singular toric variety of dimension n. If c_{1}(M) is divisible by n, then M is isomorphic to an (n − 1)-dimensional projective space bundle over ℙ^{1} of the form described in Corollary 5.6 as a toric variety.

Proof. By Proposition 5.2 and Lemma 5.3 the fan Δ(M) associated with M has n + 2 1-dimensional cones and 2n n-dimensional cones. Moreover the number of 2-dimensional cones is \frac{1}{2}n(n + 3) in case n ≥ 3. By Lemma 5.7 Δ(M) is equivalent to that of a ℙ^{n−1}- bundle over ℙ^{1} or ℙ^{1}- bundle over ℙ^{n−1}. Among such manifolds those with c_{1} divisible by n are of the form given in Corollary 5.6.

6. Remarks on singular case

So far we defined equivariant elliptic genus for complete non-singular multi-fans, and proved its rigidity and vanishing. We can define equivariant elliptic genus for general complete simplicial multi-fan eqipped with a set of edge vectors for 1-dimensional cones. However it seems rigiditiy property can not be expected in general case. In the sequel we shall briefly discuss these phenomena.

Using the θ function in Section 4 we set

ϕ_{α,y}(λ) = \frac{θ_{αy}(λ)}{θ_{−α}(λ)}.

Similarly to (4) we have

ϕ_{α,y}(qλ) = −y^{−1}ϕ_{α,y}(λ) or ϕ_{α,y}(q^{−1}λ) = −yϕ_{α,y}(λ). (14)

Let Δ = (Σ, C, w^{±}) be a complete simplicial multi-fan, and V = {v_{i} | i ∈ Σ^{(1)}} a set of prescribed edge vectors. We do not assume that v_{i}’s are primitive. For I ∈ Σ^{(n)} we define L_{I} to be the submodule of L generated by {v_{i} | i ∈ I}. Let L_{I}^{∗} be the dual lattice of L_{I}. We identify L_{I}^{∗} with the lattice in V^{∗} = L^{∗} ⊗ ℚ given by

L^{∗} = {u ∈ V^{∗} | ⟨v, u⟩ ∈ ℤ, for any v ∈ L_{I}}.

For g ∈ L/L_{I} and u ∈ L_{I}^{∗} we define

χ_{I}(u, g) = e^{2π\sqrt{−1}⟨v,u⟩},

where v ∈ L is a representative of g. If one fixes u, g ↦ χ_{I}(u, g) gives a character of the group L/L_{I}. We set

φ_{y}(t) = ∑_{I∈Σ^{(n)}} \frac{w(I)}{|L/L_{I}|} ∑_{g∈L/L_{I}} ∏_{i∈I} ϕ_{χ_{I}(u_{i}^{I},g)^{−1},y}(t^{−u_{i}^{I}}), (15)

and call it the equivariant elliptic genus of the pair (Δ, V) of multi-fan and prescribed edge vectors. Let v ∈ H_{2}(BT) be a generic vector such that ⟨u_{i}^{I}, v⟩ ∈ ℤ for all i ∈ I and I ∈ Σ^{(n)}. We set

φ_{y}^{v}(t) = ∑_{I∈Σ^{(n)}} \frac{w(I)}{|L/L_{I}|} ∑_{g∈L/L_{I}} ∏_{i∈I} ϕ_{χ_{I}(u_{i}^{I},g)^{−1},y}(t^{−⟨u_{i}^{I},v⟩}).

If (−y)^{N} = 1, −y ≠ 1, then φ_{y}^{v}(t) is an elliptic function on ℂ^{∗}/q^{N}, because ϕ_{α,y}(t) is such a function by (14). In this case φ_{y}^{v}(t) is called elliptic genus of level N.

The folloing proposition is a generalization of Lemma 4.1.

Proposition 6.1. Let φ_{y}(t) = ∑_{n=0}^{∞} φ_{y,n}(t)q^{n} be the expansion into power series, then φ_{y,n}(t) belongs to R(T) ⊗ ℂ.

The proof of Proposition 6.1 goes in a way similar to that of Lemma 4.1. We define the equivariant cohomology with real coefficients H_{T}^{∗}(Δ; ℝ) of a complete simplicial multifan Δ as the face ring of the simplicial complex Σ with real coefficients as in Section 2. We regard H^{2}(BT; ℝ) as a subspace of H_{T}^{2}(Δ, ℝ) by the formula (1). Note that this definition depends not only on Δ but on V. We also define the restriction homomorphism i_{I}^{∗} : H_{T}^{∗}(Δ; ℝ) → H^{∗}(BT; ℝ) for each I ∈ Σ^{(n)} by (2). Instead of Lemma 2.1 we use

Lemma 6.2. For any x = ∑_{i∈Σ^{(1)}} c_{i}x_{i} ∈ H_{T}^{2}(Δ; ℝ), c_{i} ∈ ℤ, the element

∑_{I∈Σ^{(n)}} \frac{w(I)}{|L/L_{I}|} ∑_{g∈L/L_{I}} \frac{χ_{I}(i_{I}^{∗}(x), g)t^{i_{I}^{∗}(x)}}{∏_{i∈I}(1 − χ_{I}(u_{i}^{I}, g)^{−1}t^{−u_{i}^{I}})}

in a fractional ring containing R(T) actually belongs to R(T).

This was proved in Lemma 7.4 of [HM] with a further assumption that i_{I}^{∗}(x) ∈ H^{2}(BT; ℤ). The general case can be proved in a similar way. The formula was also given in Corollary 12.10 of [HM] when Δ is the multi-fan associated with a torus orbifold.

Let O be as in Section 2. Let α be in O. If I is in Σ^{(n)}, then α is an orbit of the action of permutation group of I on linear forms over u_{i}^{I}. We set

χ_{I}(α, g)t_{I}^{α} = ∑_{l∈α} χ_{I}(l, g)t^{l}.

We can deduce the following corollary from Lemma 6.2 just like we deduced Lemma 2.5 from Lemma 2.1.

Corollary 6.3. For any α ∈ O the expression

∑_{I∈Σ^{(n)}} \frac{w(I)}{|L/L_{I}|} ∑_{g∈N/N_{I}} \frac{χ_{I}(α, g)t_{I}^{α}}{∏_{i∈I}(1 − χ_{I}(u_{i}^{I}, g)^{−1}t^{−u_{i}^{I}})} (16)

belongs to R(T).

We are now ready to prove Proposition 6.1. We see that φ_{y,n}(t) is a linear combination with complex coefficients of the expression of the form (16). Hence it belongs to R(T)⊗ℂ by Corollary 6.3. This proves Proposition 6.1.

It would seem reasonable to define H_{T}^{2}(Δ; ℤ) to be the submodule of H_{T}^{2}(Δ; ℚ) generated by V = {v_{i} | i ∈ Σ^{(1)}}. Then H^{2}(BT; ℤ) is contained in H_{T}^{2}(Δ; ℤ). It would be possible to define divisibility by N of the first Chern class as in Section 3. However Theorem 4.3 and Corollary 4.4 do not hold in this general setting. In fact Lemma 3.1 has no meaning as it is, because u_{i}^{I}’s do not lie in H^{2}(BT; ℤ) in general. Also the proof of Lemma 3.2 breaks down in general setting, and one can not define (v, m)-equivalence relation. However in the following favorable case we get rigidity and vanishing results.

We assume that the submodules L_{I} ⊂ L = H_{2}(BT; ℤ) are independent of I ∈ Σ^{(n)}. We write L̃ = L_{I}. It coincides with a submodule of L generated by V. The dual lattice L̃^{∗} is contained in H_{T}^{2}(Δ; ℤ) as is seen from (1). The identification of L with H_{2}(BT; ℤ) can be explained as follows. We interpret T as V/L where V = L ⊗ ℝ. Then

L = π_{1}(T) = π_{2}(BT) = H_{2}(BT : ℤ).

We put Ṽ = L̃ ⊗ ℝ and T̃ = Ṽ /L̃. This determines the identificaction L̃ = H_{2}(BT̃; ℤ) and L̃^{∗} = H^{2}(BT̃; ℤ). Let p : Ṽ → V and p : T̃ → T denote the map induced by the inclusion L̃ → L. Then the kernel of p : T̃ → T is identified with H = L/L̃. To distinguish we denote by ṽ_{i} the vector v_{i} when we consider it as lying in Ṽ so that we have p(ṽ_{i}) = v_{i}. We define a new multi-fan Δ̃ = (Σ̃, C̃, w̃^{±}) by setting Σ̃ = Σ, w̃^{±} = w^{±} and letting C̃(i) to be the cone generated by ṽ_{i}. Since {ṽ_{i} | i ∈ I} is a basis of L̃ for any I ∈ Σ^{(n)}, Δ̃ is non-singular.

Let φ̃_{y}(t) = ∑_{n=0}^{∞} φ̃_{y,n}(t)q^{n} be the equivariant elliptic genus of Δ̃. We have

φ̃_{y}(t) = ∑_{I∈Σ^{(n)}} w(I) ∏_{i∈I} ϕ_{y}(t^{−u_{i}^{I}}) (17)

By Lemma 4.1 φ̃_{y,n}(t) is of the form

φ̃_{y,n}(t) = ∑_{u∈L̃^{∗}} a_{u}t^{u}

where a_{u} ∈ ℤ[y, y^{−1}]. Compairing (17) with (15) we obtain

φ_{y,n}(t) = \frac{1}{|L/L̃|} ∑_{u∈L̃^{∗}} a_{u} ∑_{g∈L/L̃} χ(u, g)t^{u}.

Since χ(u, ) is a character of L/L̃,

∑_{g∈L/L̃} χ(u, g)t^{u} = \begin{cases}t^{u} & if u ∈ L^{∗} \\ 0 & if u ∉ L^{∗}, \end{cases}

we see that

φ_{y,n}(t) = ∑_{u∈L^{∗}} a_{u}t^{u}. (18)

As a byproduct we obtain Proposition 6.4. Let Δ be a complete simplicial multi-fan and V a set of prescribed edge vectors such that L_{I} is independent of I ∈ Σ^{(1)}. Assume that c_{1}(Δ) is divisible by N in the sense that c_{1}^{T} (Δ) = ∑_{i∈Σ^{(1)}} x_{i} ∈ H_{T}^{2}(Δ; ℤ) is written in the form

c_{1}^{T} (Δ) = Nx + u, x ∈ H_{T}^{2}(Δ), u ∈ L̃^{∗},

then the equivariant elliptic genus φ_{y} of the pair (Δ, V) vanishes for (−y)^{N} = 1, −y ≠ 1. In particular the T_{y}-genus T_{y}(Δ) vanishes for (−y)^{N} = 1, ≠ 1.

Proof. The divisibility of c_{1}(Δ) by N eventually means that c_{1}(Δ̃) is divisible by N. By Corollary 4.4 all the a_{u} vanish for (−y)^{N} = 1, −y ≠ 1. Hence, by (18), φ_{y} of the pair (Δ, V) vanishes for (−y)^{N} = 1, −y ≠ 1.

Remark. In the situation of Proposition 6.4 we constructed a new non-singular multi-fan from the pair (Δ, V). We may call it the (ramified) covering of Δ with respect to V. Conversely given a complete non-singular multi-fan Δ̃ = (Σ̃, C̃, w̃^{±}) with C̃ : Σ̃^{(1)} → H_{2}(BT̃; ℝ) and a sublattice L of H_{2}(BT̃; ℝ) such that H_{2}(BT̃; ℤ) ⊂ L we can construct in an obvious way a new complete simplicial multi-fan Δ and a set of edge vectors V such that the covering of Δ with respect to V coincides with Δ. Geometric picture of this construction is making quotient torus orbifold M/H of a torus manifold M by a subgroup H of the torus T acting on M.

References

[BT] R. Bott and C. Taubes, On the rigidity theorem of Witten, J. Amer. Math. Soc., 2 (1989), 137–186.

[F] W. Fulton, Introduction to Toric Varieties, Annals Math. Studies, No.131, Princeton UP, 1993.

[HM] A. Hattori and M. Masuda, Theory of multi-fans, Mathematical Preprint Series, Osaka City University, 2001.

[H] F. Hirzebruch, Elliptic genera of level N for complex manifolds, Differential Geometrical Methods in Theoretical Phisics, Kluwer, 1988, pp. 37–63; also reproduced with corrections and improvements in F. Hirzebruch, T. Berger and R. Jung, Manifolds and Modular Forms, Aspects of Mathematics, vol. E20, Vieweg, 1992.

[HS] F. Hirzebruch and P. Slodowy, Elliptic Genera, involutions, and homogenious spin manifolds, Geometriae Dedicata, 35 (1990), 309–343.

[M] M. Masuda, Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J., 51 (1999), 237–265.

[W] E. Witten, The index of the Dirac operator in loop space, Elliptic Curves and Modular Forms in Algebraic Geometry, Lecture Notes in Math., Springer, 1988, pp.161–181.

Graduate School of Mathematical Science, University of Tokyo, Tokyo, Japan; Department of Mathematics, Osaka city University, Osaka, Japan

  E-mail address: hattori@math.meiji.ac.jp; masuda@sci.osaka-cu.ac.jp