DUALITY OF WEIGHTS, MIRROR SYMMETRY

AND ARNOLD’S STRANGE DUALITY

Masanori KOBAYASHI

Department of Mathematics, Tokyo Institute of Technology

Introduction.

The hypersurfaces in weighted projective spaces often appear as important examples in the context of mirror symmetry. The Batyrev’s idea using polar duality for reflexive polytopes in toric geometry[5] provides many examples which are not weighted hypersurfaces.

In this paper, first we describe the relation between polar duality in toric geometry and duality of weighted hypersurfaces.

As an application, we will show that Arnold’s strange duality for the fourteen exceptional unimodal hypersurface singularities[1] reduces to polar duality.

K. Saito defined the weight systems and used effectively for isolated hypersurface singularities with a ℂ^{∗}-action[32]. Those singularities correspond to weight systems as follows; take a germ of analytic function f : ℂ^{n} → ℂ with f(0) = 0, which determines a germ of hypersurface singularity ({f = 0}, 0) ⊂ (ℂ^{n}, 0). The function f can be expressed as a weighted homogeneous polynomial of n variables (x_{1}, …, x_{n}) after some suitable analytic change of coordinates near the origin if and only if there is a holomorphic tangent vector field D such that Df = f[31]. In this case, we can assign to f a weight system W = (wt(x_{1}), …, wt(x_{n}); wt(f)).

We associate some hypersurface singularity with ℂ^{∗}-action a compact complex surface with trivial canonical sheaf in the following manner.

Let X_{0} ⊂ ℂ^{n} be a hypersurface singularity with ℂ^{∗}-action whose weights are all positive. Such X_{0} is known to be an algebraic variety defined by a weighted homogeneous polynomial[26].

Let X_{t} be its Milnor fiber (if it exists; e.g. when X_{0} is an isolated singularity). We will not restrict ourselves to isolated singularities. Let X̄ be the natural compactification of X_{t} in ℙ(1, a_{1}, …, a_{n}) by sending the points (x_{1}, …, x_{n}) of ℂ^{n} to (1 : x_{1} : ⋯: x_{n}).

We treat only the case that total degree h is greater than the sum of the weights ∑_{i=1}^{n} a_{i} here. We denote the difference h − ∑_{i=1}^{n} a_{i} by a_{0}.

Let X̂ be the image of X̄ by the ℤ/a_{0}ℤ-quotient: ℙ(1, a_{1}, …, a_{n}) → ℙ(a_{0}, a_{1}, …, a_{n}). By the adjunction formula, the canonical sheaf of X̂ is trivial. We denote a crepant resolution of X̂ by X if it exists, in which case X̂ has only canonical Gorenstein singularities.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 14 J 17, 32 S 25; Secondary 14 M 25.

Partially supported by Fûjukai Foundation and Japan Association for Mathematical Sciences.

In the case X_{0} is an isolated singularity, it is known that the Milnor lattice or exceptional lattice of resolution and the exceptional lattice at infinity enjoy some dual property[21].

We will state how polar duality of polytopes related to X descends to the duality of weight systems up to Chapter 3.

The duality of weights partly suggests why [10] produced a mirror symmetric phenomena using only a resolution of weighted hypersurface in weighted ℙ^{4}. In fact, recently it is shown that those examples in weighted 4-spaces correspond to some reflexive polytopes [9].

When n = 3, X is a K3 surface. First we assume X is a K3 surface corresponding to one of the fourteen exceptional unimodal isolated hypersurface singularities[27][15].

Aspinwall and Morrison[3] explains why mirror symmetry for algebraic K3 surface is said to correspond to Arnold’s strange duality. The transcendental lattice T_{X} decomposes as L_{G} ⊕ U, where the dual graph of L_{G} is a tree of (−2)-elements with three branches and U is the even unimodular hyperbolic lattice of rank 2. We will write L_{D} := S_{X} in this case. Then the whole cohomology ring decomposes as H^{∗}(X, ℂ) = H^{0,0} ⊕ (L_{G} ⊕ U ⊕ L_{D})_{ℂ}⊕H^{2,2}, where the subscript_{ℂ} means the tensor product by ℂ. The mirror map should interchange (L_{G})_{ℂ} and (L_{D})_{ℂ}, H^{0,0} ⊕ H^{2,2} and U_{ℂ}, respectively, since the vector space (L_{G})_{ℂ} corresponds to the tangent space of the deformation space of complex structures with a fixed Picard number and (L_{D})_{ℂ} corresponds to a deformation of complexified algebraic Kähler structures.

Roan[29] applied orbifold construction for four of the dual pairs and constructed nonlinear coordinate changes for them. Borcea[8] used Batyrev’s polar dual construction for the reflexive pyramids and calculated some of dual weights.

We shall give a proof that polar dual of polytopes corresponding Arnold’s fourteen exceptional unimodal hypersurface singularities gives corresponding dual singularity.

More generally, take a K3 surface X which is the minimal resolution of toric hypersurface with only rational double points. Let L_{D} be the restriction of the toric divisors of the ambient space and L_{0} be the orthogonal complement L_{D}^{⊥} in S_{X}. then (S_{X})_{ℚ} decomposes as a direct sum (L_{D})_{ℚ} ⊕ (L_{0})_{ℚ}.

The reason why the full algebraic lattice and L_{G} does not interchange is that the induced complexified Kähler class is trivial on L_{0}-part while embedded deformation has the full dimension. This L_{0}-part is a generalization of H_{n}^{1,1} in [29], where Roan showed a similar mirror property for a quotient construction. Batyrev pointed out in a non-published version of [5] that for general Δ, rk L_{D}(Δ) + rk L_{D}(Δ^{∗}) can be smaller than 20. For the 14 exceptional unimodal singularities, we found polytopes for which the sum above equals 20.

Since the mirror of generic marked K3 surface seems to be again a generic one, mirror phenomena for special marked K3 surfaces are of particular interest, where ‘special’ means e.g. the surface is algebraic and its (complexified) Kähler class is also ample.

For general K3 surfaces L_{0} is defined as the orthogonal complement of the fixed lattice by the automorphism group which fixes H^{2,0} and the (complexified) Kähler class. We show this group is finite using Nikulin’s results on the automorphism of a K3 surface and Torelli’s theorem. H. Tsuji kindly informed the author that the finiteness also follows from Yau’s Theorem.

We refer to [3][18] and their reference for an introduction to some background and [23][37] for general information about mirror symmetry. A preprint by Dolgachev [14] includes relevant general results and examples using lattice theory.

In Chapter 1, we shall formulate a dual correspondence of reflexive pairs in ℚ-Fano toric projective varieties after reviewing some results of toric geometry. In Chapter 2, we prepare the language of weight systems according to K. Saito and introduce our notion of duality of weight systems. Our duality partly coincides with Saito’s duality. In Chapter 3, we state the relation between duality of weight systems and Batyrev’s toric mirror symmetry using polar polytopes. We found that our construction is partly a generalization of those of Batyrev[5], Borcea[8] and Berglund-Katz[7] for reflexive simplices and pyramids. In Chapter 4, we review some general results of K3 surfaces, and we argue some lattice theory and automorphism. We also construct a monomial divisor mirror map of Aspinwall-Greene-Morrison in the case of K3 surfaces as toric hypersurfaces and prove Arnold’s strange duality. We add some examples which are not exceptional unimodal isolated singularities.

Acknowledgement

The author expresses his hearty gratitude to Y. Kawamata for his continuous warm encouragement, F. Campana, D.-T. Lê, H. Popp, M. Reid, B. Teissier, C. Weber and P. M. H. Wilson for their excellent hospitality, stimulus discussion and valuable suggestions.

He also thanks to M. Furushima, F. Michele, I. Nakamura and J. Stevens for useful knowledge of singularity theory, to K. Saito for a very interesting introduction of his theory of weight systems, to V. V. Batyrev, D. R. Morrison, Y. Ruan and A. Tyurin for useful information and to H. Ooguri, Y. Shimizu for warm encouragement.

Notation.

ℕ : the monoid of nonnegative integers.

ℤ_{+} : the semigroup of positive integers.

ℤ : the ring of integers.

ℚ, ℝ, ℂ : the field of rational, real, complex numbers.

𝕋 : n-dimensional complex torus (ℂ^{∗})^{n}.

1. Polar duality.

1.1 Reflexive polytopes.

We fix here some notations about geometry of convex body and review some results. We refer [25] for generality of toric geometry and geometry of convex bodies.

Let M be a free ℤ-module of rank n. We naturally imbed M in the ℚ-vector space M_{ℚ} := M ⊗_{ℤ} ℚ. Similarly for the dual module N := M^{∨} = Hom(M, ℤ). We denote the natural pairing by ( , ) : M × N → ℤ, which we also use for its ℚ-extensions.

Definition 1.1.1. A subset Δ in M_{ℚ} is called a convex polytope if it is a convex hull of a finite subset of M_{ℚ}. Δ is integral if the finite subset can be taken from M. A face of Δ is a nonempty intersection with a hyperplane whose closed half space contains whole Δ. We denote the k-skeleton of Δ by Δ^{[k]}, which is the union of all faces of Δ whose dimension is no more than k. We call a codimension-one face a facet. We denote the nubers of points |M ∩ Δ| and |M ∩ Int Δ| by l(Δ) and l^{∗}(Δ), respectively.

Remark 1.1.2. We use those words above similarly for subsets in N_{ℚ}. We shall sometimes omit the word “convex” in this paper since we always assume so for polytopes. A face of a polytope Δ is also a polytope.

Definition 1.1.3. Let K, K_{1} and K_{2} be subsets of M_{ℚ} and c ∈ ℚ. cK := {cx ∈ M_{ℚ}|x ∈ K}, K_{1} + K_{2} := {x_{1} + x_{2} ∈ M_{ℚ}|x_{i} ∈ K_{i} for i = 1, 2}.

Definition 1.1.4. K be a subset in M_{ℚ}. The polar dual of K is the following subset of N_{ℚ}: K^{∗}={y ∈ N_{ℚ}| (x, y) ≥ −1 for all x ∈ K}.

Lemma 1.1.5.

(1) K^{∗} is a convex set containing 0. If K_{1} ⊂ K_{2} then K_{1}^{∗} ⊃ K_{2}^{∗}.

(2) If Δ is an n-dimensional convex polytope with 0 ∈ Int Δ, then all the same for Δ^{∗}. Moreover, Δ^{∗∗} = Δ by natural identification M^{∨∨} = M.

Definition 1.1.6. [11] For an n-dimensional polytope Δ in M_{ℚ}, define an n-dimensional projective variety

ℙ_{Δ,M} = ℙ_{Δ} := Proj ⨁_{k=0}^{∞} ℂ (kΔ ∩ M) T^{k},

where ℂ (kΔ ∩ M) denotes the ℂ-vector space generated the elements of M in kΔ, and the multiplication comes from the addition of M. (ℙ_{Δ}, O(1)) is the polarized variety associated to the polytope Δ.

Definition 1.1.7. [5] Let Δ be an n-dimensional integral polytope with 0 ∈ Int Δ. Δ is said to be reflexive if one of the following equivalent conditions is satisfied:

(1) Δ^{∗} is also an integral polytope;

(2) there exists a finite subset {y_{1}, …, y_{k}} in N such that Δ = {x ∈ M_{ℚ} | (x, y_{i}) ≥ −1 for 1 ≤ i ≤ k};

(3) for each facet δ of Δ, there exists an integral element y ∈ N such that δ ⊂ {x ∈ M_{ℚ} | (x, y) = −1};

(4) for each facet δ, there are no points of M between the origin and the hyperplane containing δ.

If Δ is reflexive, Δ^{∗} is reflexive. We cite here some results in [5].

Theorem 1.1.8. [5 Theorem 4.1.9] Let Δ be an n-dimensional integral polyhedron in M_{ℚ}, ℙ_{Δ} the corresponding n-dimensional projective toric variety, and F(Δ) the family of projective Δ-regular hypersurfaces \overline{Z}_{f} in ℙ_{Δ}. Then the following conditions are equivalent:

(1) the family F(Δ) of Δ-regular hypersurfaces in ℙ_{Δ} consists of Calabi-Yau varieties with canonical singularities;

(2) the ample invertible sheaf O_{Δ}(1) on the toric variety ℙ_{Δ} is anticanonical(i.e. ℙ_{Δ} is a toric Fano variety with Gorenstein singularities);

(3) Δ contains only one integral point m_{0} in its interior, and (Δ−m_{0}, M) is a reflexive pair.

Theorem 1.1.9. [5 Theorem 4.4.3] For any reflexive polyhedron Δ of dimension n ≥ 4, the Hodge number h^{n−2,1}(Ẑ_{f}) of a MPCP-desingularization of a Δ-regular Calabi-Yau hypersurface \overline{Z}_{f} ⊂ ℙ_{Δ} equals the Picard number h^{1,1}(Ẑ_{g}) of a MPCP-desingularization of a Δ^{∗}-regular projective Calabi-Yau hypersurface \overline{Z}_{g} ⊂ ℙ_{Δ^{∗}} corresponding to the dual reflexive polyhedron Δ^{∗}.

In the above two theorems, “polyhedron” means “polytope” in our paper. We reserve the word “polyhedron” for a three dimensional polytope. MPCP-desingularization is a maximal projective crepant partial toric embedded resolution.

1.2. Dual correspondence for toric projective varieties.

In this section, we shall treat a family of anticanonical members in a ℚ-Fano toric variety ℙ_{Δ} where Δ may not be integral. We will think not only the quotient family corresponding to the same Δ but also specialization corresponding to a sub-Newton polytope.

Definition 1.2.1. Let P_{n} be the set of n-dimensional pairs (Δ, M) where M is a free ℤ-module of rank n and Δ is a n-dimensional polytope in M_{ℚ} which contains 0 in its interior, such that ℙ_{Δ,M} is ℚ-Fano and Δ represents anticanonical divisor.

We usually assume that n ≥ 2. Let (\overline{Δ}, \overline{M}) and (\overline{∇}, \overline{N}) be two elements of P_{n}. We denote the dual groups (\overline{M})^{∨} by \underline{N} and (\overline{N})^{∨} by \underline{M}. We also denote the polar duals \overline{Δ}^{∗} and \overline{∇}^{∗} by \underline{∇} and \underline{Δ}, respectively.

Definition 1.2.2. A ℚ-linear isomorphism σ: \underline{M}_{ℚ} → \overline{M}_{ℚ} such that σ(\underline{Δ}) ⊂ \overline{Δ}, is called a correspondence between (\overline{Δ}, \overline{M}) and (\overline{∇}, \overline{N}) if σ(\underline{Δ}) ∩ \overline{M} generates \overline{M} and \underline{Δ} has a nonempty intersection with each facet of \overline{Δ}.

If σ(\underline{M}) = \overline{M} and ^{t}σ is correspondence, we call it a dual correspondence.

We shall identify \underline{M}_{ℚ} and \overline{M}_{ℚ}, \overline{N}_{ℚ} and \underline{N}_{ℚ} by σ and ^{t}σ, respectively. We shall sometimes omit σ.

When (\overline{Δ}, \overline{M}) and (\overline{∇}, \overline{N}) have a dual correspondence σ, we will use M = \overline{M} = σ(\underline{M}) and N = \overline{N} = ^{t}σ(\underline{N}).

A basic observation is the following

Proposition 1.2.3. Assume there exists a dual correspondence between two elements (\overline{Δ}, M) and (\overline{∇}, N) of P_{n}. Let (Δ, M) be a reflexive pair such that \underline{Δ} ⊂ Δ ⊂ \overline{Δ}.

Then there is a natural birational map Φ: ℙ\overline{Δ} −−→ ℙ_{Δ} coming from the inclusion between the polytopes. A general anticanonical member of ℙ_{Δ} is birational to the corresponding member of ℙ\overline{Δ}.

Similarly, from \underline{∇} ⊂ Δ^{∗} ⊂ \overline{∇}, a general anticanonical member of ℙ_{Δ^{∗}} is birational to that of ℙ\overline{∇}.

Proof. We denote the sublinear system of the anticanonical linear system of ℙ\overline{Δ} corresponding to the polytope Δ by Λ(Δ).

By definition, any nonempty nonzero subsystem Λ is free from base points on the ndimensional orbit of 𝕋.

Since the set Δ ∩ M generates the whole M, Λ(Δ) defines a birational map \overline{φ}: ℙ\overline{Δ} −−→ ℙ^{l(Δ)−1}. The image V is the projective variety Proj ⨁_{k=0}^{∞} (∑_{i=1}^{k}(Δ ∩ M)), which is associated to the graded algebra generated by Δ ∩ M.

Each facet of \overline{Δ} corresponds to an (n − 1)-dimensional orbit of 𝕋 on ℙ\overline{Δ}, which has nonempty intersection with Δ, since Δ contains \underline{Δ}. Thus Λ(Δ) is free from base components and the birational transform of a hyperplane section of V in ℙ\overline{Δ} is a member of Λ(Δ). In particular, a general member of Λ(Δ) is an irreducible divisor if n > 1.

On the other hand, the anticanonical complete linear system on ℙ_{Δ} is free from base points, since Δ is an integral polytope, and defines a morphism φ to the same V, which is birational by the same reason as above. Take Φ := φ^{−1} ∘ \overline{φ}. □

Remark 1.2.4. Thus, in the situation above, if a Calabi-Yau variety Z̄_{Δ} is a specialization of Z̄_{Δ′}, then Z̄_{Δ^{∗}} is a generalization of Z̄_{Δ′}^{∗}. When you specialize Z̄, the singularity goes bad. As a result, we have more Kähler moduli and less complex structure moduli for the desingularization Z̃_{Δ}, and contrary for Z̃_{Δ^{∗}}. We hope this relation holds also outside toric varieties.

Remark 1.2.5. ℙ_{Δ} is isomorphic to V if n ≤ 3. Note that we have unique minimal models in this case.

Proposition 1.2.6. Let Δ be a reflexive polytope of dimension at most three, and (ℙ_{Δ}, O(1)) be the polarized variety associated to it. Then O(1) is simply generated and defines projective normal embedding in ℙ^{l(Δ)−1}.

Proof. The proof follows from the following

Proposition 1.2.7. Let Δ be a reflexive polytopes in M_{ℚ} whose dimension is at most three, and k be an positive integer. Then for any integral point P ∈ kΔ, there exist integral points P_{i} ∈ Δ (i = 1, …, k) with P = P_{1} + ⋯ + P_{k}.

Proof. We assume n = 3, since other cases are similar and easier. For each 2-dimensional face δ of Δ, we have a subdivision by integral simplices (triangles) such that

(1) each simplex is elementary, that is, it does not contain points of M other than the vertices,

(2) each simplex has common points with other simplices only on its edges and

(3) the union of all simplices covers δ.

We note that an elementary simplex of dimension two is regular. That means, if we denote the vertices of the simplex by Q_{0}, Q_{1} and Q_{2} and the plane containing δ by α, Q_{1} − Q_{0} and Q_{2} − Q_{0} spans α ∩ M. Let C be the closed cone over the simplex with apex at 0. Then any integral point in C can be written as an integral combination of Q_{i}’s, since Δ is reflexive.

Remark 1.2.8. In higher dimensions, there exist elementary non regular simplices.

In general for an ample divisor A on an algebraic K3 surface, it is known that 3A is very ample.

Corollary 1.2.9. For a general toric hypersurface Z̄ of dimension 1 or 2, the induced morphism Z̄ → ℙ^{l(Δ)−2}. is a projectively normal embedding.

Proof. H^{1}(ℙ_{Δ}, O(k)) vanishes for k ≥ 0. □

In Chapter 3, we shall apply these notions to the case of weighted projective spaces with one particular choice of a homogeneous coordinate.

DUALITY OF WEIGHTS, MIRROR SYMMETRY AND ARNOLD’S STRANGE DUALITY 7

2. Duality for weight systems. 2.1. Weighted projective space.

We fix here some notations. For general properties about weighted projective spaces, see for example, [22][13][17]. Definition 2.1.1. Let a_{0}, …, a_{n} be fixed positive integers. We denote by ℙ(a_{0}, …, a_{n}) the projective variety Proj ℂ[x_{0}, …, x_{n}] where the degree of x_{i} is a_{i}. We call it a weighted projective space of weight (a_{0}, …, a_{n}).

For a positive integer k, there is a natural isomorphism ℙ(a_{0}, …, a_{n}) ≅ ℙ(ka_{0}, …, ka_{n}). Definition 2.1.2. A weight (a_{0}, …, a_{n}) is reduced if gcd(a_{0}, …, a_{n}) = 1.

We assume the weight is reduced. Let k be gcd(a_{1}, …, a_{n}). Then ℙ(a_{0}, a_{1}, …, a_{n}) ≅ ℙ(a_{0}, a_{1}/k, …, a_{n}/k), which leads to the following Definition 2.1.3. A reduced weight (a_{0}, …, a_{n}) is well-formed if for all i, gcd(a_{j})_{j≠i} = 1.

We shall usually treat only well-formed weights. 2.2. Weight systems.

We refer [32] for general statements of weight systems, especially for n = 3. Definition 2.2.1. A (n+1)-uple of positive integers W = (a_{1}, …, a_{n}; h) is called a system of weights or simply a weight system. We always assume that h ∈ ∑_{i=1}^{n} a_{i}. We call the integers a_{i} as weights of W and the last weight h the degree of W. Definition 2.2.2. W is said to be reduced if gcd(a_{1}, …, a_{n}, h) = 1 holds. Remark 2.2.3. W is reduced then gcd(a_{1}, …, a_{n}) = 1, since h ∈ ∑_{i=1}^{n} a_{i}. Definition 2.2.4. W = (a_{1}, …, a_{n}; h) and W′ = (a′_{1}, …, a′_{n}; h′) are equivalent if for some rational number k and a permutation σ ∈ 𝔖_{n}, ka_{σ(i)} = a′_{i} (1 ≤ i ≤ n) and kh = h′ hold.

For each equivalence class of weight systems there is exactly one reduced weight system satisfying a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{n}, which we usually use in this paper. 2.3. Duality for weight systems.

Let n be a positive integer. Take a weight system W_{a} = (a_{1}, …, a_{n}; h). Assume that h ∈ ∑_{i=1}^{n} ℕa_{i} as always, and that W_{a} is reduced, for the sake of brevity. We denote the integral vector ^{t}(a_{1}, …, a_{n}) by a. Definition 2.3.1. a_{0} := h − ∑_{i=1}^{n} a_{i}.

Assume also that a_{0} ≠ 0.

For a rational monomial X_{0}^{m_{0}} ⋯ X_{n}^{m_{n}} of degree h, we assign an n-tuple of integers (α_{1}, …, α_{n}) = (m_{1} − 1, …, m_{n} − 1). Note that X_{0} ⋯ X_{n} corresponds to the origin. The whole such n-tuples constitute a set Definition 2.3.2.

M(W_{a}) := {(α_{1}, …, α_{n}) ∈ ℤ^{⊕n} | ∑_{i=1}^{n} a_{i}(α_{i} + 1) ≡ h mod a_{0} } .

Lemma 2.3.3. M(W_{a}) is a subgroup of ℤ^{⊕n} of index |a_{0}|.

Proof. This follows from M(W_{a}) = {(α_{1}, …, α_{n}) ∈ ℤ^{⊕n} | ∑_{i=1}^{n} a_{i}α_{i} ≡ 0 mod a_{0}} and gcd(a_{1}, …, a_{n}) = 1. □

Remark 2.3.4. If W_{a} and W_{b} are equivalent, M(W_{a}) and M(W_{b}) are the same subset of ℤ^{⊕n} up to the permutation of the coordinates. We sometimes abbreviate this set as M in this chapter.

Let C = (c_{ij}) be an n × n-matrix whose elements are nonnegative integers. Let B be the n × n-matrix (c_{ij} − 1).

Lemma 2.3.5. Assume Ca = ^{t}(h, …, h). Then

(1) (det C)/h = (det B)/a_{0}, and this is an integer.

(2) The following three conditions are equivalent: (a) {(c_{i1} − 1, …, c_{in} − 1)| 1 ≤ i ≤ n} is a basis of M, (b) | det B| = |a_{0}| and (c) | det C| = h.

Proof. Note that for 1 ≤ i ≤ n, (c_{i1} − 1, …, c_{in} − 1) ∈ π(a_{0}) ∩ M and (c_{i1}, …, c_{in}) ∈ π(h), where π(t) is the hyperplane defined by ∑_{i=1}^{n} a_{i}α_{i} = t. The rest is clear. □

Remark 2.3.6. We will say a few words for the case W_{a} is not reduced. Let d_{0} be gcd(a_{1}, …, a_{n}). Then M is a subgroup of index |a_{0}|/d_{0}, and the row vectors of B is a basis ⟺ | det B| = |a_{0}|/d_{0} ⟺ | det C| = h/d_{0}.

Let W_{a} = (a_{1}, …, a_{n}; h) and W_{b} = (b_{1}, …, b_{n}; k) be two weighted systems.

Definition 2.3.7. An integer matrix C ∈ M_{n}(ℕ) is said to be a weighted magic square of weight (W_{a}; W_{b}) if C^{t}(a_{1}, …, a_{n}) = ^{t}(h, …, h) and (b_{1}, …, b_{n})C = (k, …, k).

Remark 2.3.8. In the case a_{1} = ⋯ = a_{n} = b_{1} = ⋯ = b_{n} = 1 and h = k, C is called a integer stochastic matrix or magic square, which appears in classical combinatorics theory[35].

Definition 2.3.9. A weighted magic square C is primitive if | det C| = h/ gcd(a_{1}, …, a_{n}) = k/ gcd(b_{1}, …, b_{n}).

Remark 2.3.10. C is primitive if and only if the row vectors of B span M(W_{a}) and the column vectors span M(W_{b}).

Definition 2.3.11. When there exists a primitive weighted magic square C of weight (W_{a}; W_{b}), we say W_{a} and W_{b} are dual with respect to C.

Dual weight systems are strongly dual if all rows and columns of C contain 0.

Remark 2.3.12. If W_{a} and W_{b} are reduced dual weights, it follows that h = k, ∑_{i=1}^{n} a_{i} = ∑_{i=1}^{n} b_{i} and a_{0} = b_{0}.

A permutation on weights (a_{1}, …, a_{n}) (resp. (b_{1}, …, b_{n})) interchanges the corresponding columns (resp. rows) of C.

From this definition, one can calculate the dual weights.

The following Proposition is handy for the calculation.

Proposition 2.3.13. Let W_{a} = (a_{1}, …, a_{n}; h) and W_{b} = (b_{1}, …, b_{n}; h) be dual weight systems with a_{0} be h − ∑_{i=1}^{n} a_{i} > 0 and C = (c_{ij}) be the corresponding weighted magic square. Let Θ be the (n − 1)-simplex in M(W_{a})_{ℚ} with vertices {(c_{i1} − 1, …, c_{in} − 1)|1 ≤ i ≤ n}. Then

(1) ∑_{j=1}^{n} c_{ij}a_{j} = h for 1 ≤ i ≤ n,

(2) \frac{a_{0}}{h − a_{0}} (1, …, 1) ∈ Int Θ and

(3) Θ is elementary, i.e., Θ ∩ M(W_{a}) are the set of all vertices of Θ.

Conversely, for a given (a_{1}, …, a_{n}) ∈ (ℤ)^{⊕n}, if there exists a (n−1)-simplex Θ with vertices {(c_{i1} − 1, …, c_{in} − 1)|1 ≤ i ≤ n} which satisfies (1),(2) and (3) above, then there exists a weight system W_{b} = (b_{1}, …, b_{n}; h) such that C is a weighted magic square of weight (W_{a}; W_{b}). If W_{b} is reduced, then it is a dual weight system.

Proof. The assertions (1) and (3) are trivial. The remaining (2) is equivalent to that the origin sits inside of the n-simplex Δ which is the cone over Θ with apex (−1, …, −1). This is equivalent to that all the coefficients of the linear relation between the vertices of Δ has a same sign. Since we can take nothing but b_{i}’s as coefficients: b_{0}^{t}(−1, …, −1) + ∑_{i=1}^{n} b_{i}^{t}(c_{i1} − 1, …, c_{in} − 1) = 0, (2) follows.

We will show the converse. From the argument above, b_{i}’s are determined as positive integers up to ratio. We will take the reduced b_{i}’s at first. We define k := ∑_{i=0}^{n} b_{i} and W_{b} := (b_{1}, …, b_{n}; k). Then C is a weighted magic square of weight (W_{a}; W_{b}). We note that | det C| is a multiple of k, and by (3), | det C| = h. Thus multiplying some integer to b_{i}’s, we can achieve that h = k. In particular, if this new W_{b} is reduced then C is primitive since k = | det C|. □

Definition 2.3.14. A weight system W is self-dual if W and W are dual.

Remark 2.3.15. If one can take a symmetric matrix as C, then W is self-dual. In general, there may be several dual weights as shown in the Proposition below. Similarly, self-duality does not mean that it is the only dual weight. They are determined completely by a finite check.

2.4. Examples.

First we show examples of case a_{0} = −1, which is hopefully to be generalized as a mirror of log Calabi-Yau manifolds.

Proposition 2.4.1. A_{l−1} = {(1, k, l − k; l), 1 ≤ k ≤ l/2)} (l ≥ 2) are closed under duality for each l. D_{l+1} = (2, l − 1, l; 2l) (l ≥ 3), E_{6} = (3, 4, 6; 12), E_{7} = (4, 6, 9; 18) and E_{8} = (6, 10, 15; 30) are self-dual.

Proof. Case A_{l−1}.

For an arbitrary positive integer j such that jk < l, we can take C as ( \begin{matrix}l − jk & j & 0 \\ k & 0 & 1 \\ 0 & 1 & 1 \end{matrix} ).

Then the dual weight is (1, j, l − j; l). These are the only choice for C up to permutation.

For D_{l+1}(l:odd), D_{l+1}(l:even), E_{6}, E_{7} and E_{8}, take the following matrices as C:

( \begin{matrix}(l + 1)/2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} ) , ( \begin{matrix}l/2 & 0 & 1 \\ 0 & 0 & 2 \\ 1 & 2 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 3 & 0 \\ 3 & 1 & 0 \\ 0 & 0 & 2 \end{matrix} ) and ( \begin{matrix}5 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{matrix} ) .

They are all symmetric and unique up to the permutation of the rows. They are strongly dual. □

Next, we will treat the case of the exceptional unimodal singularities[1]. They are all weighted homogeneous with a_{0} = 1.

Theorem 2.4.2. Let W = (a_{1}, a_{2}, a_{3}; h) be a weight system which corresponds to one of the 14 exceptional unimodal singularities. Then W has a unique dual weight system W^{∗} up to equivalence. And W^{∗} is the weight system which corresponds to Arnold’s strange duality.

Proof. We list below the choice of C. These are, in fact, unique choice for primitive C up to permutation. They are automatically strongly dual.

Let us take coordinate as x, y and z. In the column of C, each monomial x^{α}y^{β}z^{γ} means

a row vector (α, β, γ) of C. For example, in the case E_{14}, C = ( \begin{matrix}4 & 0 & 1 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{matrix} ), which is

nothing but the transposed matrix ^{t}C for Q_{10}.

We list also C_{0}, which shows the vertices of the maximum Newton polytope of polynomials of degree h of x, y and z, and the Gabrielov numbers and the Dolgachev numbers[1].

Table of exceptional unimodal isolated hypersurface singularities.

class a_{1} a_{2} a_{3} h  C_{0}  C  Gab Dol
E_{12} 6 14 21 42 x^{7} y^{3} z^{2} x^{7} y^{3} z^{2} 2 3 7 2 3 7
E_{13} 4 10 15 30 x^{5}y y^{3} z^{2} x^{5}y y^{3} z^{2} 2 3 8 2 4 5
Z_{11} 6 8 15 30 x^{5} xy^{3} z^{2} x^{5} xy^{3} z^{2} 2 4 5 2 3 8
E_{14} 3 8 12 24 x^{8} y^{3} z^{2} x^{4}z y^{3} z^{2} 2 3 9 3 3 4
Q_{10} 6 8 9 24 x^{4} y^{3} xz^{2} x^{4} y^{3} xz^{2} 3 3 4 2 3 9
Z_{12} 4 6 11 22 x^{4}y xy^{3} z^{2} x^{4}y xy^{3} z^{2} 2 4 6 2 4 6
W_{12} 4 5 10 20 x^{5} y^{4} z^{2} x^{5} z^{2} y^{2}z 2 5 5 2 5 5
Z_{13} 3 5 9 18 x^{6} xy^{3} z^{2} x^{3}z xy^{3} z^{2} 2 4 7 3 3 5
Q_{11} 4 6 7 18 x^{3}y y^{3} xz^{2} x^{3}y y^{3} xz^{2} 3 3 5 2 4 7
W_{13} 3 4 8 16 x^{4}y y^{4} z^{2} x^{4}y z^{2} y^{2}z 2 5 6 3 4 4
S_{11} 4 5 6 16 x^{4} xz^{2} y^{2}z x^{4} xz^{2} y^{2}z 3 4 4 2 5 6
Q_{12} 3 5 6 15 x^{5} y^{3} xz^{2} x^{3}z y^{3} xz^{2} 3 3 6 3 3 6
S_{12} 3 4 5 13 x^{3}y xz^{2} y^{2}z x^{3}y xz^{2} y^{2}z 3 4 5 3 4 5
U_{12} 3 4 4 12 x^{4} y^{3} z^{3} x^{4} y^{2}z yz^{2} 4 4 4 4 4 4

□ Remark 2.4.3. K. Saito defines a duality of weights using a duality of poset diagrams coming from the eigenvalues of the monodromy of minimally elliptic singularities[34]. His duality also reproduces Arnold’s strange duality and he also computed the dual weights for the 49 weight systems corresponding to the minimally elliptic singularities which are not simple elliptic. His results include the uniqueness of duality for isolated singularities.

As shown below, our duality for n = 3 coincides with Saito’s in the case (0) the fourteen unimodal singularities and (3) a_{0} ≥ 2. For cases (2) a_{0} = 1 and modality m is more than one, our duality gives new dual weights, where nonisolated singularities appear as dual partners of isolated singularities. These of course also correspond to the polar duality, as shown in Chapter 5.

It is natural to treat the simple elliptic cases (1) for n = 2, since these correspond to weighted elliptic curves. Their list is the same as that of the weighted 2-spaces which correspond to reflexive polytopes. The weights are self-dual for n = 2.

Notice that the singularities of dual weights are not dual itself as Arnold’s strange duality but its (explicit) specializations enjoy the duality as shown in the later Chapters.

Proposition 2.4.4.

(1) (1, 1; 3), (1, 2; 4), (2, 3; 6) are self-dual.

(2) (2, 2, 3; 8), (2, 2, 5; 10), (2, 3, 4; 10), (2, 4, 7, 14) and (2, 6, 9, 18) are self-strongly dual; (2, 3, 6; 12) and (2, 4, 5; 12) are strongly dual. (2, 3, 3; 9) has no strongly dual weight.

(3) Duality of weights for minimally elliptic weight systems with a_{0} > 1 coincides with Saito’s duality[34].

Proof. Here we give the full list of primitive C up to permutation. The notation of classes is after [1] and [19].

(1) Simple elliptic singularities.

cl. C  a_{1} a_{2} h dual
P_{8} x^{2}y xy^{2} 1 1 3 1 1
X_{9} y^{2} x^{2}y 1 2 4 1 2
J_{10} x^{3} y^{2} 2 3 6 2 3

(2) Minimal elliptic singularities with a_{0} = 1 and m > 1.

cl.  C  a_{1} a_{2} a_{3} h dual
V_{1,0} = V_{15} xz^{2} yz^{2} x^{2}y^{2} 2 2 3 8 2 2 3
x^{2}y^{2} xy^{3} xz^{2}  * 1 2 4
N_{16} x^{3}y^{2} x^{2}y^{3} z^{2}  2 2 5 10 2 2 5
U_{1,0} = U_{14} - 2 3 3 9 -
y^{2}z x^{3}y yz^{2}  * 1 3 4
S_{1,0} = S_{14} y^{2}z x^{2}y^{2} xz^{2} 2 3 4 10 2 3 4
xz^{2} x^{3}z y^{2}z  * 1 3 5
W_{1,0} = W_{15} y^{2}z x^{3}y^{2} z^{2}  2 3 6 12 2 4 5
z^{2} x^{3}z y^{2}z  * 1 4 6
Q_{2,0} = Q_{14} y^{3} x^{2}y^{2} xz^{2} 2 4 5 12 2 3 6
Z_{1,0} = Z_{15} xy^{3} x^{3}y^{2} z^{2}  2 4 7 14 2 4 7
J_{3,0} = J_{16} y^{3} x^{3}y^{2} z^{2}  2 6 9 18 2 6 9

The * signifies that the dual weight is not strongly dual, hence the singularity corresponding to the smaller triangle is reducible. Other weight systems are strongly dual.

(3) Minimal elliptic singularities with a_{0} > 1.

\begin{matrix}cl. & & C & & a_{1} a_{2} a_{3} & h & dual \\ V′_{18} & & - & & 3 3 4 & 12 & - \\ U_{16} & x^{5} & y^{2}z & yz^{2} & 3 5 5 & 15 & 3 5 5 \\ S_{16} & x^{4}y & xz^{2} & y^{2}z & 3 5 7 & 17 & 3 5 7 \\ W_{17} & x^{5}y & z^{2} & y^{2}z & 3 5 10 & 20 & - - - \\ Q_{16} & x^{4}z & y^{3} & xz^{2} & 3 7 9 & 21 & 3 7 9 \\ Z_{17} & x^{4}z & xy^{3} & z^{2} & 3 7 12 & 24 & - - - \\ E_{18} & x^{5}z & y^{3} & z^{2} & 3 10 15 & 30 & - - - \\ _{2}V_{18}^{∗} & xy^{3} & x^{3}z & yz^{2} & 4 5 7 & 19 & 4 5 7 \\ _{1}V_{18}^{∗} & x^{3}z & y^{4} & xz^{2} & 4 5 8 & 20 & 4 5 8 \\ N_{19} & x^{3}z & xy^{4} & z^{2} & 4 5 12 & 24 & - - - \\ S_{17} & x^{6} & xz^{2} & y^{2}z & 4 7 10 & 24 & - - - \\ W_{18} & x^{7} & y^{2}z & z^{2} & 4 7 14 & 28 & 4 7 14 \\ Q_{17} & x^{5}y & y^{3} & xz^{2} & 4 10 13 & 30 & - - - \\ Z_{18} & x^{6}y & xy^{3} & z^{2} & 4 10 17 & 34 & 4 10 17 \\ E_{19} & x^{7}y & y^{3} & z^{2} & 4 14 21 & 42 & - - - \end{matrix} \begin{matrix}_{3}V_{19}^{∗} & y^{4} & x^{3}z & yz^{2} & 5 6 9 & 24 & - - - \\ _{2}N_{20} & y^{5} & x^{3}z & z^{2} & 5 6 15 & 30 & - - - \\ V′_{20} & x^{3}z & z^{3} & xy^{3} & 6 7 9 & 27 & 6 7 9 \\ _{2}V_{19}^{∗} & x^{5} & xy^{3} & yz^{2} & 6 8 11 & 30 & - - - \\ _{1}V_{19}^{∗} & x^{4}y & y^{4} & xz^{2} & 6 8 13 & 32 & 4 7 16 \\ _{1}N_{20} & x^{5}y & xy^{4} & z^{2} & 6 8 19 & 38 & 6 8 19 \\ Q_{18} & x^{8} & y^{3} & xz^{2} & 6 16 21 & 48 & 3 16 24 \\ Z_{19} & x^{9} & xy^{3} & z^{2} & 6 16 27 & 54 & 4 18 27 \\ E_{20} & x^{11} & y^{3} & z^{2} & 6 22 33 & 66 & 6 22 33 \\ V′_{21} & z^{3} & y^{4} & x^{3}z & 8 9 12 & 36 & 8 9 12 \\ V_{20}^{∗} & y^{4} & x^{5} & yz^{2} & 8 10 15 & 40 & 5 8 20 \\ N_{21} & y^{5} & x^{5}y & z^{2} & 8 10 25 & 50 & 8 10 25 \end{matrix} □

Remark 2.4.5. The four weight systems in (3): (6, 8, 13; 32), (6, 16, 21; 48), (6, 16, 27; 54) and (8, 10, 15; 40) are not in the list of Reid[28][17], hence do not correspond to K3 surface with cyclic quotient singularities.

For the weights with some C but without a dual weight (e.g. W_{17}), C does not satisfy primitivity for the dual weight. On the contrary, they enjoy a dual correspondence defined in Chapter 1, except (5, 6, 15; 30).

All the dual weights are strongly dual.

Remark 2.4.6. There are 41 weight systems with a_{0} = 1 in Reid’s list and 19 of them are not minimally elliptic. A similar calculation shows that all the duality for them are as follows; (1, 1, 1; 4), (1, 1, 2; 5), (1, 2, 2; 6), (1, 2, 3; 7) and (1, 2, 4; 8) are self-dual; (1, 1, 3; 6) and (1, 2, 2; 6) are dual; (1, 2, 4; 8), (1, 3, 4; 9), (1, 3, 5; 10) and (1, 4, 6; 12) have dual weight systems which are minimally elliptic as in Proposition 2.4.4 (2); and all the ten weight systems with a_{1} > 1 do not have dual weight systems.

3. Relation of duality of weight systems and polar duality.

We shall show in this chapter that the duality of weight for a_{0} = 1 is a special case of dual correspondence defined in Chapter 1. We will always choose one coordinate for the compactification parameter.

Let W_{a} = (a_{1}, …, a_{n}; h) be a weight system with h ∈ ∑_{i=1}^{n} ℕa_{i} and a_{0} := h − ∑_{i=1}^{n} a_{i} as usual. We assume a_{0} > 0 in this section.

Let ℙ(a) be the weighted projective space of weight (a_{0}, a_{1}, …, a_{n}) and X_{0}, …, X_{n} be the natural homogeneous coordinates of ℙ(a) induced by the definition. Then the n-dimensional complex torus 𝕋 is embedded equivariantly in ℙ(a) as the locus {F_{0} := ∏_{i=0}^{n} X_{i} ≠ 0}.

Let M(W_{a}) be the group defined in Section 3. M(W_{a}) is nothing but the abelian group of exponents of rational monomials, which appears in the context of toric construction. For a point P = (α_{1}, …, α_{n}) on M(W_{a}), we write the corresponding monic monomial by F_{P} = ∏_{i=0}^{n} X_{i}^{α_{i}+1}, where α_{0} is determined by ∑_{i=0}^{n} a_{i}α_{i} = 0.

Definition 3.1. Let \overline{Δ}(a) be the n-simplex in M(W_{a})_{ℚ} which is the convex hull of the following vertices: (−1+h/a_{1}, −1, …, −1), …, (−1, …, −1, −1+h/a_{n}) and (−1, …, −1).

(\overline{Δ}, \overline{M}) = (\overline{Δ}(a), M(W_{a})) is an element of P_{n} with ℙ_{\overline{Δ},\overline{M}} ≅ ℙ(a).

Lemma 3.2. Let (\overline{Δ}, M) be as above. Then its dual (\overline{Δ}^{∗}, M^{∗}) is integral, homothetic to \overline{Δ} and the vertices generate M^{∗}.

Proof. For the sake of simplicity, we assume that W_{a} is reduced. Recall that M = {(α_{1}, …, α_{n})| ∑_{i=1}^{n} a_{i}α_{i} ≡ 0 mod a_{0}}. Let M′ := ℤ^{⊕n}, which contains M as a ℤsubmodule of index a_{0}. By simple computation, the dual of (\overline{Δ}, M′) is an n-simplex with vertices v_{1} := (1, 0, …, 0), …, v_{n} := (0, …, 0, 1) and v_{0} := (−a_{1}/a_{0}, …, −a_{n}/a_{0}) in (M′)_{ℚ}^{∗}, with a relation ∑_{i} a_{i}v_{i} = 0 which shows the required homothety. Thus v_{1}, …, v_{n} generates (M′)^{∗}. Since (M′)^{∗} is a submodule of M^{∗}, Those n vertices are also contained in M^{∗}. On the other hand, v_{0} also belongs to M^{∗} since ∑_{i=1}^{n}(−a_{i}/a_{0})α_{i} ∈ ℤ for any (α_{1}, …, α_{n}) ∈ M.

The order of v_{0} in M^{∗}/(M′)^{∗} is exactly a_{0} since (a_{1}, …, a_{n}) is reduced. Thus the last statement follows. □

Remark 3.3. On the contrary, an n-simplex satisfying the conclusion of Lemma 3.2 is isomorphic to \overline{Δ}^{∗}.

By the previous proof, the submodule (M′)^{∗} generated by v_{1}, …, v_{n} has index a_{0} in M. Let (\overline{∇}, \overline{N}) = (\overline{Δ}(b), M(W_{b})) be another pair. We assume W_{b} is also reduced for the

sake of simplicity.

We restrict here the correspondence σ in Chapter 1 as follows: (*)

(1) σ sends the vertex of (\overline{Δ})^{∗} which corresponds to the divisor \{X_{0} = 0\} to the vertex of \overline{∇} which corresponds to the point (1 : 0 : ⋯: 0) (i.e., X_{1} = ⋯ = X_{n} = 0).

(2) same for ^{t}σ.

Proposition 3.4. Assume that (\overline{Δ}, \overline{M}) and (\overline{∇}, \overline{N}) has a dual correspondence σ which satisfies the condition (*) above.

Let (c_{i1} − 1, …, c_{in} − 1) (1 ≤ i ≤ n) be the vertices except (−1, …, −1) of σ(\overline{∇}^{∗}) in \overline{M}. Let B be the n × n-matrix (c_{ij} − 1).

Then

(1) a_{0}|h, b_{0}|h and | det B| = a_{0}b_{0},

(2) If a_{0} = b_{0} = 1, W_{a} and W_{b} are dual weight systems.

Proof. (1) By σ(\overline{N}^{∗}) = \overline{M} and σ(−b_{1}/b_{0}, …, −b_{n}/b_{0}) = (−1, …, −1), (−1, …, −1) ∈ \overline{M}. Thus a_{0}|h. Similarly for b_{0}|h. Since σ(1, 0, …, 0), …, σ(0, …, 0, 1) generates a submodule of index b_{0} in \overline{M}, | det B| = a_{0}b_{0}.

(2) Let C be (c_{ij}). By condition (*), C is a weighted magic square of weight (W_{a}; W_{b}) and primitive by | det B| = 1 using Lemma 2.3.5. □

Thus we have Theorem 3.5. Assume two weight systems W_{a} and W_{b} with a_{0} = b_{0} = 1 are dual. Then

(1) there exists an identification M = (\overline{N})^{∗} = \overline{M} such that \overline{∇}^{∗} is an integral simplex and a subcone of \overline{Δ} with a common apex (−1, …, −1) and a common plane π = \{∑_{i=1}^{n} a_{i}α_{i} = a_{0}\} of facet. The vertices of \overline{∇}^{∗} on π generates M. Similarly for the induced identification N = (\overline{M})^{∗} = \overline{N}.

(2) If a linear subsystem of anticanonical divisor of ℙ(a) corresponds to some reflexive polytope Δ such that \overline{∇}^{∗} ⊂ Δ ⊂ \overline{Δ}, then the family corresponds to Δ^{∗} is a linear subsystem of anticanonical divisor of ℙ(b). This relation is inclusion-reversing with respect to polytopes.

Remark 3.6. The facet above is the Newton polytope for an (n − 1)-dimensional hypersurface singularity X_{0} with ℂ^{∗}-action. Z̄ is the toric compactification in ℙ(a) of a deformation Z of X_{0}.

Remark 3.7. Let Δ be the full Newton polytope of ℙ(a), that is the convex hull of all vertices corresponding to the monomial anticanonical divisors. We have the associated projective variety ℙ_{Δ} as usual. In general, Δ does not contain all the generators of the ring of regular functions of ℙ(a). Thus ℙ(a) and ℙ_{Δ} are not isomorphic in general.

Remark 3.8. By [5 Theorem 5.4.5, Corollary 5.4.6], for a reflexive simplex Δ = \overline{Δ}, that is, for the case a_{i}|h for all i, ℙ_{Δ} is a weighted projective variety whose weight is the coefficients (a_{0}, …, a_{n}) of the unique linear relation ∑_{i=0}^{n} a_{i}P_{i} = 0 (a_{i} ∈ ℤ_{+}, gcd(a_{0}, …, a_{n}) = 1) among the vertices {P_{i}}. Moreover, the polar dual family of deformation of Fermat-type hypersurfaces is the quotient of a subfamily by π_{1}(Δ, M) [5 Corollary 5.5.6].

Remark 3.9. Borcea’s construction[8] does not assume the property (*) which is coming from singularity theory. As he has pointed out, for a given reflexive polytope, its realization as a family of weighted hypersurface has some choice. Nevertheless, for the fourteen exceptional singularities, our duality of weights is unique and compatible with that of usual singularity theory.

4. Application for n = 3.

4.1. K3 surfaces.

We refer the reader [6] for more complete reference about K3 surfaces and [3] for general arguments about string theory on K3 surfaces.

Let X be K3 surface, that is, a compact complex surface with trivial canonical bundle and irregularity h^{1}(O_{X}) = 0. It is well known that any K3 surface is a simply connected, Kähler and symplectic manifold. The only nonzero Hodge numbers of X are h^{0,0} = h^{2,0} = h^{0,2} = h^{2,2} = 1 and h^{1,1} = 20.

First we recall here some notions about lattices.

Definition 4.1.1. Let L be a free ℤ-module of finite rank with a symmetric bilinear ℤvalued pairing ( , ). Such an L is called a lattice. L is even if for each x ∈ L, (x, x) is an even integer, and unimodular if the discriminant is ±1. The index of L is the triplet (λ_{+}, λ_{0}, λ_{−}) of the numbers of positive, null and negative eigenvalues of the corresponding matrix. L is said to be positive definite, negative definite, definite, indefinite, respectively, if the corresponding matrix is so.

Since π_{1}(X) = {1}, we can identify H^{2}(X, ℤ) and its natural image in H^{2}(X, ℂ). H^{2}(X, ℤ) is naturally regarded as a lattice by the cup product, and it is an even unimodular lattice of index (3, 0, −19), so is isomorphic to L := (−E_{8})^{⊕2} ⊕U^{⊕3}. Here, −E_{8} is the even unimodular negative definite lattice of rank 8, whose matrix is (−1) times that of the lattice corresponding to the Dynkin diagram E_{8}. U is the even unimodular indefinite

lattice of rank 2, whose matrix is ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).

In general, the algebraic lattice S_{X} and the transcendental lattice T_{X} of a compact complex surface X are defined as sublattices in (the image of) H^{2}(X, ℤ) as S_{X} := H^{1,1}(X) ∩ H^{2}(X, ℤ) and T_{X} := S_{X}^{⊥} in H^{2}(X, ℤ). Here ⊥ denotes the orthogonal complement. The rank of S_{X} is called the Picard number of X and denoted by ρ(X).

Ruan and Tian have constructed the quantum cohomology ring for semipositive symplectic manifolds in [30]. As a graded module, the quantum cohomology ring is same as the classical cohomology ring, but in general, the multiplicative structures are different. The mirror map μ is usually thought to be an involution of moduli of Calabi-Yau manifolds with Kähler structure which induces an isomorphism between the cohomology groups of the Calabi-Yau manifolds exchanged by μ, interchanging so-called the A-model intersection number described by the quantum cohomology ring and the B-model intersection number coming from Gauss-Manin connection.

The following is an easy modification of [30 Example 8.4].

Proposition 4.1.2. Let X be a compact Kähler surface with trivial canonical bundle. Then the quantum cohomology ring of X is isomorphic to the classical cohomology ring of X.

Thus for a K3 surface or a 2-dimensional complex torus, the A-model intersection number is the classical intersection product. On the other hand, B-model intersection number for those surfaces are also the intersection product by Hodge theory. Thus we may assume that a mirror map preserves the usual cup product.

4.2. Fixed set of the mirror map.

First we shall prove some finiteness result of a special automorphism group of marked K3 surface.

Theorem 4.2.1. Let X be a K3 surface with a Kähler class ω, L be the second integral cohomology lattice H^{2}(X, ℤ) and G be a subgroup of AutX. Assume that the induced action of G on L fixes the cohomology class of a nonzero holomorphic 2-form and ω. Then

(1) The fixed lattice L^{G} contains T_{X}.

(2) The orthogonal complement of (L^{G})^{⊥} in L is a negative definite sublattice of S_{X}.

(3) G is a finite group.

Remark 4.2.2. If the algebraic dimension a(X) of X is not one, it is essentially proved in [24].

Remark 4.2.3. ω can be assumed to be complexified Kähler class, since the action of automorphism preserves the real structure of L.

First we note the following elementary fact.

Lemma 4.2.4. Let L be a lattice and K be a subset of L. Let ϕ be an isometry on L. (1) If ϕ(K) = K then ϕ(K^{⊥}) = K^{⊥}.

(2) If ϕ acts as identity on K, then for any x ∈ L, ϕ(x) − x ∈ K^{⊥}.

Corollary 4.2.5. Let X be a compact Kähler surface and g ∈ AutX. Then g^{∗}(S_{X}) = S_{X} and g^{∗}(T_{X}) = T_{X}.

Proof of Theorem. First note that the kernel of the multiplication of a nonzero holomorphic two form Ω on H^{2}(X, ℝ) is H^{1,1} ∩ H^{2}(X, ℝ).

Take an arbitrary element x ∈ T_{X} and g ∈ G . Then y := g^{∗}x−x ∈ T_{X}. By the previous lemma, y ∈ T_{X} ∩ Ω^{⊥} ∩ ω^{⊥} = T_{X} ∩ S_{X} ∩ ω^{⊥}. Since y ∈ T_{X} ∩ S_{X}, we have (y, y) = 0. On the other hand, since S_{X} ∩ ω^{⊥} is negative definite by the signature theorem, one has y = 0. Thus g^{∗}x = x. This proves (1).

(2) follows from (L^{G})^{⊥} ⊂ (T_{X})^{⊥} ∩ Ω^{⊥} ∩ ω^{⊥} = S_{X} ∩ ω^{⊥}.

Next we shall prove (3).

Since (L^{G})^{⊥} is negative definite and finitely generated, the isometry group O((L^{G})^{⊥}) is a finite group.

Let j : G → O((L^{G})^{⊥}) be the natural group homomorphism. We claim that j is injective.

Suppose the induced action of an element g ∈ G on (L^{G})^{⊥} is identity. For an arbitrary x ∈ L, y := g^{∗}x − x belongs to (L^{G})^{⊥}. Thus g^{∗}y = y. Then for an integer n, we have (g^{n})^{∗}x = x + ny and since g^{∗} is isometry, ((g^{n})^{∗}x, (g^{n})^{∗}x) = (x, x). This leads to y = 0 by definiteness of (L^{G})^{⊥}. Thus g^{∗} is the identity on L.

Since g^{∗} is, by definition, an effective Hodge isometry on L, g is the identity element in AutX by Torelli’s theorem. □

Remark 4.2.6. The structure of G can be found in the list of [24] once one knows that G is a finite algebraic automorphism group. Also, (L^{G})^{⊥} does not contain square (−2)-elements.

Corollary 4.2.7. L^{G}+(L^{G})^{⊥} is a sublattice in L of finite index and L_{ℚ} = (L^{G})_{ℚ}⊕(L^{G})_{ℚ}^{⊥}.

Proof. For any x ∈ L, there is an orthogonal decomposition whose coefficients are in \frac{1}{|G|}ℤ:

x = \frac{1}{|G|}(x_{1} ⊕ x_{2}), x_{1} = ∑_{g∈G} g^{∗}x ∈ (L^{G}), x_{2} = ∑_{g∈G}(x − g^{∗}x) ∈ (L^{G})^{⊥}. □

Take an arbitrary h ∈ O((L^{G})^{⊥}). For arbitrary element x in L, we have a decomposition x = x_{1} + x_{2} in L_{ℚ}, where x_{1} ∈ L^{G} and x_{2} ∈ (L^{G})^{⊥}. We define an action of O((L^{G})^{⊥}) on L_{ℚ} by h(x) := x_{1} + h(x_{2}).

Proposition 4.2.8. If h(L) = L holds, there exists a unique g ∈ AutX which satisfies: g^{∗}ω = ω, g^{∗}Ω = Ω and h = g^{∗}.

Proof. Since ω and Ω belong to (L^{G})_{ℂ}, h is effective Hodge isometry on L. Thus the result follows from Torelli’s theorem.

Remark 4.2.9. For an arbitrary choice of ω and Ω in period domain, there exists a unique K3 surface X with complexified Kähler class ω by the surjectivity of period map. The marking is unique up to the finite automorphism group of X fixing ω, since G is canonically isomorphic to the subgroup of O(L) which fixes ω and Ω.

Next we define a sublattice ⟨ω⟩ such that ω is ‘general’ in (⟨ω⟩)_{ℂ}.

Definition 4.2.10. For a fixed α ∈ L_{ℂ}, we denote by ⟨α⟩ the minimum primitive sublattice of L such that (⟨α⟩)_{ℂ} contains α.

Remark 4.2.11. In fact, ⟨α⟩ is determined by the set of all rational equalities which the coefficients of α satisfy, for an appropriate basis of L. When α ∈ L, ⟨α⟩ is nothing but ℤα. This definition works for an arbitrary torsion-free ℤ-module L and its extension of coefficients by any field of characteristic 0.

⟨ω⟩ is independent of the choice of the complex structure X.

Lemma 4.2.12.

(1) ⟨α + β⟩ ⊂ ⟨α⟩ + ⟨β⟩.

(2) ⟨cα⟩ = ⟨α⟩ for c ∈ ℂ^{∗}.

(3) ⟨α + iβ⟩ = ⟨α, β⟩ for α, β ∈ ℝ.

(4) Let {x_{i}} be a basis of ⟨α⟩. If α = ∑_{i} a_{i}x_{i} (a_{i} ∈ ℂ) then a_{i}’s are linearly independent over ℚ.

(5) α^{⊥} coincides with ⟨α⟩^{⊥} in L.

Proof. Immediate from the definition.

Proposition 4.2.13. α ∈ L^{G} if and only if ⟨α⟩ is contained in L^{G}.

Proof. Take a ℤ-basis {x_{i}} of ⟨α⟩ and write α as an ℝ-linear combination ∑_{i} a_{i}x_{i}. From the minimality of ⟨α⟩, a_{i}’s are linearly independent over ℚ. Let g be an element of G. Then we have α = ∑_{i} a_{i}g^{∗}x_{i}. The lattice generated by {g^{∗}x_{i}} is also primitive, thus it coincides with ⟨α⟩ from the minimality. We can write g^{∗}x_{i} as a ℤ-combination ∑_{j} b_{ij}x_{j}. From the equality ∑_{i} a_{i}g^{∗}x_{i} = α = ∑_{j} a_{j}x_{j} and torsion-freeness of L, we get b_{ij} = δ_{ij}. Thus g^{∗}x_{i} = x_{i} and we are done.

Remark 4.2.14. Thus we have L^{G} ⊃ ⟨ω, Ω⟩. By adding ω a small general element in V := (L^{G})_{ℂ} ∩ ((⟨ω, Ω⟩)^{⊥})_{ℂ}, one can arrange as L^{G} = ⟨ω, Ω⟩ holds, since V is rationally defined and V ∩ L is negative definite by index theorem.

Remark 4.2.15. The lattice L_{D} in the next section is nothing but ⟨ω⟩ when ω is the restriction of the cohomology class of a ‘general’ ample ℝ-divisor of an embedded variety, and the lattice L_{0} is ⟨ω⟩^{⊥} ∩ ⟨Ω⟩^{⊥} in L, and in fact, (L^{G})^{⊥} in L.

When ω is general in S_{X}, ⟨ω⟩ = S_{X} and L_{0} = 0.

4.3. Mirror symmetry for K3 toric hypersurfaces.

In this section, we will treat the algebraic K3 surfaces which are the minimal resolutions of toric hypersurfaces.

Definition 4.3.1. A polyhedron is a polytope of dimension three.

For a given reflexive polyhedron Δ, we use Θ and Γ for 2-and 1-dimensional faces of Δ. The dual face Γ^{∗} of Γ is a 1-dimensional face of Δ^{∗}, so the summation over Γ is same as that over Γ^{∗}.

We denote by Z the intersection of a Δ-regular anticanonical section of ℙ_{Δ} and the n-dimensional orbit of the torus, and by Z̄ its closure, namely the section itself. Z̄ has only canonical Gorenstein singularities by Theorem 1.1.8 due to Batyrev. Let Z̃ be the minimal resolution of Z̄. X = Z̃ is an algebraic K3 surface.

We define the following sublattices of H^{2}(X, ℤ) for toric hypersurface K3 surface X.

Definition 4.3.2.

(1) L_{D} := Im(H^{1,1}(ℙ̃_{Δ}) → H^{1,1}(X)) ∩ H^{2}(X, ℤ),

(2) L_{0} := L_{D}^{⊥} in S_{X}, where ℙ̃_{Δ} is a toric resolution of ℙ_{Δ}.

From a result of [12][4][2], we get

Proposition 4.3.3. Let Δ be three-dimensional reflexive polytope and Z̃ be the K3 surface which is the minimal resolution of a Δ-regular anticanonical section of ℙ_{Δ}. Then

(1) L_{D}(Z̃) = ℤ[Δ^{∗[1]} ∩ N]/N,

(2) L_{0}(Z̃) = ⨁_{dim Γ=1} ℤ(Int Γ ∩ M) × (Int Γ^{∗} ∩ N),

(3) rk TZ̃ − 2 = l(Δ^{[1]}) − 3.

Especially,

(4) rk L_{D}(Z̃) = l(Δ^{∗[1]}) − 3,

(5) rk L_{0}(Z̃) = ∑_{dim Γ=1} l^{∗}(Γ^{∗})l^{∗}(Γ).

Proof. By definition we can follow the same argument of calculation of H_{toric}^{1,1} in [2], one can show that L_{D}(Z̃) = ℤ[Δ^{∗[1]} ∩ N]/N and in particular, rk L_{D}(Z̃) = l(Δ^{∗[1]}) − 3.

By [12 5.11], we have dim H^{2}(Z) = l(Δ^{[1]}) − 1, which is generated by vanishing cycles whose image is nothing but rk TZ̃.

The toric compactification 𝕋 → ℙ_{Δ} adds a divisor for each vertice of Δ^{∗}.

A toric resolution of the ambient space produces the divisors corresponding to the integral points P of Δ^{∗} − (Δ^{∗(0)} ∪ {0}). We denote by C_{P} the support of the center of blowing up in ℙ_{Δ} corresponding to P. If P is in some interior of a 2-dimensional face of Δ^{∗}, C_{P} is a point which is disjoint from Z̄, since Z̄ is Δ-regular.

If P is in some Int Γ^{∗}, C_{P} is a curve in ℙ_{Δ} which intersects normally with Z̄ at (l^{∗}(Γ)+1) points. Since each toric blowing-up corresponding to P creates one ambient divisor, its orthogonal complement increases L_{0} by l^{∗}(Γ) divisors. Thus we have the description of L_{0} and rk L_{0}(Z̃) = ∑_{Γ} l^{∗}(Γ)l^{∗}(Γ^{∗}). □

Remark 4.3.4. In fact, we can easily determine the lattices L_{D} and L_{0} from the data (Δ, M). For instance, the dual graph of L_{D} is obtained from Δ^{∗[1]} by removing three integral points which consist a basis of N. Each remaining vertex is a (−2)-curve and each remaining integral point on a edge Γ is a sum of (l^{∗}(Γ^{∗}) + 1) disjoint (−2)-curves. Each edge in Γ joining integral points shows the intersection number of the two end-points is (l^{∗}(Γ^{∗}) + 1).

We conjecture that the same rule of intersection product holds for L_{G} and Δ^{[1]}.

Corollary 4.3.5. For a fixed reflexive polyhedron Δ, rk TZ̃, L_{D}(Z̃) and L_{0}(Z̃) are independent of the choice of Δ-regular hypersurface Z.

We will thus write them as L_{D}(Δ) and L_{0}(Δ), respectively.

Corollary 4.3.6. rk T (Δ) − 2 = rk L_{D}(Δ^{∗}), L_{0}(Δ) = L_{0}(Δ^{∗}).

Remark 4.3.7. Since the rank of H^{2} is 22, rk T (Δ) + rk L_{D}(Δ) + rk L_{0}(Δ) = 22. That formula gives a relation ∑_{Γ}(l^{∗}(Γ) + 1)(l^{∗}(Γ^{∗}) + 1) = 24.

We have rk T (Δ) − 2 = l(Δ^{[1]}) − 3 = rk(Δ^{[1]} ∩ M)/M, which is the expected dimension for H_{poly}^{1,1} in [2].

Remark 4.3.8. In the case of Arnold’s strange duality, the lattice structure of L_{G}(Δ) is isomorphic to that of L_{D}(Δ^{∗}).

In general, L_{G} ⊕ U is not isomorphic as a lattice over ℤ to the Milnor lattice but it seems so over \frac{1}{|G|}ℤ. This may be for a future investigation.

Theorem 4.3.9. Let X_{0} be one of the fourteen exceptional unimodal isolated hypersurface singularities and W_{a} = (a_{1}, a_{2}, a_{3}; h) be the corresponding weight system.

(1) There exists a unique dual weight system W_{b}.

(2) W_{b} coincides with the weight system of the dual singularity in the sense of Arnold.

(3) The full Newton polyhedron, namely the convex hull of \overline{Δ}(a) ∩ M(W_{a}) in M(W_{a})_{ℚ} is a reflexive polyhedron containing (\overline{∇})^{∗} with the dual correspondence above.

(4) Let Δ be any reflexive polyhedron in \overline{Δ}(a) which contains (\overline{∇})^{∗}. The intersection with Δ and the plane ∑_{i=1}^{3} a_{i}α_{i} = 1 is a facet which determines the equivalent singularity to X_{0}.

(5) Δ^{∗} is a reflexive polyhedron in \overline{∇} containing (\overline{Δ})^{∗}.

(6) There exists a Δ such that L_{0}(Δ) = 0.

Proof. (1)(2)(5) is done in previous chapters. (3) is well-known and (4) is easy case by case check. Here we show an example of choices of Δ with L_{0} = 0. For dual singularities, these reflexive polytopes are dual to each other. For self-dual singularities, these examples satisfy Δ^{∗} ≅ Δ. These choices of Δ are not unique in general.

\begin{matrix}class & a_{1} & a_{2} & a_{3} & h & Δ \\ E_{12} & 6 & 14 & 21 & 42 & W^{42}, X^{7}, Y^{3}, Z^{2} \\ E_{13} & 4 & 10 & 15 & 30 & W^{30}, W^{6}X^{6}, X^{5}Y, Y^{3}, Z^{2} \\ Z_{11} & 6 & 8 & 15 & 30 & W^{30}, W^{6}Y^{3}, X^{5}, XY^{3}, Z^{2} \\ E_{14} & 3 & 8 & 12 & 24 & W^{24}, W^{6}X^{6}, X^{4}Z, Y^{3}, Z^{2} \\ Q_{10} & 6 & 8 & 9 & 24 & W^{24}, W^{6}Z^{2}, X^{4}, Y^{3}, XZ^{2} \\ Z_{12} & 4 & 6 & 11 & 22 & W^{22}, W^{6}X^{4}, W^{4}Y^{3}, X^{4}Y, XY^{3}, Z^{2} \\ W_{12} & 4 & 5 & 10 & 20 & W^{20}, W^{10}Y^{2}, W^{2}X^{2}Y^{2}, X^{5}, Y^{2}Z, Z^{2} \\ Z_{13} & 3 & 5 & 9 & 18 & W^{18}, W^{6}X^{4}, W^{3}Y^{3}, X^{3}Z, XY^{3}, Z^{2} \\ Q_{11} & 4 & 6 & 7 & 18 & W^{18}, W^{6}X^{3}, W^{4}Z^{2}, X^{3}Y, Y^{3}, XZ^{2} \\ W_{13} & 3 & 4 & 8 & 16 & W^{16}, W^{4}X^{4}, W^{4}Y^{3}, X^{4}Y, Y^{4}, Z^{2} \\ S_{11} & 4 & 5 & 6 & 16 & W^{16}, W^{6}Y^{2}, W^{4}Z^{2}, X^{4}, XZ^{2}, Y^{2}Z \\ Q_{12} & 3 & 5 & 6 & 15 & W^{15}, W^{6}X^{3}, W^{3}Z^{2}, X^{3}Z, XZ^{2}, Y^{3} \\ S_{12} & 3 & 4 & 5 & 13 & W^{13}, W^{4}X^{3}, W^{3}Z^{2}, W Y^{3}, X^{3}Y, XZ^{2}, Y^{2}Z \\ U_{12} & 3 & 4 & 4 & 12 & W^{12}, W^{4}Y^{2}, W^{4}Z^{2}, X^{4}, Y^{2}Z, Y Z^{2} \end{matrix} □

Remark 4.3.10. One should choose the polyhedron rather carefully, since the restricted Kähler class ω of a general Kähler class of ℙ̃_{Δ} is sometimes not general in (S_{X})_{ℝ}. Namely, there might be some non-zero (non-effective) integral algebraic 2-cocycle α such that ω ∪ α = 0. This is where L_{0} appears. Thus it is necessary to resort to a subpolytope of the full Newton polytope.

4.4. Examples.

We give some illuminating examples other than exceptional unimodal isolated singularities.

Example 4.4.1. Let us think about a pair of dual weight systems W_{a} = (2, 3, 6; 12) and W_{b} = (2, 4, 5; 12). In these cases a_{0} = b_{0} = 1. We will denote the homogeneous coordinates by W = X_{0}, X = X_{1}, Y = X_{2} and Z = X_{3} and the inhomogeneous ones by small letters. A polytope is represented by its vertex set (vertices).

The weighted magic square C is ( \begin{matrix}0 & 2 & 1 \\ 3 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} ) whose determinant is (−12). Thus B =

( \begin{matrix}−1 & 1 & 0 \\ 2 & 1 & −1 \\ −1 & −1 & 1 \end{matrix} ), −B^{−1} = ( \begin{matrix}0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 3 \end{matrix} ).

We use (α_{1}, …, α_{n}) for the coordinates coming from M(W_{a}) = ℤ^{⊕n} and [β_{1}, …, β_{n}] for those from M(W_{b}). Then [β_{1}, …, β_{n}]B = (α_{1}, …, α_{n}) holds for the vertices in M_{ℚ} and the hyperplanes in N_{ℚ}.

\overline{Δ} = ⟨W^{12}, X^{6}, Y^{4}, Z^{2}⟩. The coordinates of the vertices are : (−1, −1, −1), (5, −1, −1), (−1, 3, −1), (−1, −1, 1); [−2, −3, −5], [−2, 2, 1], [2, 0, −1], [0, 0, 1], respectively. This is a reflexive simplex. In fact, the dual is given by (\overline{Δ})^{∗} = ⟨W^{12}, Y^{3}, X^{2}Y^{2}, XZ^{2}⟩.

Let X_{0} be the singularity {x^{6} + y^{4} + z^{2} = 0} in ℂ^{3}. This is an isolated hypersurface singularity with Milnor number μ = 15. In the table of Arnold[1], X_{0} belongs to class W_{1,0}.

Z = {x^{6} + y^{4} + z^{2} + 1 = 0} is a smooth surface in ℂ^{3} thus a Milnor fiber of X_{0}.

Ebeling [16] has calculated Milnor lattices of isolated bimodal singularities. The Milnor lattice H_{2}(Z) is a direct sum of U and the lattice L_{G}(\overline{Δ}) represented by Fig. 1,

(Fig. 1)

where circles are vanishing cycles.

The resolution graph of X → Z̄ is as in Fig. 2.

(Fig. 2)

where a circle represents a smooth rational curve with self-intersection (−2) and the central curve is {W = 0}. L_{D} is generated by the central curve and three (−4)-elements a+a′, b+b′ and c+c′. L_{0} is generated by three (−4)-elements e_{1} := a−a′, e_{2} := b−b′ and e_{3} := c−c′. The set of all the (−4)-elements of L_{0} are {±e_{1}, ±e_{2}, ±e_{3}, ±(e_{1} + e_{2})} and the orthogonal group O(L_{0}) keeps this set. The elements of O(L_{0}) which extend to elements of O(L) as in 4.2 are : id, σ(e_{1} ↦ e_{1}, e_{2} ↦ e_{2}, e_{3} ↦ −e_{3}), τ(e_{1} ↦ −e_{1}, e_{2} ↦ −e_{2}, e_{3} ↦ e_{3}) and στ. Thus G ≅ ℤ/2ℤ × ℤ/2ℤ and in fact each isomorphism is realized by a multiplication of coordinates: σ(W : X : Y : Z) = (iW : X : iY : −Z), τ(W : X : Y : Z) = (ζ^{2}W : ζX : Y : −Z) where i := exp(πi/2) and ζ := exp(πi/3).

Next think about the dual polyhedron \overline{Δ}^{∗}. The singularity \{y^{3} + x^{2}y^{2} + xz^{2} = 0\} has a singular curve {x = 0}. We have a Milnor fiber {y^{3} + x^{2}y^{2} + xz^{2} + 1 = 0} and its weighted compactification Z̄ = {Y^{3} + X^{2}Y^{2} + XZ^{2} + W^{12} = 0} in ℙ(1, 2, 4, 5). In this case, the restriction of the divisor {X = 0} decomposes as a sum of three divisors E_{i} : {X = Y + ζ^{i}W^{4} = 0} (i = 0, 1, 2) where ζ is a nontrivial cubic root of unity. The other intersections with singular 2-dimensional strata of ℙ are irreducible.

The singular set of Z̄ consists of one D_{8} at {W = Y = Z = 0}, one A_{4} at {W = X = Y = 0} and one A_{1} at {W = X^{2} +Y = Z = 0}. The resolution graph at infinity (including the divisor W = 0) becomes as in Fig. 3.

(Fig.3)

This graph is one longer than that of the L_{G}(\overline{Δ}). L_{0} is generated by a−a′, E_{0} −E_{1} and E_{1} −E_{2}. L_{D} is generated by a+a′ and other circles. Change the base as a+a′ → a′ +a+b (other circles are same), we have the same graph as the L_{G}(\overline{Δ}).

Let us take other polyhedra. In our duality, every reflexive polyhedron in M_{ℚ} should contain \overline{∇}^{∗} = ⟨W^{12}, X^{3}Y^{2}, Y^{2}Z, Z^{2}⟩, or rather ∇^{∗} = ⟨W^{12}, Y^{2}W^{6}, X^{3}Y^{2}, Y^{2}Z, Z^{2}⟩, where ∇ is the maximum Newton polyhedron in N, that is the convex hull of \overline{∇} ∩ N.

Take Δ = ⟨W^{12}, W^{3}Y^{3}, W^{2}X^{5}, W XY^{3}, X^{3}Y^{2}, X^{3}Z, Y^{2}Z, Z^{2}⟩, which has the same L_{G} as \overline{Δ} and L_{0}(Δ) = 0. Thus Δ represents a Milnor fiber of the singularity of class W_{1,0} and the dual graph of L_{D} coincides with the resolution graph as above. Its dual Δ^{∗} is ⟨W^{12}, W^{7}Z, W^{6}XY, W^{5}XZ, W^{3}Y Z, X^{2}Y^{2}, Y^{3}, XZ^{2}⟩.

Example 4.4.2. The weight system W = (2, 3, 4; 10) is self-dual. Take the following six reflexive polyhedra (cf. Fig. 4):

Δ_{1} = ⟨W^{10}, W^{2}X^{4}, X^{3}Z, X^{2}Y^{2}, XZ^{2}, Y^{2}Z, W^{2}Z^{2}, W Y^{3}⟩ ,
Δ_{2} = ⟨W^{10}, W^{4}X^{3}, W X^{3}Y, X^{2}Y^{2}, XZ^{2}, Y^{2}Z, W^{2}Z^{2}, W Y^{3}⟩ ,
Δ_{3} = ⟨W^{10}, W^{6}X^{2}, W^{2}X^{2}Z, X^{2}Y^{2}, XZ^{2}, Y^{2}Z, W^{2}Z^{2}, W Y^{3}⟩ ,
Δ_{4} = ⟨W^{10}, W^{6}X^{2}, W^{2}X^{2}Z, X^{2}Y^{2}, XZ^{2}, Y^{2}Z, W^{6}Z, W^{4}Y^{2}⟩ ,
Δ_{5} = ⟨W^{10}, W^{8}X, W^{3}X^{2}Y, X^{2}Y^{2}, XZ^{2}, Y^{2}Z, W^{6}Z, W^{4}Y^{2}⟩ and
Δ_{6} = ⟨W^{10}, W^{4}XZ, X^{2}Y^{2}, XZ^{2}, Y^{2}Z, W^{6}Z, W^{4}Y^{2}⟩ .

These polyhedra all satisfy L_{0}(Δ_{i}) = 0 and Δ_{i}^{∗} ≅ Δ_{7−i}. Δ_{1} corresponds to the Milnor fiber of the singularity of class S_{1,0}.

The dual graphs of L_{D}(Δ_{i}) is as in Fig. 5.

(Fig. 5)

There are inclusions of reflexive polyhedra: Δ_{i} ⊃ Δ_{j} for i < j and thus specializations of the families of weighted hypersurfaces. As the family specializes, 2A_{1} singularities at infinity get ‘closer’ to A_{3}, D_{4}, D_{5} and finally to D_{6} singularities.

Example 4.4.3. For the weight system W = (2, 6, 9; 18), ten reflexive polytopes Δ_{i} = ⟨W^{18}, W^{18−2i}X^{i}, X^{3}Y^{2}, Y^{3}, Z^{2}⟩ satisfy the relation Δ_{i}^{∗} ≅ Δ_{9−i}.

Example 4.4.4. Let us think families of quartic surfaces in ℙ^{3}; W = (1, 1, 1; 4). The full Newton polytope \overline{Δ} itself is reflexive with L_{0} = 0 and its dual \overline{Δ}^{∗} can be embedded in \overline{Δ} as ⟨W^{4}, X^{2}Y Z, XY^{2}Z, XY Z^{2}⟩. The computation of lattices is in [14].

Let us make a self-dual polyhedron. An easy choice Δ = ⟨W^{4}, X^{2}Y^{2}, Y^{2}Z^{2}, Z^{2}X^{2}⟩ is a self-dual polyhedron with the same correspondence σ as in the case of \overline{Δ}, but the rank ρ of algebraic lattice is thirteen. Another choice Δ′ = ⟨W^{4}, X^{4}, X^{2}Y^{2}, XY^{2}Z, XY Z^{2}, Z^{2}X^{2}⟩ is self-dual and satisfies L_{0} = 0 and ρ = 10. o o ∣ ∣

o-o-o-o-o-o-o-o-o-o-o Fig. 1

o c ∣ o-o-o-o-o ∣ a b o b′ a′ c′

Fig. 2

a′o o

∣ ∣ o-o-o-o-o-o-o-o-o-o-o-o a b c

Fig. 3

o ∣ o ∣

o-o-o-o-o ∣ o Δ_{1} o ∣

o o ∣ ∣

o-o-o-o-o-o

Δ_{2} o ∣

o o ∣ ∣

o-o-o-o-o-o-o

Δ_{3} and Δ_{4} o ∣ o o ∣ ∣ o-o-o-o-o-o-o-o

Δ_{5}

o ∣

o o ∣ ∣ o-o-o-o-o-o-o-o-o

Δ_{6}

Fig. 5 Δ_{1} Δ_{6}

Δ_{2} Δ_{5}

Δ_{3} Δ_{4}

Fig. 4. W = (2, 3, 4; 10). The black circle represents the origin XY ZW.

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Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro, Tokyo 152 Japan;

E-mail address: masanori@math.titech.ac.jp