Calabi-Yau Threefolds of Quasi-Product Type

Keiji Oguiso

Received: March 20, 1996

Communicated by Thomas Peternell

Abstract. According to the numerical Iitaka dimension ν(X, D) and c_{2}(X)· D, fibered Calabi-Yau threefolds Φ_{|D|} : X → W (dim W > 0) are coarsely classified into six different classes. Among these six classes, there are two peculiar classes called of type II_{0} and of type III_{0} which are characterized respectively by ν(X, D) = 2 and c_{2}(X) · D = 0 and by ν(X, D) = 3 and c_{2}(X) · D = 0. Fibered Calabi-Yau threefolds of type III_{0} are intensively studied by Shepherd-Barron, Wilson and the author and now there are a satisfactory structure theorem and the complete classification. The purpose of this paper is to guarantee a complete structure theorem of fibered Calabi-Yau threefolds of type II_{0} to finish the classification of these two peculiar classes. In the course of proof, the log minimal model program for threefolds established by Shokurov and Kawamata will play an important role. We shall also introduce a notion of quasi-product threefolds and show their structure theorem. This is a generalization of the notion of hyperelliptic surfaces to threefolds and will have other applicability, too.

1991 Mathematics Subject Classification: Primary: 14J, secondary 14D.

Introduction

Let us start this introduction by recalling a global picture of fibered Calabi-Yau threefolds known at the present and then state the Main Theorem precisely.

Throughout this paper, by a Calabi-Yau threefold, we mean a normal projective complex threefold X with only ℚ−factorial terminal singularities (so that isolated) and with O_{X}(K_{X}) ≃ O_{X} and π_{1}^{alg}(X) = {1}. The last condition is equivalent to π_{1}^{alg}(X − Sing X) = {1}, because the local fundamental group of three dimensional terminal Gorenstein singularities is trivial ([Kw3]). This also implies h^{1}(O_{X}) = 0 ([O1]). We define

c_{2}(X) · D := c_{2}(X′) · ν^{∗}(D)

for any resolution ν : X′ → X of Sing (X).

It is known by Miyaoka that c_{2}(X) · D is non-negative if D is nef ([Mi]).

A surjective morphism Φ : X → W is called a fibered Calabi-Yau threefold if X is a Calabi-Yau threefold, W is a normal projective variety (of positive dimension) and Φ has connected fibers. Note that Φ is nothing but Φ_{|D|} if D is the pull back of (any) very ample divisor H on W.

Fibered Calabi-Yau threefolds Φ_{|D|} : X → W are divided into six classes by the numerical invariants ν(X, D) and c_{2} · D:

Type I_{0} : ν(X, D) = 1 and c_{2} · D = 0; Type I_{+} : ν(X, D) = 1 and c_{2} · D > 0; Type II_{0} : ν(X, D) = 2 and c_{2} · D = 0; Type II_{+} : ν(X, D) = 2 and c_{2} · D > 0; Type III_{0} : ν(X, D) = 3 and c_{2} · D = 0; Type III_{+} : ν(X, D) = 3 and c_{2} · D > 0.

The following (more or less tautological) coarse classification is proved in [O1].

Theorem 1 ([O1]). Each class of fibered Calabi-Yau threefolds Φ(= Φ_{|D|}) : X → W defined above is characterized as follows.

Type I_{0}: General fibers are smooth Abelian surfaces and W = ℙ^{1}, Type I_{+}: General fibers are smooth K3 surfaces and W = ℙ^{1}, Type II_{0}: General fibers are smooth elliptic curves and W is a normal projective rational surface with only quotient singularities and with K_{W} ≡ 0, Type II_{+}: General fibers are smooth elliptic curves and W is a normal projective rational surface with only quotient singularities and with K_{W} + Δ ≡ 0 for some non-zero effective ℚ-divisor Δ such that (W, Δ) is klt, Type III_{0}: Φ is a birational morphism and W is a normal projective threefold with only canonical singularities and with O_{W} (K_{W}) ≃ O_{W} and c_{2}(W)(:= Φ_{∗}c_{2}(X)) = 0 as a linear form on P ic(W), Type III_{+}: Φ is a birational morphism and W is a normal projective threefold with only canonical singularities and with O_{W} (K_{W}) ≃ O_{W} and c_{2}(W) ≠ 0.

Moreover, if Φ : X → W is a fibered Calabi-Yau threefold of type II_{0} and H is a general very ample divisor on W, then the induced elliptic surface Φ^{−1}(H) → H has no singular fibers while Φ^{−1}(H) → H has at least one singular fiber composed of rational curves if Φ : X → W is of type II_{+}.

Theorem 1 shows that fibered Calabi-Yau threefolds of type III_{0} or of type II_{0} have rather special nature.

The following two theorems give a complete picture of fibered Calabi-Yau threefolds of type III_{0}.

Theorem 2 ([SW]). Let Φ : X → \overline{X} be a fibered Calabi-Yau threefold of type III_{0}. Then, there exist an Abelian threefold A and a finite Gorenstein automorphism group G of A such that

(1) A^{[G]} is a non-empty finite set, and

(2) \overline{X} = A/G.

Theorem 3 ([O3]). Two fiber spaces Φ_{3} : X_{3} → \overline{X_{3}} and Φ_{7} : X_{7} → \overline{X_{7}} defined in the following (1) and (2) are fibered Calabi-Yau threefolds of type III_{0}.

(1) Let E_{ζ_{3}} be the elliptic curve with period ζ_{3} := exp (2πi/3). Setting \overline{X_{3}} := E_{ζ_{3}}^{3}/⟨diag (ζ_{3}, ζ_{3}, ζ_{3})⟩, we define Φ_{3} : X_{3} → \overline{X_{3}} to be a unique crepant (toric) resolution of \overline{X_{3}}.

(2) Let A_{7} be the Jacobian threefold of the Klein quintic curve C := (x_{0}x_{1}^{3} + x_{1}x_{2}^{3} + x_{2}x_{0}^{3} = 0) ⊂ ℙ_{[x_{0}:x_{1}:x_{2}]}^{2} and g_{7} the automorphism of A_{7} induced by the automorphism of C given by [x_{0} : x_{1} : x_{2}] ↦ [x_{0}^{1} : x_{1}^{2} : x_{2}^{4}]. Setting \overline{X_{7}} := A_{7}/⟨g_{7}⟩, we define Φ_{7} : X_{7} → \overline{X_{7}} to be a unique crepant (toric) resolution of \overline{X_{7}}.

Conversely, any fibered Calabi-Yau threefold of type III_{0} is isomorphic to either Φ_{3} : X_{3} → \overline{X_{3}} or Φ_{7} : X_{7} → \overline{X_{7}} as fiber spaces.

In particular, there are exactly two fibered Calabi-Yau threefolds of type III_{0} and both of them are smooth and rigid.

Now it is interesting to study another peculiar class of fibered Calabi-Yau threefolds called of type II_{0}.

Base surfaces W of fibered Calabi-Yau threefolds Φ : X → W of type II_{0} are classified into two classes by the global canonical covering π : T → W, for which we have either

(1) T is a smooth Abelian surface, or

(2) T is a (projective) K3 surface with only Du Val singularities. In case (1) (resp. (2)), a fibered Calabi-Yau threefold Φ : X → W of type II_{0} is called of type II_{0}A (resp. of type II_{0}K).

The following theorem gives a complete classification of fibered Calabi-Yau threefolds of type II_{0}A.

Theorem 4 ([O2]).

(1) Let Φ_{3} : X_{3} → E_{ζ_{3}}^{3}/diag (ζ_{3}, ζ_{3}, ζ_{3}) be as in Theorem 3 and p : X_{3} → E_{ζ_{3}}^{2}/diag (ζ_{3}, ζ_{3}) the natural map given by the composite of Φ_{3} and the natural projection p_{12} : E_{ζ_{3}}^{3}/diag (ζ_{3}, ζ_{3}, ζ_{3}) → E_{ζ_{3}}^{2}/diag (ζ_{3}, ζ_{3}). Then, any composite of flops f : X_{3} · ·· → X′_{3} along curves in p^{−1}(Sing (E_{ζ_{3}}^{2}/diag (ζ_{3}, ζ_{3}))) gives a fibered Calabi-Yau threefolds p ∘ f^{−1} : X′_{3} → E_{ζ_{3}}^{2}/diag (ζ_{3}, ζ_{3}) of type II_{0}A. In this case, E_{ζ_{3}}^{2} is nothing but the global canonical cover of the base surface E_{ζ_{3}}^{2}/diag (ζ_{3}, ζ_{3}).

(2) Conversely, every fibered Calabi-Yau threefolds of type II_{0}A is obtained by the above process up to isomorphisms as fiber spaces. In particular, every fibered Calabi-Yau threefolds of type II_{0}A is smooth and rigid. Moreover, there are exactly 14 different fibered Calabi-Yau threefolds of type II_{0}A up to isomorphism as fiber spaces.

The purpose of this paper is to show the following structure theorem of fibered Calabi-Yau threefolds of type II_{0}K. This theorem tells us how to construct all the fibered Calabi-Yau threefolds of type II_{0}K.

Main Theorem. Let us prepare (i) a smooth elliptic curve E with a fixed origin 0, (ii) a projective K3 surface S with only Du Val singularities and its minimal

resolution μ : S′ → S, and (iii) two groups

G ∈ {{1}, ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{5}, ℤ_{6}, ℤ_{7}, ℤ_{8}, (ℤ_{2})^{2}, (ℤ_{3})^{2}, (ℤ_{4})^{2}, ℤ_{2} × ℤ_{4}, ℤ_{2} × ℤ_{6}}, and ⟨g⟩ ≃ ℤ_{I} ∈ {ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}},

such that G̃ := G ⋊ ⟨g⟩ (semi-direct product) acts faithfully on both E and S (and then on S′ and E × S′) in such a way that

(iv) G ∋ a : E × S′ → E × S′, (x, y) ↦ (x + a_{E}, a_{S′}(y)) with a_{E} ∈ (E)_{ord (a)} and

a_{S′}^{∗}ω_{S′} = ω_{S′}, where ω_{S′} is a nowhere vanishing regular 2 form on S′,

(v) g : E × S′ → E × S′, (x, y) ↦ (ζ_{I}^{−1}x, g_{S′}(y)) with g_{S′}^{∗}ω_{S′} = ζ_{I}ω_{S′}, and (vi) (S′)^{[G̃]} ⊂ Exc(μ) except for finitely many points in (S′)^{[G̃]}, that is, (S)^{[G̃]} is a

finite set. Note that G̃ is a finite Gorenstein automorphism group of E × S′. Let

ν : Y (E, S, G̃) → (E × S′)/G̃

be a crepant resolution (whose existence is now guaranteed by Roan [Ro]) and

p : Y (E, S, G̃) → S/G̃

the natural projection given by the composite of ν : Y (E, S, G̃) → (E × S′)/G̃, p_{2} : (E × S′)/G̃ → S′/G̃, and μ/G̃ : S′/G̃ → S/G̃.

Then,

(1) any composite of flop f : Y (E, S, G̃) · ·· → Y′ along curves in p^{−1}(Sing (S/G̃)) gives a fibered Calabi-Yau threefold p∘f^{−1} : Y′ → S/G̃ of type II_{0}K provided that π_{1}^{alg}(Y) = {1}. In this case S/G gives the global canonical cover of the base space S/G̃.

(2) Conversely, every fibered Calabi-Yau threefold of type II_{0}K is obtained by the above process for some triplet (E, S, G̃) satisfying the conditions (i)-(vi) up to isomorphisms as fiber spaces. In particular, every fibered Calabi-Yau threefold of type II_{0}K is smooth.

This together with Theorems 2, 3 and 4 will complete the structure theorem of the two peculiar classes of fibered Calabi-Yau threefolds called of types II_{0} and III_{0}.

Remark. Investigating the actions of G and ⟨g⟩ on E, we easily see that (1) G̃ is uniquely determined by G and ⟨g⟩ as an abstract group, and

(2) among 52 possibilities of (G, ⟨g⟩) in the Main Theorem, the following 18 combinations do not occur: (ℤ_{4}, ℤ_{3}), (ℤ_{5}, ℤ_{3}), (ℤ_{6}, ℤ_{3}), (ℤ_{8}, ℤ_{3}), (ℤ_{2} × ℤ_{6}, ℤ_{3}), (ℤ_{2} × ℤ_{8}, ℤ_{3}), (ℤ_{3}, ℤ_{4}), (ℤ_{4}, ℤ_{4}), (ℤ_{6}, ℤ_{4}), (ℤ_{7}, ℤ_{4}), (ℤ_{2} × ℤ_{8}, ℤ_{4}), (ℤ_{2}, ℤ_{6}), (ℤ_{4}, ℤ_{6}), (ℤ_{5}, ℤ_{6}), (ℤ_{6}, ℤ_{6}), (ℤ_{8}, ℤ_{6}), (ℤ_{2} × ℤ_{6}, ℤ_{6}), (ℤ_{2} × ℤ_{8}, ℤ_{6}).

Remark. There are examples of non-rigid fibered Calabi-Yau threefolds of type II_{0}K and the number of fibered Calabi-Yau threefolds of type II_{0}K is not finite any more ([O1]).

Remark. It is interesting to compare Theorems 2, 3, 4 and main theorem with the so called Bogomolov decomposition theorem (see for example [Bo]). These look very similar, while our proof is free from the Bogomolov decomposition theorem.

The Main Theorem and Theorem 4 immediately imply

Corollary. Let Φ : X → W is a fibered Calabi-Yau threefold of type II_{0}. Then the global canonical index of W is either 2, 3, 4 or 6.

Corollary. Let Φ : X → W be a fibered Calabi-Yau threefold of type II_{0}K (resp. of type II_{0}A). Then, there is a composite of flops Y → W of Φ : X → W over W such that Y has at least two different fiber space structures, Y → W of type II_{0}K (resp. of type II_{0}A) and Y → ℙ^{1} of type I_{+} (resp. of type I_{0}).

Very little is known for a fibered Calabi-Yau threefold of type I_{0}, that is, a Calabi-Yau threefold with an Abelian fibration. However, our main theorem and Theorem 4 show

Corollary. Let X be a Calabi-Yau threefold with at least two different Abelian fibrations. Then, X is a Calabi-Yau threefold described as in either the Main Theorem (2) or Theorem 4(2). In particular, X is smooth and birational to either a quotient of an Abelian threefolds or that of the product of a K3 surface and an elliptic curve.

In fact, if Φ_{|D_{i}|} : X → ℙ^{1} (i = 1, 2) are two different Abelian fibrations on X, then Φ_{|m(D_{1}+D_{2})|} : X → W is of type II_{0} for some m.

The outline of this paper is as follows.

In section 1, we introduce the notion of quasi-product threefolds ((1.1)) and show their structure theorem ((1.3)). This plays an important role for our proof of the Main Theorem.

Sections 2 - 4 are devoted to prove the Main Theorem. Since Main Theorem (1) is quite clear, we prove only Main Theorem (2).

Let Φ_{T} : X_{T} := X ×_{W} T → T be the base change of a fibered Calabi-Yau threefold Φ : X → W of type II_{0}K to the global canonical cover π : T → W. Since Φ always has a two dimensional fibers ([O1]), X_{T} has very bad singularities and Φ_{T} itself is a very complicated map in general.

In section 2, we apply the log minimal model program established by Shokurov and Kawamata or Kollár et al. [Sh] and [Kw4] (also [Ko3]) to find a good birational (canonical) model f : Z → T of Φ_{T} : X_{T} → T over T such that

(1) Gal(T/W) := ⟨g⟩ acts regularly on f : Z → T and

(2) Φ : X → W is birational to the quotient (f : Z → T)/⟨g⟩. Moreover applying the result in section 1, we show that there are a smooth elliptic curve E, a normal projective surface S which is either an Abelian surface or a K3 surface with only Du Val singularities, and a finite automorphism group G of the fiber space p_{2} : E × S → S such that (f : Z → T) = (p_{2} : E × S → S)/G.

In section 3, we show that the action of ⟨g⟩ on f : Z → T lifts to that on its covering p_{2} : E × S → S in an equivariant way. This is a rather special phenomenon, because a composite of Galois extensions is not Galois in general.

Till section 3, the main part of our proof of the Main Theorem is completed. It remains only to show the impossibility for S to be a smooth Abelian surface. This problem is treated in section 4. This requires our assumption π_{1}^{alg}(X) = {1} and forces rather minute analysis of automorphism groups of an Abelian surface.

Acknowledgement

This work was completed during the author’s stay at Johns Hopkins University in October 1995. The author would like to express his best thanks to Professors Y. Kawamata and V. Shokurov for their invitation. The author would like to express his thanks to JSPS for financial support during his stay.

Notation and Convention

Throughout this paper, we work over the complex number field ℂ.

We will employ standard notion and notation in minimal model program ([KMM] or [Ko3]) freely.

By a minimal threefold, we mean a normal projective threefold V with only ℚ−factorial terminal singularities and with nef canonical (Weil) divisor K_{V}.

A surjective morphism Φ : V → W is said to be relatively minimal if V has only ℚ−factorial terminal singularities and the canonical divisor K_{V} is relatively nef with respect to Φ.

We often use the notion of klt (Kawamata log terminal) given in [Ko3]. This is same as the notion of log terminal in [KMM].

By a fiber space on a normal projective variety V, we mean a surjective morphism Φ : V → W to a normal projective variety W with connected fibers. Note that Φ is not equi-dimensional in general. By Φ^{−1}(w) (w ∈ W), we denote the scheme theoretic fiber over w. We denote its reduction by Φ^{−1}(w)_{red}. This is in some sense a set theoretical fiber.

Two fiber spaces Φ : V → W and Φ′ : V′ → W′ are said to be isomorphic if there are isomorphisms F : V → V′ and f : W → W′ such that Φ′ ∘ F = f ∘ Φ.

For two morphisms Φ : V → W and π : T → W, we sometimes denote natural morphisms V ×_{W} T → T and V ×_{W} T → V by Φ_{T} : V_{T} → T and π_{V} : V_{T} (= T_{V}) → V respectively.

The primitive n−th root of unity exp(2πi/n) is denoted by ζ_{n}.

We denote the cyclic group of order n by ℤ_{n}.

The elliptic curve with period τ ∈ ℍ is written as E_{τ}.

The n−torsion group of an Abelian variety A with origin 0 is denoted by (A)_{n}. By global coordinates around a point P of an n−dimensional Abelian variety A, we mean those of its universal cover ℂ^{n} or, equivalently, those of the tangent space T_{A,P}.

For a faithful group action of G on a variety V, we set

V^{[G]} := {x ∈ V | ∃g ∈ G − {1}, g(x) = x},

while,

H^{G} := {v ∈ H | ∀g ∈ G, g^{∗}(v) = v}

for any cohomology group H of V.

Similarly, for an automorphism g of a variety V, we set

V^{g} := {x ∈ V | g(x) = x}.

An equivariant action of a finite group G on a fibration Φ : V → W induces a new fibration Φ(mod G) : X/G → W/G. We sometimes abbreviate this fibration by (Φ : V → W)/G.

We say that G acts on Φ : V → W over W if the action of G is equivariant and is trivial on W.

An automorphism group G of a variety V with O_{V} (K_{V}) ≃ O_{V} is called Gorenstein if the action of G on H^{0}(V, O_{V} (K_{V})) is trivial, that is, all elements g of G satisfy g^{∗}ω_{V} = ω_{V} for a generator ω_{V} of H^{0}(V, O_{V} (K_{V})).

For the automorphism group Aut (V) of a variety V and a subset B in V, we often consider the subgroup {g ∈ Aut (V) | g(B) = B}. We denote this group by Aut (X, B). For example, if A is an Abelian variety with origin 0, then Aut (A, {0}) is nothing but the so called Lie automorphism group of A.

§1. Quasi-product threefolds

In this preliminary section, we shall introduce the notion of quasi-product threefolds and prove their structure theorem (Theorem (1.3)). This is a rather wide generalisation of the notion of hyperelliptic surfaces to threefolds.

Definition (1.1). A normal projective threefold V with only rational singularities is called a quasi-product threefold with distinguished morphisms a and f if

(1) V has a fiber space structure a : V → A over a smooth elliptic curve A,

(2) V has a fiber space structure f : V → T over a normal projective surface T with only rational singularities and with H^{1}(O_{T}) = 0 such that f^{−1}(t)_{red} is a smooth elliptic curve for any t ∈ T, and that f^{−1}(t) itself is smooth except at most finitely many points t ∈ T.

Example (1.2). Let S be a normal projective surface with only rational singularities and E a smooth elliptic curve. Assume that a finite group of translations G of E acts faithfully on S in such a way that S^{[G]} is finite and H^{1}(O_{S})^{G} = 0. Then the quotient threefold (E × S)/G is a quasi-product threefold with distinguished morphisms p_{1} : (E × S)/G → E/G and p_{2} : (E × S)/G → S/G.

Conversely, we shall show

Theorem (1.3). Let V be a quasi-product threefold with two distinguished morphisms a : V → A and f : V → T. Let S be a general fiber of a.

Then, there exist an elliptic curve E and a finite subgroup G ⊂ E, that is, a finite group of translations of E (and then is isomorphic to either ℤ_{m} or ℤ_{n} × ℤ_{m} with (n|m)) such that

(1) there is an injective homomorphism ι : G → Aut (S),

(2) V = (E × S)/G under the (free) action of G on E × S defined by

G ∋ g : E × S ∋ (u, v) ↦ (u + g, ι(g)v) ∈ E × S, (3) two distinguished morphisms a : V → A and f : V → T are given by the

natural projections

p_{1} : (E × S)/G → E/G

and

p_{2} : (E × S)/G → S/ι(G)

respectively.

As a result, S can be replaced by any fiber of a. We set G_{S} := ι(G)(≃ G). Moreover, if O_{V} (K_{V}) ≃ O_{V}, then,

(4) any fiber S of a is either a K3 surface with only Du Val singularities or a smooth Abelian surface,

(5) G_{S} is a finite Gorenstein automorphism of S,

(6) if S is a K3 surface with only Du Val singularities, then S^{[G_{S}]} is a non-empty finite set and G_{S}(≃ G) is isomorphic to either one of the following groups; {1}, ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{5}, ℤ_{6}, ℤ_{7}, ℤ_{8}, ℤ_{2} ×ℤ_{2}, ℤ_{2} ×ℤ_{4}, ℤ_{2} ×ℤ_{6}, ℤ_{3} ×ℤ_{3}, or ℤ_{4} ×ℤ_{4},

(7) if S is a smooth Abelian surface, then S^{[G_{S}]} is a non-empty finite set and G_{S}(≃ G) is isomorphic to either one of the following groups; {1}, ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}, or ℤ_{2} × ℤ_{2}, ℤ_{2} × ℤ_{4}, ℤ_{3} × ℤ_{3}. In addition, if G_{S} ≃ ℤ_{m}, then G_{S} ⊂ Aut (S, {0}) for an appropriate origin 0 of S, while, if G_{S} ≃ ℤ_{n} × ℤ_{m} (n|m), then ℤ_{n} ⊂ (S)_{n} and ℤ_{m} ⊂ Aut (S, {0}) for an appropriate origin 0 of S. Moreover, Sing (S/G_{S}) is described as follows for each G_{S} ([Kt]). (G_{S}, Sing (S/G_{S})) = (ℤ_{2}, 16A_{1}), (ℤ_{2} × ℤ_{2}, 16A_{1}), (ℤ_{3}, 9A_{2}), (ℤ_{3} × ℤ_{3}, 9A_{2}) (ℤ_{4}, 4A_{3} + 6A_{1}), (ℤ_{2} × ℤ_{4}, 4A_{3} + 6A_{1}), (ℤ_{6}, A_{5} + 4A_{2} + 5A_{1}).

Remark. Let ν : S′ → S be the minimal resolution of S. Then G induces an equivariant free action on id × ν : E × S′ → E × S. The induced morphism (E × S′)/G → (E × S)/G gives a resolution of (E × S)/G.

Remark. Our proof given here basically follows the argument of Bombieri and Mumford for hyperelliptic surfaces([BM]). However, since we work at threefolds, we should keep the following two essential differences in mind:

(1) f may not be flat over T,

(2) three dimensional relatively minimal models are not unique among their birational models (even if they exist) so that rational actions on a relatively minimal model are not necessarily regular in general.

Proof. Set B := {t ∈ T | either f^{−1}(t) is not reduced or T is singular at t}, and denote C_{t} := f^{−1}(t)(t ∈ T) and S_{x} := a^{−1}(x) (x ∈ A). By our assumption, B is a finite set. Let us fix a general point 0 ∈ A and regard this point as an origin of A. Set S := S_{0}. Then S is a normal surface with only rational singularities. Put n := (C_{t} · S). This is independent of t ∈ T − B (because T − B is smooth and f|_{f^{−1}(T −B)} is a smooth morphism over T − B.)

Claim (1.4). a_{t} := a|_{C_{t}} : C_{t} → A is surjective for each t ∈ T − B. In particular, a_{t} is an isogeny of elliptic curves of degree n := (C_{t} · S) for each t ∈ T − B (and then n > 0).

Proof of Claim (1.4). Assume the contrary that a(C_{t}) is a point on A for some t ∈ T −B. Then, a(C_{t′}) must be a point for every t′ ∈ T −B because f is flat over T −B. Thus, a induces a morphism \overline{a} : T − B → A. This gives a rational map \overline{a} : T · · → A with a = \overline{a}∘f. Let T′ → T be a resolution of both singularities of T and indeterminacy of \overline{a}. Since T has only rational singularities, we have h^{1}(O_{T′}) = h^{1}(O_{T}) = 0. Thus, \overline{a} ∘ ν(T′) is a point. Hence \overline{a} is a morphism and \overline{a}(T) is a point. Then, a(V) would be a point because a = \overline{a} ∘ f. But this contradicts the surjectivity of a. q.e.d. for (1.4).

Let t be an arbitrary point on T − B. Then, by (1.4), A acts on C_{t} via the composite of the group homomorphism A ≃ P ic^{0}(A) → P ic^{0}(C_{t}) given by a_{t}^{∗} and the natural action of P ic^{0}(C_{t}) on C_{t}. More concretely, this action is written as

A ∋ x : C_{t} ∋ P ↦ P + x_{1} + ... + x_{n} − 0_{1} − ... − 0_{n} ∈ C_{t},

where {x_{1}, ..., x_{n}} := a_{t}^{−1}(x) = C_{t} ∩ S_{x} and {0_{1}, ..., 0_{n}} := a_{t}^{−1}(0) = C_{t} ∩ S. Note that f has a local section over T − B. Thus, gluing these together, we get a regular action of A on ∪_{t∈T −B}C_{t} = f^{−1}(T − B) over T − B. This gives a rational action on V over T. But, since the possible indeterminacy f^{−1}(B) of this action on V consists of elliptic curves (then no rational curves) and since V has only rational singularities, this action of A on V must be regular. Let us denote this action by σ : A × V → V. By construction, σ stabilizes each fiber of f. Set τ := σ|_{A×S} : A × S → V. Since a_{t} is an isogeny, we have

a_{t}(P + x_{1} + ... + x_{n} − 0_{1} − ... − 0_{n}) = a_{t}(P) + nx

for t ∈ T − B and x ∈ A. So, once we define a new action of A on A by

A ∋ x : A → A; y ↦ y + nx,

that is, by n × (translation), then A induces an equivariant action on the fibration V − f^{−1}(B) → A. By the same reason as before, this action of A is extended to an equivariant regular action on the whole space a : V → A.

By definition, we have x(S)(= x(S_{0})) = S_{nx} (x ∈ A). In particular, τ : A×S → V is surjective. Moreover, the action of the n−torsion group (A)_{n} of A on V stabilizes S = S_{0}. This induces a group homomorphism ι : (A)_{n} → Aut (S).

The following claim ([BM]) is now proved formally.

Claim (1.5). Let (x, v) and (x′, v′) be points on A × S. Then, the following (1) and (2) are equivalent to one another.

(1) τ(x, v) = τ(x′, v′),

(2) (x, v) and (x′, v′) are in the same orbit of the action

(A)_{n} ∋ k : A × S → A × S; (x, v) ↦ (x − k, ι(k)v).

Proof of Claim (1.5). Since τ(x − k, ι(k)v) = σ(x − k, σ(k, v)) = σ(x − k + k, v) = τ(x, v), (2) implies (1). We prove the converse. Since τ(x, v) ∈ S_{nx} and τ(x′, v′) ∈ S_{nx′}, it follows that nx = nx′, or equivalently, k := x − x′ ∈ (A)_{n}. We may show that ι(k)(v) = v′. Using τ(x, v) = τ(x′, v′), that is, σ(x, v) = σ(x′, v′), we calculate

v′ = σ(−x′, σ(x′, v′)) = σ(−x′, σ(x, v)) = σ(x − x′, v).

This is nothing but the desired equality, ι(k)(v) = v′. q.e.d. for (1.5).

By (1.5), we get V = (A × S)/(A)_{n}. Moreover, just by construction, we see that f : (A × S)/(A)_{n} → T factors through the natural projection p_{2} : (A × S)/(A)_{n} → S/(A)_{n}. In fact, f factors through p_{2} at least over T − B. But, since B is finite and S/(A)_{n} is normal, this is so over the whole T. Let μ : S/(A)_{n} → T be the induced morphism. Since both f and p_{2} have only one dimensional connected fibers, μ must be a finite birational morphism. Thus, by the Zariski main theorem, μ is isomorphism and then f = p_{2} under the identification T = S/(A)_{n}. Similarly, a : (A×S)/(A)_{n} → A factors through p_{1} : (A × S)/(A)_{n} → A/(A)_{n} = A. Now the equality a = p_{2} is shown by the same argument as before.

It only remains to make ι injective to complete the first half part of (1.3). But this is done as follows. Let G = (A)_{n}/Ker ι. Then, (A × S)/(A)_{n} = (A/(Ker ι) × S)/G and A/(A)_{n} = (A/Ker ι)/G, in which G acts on translation group of an elliptic curve A/Ker ι. Now replacing A, (A)_{n} and ι by E = A/(Ker ι), G, and the injection ι ∘ (−1) : G → Aut (S), we are done. Here we will compose (−1) only to change the sign − in (1.5) into + as in (1.3).

From now on, we shall prove the latter half part of (1.3). It is obvious that S is either a K3 surface with only Du Val singularities or a smooth Abelian surface. Moreover, since G acts on E as a translation group and O_{V} (K_{V}) ≃ O_{V}, it follows that G_{S} must be a Gorenstein automorphism group of S. In the rest we denote G_{S} simply by G if no confusion seems to arise.

Assume first that S is a K3 surface with only Du Val singularities. Let S′ → S be the minimal resolution of S. Then G gives a commutative Gorenstein action on S′. Now the result follows from the Nikulin’s classification ([Ni]). Note that two groups (ℤ_{2})^{3} and (ℤ_{2})^{4} in his list are excluded because G is isomorphic to either ℤ_{n} or ℤ_{n} × ℤ_{m} (n|m).

Finally, assuming that S is a smooth Abelian surface, we show that G satisfies the condition in (1.3)(7). Since G is a finite Gorenstein automorphism group of S with T = S/G and since h^{1}(T, O_{T}) = 0, it follows that S^{[G]} is a non-empty finite set. Choose an appropriate origin 0 of S and identify S with its translation automorphism group. Set Aut^{0}(S) := {σ ∈ Aut (S)|σ^{∗}ω_{S} = ω_{S}}, Aut^{0}(S, {0}) := {σ ∈ Aut^{0}(S)|σ(0) = 0}, where ω_{S} is a non-zero global regular two form on S. Then, Aut^{0}(S) = S ⋊ Aut^{0}(S, {0}) and G ⊂ Aut^{0}(S). Identifying Aut^{0}(S, {0}) = Aut^{0}(S)/S, we denote the natural projection by p : Aut^{0}(S) → Aut^{0}(S, {0}). If we choose global coordinates around 0, we can explicitly write down the action of g ∈ Aut^{0}(S) in its affine form

g(x) = M_{g}x + t_{g}, M_{g} ∈ SL(2, ℂ), t_{g} ∈ S.

Then p is nothing but the map taking the matrix part, that is, g ↦ M_{g}. It follows from this expression that

(1) as an abstract group, p(G) is independent of the choice of an origin of S,

(2) a finite Gorenstein automorphism g ∈ Aut^{0}(S) has a fixed point if and only if g is not a translation.

On the other hand, Katsura’s classification ([Kt]) of possible finite subgroups of Aut^{0}(S, {0}) shows that the commutative group p(G) is isomorphic to either ℤ_{2}, ℤ_{3}, ℤ_{4} or ℤ_{6}.

Thus we can choose g ∈ G and 0 ∈ S such that p(g) generates p(G) and g(0) = 0. From now on, we regard this point 0 as the origin of S.

Claim (1.6).

(1) H := Ker(p) consists of translations in G, that is, H ⊂ S,

(2) ⟨g⟩ ≃ p(G).

(3) G is isomorphic to H × ⟨g⟩.

(4) H is a subgroup of S^{g} (under the inclusion H ⊂ S).

Proof of (1.6). The assertion (1) follows from M_{h} = id for h ∈ H. By definition, p|_{⟨g⟩} : ⟨g⟩ → p(G) is surjective group homomorphism. Let h be an element of Ker(p|_{⟨g⟩}). Then, h(0) = 0 and h ∈ H. Combining this with (1), we get h = id. Thus, p|_{⟨g⟩} is isomorphism. This shows that G is a semi-direct product of H and ⟨g⟩. Since G is commutative, this must be the direct product. The last statement now directly follows from the relation gh = hg (h ∈ H). q.e.d. of (1.6).

Claim (1.7). According to ord (g) = 2, 3, 4, 6, S^{g} is isomorphic to (ℤ_{2})^{4}, (ℤ_{3})^{2}, (ℤ_{2})^{2} and {0}.

Proof of (1.7). If ord (g) = 2, then S^{g} = (S)_{2}. Since (S)_{2} ≃ (ℤ_{2})^{4}, we are done.

Assume that ord (g) = 3. Then, using appropriate global coordinates (x, y) around 0, we can write g = diag (ζ_{3}, ζ_{3}^{−1}). In particular, 1+g+g^{2} = 0. Thus, 3p = p+p+p = p + g(p) + g^{2}(p) = (1 + g + g^{2})(p) = 0 for p ∈ (S)^{g}. Hence S^{g} ⊂ (S)_{3} and S^{g} ≃ (ℤ_{3})^{k} for some non negative integer k. On the other hand, by the Lefschetz fixed point formula, we have ♯S^{g} = ∑_{i=0}^{4}(−1)^{i}tr(g^{∗}|H^{i}(S, ℂ)). Recall that

H^{1}(S, ℂ) = ℂdx ⊕ ℂdy ⊕ ℂd\overline{x} ⊕ ℂd\overline{y},

and

H^{i}(S, ℂ) = ∧^{i}H^{1}(S, ℂ).

Now an explicit calculation based on g = diag (ζ_{3}, ζ_{3}^{−1}) shows tr(g^{∗}|H^{0}(S, ℂ)) = 1, −2, 3, −2, 1 according to i = 0, 1, 2, 3, 4. Thus, ♯S^{g} = 9. This implies S^{g} ≃ (ℤ_{3})^{2}.

Assume that ord(g) = 4. Since S^{g} ⊂ S^{g^{2}} ≃ (ℤ_{2})^{4}, it follows that S^{g} ≃ (ℤ_{2})^{k} for some non negative integer k. As in the case of ord (g) = 3, we can choose appropriate global coordinates (x, y) around 0 such that g = diag (ζ_{4}, ζ_{4}^{−1}). Then, again using the Lefschetz fixed point formula, we calculate ♯S^{g} = 4. This implies S^{g} ≃ (ℤ_{2})^{2}.

Finally assume that ord (g) = 6. Then, it follows from the previous observation that S^{g} ⊂ S^{g^{2}} ∩ S^{g^{3}} ⊂ (S)_{2} ∩ (S)_{3} = {0}. q.e.d. of (1.7).

Now Claims (1.6), (1.7) and the fact that G is a finite Abelian group of the form ℤ_{n} or ℤ_{n} ×ℤ_{m} (n|m) together with the fundamental theorem on finite Abelian groups imply the assertion (1.3)(7).

The only remaining problem is to study Sing (S/G) for each G. If G is isomorphic to ℤ_{m}, the result follows from Katsura’s table ([Kt]). Next, consider the case when ℤ_{n} × ℤ_{m} for some n and m (with n|m). Since S/G ≃ (S/ℤ_{n})/ℤ_{m} and since (S/ℤ_{n}) is again an Abelian surface, the assertion follows from the first case.

Now we are done. Q.E.D. of (1.3).

§2. Good model over the global canonical covering

Let us fix a fibered Calabi-Yau threefold Φ : X → W of type II_{0}K. Define I := min{n ∈ ℕ|O_{W} (nK_{W}) ≃ O_{W} } and denote the global canonical cover of W by π : T → W ([Kw1, Z]). By our assumption, T is a projective K3 surface with only Du Val singularities. Set W_{0} := W − Sing (W). It is well known by [Kw1, Z] that π : T → W is a cyclic Galois covering of order I(W) and is étale over W_{0}. Moreover, there is a generator g of the Galois group Gal(T/W) such that g^{∗}ω_{T} = ζ_{I}ω_{T}, where ω_{T} is a nowhere vanishing regular two form on T, that is, a generator of H^{0}(O_{T} (K_{T})).

We fix these notation till the end of Section 4.

Set Φ_{T} : X_{T} := X ×_{W} T → T. Then, the Galois group Gal(T/W) = ⟨g⟩ acts on this fibration by g : (x, y) ↦ (x, g(y)) and induces an isomorphism

(Φ : X → W) ≃ (Φ_{T} / : X_{T} → T)/⟨g⟩.

However, X_{T} itself has very bad singularities in general.

The goal of this section is to prove the following

Key Lemma (2.1). There is a normal projective threefold Z such that (1) Z has only ℚ−factorial canonical singularities with O_{Z}(K_{Z}) ≃ O_{Z},

(2) Z is a quasi-product threefold ((1.1)) with two distinguished morphisms f : Z → T and a : Z → A, where the latter map is the Albanese morphism of Z (see [Kw2] for the definition of the Albanese variety and the Albanese morphism for varieties with rational singularities), and

(3) there is a regular action of the Galois group of ⟨g⟩ on the fibration f : Z → T such that W = T/⟨g⟩ and (Φ : X → W) is birational to (f : Z → T)/⟨g⟩ over W = T/⟨g⟩. Moreover, these are isomorphic over W − Sing (W).

The plan of proof of Key Lemma is as follows. First, applying the log minimal model program, we find a birational model f : Z → T of Φ_{T} : X_{T} → T with property (1) in (2.1). Then, we check that f : Z → T also satisfies (2) and (3).

In order to carry out this plan, we start by observing some general lemmas.

Proposition (2.2). Let φ : V → S be a surjective morphism from a normal projective ℚ−factorial threefold V to a normal projective surface S. Let {E_{i}}_{i∈I} be the set of all two-dimensional irreducible components in fibers of φ. Set E = Σ_{i∈I}E_{i}. Assume that

(1) V is not covered by rational curves, (2) K_{V} = Σ_{i∈I}a_{i}E_{i} (as a Weil divisor on V) for some a_{i} ∈ ℤ_{≥0},

(3) (V, ϵE) is klt for some positive small rational number ϵ. Then, there are a normal projective threefold V^{(n)} and a surjective morphism φ^{(n)} : V^{(n)} → S such that

(4) V^{(n)} has only ℚ−factorial canonical singularities with O_{V^{(n)}}(K_{V^{(n)}}) ≃ O_{V^{(n)}},

(5) φ^{(n)} : V^{(n)} → S is birational to φ : V → S over S and is isomorphic except over a finite set φ(E), and

(6) φ^{(n)} : V^{(n)} → S is an equi-dimensional elliptic fibration.

Proof. First, we remark

Claim (2.3). K_{V} + ϵE is not nef unless E = 0 as a divisor.

Proof of (2.3). Let H be a general very ample divisor on V. Then H is a normal surface and the restriction φ|_{H} : H → S is surjective. Since (K_{V} +ϵE)|_{H} ≡ Σ_{i∈I}(a_{i} + ϵ)E_{i}|_{H} and since E_{i}|_{H} are contracted by φ_{H}, we get

((K_{V} + ϵE)^{2} · H) = ((K_{V} + ϵE)|_{H})^{2} = (Σ_{i∈I}(a_{i} + ϵ)E_{i}|_{H})^{2} < 0

unless E = 0. q.e.d. of (2.3).

Let us apply the log minimal model program for a klt divisor K_{V} + ϵE. If E ≠ 0, then K_{V} + ϵE is not nef by (2.3). Thus, there is a log extremal ray R such that (K_{V} + ϵE) · C < 0 for any curve C belonging to R. Let cont_{R} : V → W be the contraction morphism associated to R. This is a birational morphism by our assumption (1). Since 0 > (K_{V} + ϵE) · C = Σ(a_{i} + ϵ)(E_{i} · C), there is a prime divisor E_{i} such that E_{i} · C < 0. This implies C ⊂ E_{i}. Thus cont_{R} is defined over S. Let ϕ : W → S be the induced morphism.

If cont_{R} is a divisorial contraction, setting V^{(1)} := W, φ^{(1)} := ϕ and changing E by its strict transform E^{(1)} on V^{(1)}, we see that φ^{(1)} : V^{(1)} → S and E^{(1)} satisfy all the assumptions in (2.2) (without any change of coefficients).

If cont_{R} is a small contraction, then we apply a log flip for cont_{R} to get cont_{R}^{+} : V^{+} → W.

The existence of log flips for threefolds is guaranteed by [Sh].

Now, setting V^{(1)} := V^{+}, φ^{(1)} := ϕ∘cont_{R}^{+} and changing E by its strict transform E^{(1)} on V^{(1)}, we see that φ^{(1)} : V^{(1)} → S and E^{(1)} also satisfy all the assumptions in (2.2).

Putting V^{(0)} := V, φ^{(0)} := φ and E^{(0)} := E and repeating this process, say, for n(≥ 0) times, we finally get φ^{(n)} : V^{(n)} → S and the strict transform E^{(n)} of E to V^{(n)} such that

(1) φ^{(n)} : V^{(n)} → S and E^{(n)} satisfy all the assumptions in (2.2), and

(2) K_{V^{(n)}} + ϵE^{(n)} is nef.

This is due to the termination of log flips for threefolds shown by [Kw4].

Then E^{(n)} = 0 by (2.3). This implies the equi-dimensionality of φ^{(n)}. Note that all modifications are done over φ(E). Thus φ^{(n)} : V^{(n)} → S and φ : V → S coincide over S − φ(E). Set V_{0} := V − E − Sing (V). Then the assumption (2) implies O_{V_{0}}(K_{V_{0}}) ≃ O_{V_{0}}. Let ν : V · · → V^{(n)} be the birational map obtained by the above process. Since ν|_{V_{0}} : V_{0} → ν(V_{0}) is an isomorphism, we have O_{ν(V_{0})}(K_{ν(V_{0})}) ≃ O_{ν(V_{0})}.

Since the codimension of V^{(n)} − ν(V_{0}) in V^{(n)} is at least two by E^{(n)} = 0 and since V^{(n)} is normal, this isomorphism gives O_{V^{(n)}}(K_{V^{(n)}}) ≃ O_{V^{(n)}}. Note that V^{(n)} has only rational singularities, because (V^{(n)}, E^{(n)}) = (V^{(n)}, 0) is klt. Thus V^{(n)} has only rational Gorenstein singularities, that is, canonical singularities of index one. Now the remaining assertion is obvious. Q.E.D. of(2.2).

The next two lemmas are concerned with singular fibers of certain elliptic threefolds.

Lemma (2.4). Let φ : V → S be a fiber space such that

(1) V is a normal projective threefold with only ℚ−factorial terminal singularities and with K_{V} ≡ 0,

(2) S is a normal projective surface with only quotient singularities and with K_{V} ≡ 0.

Then, φ^{−1}(s) is a smooth elliptic curve if s ∈ S − Sing (S). In particular, φ is a smooth morphism over S − Sing (S).

Proof. We make use of the following theorem due to Nakayama.

Theorem (2.5)([Na1 also Na2]). Let f : V_{Δ^{2}} → Δ^{2} be a relatively minimal projective elliptic fibration over a two-dimensional (small) polydisk

Δ^{2} := {(x, y) ∈ ℂ^{2} | |x| < ϵ, |y| < ϵ}.

Assume that f has (singular) fibers of type I_{a} (a ≥ 0) over (x = 0)−{(0, 0)} and those of type I_{b} (b ≥ 0) over (y = 0) − {(0, 0)}. (Here we employed Kodaira’s notation.) Then f^{−1}((0, 0)) is a (singular) fiber of type I_{a+b}. In particular, if f is smooth over Δ^{2} − {(0, 0)}, then f^{−1}((0, 0)) is a smooth elliptic curve and f is a smooth morphism over the whole Δ^{2}.

First, we show

Claim (2.6). φ : V → S is an elliptic fibration and has singular fibers only over a finite set of points of S.

Proof of (2.6). Note that a general fiber of φ is a smooth elliptic curve. Let H be a general very ample divisor on S. Set V_{H} := φ^{−1}(H). Since V has only isolated singularities and since H is general, we may assume that H∩(Sing (S)∪φ(Sing (V))) = ϕ and both H and V_{H} are smooth. Let φ|_{V_{H}} : V_{H} → H be the induced elliptic fibration. Using the adjunction formula, we calculate K_{H} ≡ H|_{H} and K_{V_{H}} = (K_{V} + V_{H})|_{V_{H}} ≡ φ^{∗}(K_{H}). Comparing this with the canonical bundle formula of an elliptic surface (for example see [BPV]), we find that φ|_{V_{H}} is a smooth morphism. This implies the result. q.e.d of (2.6).

Let s ∈ S be an arbitrary smooth point of S and take a sufficiently small polydisk Δ^{2} ⊂ S around s. By (2.6), φ is smooth over Δ^{2} − {s}. Now applying (2.5) for an elliptic fibration φ|_{φ^{−1}(Δ^{2})} : φ^{−1}(Δ^{2}) → Δ^{2}, we get (2.4). Q.E.D. of (2.4).

Lemma (2.7). Let φ : V → S be a fiber space such that

(1) V is a normal projective threefold with only canonical singularities and with O_{V} (K_{V}) ≃ O_{V},

(2) S is a normal projective surface with only Du Val singularities and with O_{S}(K_{S}) ≃ O_{S},

(3) φ is an equi-dimensional fibration, and

(4) φ is smooth except over a finite set of points of S. Then, the reduction of each fiber φ^{−1}(s)_{red} (s ∈ S) is a smooth elliptic curve. Moreover, if s is a smooth point of S, then, φ^{−1}(s) itself is a smooth elliptic curve. In particular, φ is a smooth morphism over S − Sing (S).

Proof. Let s ∈ S be an arbitrary point of S. Since S has only Du Val singularities, we can choose a small neighborhood U around s such that

U = Δ^{2}/G, s = (0, 0)(mod G).

Here Δ^{2} is a two dimensional small polydisk and G is a finite Gorenstein automorphism group of Δ^{2} each of whose element fixes only the origin (0, 0). We may also assume by (4) that φ is smooth over U − {s}.

Letting φ_{U} : V_{U} → U be the restriction of φ, we consider the fiber product

φ_{Δ^{2}} : V_{Δ^{2}} := V_{U} ×_{U} Δ^{2} → Δ^{2}.

Since Δ^{2} → U is étale over U − {s} and φ_{U} is smooth over U − {s}, it follows that φ_{Δ^{2}} : V_{Δ^{2}} → Δ^{2} is smooth over Δ^{2} − {(0, 0)}.

Take a resolution ν : V^{(1)} → V_{Δ^{2}} of V_{Δ^{2}} and set φ^{(1)} := φ ∘ ν : V^{(1)} → Δ^{2}. Note that φ and φ^{(1)} coincide over Δ^{2} − {(0, 0)}.

Applying a relatively minimal model program with respect to K_{V^{(1)}} over Δ^{2} ([Mo]), we get a relatively minimal model

φ^{(2)} : V^{(2)} → Δ^{2}

of φ^{(1)} : V^{(1)} → Δ^{2}. Since each fiber of φ^{(1)} over Δ^{2} − {(0, 0)} is a smooth elliptic curve, φ^{(2)} coincides with φ^{(1)} (and then φ_{Δ^{2}}) over Δ^{2} − {(0, 0)}. This together with (2.5) implies that (φ^{(2)})^{−1}((0, 0)) is also a smooth elliptic curve and that φ^{(2)} is smooth over whole Δ^{2}. In particular, V^{(2)} is also smooth. Since φ_{Δ^{2}} and φ^{(2)} are birational over Δ^{2}, the natural action of G on φ_{Δ^{2}} : V_{Δ^{2}} → Δ^{2} induces a rational action on

φ^{(2)} : V^{(2)} → Δ^{2}.

On the other hand, since each fiber of φ^{(2)} is an elliptic curve, it follows that φ^{(2)} is a unique relatively minimal model. Thus this action of G on φ^{(2)} : V^{(2)} → Δ^{2} is regular and induces

\overline{φ^{(2)}} : V^{(2)}/G → Δ^{2}/G = U.

This is birational to φ_{U} : V_{U} → U over U and is isomorphic over U − {s}. Denote this birational map over U by

μ : V_{U} · · → V^{(2)}/G.

Then, μ gives an isomorphism

V_{U} − φ_{U}^{−1}(s) ≃ V^{(2)}/G − (\overline{φ^{(2)}})^{−1}(s).

Since O_{V_{U} −φ_{U}^{−1}(s)}(K_{V_{U}}) ≃ O_{V_{U} −φ_{U}^{−1}(s)} by our assumption (1) and since (\overline{φ^{(2)}})^{−1}(s) is of codimension two in a normal variety it follows that

O_{V^{(2)}/G}(K_{V^{(2)}/G}) ≃ O_{V^{(2)}/G}.

This shows that the action of G on V^{(2)} is Gorenstein. Since each element of G fixes the origin (0, 0) of Δ^{2}, G stabilizes a smooth elliptic curve E := (φ^{(2)})^{−1}((0, 0)). Since G is also Gorenstein on Δ^{2}, so is on E. That is, G acts on E as a translation group. Thus (\overline{φ^{(2)}})^{−1}(s)_{red} = E/G is a smooth elliptic curve.

Now, in order to complete the first part of (2.7), it is enough to show that μ : V_{U} · · → V^{(2)}/G is actually an isomorphism. But, now, this immediately follows from the facts that V_{U} has only rational singularities and that V^{(2)}/G is ℚ−factorial.

If s is a smooth point of S, then we can take G = {1} and then V_{U} = V^{(2)} over U = Δ^{2}. This implies the last half of (2.7). Q.E.D. of (2.7).

The next lemma is a slight generalization of Kollár’s result (in the three dimensional case), which should be known by specialists. However, because of the lack of suitable references, we give here a brief proof based on the Kollár’s original result.

Lemma (2.8). Let φ : V → S be a fiber space such that (1) V is a normal projective threefold with only canonical singularities,

(2) S is a normal surface with only Du Val singularities. Let ω_{V} and ω_{S} be the dualizing sheaves on V and S. Then, R^{1}φ_{∗}ω_{V} ≃ ω_{S}. Assume furthermore that

(3) O_{V} (K_{V}) ≃ O_{V} and

(4) S is a K3 surface with only Du Val singularities. Then h^{1}(O_{V}) = 1.

Remark. Kollár proved the first part of (2.8) under the assumption that both V and S are smooth ([Ko1]).

Proof. We want to reduce our proof to the smooth case.

Consider the following commutative diagram,

V′ ⟶^{ν} V
Φ↓  ↓^{φ}
S′ ⟶_{μ} S

where μ : S′ → S is the minimal resolution of S′ and ν : V′ → V is a resolution of both the singularities of V and indeterminacy of μ^{−1} ∘ φ.

Then R^{i}ν_{∗}ω_{V′} = 0 for i > 0. Moreover, ν_{∗}ω_{V′} = ω_{V} because V has only canonical singularities. Thus, from the Leray spectral sequence

R^{p}φ_{∗}(R^{q}ν_{∗}ω_{V′}) ⇒ R^{p+q}(φ ∘ ν)_{∗}ω_{V′}

we get

R^{p}φ_{∗}ω_{V} ≃ R^{p}(φ ∘ ν)_{∗}ω_{V′} ≃ R^{p}(μ ∘ Φ)_{∗}ω_{V′}.

In particular,

R^{1}φ_{∗}ω_{V} ≃ R^{1}(μ ∘ Φ)_{∗}ω_{V′}.

On the other hand, the edge sequence of another Leray spectral sequence

R^{p}μ_{∗}(R^{q}Φ_{∗}ω_{V′}) ⇒ R^{p+q}(μ ∘ Φ)_{∗}ω_{V′}

gives an exact sequence

0 → R^{1}μ_{∗}(Φ_{∗}ω_{V′}) → R^{1}(μ ∘ Φ)_{∗}ω_{V′} → μ_{∗}(R^{1}Φ_{∗}ω_{V′}) → R^{2}μ_{∗}(Φ_{∗}ω_{V′}).

Note that R^{2}μ_{∗}(Φ_{∗}ω_{V}) = 0 and that R^{1}μ_{∗}(Φ_{∗}ω_{V′}) is a torsion sheaf, because μ : S′ → S is a birational morphism between surfaces.

On the other hand, since V′ is smooth, R^{1}(μ ∘ Φ)_{∗}ω_{V′} is a torsion free sheaf by [Ko1]. Then, chasing the above exact sequence, we get

R^{1}μ_{∗}(Φ_{∗}ω_{V′}) = 0

and

R^{1}(μ ∘ Φ)_{∗}ω_{V′} ≃ μ_{∗}(R^{1}Φ_{∗}ω_{V′}).

Since V′ and S′ are smooth, Kollár’s original result implies

R^{1}Φ_{∗}ω_{V′} ≃ ω_{S′}.

Thus,

R^{1}(μ ∘ Φ)_{∗}ω_{V′} ≃ μ_{∗}ω_{S′}.

Moreover, since S has only canonical singularities, it follows that

μ_{∗}ω_{S′} ≃ ω_{S}.

Thus,

R^{1}(μ ∘ Φ)_{∗}ω_{V′} ≃ ω_{S}.

Combining these, we get

R^{1}(μ ∘ Φ)_{∗}ω_{V′} ≃ ω_{S}.

This completes the proof of the first part.

We show the second part. Since ω_{V} ≃ O_{V} and ω_{S} ≃ O_{S}, the first part of (2.8) gives

R^{1}φ_{∗}O_{Z} ≃ O_{S}.

Substituting this into the edge sequence of the Leray spectral sequence

H^{p}(R^{q}φ_{∗}O_{V}) ⇒ H^{p+q}(O_{V}),

we get an exact sequence

0 → H^{1}(O_{S}) → H^{1}(O_{V}) → H^{0}(O_{S}).

This implies

h^{1}(O_{V}) ≤ h^{1}(O_{S}) + h^{0}(O_{S}) = 0 + 1 = 1.

We show that h^{1}(O_{V}) ≥ 1. Considering the pullback of the regular two forms by Φ and using Hodge theory, we calculate

h^{2}(O_{V′}) = h^{2,0}(V′) ≥ h^{2,0}(S′) = 1.

On the other hand, using the fact that V has only rational singularities and the Serre duality, we see that

h^{2}(O_{V′}) = h^{2}(O_{V}) = h^{1}(O_{V}).

Combining these, we get the desired inequality h^{1}(O_{V}) ≥ 1. Q.E.D. of (2.8).

We return back to Key Lemma (2.1). This is now proved by a simple combination of the previous lemmas.

Proof of Key Lemma.

Set W_{0} := W − Sing (W) as before and denote the restrictions of Φ : X → W and π : T → W to W_{0} by

Φ_{0} : X_{0} := Φ^{−1}(W_{0}) → W_{0}

and

π_{0} : T_{0} := π^{−1}(W_{0}) → W_{0}.

Note that Φ_{0} is a smooth morphism by (2.4) and π_{0} is an étale morphism by definition. We consider the Cartesian product defined by Φ and π

X_{T} := X ×_{W} T ⟶^{π_{X}} X
Φ_{T} ↓  ↓^{Φ}
T  ⟶_{π} W

and its restriction over W_{0}

(X_{T})_{0} := X_{0} ×_{W_{0}} T_{0} ⟶ X_{0}
↓  ↓
T_{0}  ⟶ W_{0}

Since W_{0} is smooth and since each morphism in the second diagram is smooth or étale, it follows that

Sing (X) ⊂ Φ^{−1}(W − W_{0}),

and

Sing (X_{T}) ⊂ π_{X}^{−1}(Sing (X)) ⊂ (π_{X} ∘ Φ)^{−1}(W − W_{0}) = Φ_{T}^{−1}(T − T_{0}).

In what follows, we apply several birational modifications on the first diagram keeping everything in the second diagram invariant.

Since all singularities in the first diagram are supported over W − W_{0}, we find a commutative diagram

X′_{T} ⟶^{π′_{X}} X′
ν_{X_{T}} ↓  ↓^{ν_{X}}
X_{T} ⟶_{π_{X}} X

such that (1) X′ and X′_{T} are smooth,

(2) ν_{X} : X′ → X is a birational modification only over W − W_{0}, and that

(3) ν_{X_{T}} : X′_{T} → X_{T} is a birational modification only over T − T_{0}.

Let {E_{i}}_{i∈I} be the set of all the two dimensional irreducible components of fibers of Φ′_{T} := Φ_{T} ∘ ν_{X_{T}} : X′_{T} → X_{T} → T. Set E := Σ_{i∈I}E_{i}. By construction, E is supported only over T − T_{0}.

Claim (2.10).

(1) X′_{T} is not covered by rational curves.

(2) K_{X′_{T}} = Σ_{i∈I}a_{i}E_{i} for some non-negative integers a_{i}.

(3) (X′_{T} , ϵE) is klt if ϵ > 0 is sufficiently small.

Proof of (2.10). The assertions (1) and (3) are clear. We show the assertion (2). Since X has only terminal singularities, Sing (X) ⊂ X − X_{0}, and K_{X} = 0 as a divisor, we see that

K_{X′} = Σc_{j}E′_{j},

where c_{j} are some positive integers and E′_{j} are some irreducible divisors supported in ν_{X}^{−1}(X − X_{0}).

On the other hand, since π′_{X_{T}} : X′_{T} → X′ ramifies only at E, the ramification formula gives

K_{X′_{T}} = (π′_{X_{T}})^{∗}(K_{X′}) + Σ_{i∈I}b_{i}E_{i},

for some non-negative integers b_{i}. Since (π′_{X_{T}})^{∗}E′_{i} are effective divisors supported in E, substituting the first equality into the second, we get the result. q.e.d. of (2.10).

Now we can apply (2.2) for Φ′_{T} : X′_{T} → T to get a fiber space f : Z → T such that

(1) Z has only ℚ−factorial canonical singularities with O_{Z}(K_{Z}) ≃ O_{Z},

(2) f : Z → T is birational to Φ_{T} : X_{T} → T over T and is isomorphic over T_{0},

(3) f : Z → T is an equi-dimensional elliptic fibration.

Recall that T is a K3 surface with only Du Val singularities, and that Φ_{T} is smooth over T_{0}. Now using (2.7) and (2.8), we see that

(4) f^{−1}(t)_{red} is a smooth elliptic curve for each t ∈ T, (5) f^{−1}(t) itself is smooth if t is a smooth point of T (in particular, if t ∈ T_{0}),

(6) h^{1}(O_{Z}) = 1.

Thus, it follows from (1) and (6) and [Kw2] that

(7)A := Alb (Z) is a smooth elliptic curve and the Albanese morphism a : Z → A is a fiber space.

By (2), the natural action of ⟨g⟩ on Φ_{T} : X_{T} → T induces a rational action of G on f : Z → T which is regular over T_{0}. By virtue of (1) and (4), we can apply the same argument as in the last part of the proof of (2.7) to conclude

(8) ⟨g⟩ induces a regular action on f : Z → T and

(9) (f : Z → T)/⟨g⟩ is birational to Φ : X → W and is isomorphic over W_{0} = T_{0}/⟨g⟩.

Now these statements (1) - (9) imply the Key Lemma. Q.E.D. of Key Lemma.

§3. Lifting the group action on a fiber space to its covering

In this section, we continue to employ the same notation given at the beginning of Section 2.

Let f : Z → T be the quasi-product threefold found in (2.1) for a fibered Calabi-Yau threefold Φ : X → W of type II_{0}K.

Then (f : Z → T) ≃ (p_{2} : E × S → S)/G, where

(1) E is a smooth elliptic curve,

(2) S is either a (projective) K3 surface with only Du Val singularities or a smooth Abelian surface, given as (any) fiber of the Albanese morphism a : Z → A,

(3) G is a finite commutative Gorenstein automorphism group of E × S as is described in Theorem (1.3).

We want to lift the action of ⟨g⟩ on f : Z → T to one on p_{2} : E × S → S in an equivariant way.

Lemma (3.1). There is a point 0 on A such that ⟨g⟩ stabilizes a^{−1}(0).

Proof. Since the Albanese morphism is an intrinsically and uniquely defined object, ⟨g⟩ acts on the Albanese morphism a : Z → A. This induces a fibration

\overline{a} : Z/⟨g⟩ → A/⟨g⟩.

On the other hand, since X and Z/⟨g⟩ are birational and since both of them have only rational singularities, it follows that h^{1}(O_{Z/⟨g⟩}) = h^{1}(O_{X}) = 0. This implies A/⟨g⟩ = ℙ^{1}. Thus, A^{⟨g⟩} ≠ ϕ. Since A is an elliptic curve, this is equivalent to A^{g} ≠ ϕ. Hence we can choose such a point 0 in A^{g}. Q.E.D. of (3.1).

Let us take a^{−1}(0) as S. Then g induces an action g_{S} := g|_{S} : S → S. Since g acts on the fiber space f : Z → T, ⟨g_{S}⟩ and ⟨g⟩ give an equivariant action on q_{T} := f|_{S} : S → T. Note that q_{T} is nothing but the quotient map S → T = S/G.

Lemma (3.2). g_{S}^{∗}ω_{S} = ζ_{I}ω_{S}, where ω_{S} is a nowhere vanishing regular two form on S, that is, a generator of H^{0}(S, O_{S}(K_{S})).

Proof. Let ω_{T} be a nowhere vanishing regular two form on T. Then, ω_{S} := q_{T}^{∗} ω_{T} is a nowhere vanishing regular two form on S. Thus,

g_{S}^{∗}ω_{S} = g_{S}^{∗} ∘ q_{T}^{∗} ω_{T} = q_{T}^{∗} ∘ g^{∗}ω_{T} = q_{T}^{∗} ζ_{I}ω_{T} = ζ_{I}ω_{S}.

This implies the result. Q.E.D. of (3.2).

Lemma (3.3). There is an automorphism g_{E×S} of E × S such that g_{E×S}, g_{S} and g give an equivariant action on the commutative diagram

E × S ⟶^{p_{2}} S
q′↓  ↓^{q_{T}}
Z ⟶_{f} T

where q and q′ are natural quotient maps.

Proof. Let us consider the fiber product

Z ×_{T} S ⟶^{p_{2}} S
p_{1}↓  ↓^{q_{T}}
Z ⟶_{f} T

Define the action of ⟨g′⟩ on Z ×_{T} S by

g′ : Z ×_{T} S ∋ (u, v) ↦ (g(u), g_{S}(v)) ∈ Z ×_{T} S.

Then, g′, ⟨g_{S}⟩ and ⟨g⟩ give an equivariant action on this fiber product.

By the definition of fiber product, there is a surjective morphism ν : E × S → Z ×_{T} S which factors through the quotient map q : E × S → Z = (E × S)/G and the second projection p_{2} : E × S → S.

Claim (3.4). ν : E × S → Z ×_{T} S is the normalization of Z ×_{T} S.

Proof of (3.4). Obvious. q.e.d. of (3.4).

Since normalization is an intrinsically and uniquely defined notion, the action ⟨g′⟩ on Z ×_{T} S lifts to the action ⟨g_{E×S}⟩ on E × S equivariantly with respect to ν : E × S → Z ×_{T} S. This gives a desired action on E × S. Q.E.D. of (3.3).

Corollary (3.5). ord (g_{S}) = ord (g_{E×S}) = I(:= ord (g)).

Proof. Since g_{S} is a restriction of g, it follows that ord (g_{S}) ≤ ord (g). On the other hand, since τ : S → T is surjective and since g_{S} and g induce an equivariant action on τ, we see that ord (g_{S}) ≥ ord (g). This implies ord (g_{S}) = ord (g). Now it follows from the construction of g_{E×S} that ord (g_{E×S}) = ord (g′) = ord (g). Q.E.D. of (3.5).

Define G̃ to be the subgroup of Aut (E × S) generated by G and g_{E×S} found in (3.3). Then G̃ acts on the fiber space p_{2} : E ×S → S. Thus, there is a (unique) group homomorphism ρ : G̃ → Aut (S) such that p_{2} ∘h = ρ(h)∘p_{2}. By construction, we have ρ(G) = G_{S} and ρ(g_{E×S}) = g_{S}. Corollary (3.5) shows that ρ|_{⟨g_{E×S}⟩} : ⟨g_{E×S}⟩ → ⟨g_{S}⟩ is a group isomorphism as is ρ|_{G} : G → G_{S}. Set G̃_{S} = ρ(G̃).

Lemma (3.6).

(1) G_{S} is a normal subgroup of G̃_{S}.

(2) G̃_{S} = G_{S} ⋊ ⟨g_{S}⟩.

(3) G is a normal subgroup of G̃.

(4) G̃ = G ⋊ ⟨g_{E×S}⟩.

(5) ρ : G̃ → G̃_{S} is an isomorphism.

Proof. For the assertion (1), it is enough to show that there is an h′ ∈ G_{S} such that g_{S} ∘ h = h′ ∘ g_{S} for each h ∈ G_{S}. Let s ∈ S be a point on S such that g_{S}(s) ∉ S^{G_{S}}. Using g ∘ q_{T} = q_{T} ∘ g_{S} and T = S/G_{S}, we calculate

q_{T} ∘ g_{S} ∘ h(s) = g ∘ q_{T} ∘ h(s) = g ∘ q_{T} (s) = q_{T} ∘ g_{S}(s).

Thus, for each s ∈ S, there is h_{s} ∈ G_{S} such that g_{S} ∘ h(s) = h_{s} ∘ g_{S}(s). Such an h_{s} is uniquely determined by s because g_{S}(s) ∉ S^{G_{S}}. Thus, we find a continuous map S − R → G_{S} defined by s ↦ h_{s}. Since G_{S} is discrete, the image must be one point, say h′. Then, g_{S} ∘ h = h′ ∘ g_{S} over S − g_{S}^{−1}(S^{G_{S}}). Taking the closure, we find that g_{S} ∘ h = h′ ∘ g_{S} whole over S. This finishes the proof of (1).

Applying the same argument for E ×S → (E ×S)/G = Z (instead of T = S/G_{S}), we can also show assertion (3).

We show assertion (2). By (1), we have G̃_{S}/G_{S} = ⟨g_{S}(mod G_{S})⟩. Consider the natural representation G̃_{S} on H^{0}(S, O_{S}(K_{S}))

ζ : G̃ → ℂ^{×}, h ↦ ζ(h)

defined by h^{∗}ω_{S} = ζ(h)ω_{S}. Since G_{S} is a Gorenstein automorphism group of S, this factors

\overline{ζ} : G̃_{S}/G_{S} = ⟨g_{S}(mod G_{S})⟩ → ℂ^{×}.

Since \overline{ζ}(g_{S}(mod G_{S})) = ζ(g_{S}) = ζ_{I} by (3.3), it follows that ord (g_{S}(mod G_{S})) ≥ I = ord (g_{S}). Thus, the natural surjective group homomorphism ⟨g_{S}⟩ → ⟨g_{S}(mod G_{S})⟩ must be isomorphism. This implies the assertion (2).

Finally, we show assertions (4) and (5).

By (3), we see that G̃/G ≃ ⟨g_{E×S}(mod G)⟩. Combining this with (3.5), we get

♯G̃ = (♯G) · (♯⟨g_{E×S}(mod G)⟩) ≤ (♯G) · (♯⟨g⟩).

On the other hand, by (2) and (3.5), we have

♯G̃_{S} = (♯G_{S}) · (♯⟨g_{S}⟩) = (♯G) · (♯⟨g⟩).

However, since G̃_{S} is an image of G̃, it follows that

♯G̃ ≥ ♯G̃_{S}.

Combining these three we get ♯G̃ = ♯G̃_{S}. This implies that the surjective group homomorphism ρ : G̃ → G̃_{S} is an isomorphism. Combining this together with (2), we get G̃ = G ⋊ ⟨g_{E×S}⟩. This completes the proof. Q.E.D. of (3.6).

From now on, we denote the equivariant actions G̃ and G̃_{S} on the fiber space p_{2} : E × S → S simply by G̃. We also set g̃ := g_{E×S} for consistency of notation. If no confusion seems to arise, we also identify g_{S} and G_{S} with g̃ and G (under the isomorphism ρ).

The following corollary is an immediate consequence of Lemma (3.6).

Corollary (3.7).

(f : Z → T)/⟨g⟩ = (p_{2} : E × S → S)/G̃.

Thus, the fiber space Φ : X → W is birational to (p_{2} : E × S → S)/G̃ over W = S/G̃ and is isomorphic over W_{0}.

Now this together with the next lemma and the corollary completes the proof of Main Theorem (2) modulo impossibility for S to be a smooth Abelian surface.

Lemma (3.8). Assume that S is a K3 surface with only Du Val singularities. Then, the action of g̃ on E × S is written as follows:

g̃ : E × S ∋ (x, y) ↦ (ζ_{I}^{−1}x, g_{S}(y)) ∈ E × S

for an appropriate origin 0 of E.

Proof. Since ⟨g̃⟩ acts on p_{2} : E × S → S, there is a homomorphic map

c : S → Aut (E) = E ⋊ Aut (E, {0})

defined by s ↦ (p_{1}((x, s)) ↦ p_{1}(g̃(x, s))).

On the other hand, since h^{1}(O_{S}) = 0 and S has only Du Val singularities, the Albanese variety of S is trivial. Thus c must be constant map. That is, g̃ = (g_{E}, g_{S}) for some g_{E} ∈ Aut (E). Since X is isomorphic to (E × S)/G̃ over W_{0} and since (E × S)/G̃ → W is equidimensional, O_{X}(K_{X}) ≃ O_{X} implies O_{(E×S)/G̃}(K_{(E×S)/G̃}) ≃ O_{(E×S)/G̃}. This means G̃ is a Gorenstein automorphism of E × S. In particular, so is g̃. Combining this with g_{S}^{∗}ω_{S} = ζ_{I}ω_{S}, we get g_{E}^{∗}ω_{E} = ζ_{I}^{−1}ω_{E}. In particular, E^{g_{E}} ≠ ϕ. Now, choosing the origin 0 of E in E^{g_{E}}, we get the desired expressions of g̃. This completes the proof of (3.8). Q.E.D.

Combining (3.8) and (3.7), we get

Corollary(3.9). Assume that S is a K3 surface with only Du Val singularities. Then,

(1) the global canonical index I = I(W) of W is either 2, 3, 4, or 6,

(2) if ν : S′ → S is a minimal resolution of S, then the action ⟨g̃⟩ on E × S lifts to E × S′ in an equivariant way and Φ : X → W is birational to

(p_{2} ∘ (id. × ν) : E × S′ → S)/G̃

over W = S/G̃ and is isomorphic over W_{0}.

§4. Impossibility for S to be a smooth abelian surface

We continue to employ the same notation given in the previous sections 2 and 3. In this section, we show that each surface S (found at the beginning of section 3) is not a smooth abelian surface if Φ : X → W is a Calabi-Yau threefold of type II_{0}K. This completes the proof of Main Theorem (2).

Thoughout this section, assuming the contrary that S is a smooth abelian surface, we shall derive a contradiction.

For simplicity, we denote G̃_{S}, G_{S} and g_{S} by G̃, G and g̃ respectively. Under this notation, we have T = S/G, W = T/⟨g⟩ = S/G̃ and I = ord (g) = ord (g̃). As before, we denote by q_{T} : S → T the natural quotient morphism. This has an equivariant action of ⟨g̃⟩ and ⟨g⟩. Recall also that all the possibilities of G are listed up in (1.3)(4).

The next Lemma is shown by [O2].

Lemma (4.1). I is either 2, 3, 4, 6, or 12.

By virtue of this Lemma, the next two Claims will give a contradiction.

Key Claim (4.2). I is not divided by 2.

Key Claim (4.3). I ≠ 3.

The following obvious lemma and its corollaries will be frequently used to prove these claims.

Lemma (4.4). Let q : S_{1} → S_{2} be a surjective finite morphism between normal projective surfaces with K_{S_{1}} ≡ 0 and K_{S_{2}} ≡ 0. Then q ramifies only at finitely many points.

Corollary (4.5). The quotient map S → W (= S/G̃) ramifies only at finitely many points. In particular, SG̃_{S} is a finite set.

Corollary (4.6). Let h be a non-Gorenstein involution in G̃. Then, S^{h} = ϕ. In particular, if I = 2k is even, then S^{g̃^{k}} = ϕ and Sg̃ = ϕ.

Proof. Assuming S^{h} ≠ ϕ, we take a point P in S^{h}. Since h is an involution with h^{∗}ω_{S} = −ω_{S}, it follows that h = diag (−1, 1) under appropriate coordinates (x, y) of S around P. But then h would have a fixed curve (x = 0), contradiction. q.e.d. of (4.6).

Corollary (4.7). If I is either 2, 3, or 4, then T^{g} ≠ ϕ. If I = pq where p = 2 or 4 and q = 3, then T^{g^{p}} ≠ ϕ and T^{g^{q}} ≠ ϕ. Moreover, if I is either 2 or 4, then (ϕ ≠)T^{g} ⊂ Sing (T).

Proof. Since I is the least common multiple of the local canonical indices of W, the first part of the assertion is obvious. Assume that I is either 2 or 4. The first half part shows T^{g} ≠ ϕ. Assume the contrary that there is a smooth point Q in T^{g}. Then, arguing similarly as in (4.6), we see that g^{I/2} = diag (−1, 1) under appropriate local coordinates around P. Then, g^{I/2} has a fixed curve. On the other hand, Lemma (4.4) shows T → W (= T/⟨g⟩) has no ramification divisor, contradiction. q.e.d. of (4.7).

We return back to the key claims (4.2) and (4.3).

Proof of Key Claim (4.2).

Assume the contrary that I = 2k for some integer k. We set h := g̃^{k}. Then h is a non-Gorenstein involution on S. Dividing into the following five cases, we shall derive a contradiction: Case 1. G ≃ ℤ_{3} or ℤ_{3} × ℤ_{3}, Case 2. G ≃ ℤ_{6}, Case 3. G ≃ ℤ_{2}, Case 4. G ≃ ℤ_{2} × ℤ_{2}, Case 5. G ≃ ℤ_{4} or ℤ_{2} × ℤ_{4}.

Case 1. Since g acts on the set B consisting of nine singular points of type A_{3} on T ((1.3)(4)), ⟨g̃⟩ acts on q_{T}^{−1}(B). Since ♯q_{T}^{−1}(B) is either 9 or 27, h has a fixed points. This contradicts (4.6).

Case 2. Consider the unique singular point Q of type A_{5} on T ((1.3)(4)). Then, q_{T}^{−1}(Q) consists of one point, say, P. Since g(Q) = Q, it follows that g̃(P) = P. But this contradicts (4.6).

Case 3. By (4.7), T^{g^{k}} ≠ ϕ. On the other hand, since g^{k} is a non-Gorenstein involution on T, the same argument as in (4.7) implies that T^{g^{k}} ⊂ Sing (T). Let Q ∈ T^{g^{k}}. Then Q is a singular point of type A_{1} and then q_{T}^{−1}(Q) consists of one point, say, P ((1.3)(4)). But then h(P) = P, contradiction.

Case 4. The same argument as in case 3 shows that T^{g^{k}} ≠ ϕ and T^{g^{k}} ⊂ Sing (T). Let Q ∈ T^{g^{k}}. Then, Q is a singular point of type A_{1} and q_{T}^{−1}(Q) is written as {P, r(P)} for some point P and a translation r in G ((1.3)(4)). Since h acts on this set, we have either h(P) = P or h(P) = r(P). The first equality contradicts (4.6). Consider the second case. Set h′ := r ∘ h. Then h^{∗}ω_{S} = −ω_{S}. Since the translation subgroup of G is just ⟨r⟩ and since h^{−1} ∘ r ∘ h is a translation in G (because G is a normal subgroup of G̃), it follows that h^{−1} ∘ r ∘ h ∈ ⟨r⟩ and then ⟨r, h⟩ = ⟨r⟩ × ⟨h⟩ ≃ (ℤ_{2})^{2}. Thus h′ is a non-Gorenstein involution with h′(P) = P. But this contradicts (4.6).

Case 5. We treat the following three cases separately:

Case 5a. 3|I, Case 5b. I = 4, and Case 5c. I = 2.

Case 5a. In this case, I = 6m for some integer m. Set j := g̃^{m}. This is of order 6. Since g acts on the set consisting of 4 singular points of type A_{3} on T ((1.3)(4)), j^{2} acts on the inverse image of these points. This consists of either 4 or 8 points. Thus, j^{2} has a fixed point among these points. Let P be such a fixed point. Then, j^{2}(P) = P. Since (j^{2})^{∗}ω_{S} = ζ_{3}ω_{S} and j^{2} has at most finite fixed points by (4.5), an easy coordinate calculation shows that j^{2} = diag (ζ_{3}^{2}, ζ_{3}^{2}) under appropriate global coordinates (x, y) around P. Thus, the eigen value of the matrix part of j is in {ζ_{3}, −ζ_{3}}. Thus, j has a fixed point on S, say Q. Since h = j^{3}, Q is also a fixed point of h. But this contradicts (4.6).

Case 5b. By (4.7), we can take a point Q in T^{g}. Again by (4.7) and (1.3)(4), Q is either a singular point of type A_{3} or of type A_{1}.

If Q is a singular point of type A_{3}, then q_{T}^{−1}(Q) is written as {P } (in the case when G ≃ ℤ_{4}) and {P, r(P)} for a translation r in G (in the case when G ≃ ℤ_{2} × ℤ_{4}). In the first case, we have g̃(P) = P. But this contradicts (4.6). In the second case, we have either g̃(P) = P or g̃(P) = r(P). Since r is of order two, in each case, we get h(P) = g̃^{2}(P) = P, contradiction.

If Q is a singular point of type A_{1}, then q_{T}^{−1}(Q) is written as {P, u^{2}(P)} (if G = ⟨u⟩ ≃ ℤ_{4}) and {P, u^{2}(P), r(P), r ∘ u^{2}(P)} (if G ≃ ℤ_{2} × ℤ_{4}). In the second case, r is the unique translation in G and u is some (suitable) generator of G.

In anyway, we have g̃(P) = P or g̃(P) = t(P), where t is an involution in G. Thus, h(P) = g̃^{2}(P) = P, contradiction.

Case 5c. First consider the case G = ⟨u⟩ ≃ ℤ_{4}.

Since G̃ = ⟨u⟩ ⋊ ⟨g̃⟩ is of order 8, elementary group theory shows that G̃ is isomorphic to either

(1) D_{8}, the dihedral group of order 8, or

(2) ℤ_{4} × ℤ_{2}.

Assume first that G̃ ≃ D_{8}. Then, g̃ ∘ u is a non-Gorenstein involution. Take a point Q in T^{g}. Then, Q is a singular point either of type A_{3} or of type A_{1}.

If Q is of type A_{3}, then q_{T}^{−1}(Q) = {P }, a one point set. But then g̃(P) = P, contradiction.

If Q is of type A_{1}, then q_{T}^{−1}(Q) is written as {P, u(P)} and g̃ stabilizes this set. If g̃(P) = P, then we get the same contradiction as before. If g̃(P) = u(P), then g̃∘u(P) = P. Since g̃∘u is a non-Gorenstein involution, we again get a contradiction. In any case, we found a contradiction if G̃ ≃ D_{8}.

Next consider the case when G̃ ≃ ℤ_{4}×ℤ_{2}, that is, G̃ = ⟨u⟩×⟨g̃⟩. Then ⟨u⟩ ≃ G̃/⟨g̃⟩ acts on \overline{p_{2}} : (E × S)/⟨g̃⟩ → S/⟨g̃⟩. Note that (E × S)/⟨g̃⟩ is also a smooth threefold, because S^{[⟨g̃⟩]} = ϕ by (4.6) so that (E × S)^{[⟨g̃⟩]} = ϕ.

Claim. (E × S/⟨g̃⟩)^{[⟨u⟩]} = ϕ.

Proof of Claim. Since u is of order 4, it is sufficient to show that

(E × S/⟨g̃⟩)^{u^{2}} = ϕ.

Assume the contrary that P ∈ (E × S/⟨g̃⟩)^{u^{2}}. Set \overline{p_{2}}(P) = Q. Then u^{2}(Q) = Q. Thus u^{2} acts on the fiber E_{Q} := (\overline{p_{2}})^{−1}(Q). On the other hand, the fiber of E × S → (E × S/⟨g̃⟩) over Q is written as {R, g̃(R)} and u^{2} also acts on this set. If u^{2}(R) = g̃(R), then u^{2} ∘ g̃(R) = R on S. But, since u^{2} ∘ g̃ is a non-Gorenstein involution on S, this contradicts (4.6). Thus u^{2}(R) = R. Let E_{R} be the fiber of p_{2} : E × S → S over R. Then the natural projection E × S → E × S/⟨g̃⟩ (of degree two) induces an isomorphism E_{R} ≃ E_{Q}, because E_{g̃(R)} is also mapped to E_{Q}. Since u^{2} gives an equivariant action on this isomorphism and since u^{2} acts on E_{R} as a translation of order two by (1.3), we see that u^{2} also acts on E_{Q} as a translation of order two. Thus E_{Q}^{u^{2}} = ϕ. But this is absurd, because P ∈ E_{Q} is a fixed point of u^{2}. q.e.d. of Claim.

Thus Y := ((E × S)/⟨g̃⟩)/⟨u⟩ = (E × S)/G̃ is also a smooth threefold (with O_{Y} (K_{Y}) ≃ O_{Y}). Since X is birational to Y, X is connected with Y by flops. Then X is also smooth and π_{1}(X) ≃ π_{1}(Y) ([Ko2]). Thus X has a non-trivial finite étale covering, because so does Y. But this contradicts our assumption π_{1}^{alg}(X) = {1}. Therefore, we get a contradiction even in the case G ≃ ℤ_{4}.

We consider the remaining case G = ⟨t⟩×⟨u⟩ ≃ ℤ_{2} ×ℤ_{4}. Reducing to the previous case G ≃ ℤ_{4}, we find a contradiction.

Since the translation group of G is just ⟨t⟩ and since G is a normal subgroup of G̃, the same argument as before shows ⟨t⟩ is a normal subgroup of G̃. Thus G̃/⟨t⟩ ≃ ⟨u_{1}⟩ ⋊ ⟨g̃_{1}⟩, where u_{1} := u(mod ⟨t⟩) and g̃_{1} := g̃(mod ⟨t⟩). Observe that u_{1} is of order four and g̃_{1} is of order two.

On the other hand, since ⟨t⟩ acts on p_{2} : E × S → S, we get a new fiber space

\overline{p_{2}} : (E × S)/⟨t⟩ → S/⟨t⟩,

on which ⟨u_{1}⟩ × ⟨g̃_{1}⟩ gives an equivariant action. Since ⟨t⟩ is a translation group on both E × S and S, it follows that (E × S)/⟨t⟩ is an Abelian threefold and S/⟨t⟩ is an Abelian surface. Set S_{1} := S/⟨t⟩ and V := (E × S)/⟨t⟩. Then, T = S_{1}/⟨u_{1}⟩ and W = S_{1}/⟨u_{1}, g̃_{1}⟩.

Observe that g̃_{1}^{∗}ω_{S_{1}} = −ω_{S_{1}}, u_{1}^{∗}ω_{S_{1}} = ω_{S_{1}} and that u_{1} acts on each fiber over S_{1}^{u_{1}}(≠ ϕ) as a translation of order 4. The last statement follows from (1.3) and a similar argument as is given in the last claim. Thus we can apply the same argument as in the previous case (G ≃ ℤ_{4}) for \overline{p_{2}} : (E × S)/⟨t⟩ → S/⟨t⟩ and S_{1} → T → W to get a contradiction. This finishes the proof of case 5c.

Now we have completed the proof of (4.2). Q.E.D. of (4.2).

Proof of Key Claim (4.3).

Assuming the contrary that I = 3 and dividing into the following five cases, we shall derive a contradiction.

Case 1. G ≃ ℤ_{4} or ℤ_{2} × ℤ_{4},

Case 2. G ≃ ℤ_{2} or ℤ_{2} × ℤ_{2},

Case 3. G ≃ ℤ_{6},

Case 4. G ≃ ℤ_{3},

Case 5. G ≃ ℤ_{3} × ℤ_{3}.

Case 1. Since g acts on the set of singular points of type A_{3} and since this set consists of 4 points, g has a fixed point, say Q, in this set. Then, g̃ acts on q_{T}^{−1}(Q). Since q_{T}^{−1}(Q) consists of one or two points, g̃ has a fixed point in q_{T}^{−1}(Q). Denote this point by 0. Since g̃^{∗}ω_{S} = ζ_{3}ω_{S}, g̃(0) = 0 and since g̃ has only finitely many fixed points, we can apply [CC, also O2] to get S ≃ E_{ζ_{3}}^{2} and g̃ = ζ_{3}^{2}, the scalar multiplication by ζ_{3}^{2}. On the other hand, the stabilizer of 0 in G is a cyclic group of order 4. We denote this group by ⟨u⟩. Then u = diag (ζ_{4}, ζ_{4}^{−1}) under appropriate global coordinates around 0. Set H := ⟨u, g̃⟩. Then, H ⊂ Aut (S, {0}). Moreover H is a cyclic group of order 12, because g̃ = ζ_{3}^{2} so that u ∘ g̃ = g̃ ∘ u. In particular H ∋ −1. But this is impossible by Fujiki’s classification ([Fu, Table 6]).

Case 2. Just by the same argument as in case 1, we see that g̃ has a fixed point 0 (over some singular point of type A_{1} of T) and then S = E_{ζ_{3}}^{2} and g̃ = ζ_{3}^{2}. Set Stab_{{0}}(G) = ⟨u⟩. This is a cyclic group of order two and u = diag (−1, −1) under appropriate global coordinates around 0. Thus u ∘ g̃ = g̃ ∘ u. Since G̃ gives an equivariant action on p_{2} : E × S → S, g̃ and u act on the fiber E := p_{2}^{−1}(0). Since g̃ is a Gorenstein automorphism of E × S, the matrix part of g̃ on E is ζ_{3}^{2} so that g̃ acts on E by

g̃ : E ∋ x ↦ ζ_{3}^{2}x ∈ E,

if we fix an origin 0_{E} of E in Eg̃(≠ ϕ). On the other hand, by (1.3), the action of u on E is written as

u : E ∋ x ↦ x + P ∈ E,

where P ∈ (E)_{2} − {0}. Since u ∘ g̃ = g̃ ∘ u in G̃, we calculate

g̃(x) + g̃(P) = g̃ ∘ u(x) = u ∘ g̃(x) = g̃(x) + P.

Thus, P ∈ Eg̃ = E^{ζ_{3}} ⊂ (E)_{3}. But this is impossible because (E)_{3} ∩ ((E)_{2} − {0}) = ϕ.

Case 3. Let Q be the unique singular point of type A_{5} on T. Then, q_{T}^{−1}(Q) consists of one point, say, 0. Since g(Q) = Q, it follows that g̃(0) = 0. Thus, just by the same argument as before, we get g̃ = ζ_{3}^{2}. Set Stab_{{0}}(G) = ⟨u⟩. This is a cyclic group of order 6 and u = diag (ζ_{6}, ζ_{6}^{−1}) under an appropriate global coordinates (x, y) around 0. it follows that g̃ ∘ u^{2} = diag (1, ζ_{3}). Then g̃ ∘ u^{2} has a fixed curve (y = 0), contradiction.

Case 4. Set G = ⟨u⟩. Since G̃ = ⟨u⟩⋊⟨g̃⟩ is of order 9, it follows that G̃ = ⟨u⟩×⟨g̃⟩. Let Q be a point in T^{g}. Then ♯q_{T}^{−1}(Q) is either one or three. If q_{T}^{−1}(Q) = {P }, a one point set, then g̃(P) = P. If q_{T}^{−1}(Q) = {P_{1}, P_{2}, P_{3}}, then, g̃(P_{1}) = P_{j} for some j = 1, 2, or 3. Since ⟨u⟩ acts on {P_{1}, P_{2}, P_{3}} transitively, we find that u^{i}(P_{1}) = P_{j} for some i. Set h := u^{−i} ∘ g̃. Then, h(P_{1}) = P_{1}. Note that h is of order 3 and satisfies h^{∗}ω_{S} = ζ_{3}ω_{S} and G̃ = ⟨u⟩ × ⟨h⟩. In addition, h and g give an equivariant action on q_{T} : S → T. Thus, we may replace g̃ by h in the second case. Then g̃(P_{1}) = P_{1} in each case. We regard this point P_{1} as an origin of S and denote it by 0_{S}.

Since g̃ has only isolated fixed points ((4.5)), the same argument as before shows that S = E_{ζ_{3}}^{2} and g̃ = ζ_{3}^{2}. This implies (S)g̃ ∩ (S)^{u} = ϕ. (In fact, otherwise, choosing a point P in (S)g̃ ∩ (S)^{u}, we find appropriate coordinates (x, y) around P such that u = diag (ζ_{3}, ζ_{3}^{−1}). Then, g̃∘ u = diag (1, ζ_{3}) has a fixed curve (y = 0), contradiction.)

Since G̃ is a Gorenstein automorphism of E ×S and gives an equivariant action on p_{2} : E × S → S, g̃ induces an automorphism on the fiber E := p_{2}^{−1}(0_{S}) whose matrix part is ζ_{3}^{2}. Thus E = E_{ζ_{3}} and then E × S = E_{ζ_{3}}^{3}. Moreover, choosing an origin 0_{E} of E in Eg̃, we get g̃ = ζ_{3}^{2} on E. Now, taking 0 := (0_{S}, 0_{E}) as an origin of E × S = E_{ζ_{3}}^{3}, we have g̃ = ζ_{3}^{2} on E_{ζ_{3}}^{3}. Let us consider the quotient threefolds (E_{ζ_{3}})^{3}/⟨g̃⟩ and its crepant resolution ν : Y → (E_{ζ_{3}})^{3}/⟨g̃⟩. Note that ⟨u⟩ ≃ G̃/⟨g̃⟩ acts on (E_{ζ_{3}})^{3}/⟨g̃⟩. Note also that ν is unique. (In fact, one of such ν is given by replacing each of 27 singular points of type 1/3(1, 1, 1) of (E_{ζ_{3}})^{3}/⟨g̃⟩ by ℙ^{2} and then has no flopping curves in the exceptional divisor.) Thus, ⟨u⟩ induces a regular action on Y.

Claim. ⟨u⟩ acts freely on Y.

Proof of Claim. Since ord (u) = 3, it is sufficient to show that Y^{u} = ϕ. Assume the contrary that P ∈ Y^{u}. Put Q := ν(P). Then u(Q) = Q. Denote the natural quotient map E_{ζ_{3}}^{3} → (E_{ζ_{3}})^{3}/⟨g̃⟩ by τ. Then, Q ∉ τ((E_{ζ_{3}}^{3})g̃). (In fact, otherwise, τ^{−1}(Q) = {R}(⊂ (E_{ζ_{3}}^{3})g̃), a one point set. Thus, u(R) = R and g̃(R) = R on (E_{ζ_{3}})^{3}. Set R′ := p_{2}(R). Then, u(R′) = R′ and g̃(R′) = R′, because G̃ gives an equivariant action on p_{2} : E × S → S. But this contradicts (S)g̃ ∩ (S)^{u} = ϕ.)

Thus, τ^{−1}(Q) consists of three points, say, R_{1}, R_{2} and R_{3}. Since u(Q) = Q, u acts on {R_{1}, R_{2}, R_{3}}. Since ⟨u⟩ acts freely on E_{ζ_{3}}^{3} by (1.3), we may assume without loss of generality that u(R_{1}) = R_{2}. On the other hand, {R_{1}, R_{2}, R_{3}} is the orbit space of R_{1} by ⟨g̃⟩, it follows that g̃^{i}(R_{1}) = R_{2} for some i = 1, 2. Set again R′ := p_{2}(R_{1}). Then, g̃^{i}(R′) = u(R′)(= p_{2}(R_{2})) so that u^{−1} ∘ g̃^{i}(R′) = R′. Since the matrix part of u^{−1} is diag (ζ_{3}, ζ_{3}^{−1}) under some appropriate global coordinates around R′, we calculate u^{−1} ∘ g̃^{i} = diag (1, ζ_{3}). Thus u^{−1} ∘ g̃^{i} has a fixed curve (y = 0), contradiction. q.e.d. of Claim.

By this claim Y/⟨u⟩ is a smooth threefold with O_{Y/⟨u⟩}(K_{Y/⟨u⟩}) ≃ O_{Y/⟨u⟩} and with non-trivial étale covering. On the other hand, by construction, our original Calabi-Yau threefold X is birational to Y and then is connected with Y by flops. Thus X is also smooth and π_{1}(X) ≃ π_{1}(Y) by [Ko2]. This implies that X has also non-trivial finite étale covering. But this contradicts our assumption π_{1}^{alg}(X) = {1}.

Case 5. As in case (5c) in Claim (4.2), reducing to the previous case 4, we find a contradiction. Set G = ⟨t⟩ × ⟨u⟩, where t is a translation of order 3. Since the translation group of G is just ⟨t⟩, and G is a normal subgroup of G̃, the same argument as in case 4 in Claim (4.2) implies that ⟨t⟩ is a normal subgroup of G̃. Thus, G̃/⟨t⟩ = ⟨u_{1}⟩ × ⟨g̃_{1}⟩ ≃ (ℤ_{3})^{2}, where u_{1} := u(mod ⟨t⟩) and g̃_{1} := g̃(mod ⟨t⟩).

By the way, since ⟨t⟩ acts on p_{2} : E × S → S, we get a new fiber space

\overline{p_{2}} : (E × S)/⟨t⟩ → S/⟨t⟩,

on which ⟨u_{1}⟩ × ⟨g̃_{1}⟩ gives an equivariant action. Since ⟨t⟩ is a translation group on both E ×S and S, (E ×S)/⟨t⟩ is an Abelian threefold and S/⟨t⟩ is an Abelian surface. Set S_{1} := S/⟨t⟩ and V := (E × S)/⟨t⟩. Then, T = S_{1}/⟨u_{1}⟩ and W = S_{1}/⟨u_{1}, g̃_{1}⟩. Moreover g̃_{1}^{∗}ω_{S_{1}} = ζ_{3}ω_{S_{1}} while u_{1}^{∗}ω_{S_{1}} = ω_{S_{1}}. Now applying the same argument as in case 4 for S_{1} → T → W, we find that S_{1} = E_{ζ_{3}}^{2} and g̃_{1} = ζ_{3}^{2} (after replacing g̃_{1} by u_{1}^{i} ∘ g̃_{1} so that Sg̃_{1} ; 1 ≠ ϕ and then fixing the origin 0 of S_{1} in Sg̃_{1} ; 1(≠ ϕ)). Note that ⟨u_{1}, g̃_{1}⟩ gives a Gorenstein action on V. Then letting E := \overline{p_{2}}^{−1}(0) and applying the same argument as in case 4, we see that E = E_{ζ_{3}} and the action of g̃_{1} on E is g̃_{1} = ζ_{3}^{2} (after fixing an origin 0_{E} of E in Eg̃_{1}(≠ ϕ)). Thus, regarding 0_{E} as an origin 0 of V, we get g̃_{1} = ζ_{3}^{2} under appropriate global coordinates around 0. This together with [CC also O2] implies V = E_{ζ_{3}}^{3}. Now again applying the same argument as in case 4 for \overline{p_{2}} : V → S_{1}, we finally get a contradiction that X is birational to a smooth threefold Y with non-trivial finite étale covering.

Now this completes the proof of Claim (4.3).

Now we are done. Q.E.D. of Main Theorem (2).

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Keiji Oguiso

Department of Mathematical Sciences University of Tokyo Komaba, Tokyo Japan oguiso@ms.u-tokyo.ac.jp 448

Documenta Mathematica 1 (1996)