Calabi-yau Threefolds of Quasi-product Type

According to the numerical Iitaka dimension (X; D) and c 2 (X) D, bered Calabi-Yau threefolds jDj : X ! W (dim W > 0) are coarsely classiied into six diierent 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 (X; D) = 2 and c 2 (X) D = 0 and by (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 classiication. The purpose of this paper is to guarantee a complete structure theorem of bered Calabi-Yau threefolds of type II 0 to nish the classiication 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.


Introduction
Let us start this introduction by recalling a global picture of bered 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 alg 1 (X) = f1g.The last condition is equivalent to alg 1 (X Sing X) = f1g, because the local fundamental group of three dimensional terminal Gorenstein singularities is trivial ( Kw3]).This also implies h 1 (O X ) = 0 ( O1]).We de ne c 2 (X) D := c 2 (X 0 ) (D) for any resolution : X 0 !X of Sing (X).
A surjective morphism : X !W is called a bered Calabi-Yau threefold if X is a Calabi-Yau threefold, W is a normal projective variety (of positive dimension) and has connected bers.Note that is nothing but jDj if D is the pull back of (any) very ample divisor H on W. The following (more or less tautological) coarse classi cation is proved in O1].
Type I 0 : General bers are smooth Abelian surfaces and W = P 1 , Type I + : General bers are smooth K3 surfaces and W = P 1 , Type II 0 : General bers are smooth elliptic curves and W is a normal projective rational surface with only quotient singularities and with K W 0, Type II + : General bers are smooth elliptic curves and W is a normal projective rational surface with only quotient singularities and with K W + 0 for some non-zero e ective Q-divisor such that (W; ) is klt, Type III 0 : 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 )(:= c 2 (X)) = 0 as a linear form on Pic(W), Type III + : 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 ) 6 = 0.
Moreover, if : X !W is a bered Calabi-Yau threefold of type II 0 and H is a general very ample divisor on W, then the induced elliptic surface 1 (H) !H has no singular bers while 1 (H) !H has at least one singular ber composed of rational curves if : X !W is of type II + .
Theorem 1 shows that bered 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 bered Calabi-Yau threefolds of type III 0 .
In particular, there are exactly two bered 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 bered Calabi-Yau threefolds called of type II 0 .
Base surfaces W of bered Calabi-Yau threefolds : X !W of type II 0 are classi ed into two classes by the global canonical covering : 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.
The following theorem gives a complete classi cation of bered Calabi-Yau threefolds of type II 0 A. Main Theorem.Let us prepare (i) a smooth elliptic curve E with a xed origin 0, (ii) a projective K3 surface S with only Du Val singularities and its minimal resolution : S 0 !S, and (iii) two groups G 2 ff1g; 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 g; and hgi ' Z I 2 fZ 2 ; Z 3 ; Z 4 ; Z 6 g; such that G := G o hgi (semi-direct product) acts faithfully on both E and S (and then on S 0 and E S 0 ) in such a way that (iv) G 3 a : E S 0 !E S 0 ; (x; y) 7 !(x + a E ; a S 0(y)) with a E 2 (E) ord (a) and a S 0!S 0 = !S 0, where !S 0 is a nowhere vanishing regular 2 form on S 0 , (v) g : E S 0 !E S 0 ; (x; y) 7 !( 1 I x; g S 0 (y)) with g S 0!S 0 = I !S 0, and (vi) (S 0 ) G] Exc( ) except for nitely many points in (S 0 ) G] , that is, (S) G] is a nite set.
Note that G is a nite Gorenstein automorphism group of E S 0 .Let (2) Conversely, every bered Calabi-Yau threefold of type II 0 K is obtained by the above process for some triplet (E; S; G) satisfying the conditions (i)-(vi) up to isomorphisms as ber spaces.In particular, every bered 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 bered Calabi-Yau threefolds called of types II 0 and III 0 .
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 : X !W is a bered Calabi-Yau threefold of type II 0 .Then the global canonical index of W is either 2, 3, 4 or 6.
Corollary.Let : X !W be a bered Calabi-Yau threefold of type II 0 K (resp. of type II 0 A).Then, there is a composite of ops Y !W of : X !W over W such that Y has at least two di erent ber 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 bered Calabi-Yau threefold of type I 0 , that is, a Calabi-Yau threefold with an Abelian bration.However, our main theorem and Theorem 4 show Corollary.Let X be a Calabi-Yau threefold with at least two di erent Abelian brations.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 jDij : X !P 1 (i = 1; 2) are two di erent Abelian brations on X, then jm(D1+D2)j : 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 T : X T := X W T !T be the base change of a bered Calabi-Yau threefold : X !W of type II 0 K to the global canonical cover : T !W. Since always has a two dimensional bers ( O1]), X T has very bad singularities and 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 Koll ar et al.Sh] and Kw4] (also Ko3]) to nd a good birational (canonical) model f : Z !T of T : X T !T over T such that (1) Gal(T=W) := hgi acts regularly on f : Z !T and (2) : X !W is birational to the quotient (f : Z !T)=hgi.
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 nite automorphism group G of the ber 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 hgi 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 alg 1 (X) = f1g 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 nancial support during his stay.

Notation and Convention
Throughout this paper, we work over the complex number eld 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 : 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 .
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 ber space on a normal projective variety V , we mean a surjective morphism : V !W to a normal projective variety W with connected bers.Note that is not equi-dimensional in general.By 1 (w) (w 2 W), we denote the scheme theoretic ber over w.We denote its reduction by 1 (w) red .This is in some sense a set theoretical ber.
Two ber spaces : V !W and 0 : V 0 !W 0 are said to be isomorphic if there are isomorphisms F : V !V 0 and f : W ! W 0 such that 0 F = f .For two morphisms : V !W and : T !W, we sometimes denote natural morphisms V W T !T and V W T !V by T : V T !T and V : V T (= T V ) ! V respectively.
The primitive n th root of unity exp(2 i=n) is denoted by n .We denote the cyclic group of order n by Z n .
The elliptic curve with period 2 H is written as E .
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] := fx 2 V j 9g 2 G f1g; g(x) = xg; while, H G := fv 2 H j 8g 2 G; g (v) = vg for any cohomology group H of V .
Similarly, for an automorphism g of a variety V , we set V g := fx 2 V j g(x) = xg: An equivariant action of a nite group G on a bration : V !W induces a new bration (mod G) : X=G !W=G.We sometimes abbreviate this bration by ( : V !W)=G.
We say that G acts on : 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 For the automorphism group Aut (V ) of a variety V and a subset B in V , we often consider the subgroup fg 2 Aut (V ) j g(B) = Bg: We denote this group by Aut (X; B).For example, if A is an Abelian variety with origin 0, then Aut (A; f0g) is nothing but the so called Lie automorphism group of A.

x1. 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 ber space structure a : V !A over a smooth elliptic curve A, (2) V has a ber 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 2 T, and that f 1 (t) itself is smooth except at most nitely many points t 2 T.
Example (1.2).Let S be a normal projective surface with only rational singularities and E a smooth elliptic curve.Assume that a nite group of translations G of E acts faithfully on S in such a way that S G] is nite 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 ber of a.
Then, there exist an elliptic curve E and a nite subgroup G E, that is, a nite group of translations of E (and then is isomorphic to either Z m or Z n Z m with (njm)) such that (1) there is an injective homomorphism : G ! Aut (S), (2) V = (E S)=G under the (free) action of G on E S de ned by G 3 g : E S 3 (u; v) 7 !(u + g; (g)v) 2 E S; Documenta Mathematica 1 (1996) 417{447 (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= (G) respectively.
As a result, S can be replaced by any ber of a.We set G S := (G)(' 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 di erences in mind: (1) f may not be at 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 := ft 2 Tj either f 1 (t) is not reduced or T is singular at tg, and denote C t := f 1 (t)(t 2 T) and S x := a 1 (x) (x 2 A).By our assumption, B is a nite set.
Let us x a general point 0 2 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 2 T B (because T B is smooth and fj f 1 (T B) is a smooth morphism over T B.) Claim (1.4). a t := aj Ct : C t !A is surjective for each t 2 T B. In particular, a t is an isogeny of elliptic curves of degree n := (C t S) for each t 2 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 2 T B. Then, a(C t 0) must be a point for every t 0 2 T B because f is at over T B. Thus, a induces a morphism a : T B ! A. This gives a rational map a : T !A with a = a f.Let T 0 !T be a resolution of both singularities of T and indeterminacy of a. Since T has only rational singularities, we have h 1 (O T 0) = h 1 (O T ) = 0. Thus, a (T 0 ) is a point.Hence a is a morphism and a(T) is a point.Then, a(V ) would be a point because a = 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 ' Pic 0 (A) !Pic 0 (C t ) given by a t and the natural action of Pic 0 (C t ) on C t .More concretely, this action is written as A 3 x : C t 3 P 7 !P + x 1 + ::: + x n 0 1 ::: 0 n 2 C t ; where fx 1 ; :::; x n g := a 1 t (x) = C t \ S x and f0 1 ; :::; 0 n g := a 1 t (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 t2T 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 : A V !V .By construction, stabilizes each ber of f.Set := j 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 2 T B and x 2 A. So, once we de ne a new action of A on A by A 3 x : A ! A; y 7 !y + nx; that is, by n (translation), then A induces an equivariant action on the bration 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 de nition, we have x(S)(= x(S 0 )) = S nx (x 2 A).In particular, : 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 : (A) n !Aut (S).
The following claim ( BM]) is now proved formally.
Claim (1.5).Let (x; v) and (x 0 ; v 0 ) be points on A S. Then, the following (1) and (2) are equivalent to one another.
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 nite and S=(A) n is normal, this is so over the whole T. Let : S=(A) n !T be the induced morphism.Since both f and p 2 have only one dimensional connected bers, must be a nite birational morphism.Thus, by the Zariski main theorem, is isomorphism 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 rst that S is a K3 surface with only Du Val singularities.Let S 0 !S be the minimal resolution of S. Then G gives a commutative Gorenstein action on S 0 .Now the result follows from the Nikulin's classi cation ( 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 (njm).
Finally, assuming that S is a smooth Abelian surface, we show that G satis es the condition in (1.3)(7).Since G is a nite 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 nite set.Choose an appropriate origin 0 of S and identify S with its translation automorphism group.Set Aut 0 (S) := f 2 Aut (S)j !S = !S g, Aut 0 (S; f0g) := f 2 Aut 0 (S)j (0) = 0g, where !S is a non-zero global regular two form on S.Then, Aut 0 (S) = S o Aut 0 (S; f0g) and G Aut 0 (S).Identifying Aut 0 (S; f0g) = Aut 0 (S)=S, we denote the natural projection by p : Aut 0 (S) !Aut 0 (S; f0g).If we choose global coordinates around 0, we can explicitly write down the action of g 2 Aut 0 (S) in its a ne form g(x) = M g x + t g ; M g 2 SL(2; C ); t g 2 S: Documenta Mathematica 1 (1996) 417{447 Then p is nothing but the map taking the matrix part, that is, g 7 !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 nite Gorenstein automorphism g 2 Aut 0 (S) has a xed point if and only if g is not a translation.
On the other hand, Katsura's classi cation ( Kt]) of possible nite subgroups of Aut 0 (S; f0g) 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 2 G and 0 2 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.
(3) G is isomorphic to H hgi.
(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 2 H.By de nition, pj hgi : hgi !p(G) is surjective group homomorphism.Let h be an element of Ker(pj hgi ).Then, h(0) = 0 and h 2 H. Combining this with (1), we get h = id.Thus, pj hgi is isomorphism.This shows that G is a semi-direct product of H and hgi.Since G is commutative, this must be the direct product.The last statement now directly follows from the relation gh = hg (h 2 H).q.e.d. of (1.6).
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 ( 4 ; 1 4 ).Then, again using the Lefschetz xed 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 = f0g.q.e.d. of (1.7).Now Claims (1.6), (1.7) and the fact that G is a nite Abelian group of the form Z n or Z n Z m (njm) together with the fundamental theorem on nite 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 njm).Since S=G ' (S=Z n )=Z m and since (S=Z n ) is again an Abelian surface, the assertion follows from the rst case.Now we are done.Q.E.D. of (1.3).
x2. Good model over the global canonical covering Let us x a bered Calabi-Yau threefold : X !W of type II 0 K. De ne I := minfn 2 NjO W (nK W ) ' O W g and denote the global canonical cover of W by : 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 : T !W is a cyclic Galois covering of order I(W) and is etale over W 0 .Moreover, there is a generator g of the Galois group Gal(T=W) such that g !T = I !T , where !T is a nowhere vanishing regular two form on T, that is, a generator of H 0 (O T (K T )).
We x these notation till the end of Section 4.
Set T : X T := X W T !T.Then, the Galois group Gal(T=W) = hgi acts on this bration by g : (x; y) 7 !(x; g(y)) and induces an isomorphism ( : X !W) ' ( T = : X T !T)=hgi: 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 de nition 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 hgi on the bration f : Z !T such that W = T=hgi and ( : X !W) is birational to (f : Z !T)=hgi over W = T=hgi.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 nd a birational model f : Z !T of T : X T !T with property (1) in (2.1).Then, we check that f : Z !T also satis es (2) and (3).
In order to carry out this plan, we start by observing some general lemmas.
Proposition (2.2).Let ' : V !S be a surjective morphism from a normal projective Q factorial threefold V to a normal projective surface S. Let fE i g i2I be the set of all two-dimensional irreducible components in bers of '.Set E = i2I E i .

Assume that
(1) V is not covered by rational curves, Documenta Mathematica 1 (1996) 417{447 (2) K V = i2I a i E i (as a Weil divisor on V) for some a i 2 Z 0 , (3) (V; E) is klt for some positive small rational number .Then, there are a normal projective threefold V (n) and a surjective morphism ' (n) : (5) ' (n) : V (n) !S is birational to ' : V !S over S and is isomorphic except over a nite set '(E), and ( 6) ' (n) : V (n) !S is an equi-dimensional elliptic bration.
Proof.First, we remark Claim (2.3).K V + E 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 'j H : H ! S is surjective.Since (K V + E)j H i2I (a i + )E i j H and since E i j H are contracted by ' H , we get ((K V + E) 2 H) = ((K V + E)j H ) 2 = ( i2I (a i + )E i j 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 + E. If E 6 = 0, then K V + E is not nef by (2.3).Thus, there is a log extremal ray R such that (K V + E) 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 + E) C = (a i + )(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 de ned over S. Let : W ! S be the induced morphism.
If cont R is a small contraction, then we apply a log ip for cont R to get cont + R : V + !W.
Putting V (0) := V , ' (0) := ' and E (0) := E and repeating this process, say, for n( 0) times, we nally get ' (n) : V (n) !S and the strict transform E (n) of E to V (n) such that (1) ' (n) : V (n) !S and E (n) satisfy all the assumptions in (2.2), and This is due to the termination of log ips for threefolds shown by Kw4].
Since the codimension of V (n) (V 0 ) in V (n) is at least two by E (n) = 0 and since 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 bers of certain elliptic threefolds.
Lemma (2.4).Let ' : V !S be a ber 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, ' 1 (s) is a smooth elliptic curve if s 2 S Sing (S).In particular, ' 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 2 ! 2 be a relatively minimal projective elliptic bration over a two-dimensional (small) polydisk 2 := f(x; y) 2 C 2 j jxj < ; jyj < g: Assume that f has (singular) bers of type I a (a 0) over (x = 0) f(0; 0)g and those of type I b (b 0) over (y = 0) f(0; 0)g.(Here we employed Kodaira's notation.) Then f 1 ((0; 0)) is a (singular) ber of type I a+b .In particular, if f is smooth over 2 f(0; 0)g, then f 1 ((0; 0)) is a smooth elliptic curve and f is a smooth morphism over the whole 2 .First, we show Claim (2.6).' : V !S is an elliptic bration and has singular bers only over a nite set of points of S. Proof of (2.6).Note that a general ber of ' is a smooth elliptic curve.Let H be a general very ample divisor on S. Set V H := ' 1 (H).Since V has only isolated singularities and since H is general, we may assume that H\(Sing (S) '(Sing (V ))) = and both H and V H are smooth.Let 'j VH : V H ! H be the induced elliptic bration.Using the adjunction formula, we calculate K H Hj H and K VH = (K V + V H )j VH ' (K H ). Comparing this with the canonical bundle formula of an elliptic surface (for example see BPV]), we nd that 'j VH is a smooth morphism.This implies the result.q.e.d of (2.6).
Then R i !V 0 = 0 for i > 0: Moreover, !V 0 = !V because V has only canonical singularities.Thus, from the Leray spectral sequence On the other hand, the edge sequence of another Leray spectral sequence Note that R 2 ( !V ) = 0 and that R 1 ( !V 0 ) is a torsion sheaf, because : S 0 !S is a birational morphism between surfaces.
On the other hand, since V 0 is smooth, R 1 ( ) ! V 0 is a torsion free sheaf by Ko1].Then, chasing the above exact sequence, we get R 1 ( !V 0) = 0 and R 1 ( Since V 0 and S 0 are smooth, Koll ar's original result implies R 1 !V 0 ' !S 0 : Moreover, since S has only canonical singularities, it follows that !S 0 ' !S : Combining these, we get R 1 ( This completes the proof of the rst part. We show the second part.Since !V ' O V and !S ' O S , the rst part of (2.8) gives R 1 ' O Z ' O S : Documenta Mathematica 1 (1996) 417{447 Substituting this into the edge sequence of the Leray spectral sequence we get an exact sequence Considering the pullback of the regular two forms by and using Hodge theory, we calculate h 2 (O V 0) = h 2;0 (V 0 ) h 2;0 (S 0 ) = 1: On the other hand, using the fact that V has only rational singularities and the Serre duality, we see that 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 : X !W and : T !W to W 0 by 0 : X 0 := 1 (W 0 ) ! W 0 and 0 : T 0 := 1 (W 0 ) ! W 0 : Note that 0 is a smooth morphism by (2.4) and 0 is an etale morphism by de nition.We consider the Cartesian product de ned by and X T := X W T X !X T ??y ? ? yT !W and its restriction over W 0 (X T ) 0 := X 0 W0 T 0 !X 0 ??y ??y T 0 !W 0 Since W 0 is smooth and since each morphism in the second diagram is smooth or etale, it follows that Sing (X) In what follows, we apply several birational modi cations on the rst diagram keeping everything in the second diagram invariant.Since all singularities in the rst diagram are supported over W W 0 , we nd a commutative diagram (1) X 0 and X 0 T are smooth, (2) X : X 0 !X is a birational modi cation only over W W 0 , and that (3) XT : X 0 T !X T is a birational modi cation only over T T 0 .Let fE i g i2I be the set of all the two dimensional irreducible components of bers of 0 T := T XT : X 0 T !X T !T. Set E := i2I E i .By construction, E is supported only over T T 0 .
(1) X 0 T is not covered by rational curves.
(2) K X 0 T = i2I a i E i for some non-negative integers a i . ( 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 0 = c j E 0 j ; where c j are some positive integers and E 0 j are some irreducible divisors supported in 1 X (X X 0 ).
On the other hand, since 0 XT : X 0 T !X 0 rami es only at E, the rami cation formula gives for some non-negative integers b i .Since ( 0XT ) E 0 i are e ective divisors supported in E, substituting the rst equality into the second, we get the result.q.e.d. of (2.10).Now we can apply (2.2) for 0 T : X 0 T !T to get a ber 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 T : X T !T over T and is isomorphic over T 0 , (3) f : Z !T is an equi-dimensional elliptic bration.
Recall that T is a K3 surface with only Du Val singularities, and that 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 2 T; (5) f 1 (t) itself is smooth if t is a smooth point of T (in particular, if t 2 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 ber space.By (2), the natural action of hgi on 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) hgi induces a regular action on f : Z !T and (9) (f : Z !T)=hgi is birational to : X !W and is isomorphic over W 0 = T 0 =hgi.Now these statements (1) -( 9) imply the Key Lemma.Q.E.D. of Key Lemma.
x3. 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 bered Calabi-Yau threefold : 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) ber of the Albanese morphism a : Z !A, (3) G is a nite commutative Gorenstein automorphism group of E S as is described in Theorem (1.3).
We want to lift the action of hgi 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 hgi stabilizes a 1 (0).
Proof.Since the Albanese morphism is an intrinsically and uniquely de ned object, hgi acts on the Albanese morphism a : Z ! A. This induces a bration a : Z=hgi !A=hgi: On the other hand, since X and Z=hgi are birational and since both of them have only rational singularities, it follows that h 1 (O Z=hgi ) = h 1 (O X ) = 0.This implies A=hgi = P 1 .Thus, A hgi 6 = .Since A is an elliptic curve, this is equivalent to A g 6 = .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 := gj S : S ! S. Since g acts on the ber space f : Z !T, hg S i and hgi give an equivariant action on q T := fj S : S !T. Note that q T is nothing but the quotient map S !T = S=G.Lemma (3.2). g S ! S = I !S , where !S is a nowhere vanishing regular two form on S, that is, a generator of H 0 (S; O S (K S )).
Proof.Let !T be a nowhere vanishing regular two form on T.Then, !S := q T !T is a nowhere vanishing regular two form on S. Thus, g S ! S = g S q T !T = q T g !T = q T I !T = I !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 p2 !S q 0 ??y ? ?y qT Z !f T where q and q 0 are natural quotient maps.
Proof.Let us consider the ber product Z T S p2 !S p1 ??y ? ? yqT Z !f T De ne the action of hg 0 i on Z T S by g 0 : Z T S 3 (u; v) 7 !(g(u); g S (v)) 2 Z T S: Then, g 0 , hg S i and hgi give an equivariant action on this ber product.
By the de nition of ber product, there is a surjective morphism : 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).: E S !Z T S is the normalization of Z T S.
Since normalization is an intrinsically and uniquely de ned notion, the action hg 0 i on Z T S lifts to the action hg E S i on E S equivariantly with respect to : E S !Z T S. This gives a desired action on E S. Q.E.D. of (3.3).
Proof.Since g S is a restriction of g, it follows that ord (g S ) ord (g).On the other hand, since : S !T is surjective and since g S and g induce an equivariant action on , 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 0 ) = ord (g).Q.E.D. of (3.5).
Combining these three we get ] G = ] GS : This implies that the surjective group homomorphism : G ! GS is an isomorphism.Combining this together with (2), we get G = G o hg E S i.This completes the proof.Q.E.D. of (3.6).
From now on, we denote the equivariant actions G and GS on the ber space p 2 : E S ! S simply by G.We also set g := g E S for consistency of notation.
If no confusion seems to arise, we also identify g S and G S with g and G (under the isomorphism ).
The following corollary is an immediate consequence of Lemma (3.6).
(f : Z !T)=hgi = (p 2 : E S !S)= G: Thus, the ber space : X !W is birational to (p 2 : E S !S)= G over W = S= G 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 g on E S is written as follows: g : E S 3 (x; y) 7 !( 1 I x; g S (y)) 2 E S for an appropriate origin 0 of E.
Proof.Since hgi acts on p 2 : E S !S, there is a homomorphic map c : S !Aut (E) = E o Aut (E; f0g) de ned by s 7 !(p 1 ((x; s)) 7 !p 1 (g(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, g = (g E ; g S ) for some g E 2 Aut (E).Since X is isomorphic to (E S)= G over W 0 and since This means G is a Gorenstein automorphism of E S. In particular, so is g.Combining this with g S ! S = I !S , we get g E !E = 1 I !E .In particular, E gE 6 = .Now, choosing the origin 0 of E in E gE , we get the desired expressions of g.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 : S 0 !S is a minimal resolution of S, then the action hgi on E S lifts to E S 0 in an equivariant way and : X !W is birational to (p 2 (id: ) : E S 0 !S)= G over W = S= G and is isomorphic over W 0 .
x4. 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 : 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 GS , G S and g S by G, G and g respectively.Under this notation, we have T = S=G, W = T=hgi = S= G and I = ord (g) = ord (g).As before, we denote by q T : S !T the natural quotient morphism.This has an equivariant action of hgi and hgi.Recall also that all the possibilities of G are listed up in (1.3)(4).
The next Lemma is shown by O2].
By virtue of this Lemma, the next two Claims will give a contradiction.
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 nite morphism between normal projective surfaces with K S1 0 and K S2 0. Then q rami es only at nitely many points.
Corollary (4.5).The quotient map S !W(= S= G) rami es only at nitely many points.In particular, S GS is a nite set.Corollary (4.6).Let h be a non-Gorenstein involution in G.Then, S h = .In particular, if I = 2k is even, then S gk = and S g = .
Proof.Assuming S h 6 = , we take a point P in S h .Since h is an involution with h !S = !S , it follows that h = diag ( 1; 1) under appropriate coordinates (x; y) of S around P. But then h would have a xed curve (x = 0), contradiction.q.e.d. of (4.6).
Proof.Since I is the least common multiple of the local canonical indices of W, the rst part of the assertion is obvious.Assume that I is either 2 or 4. The rst half part shows T g 6 = .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 xed curve.On the other hand, Lemma (4.4) shows T !W(= T=hgi) has no rami cation divisor, contradiction.q.e.d. of (4.7).
Assume the contrary that I = 2k for some integer k.We set h := gk .Then h is a non-Gorenstein involution on S. Dividing into the following ve 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)), hgi acts on q 1 T (B).Since ]q 1 T (B) is either 9 or 27, h has a xed points.
Case 2. Consider the unique singular point Q of type A 5 on T ((1.3)(4)).Then, q 1 T (Q) consists of one point, say, P. Since g(Q) = Q, it follows that g(P ) = P.But this contradicts (4.6).
Case 3. By (4.7), T g k 6 = .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 2 T g k .Then Q is a singular point of type A 1 and then q 1 T (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 6 = and T g k Sing (T ).Let Q 2 T g k .Then, Q is a singular point of type A 1 and q 1 T (Q) is written as fP; r(P)g 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 rst equality contradicts (4.6).Consider the second case.Set h 0 := r h.Then h !S = !S .Since the translation subgroup of G is just hri and since h 1 r h is a translation in G (because G is a normal subgroup of G), it follows that h 1 r h 2 hri and then hr; hi = hri hhi ' (Z 2 ) 2 .Thus h 0 is a non-Gorenstein involution with h 0 (P ) = P.But this contradicts (4.6).
Case 5. We treat the following three cases separately: Case 5a.3jI, Case 5b.I = 4, and Case 5c.I = 2. Case 5a.In this case, I = 6m for some integer m.Set j := gm .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 xed point among these points.Let P be such a xed point.Then, j 2 (P ) = P. Since (j 2 ) ! S = 3 !S and j 2 has at most nite xed points by (4.5), an easy coordinate calculation shows that j 2 = diag ( 2 3 ; 2 3 ) under appropriate global coordinates (x; y) around P. Thus, the eigen value of the matrix part of j is in f 3 ; 3 g.Thus, j has a xed point on S, say Q.Since h = j 3 , Q is also a xed 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 .
X is also smooth and 1 (X) ' 1 (Y ) ( Ko2]).Thus X has a non-trivial nite etale covering, because so does Y .But this contradicts our assumption alg 1 (X) = f1g.Therefore, we get a contradiction even in the case G ' Z 4 .
We consider the remaining case G = hti hui ' Z 2 Z 4 .Reducing to the previous case G ' Z 4 , we nd a contradiction.
Since the translation group of G is just hti and since G is a normal subgroup of G, the same argument as before shows hti is a normal subgroup of G. Thus G=hti ' hu 1 i o h g1 i, where u 1 := u(mod hti) and g1 := g(mod hti).Observe that u 1 is of order four and g1 is of order two.
On the other hand, since hti acts on p 2 : E S !S, we get a new ber space p 2 : (E S)=hti !S=hti; on which hu 1 i hg 1 i gives an equivariant action.Since hti is a translation group on both E S and S, it follows that (E S)=hti is an Abelian threefold and S=hti is an Abelian surface.Set S 1 := S=hti and V := (E S)=hti.Then, T = S 1 =hu 1 i and W = S 1 =hu 1 ; g1 i.
Observe that g 1 !S1 = !S1 , u 1 !S1 = !S1 and that u 1 acts on each ber over S u1 1 (6 = ) 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 p 2 : (E S)=hti !S=hti and S 1 !T !W to get a contradiction.This nishes 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 ve 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 xed point, say Q, in this set.Then, g acts on q 1 T (Q).

Since q 1
T (Q) consists of one or two points, g has a xed point in q 1 T (Q).Denote this point by 0. Since g !S = 3 !S , g(0) = 0 and since g has only nitely many xed points, we can apply CC, also O2] to get S ' E 2 3 and g = 2 3 , the scalar multiplication by 2 3 .On the other hand, the stabilizer of 0 in G is a cyclic group of order 4. We denote this group by hui.Then u = diag ( 4 ; 1 4 ) under appropriate global coordinates around 0. Set H := hu; gi.Then, H Aut (S; f0g).Moreover H is a cyclic group of order 12, because g = 2 3 so that u g = g u.In particular H 3 1.But this is impossible by Fujiki's classi cation ( Fu,Table 6]).
Case 2. Just by the same argument as in case 1, we see that g has a xed point 0 (over some singular point of type A 1 of T) and then S = E 2 3 and g = 2 3 .Set Stab f0g (G) = hui.This is a cyclic group of order two and u = diag ( 1; 1) under Documenta Mathematica 1 (1996) 417{447 appropriate global coordinates around 0. Thus u g = g u.Since G gives an equivariant action on p 2 : E S !S, g and u act on the ber E := p 1 2 (0).Since g is a Gorenstein automorphism of E S, the matrix part of g on E is 2 3 so that g acts on E by g : E 3 x 7 ! 2 3 x 2 E; if we x an origin 0 E of E in E g(6 = ).On the other hand, by (1.3), the action of u on E is written as u : E 3 x 7 !x + P 2 E; where P 2 (E) 2 f0g.Since u g = g u in G, we calculate g(x) + g(P ) = g u(x) = u g(x) = g(x) + P: Thus, P 2 E g = E3 (E) 3 .But this is impossible because (E) 3 \ ((E) 2 f0g) = .
Case 3. Let Q be the unique singular point of type A 5 on T.Then, q 1 T (Q) consists of one point, say, 0. Since g(Q) = Q, it follows that g(0) = 0. Thus, just by the same argument as before, we get g = 2 3 .Set Stab f0g (G) = hui.This is a cyclic group of order 6 and u = diag ( 6 ; 1 6 ) under an appropriate global coordinates (x; y) around 0. it follows that g u 2 = diag (1; 3 ).Then g u 2 has a xed curve (y = 0), contradiction.
Case 4. Set G = hui.Since G = huiohgi is of order 9, it follows that G = hui hgi.
Let Q be a point in T g .Then ]q 1 T (Q) is either one or three.If q 1 T (Q) = fPg, a one point set, then g(P ) = P.If q 1 T (Q) = fP 1 ; P 2 ; P 3 g, then, g(P 1 ) = P j for some j = 1; 2; or 3. Since hui acts on fP 1 ; P 2 ; P 3 g transitively, we nd that u i (P 1 ) = P j for some i.Set h := u i g.Then, h(P 1 ) = P 1 .Note that h is of order 3 and satis es h !S = 3 !S and G = hui hhi.In addition, h and g give an equivariant action on q T : S !T. Thus, we may replace g by h in the second case.Then g(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 g has only isolated xed points ((4.5)), the same argument as before shows that S = E 2 3 and g = 2 3 .This implies (S) g \ (S) u = : (In fact, otherwise, choosing a point P in (S) g \ (S) u , we nd appropriate coordinates (x; y) around P such that u = diag ( 3 ; 1 3 ).Then, g u = diag (1; 3 ) has a xed curve (y = 0), contradiction.)Since G is a Gorenstein automorphism of E S and gives an equivariant action on p 2 : E S !S, g induces an automorphism on the ber E := p 1 2 (0 S ) whose matrix part is 2 3 .Thus E = E 3 and then E S = E 3 3 .Moreover, choosing an origin 0 E of E in E g, we get g = 2 3 on E. Now, taking 0 := (0 S ; 0 E ) as an origin of E S = E 3 3 , we have g = 2 3 on E 3 3 .Let us consider the quotient threefolds (E 3 ) 3 =hgi and its crepant resolution : Y !(E 3 ) 3 =hgi.Note that hui ' G=hgi acts on (E 3 ) 3 =hgi.
Note also that is unique.(In fact, one of such is given by replacing each of 27 singular points of type 1=3(1; 1; 1) of (E 3 ) 3 =hgi by P 2 and then has no opping curves in the exceptional divisor.)Thus, hui induces a regular action on Y .
Claim.hui acts freely on Y .
Proof of Claim.Since ord (u) = 3, it is su cient to show that Y u = .Assume the contrary that P 2 Y u .Put Q := (P ).Then u(Q) = Q.Denote the natural quotient map E 3
: Y (E; S; G) !(E S 0 )= G be a crepant resolution (whose existence is now guaranteed by Roan Ro]) and p : Y (E; S; G) !S= G the natural projection given by the composite of : Y (E; S; G) !(E S 0 )= G, p 2 : (E S 0 )= G ! S 0 = G, and = G : S 0 = G ! S= G.Then, (1) any composite of op f : Y (E; S; G) !Y 0 along curves in p 1 (Sing (S= G)) gives a bered Calabi-Yau threefold p f 1 : Y 0 !S= G of type II 0 K provided that alg 1 (Y ) = f1g.In this case S=G gives the global canonical cover of the base space S= G.
and then f = p 2 under the identi cation 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 injective to complete the rst half part of (1.3).But this is done as follows.Let G = (A) n =Ker .Then, (A S)=(A) n = (A=(Ker ) S)=G and A=(A) n = (A=Ker )=G, in which G acts on translation group of an elliptic curve A=Ker .Now replacing A, (A) n and by E = A=(Ker ), G, and the injection( 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).
then, (4) any ber S of a is either a K3 surface with only Du Val singularities or a smooth Abelian surface, (5) G S is a nite Gorenstein automorphism of S, (6) if S is a K3 surface with only Du Val singularities, then S GS] is a non-empty nite set and G S (' G) is isomorphic to either one of the following groups; f1g, 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 GS] is a non-empty nite set and G S (' G) is isomorphic to either one of the following groups;