c 2011 International Press Adv. Theor. Math. Phys. 15 (2011) 1745 1787 A rigid Calabi Yau three-fold Sara Angela Filippini 1 and Alice Garbagnati 2 1 Dipartimento di Scienze e Alta Tecnologia, Università dell Insubria, via Valleggio 11, I-22100 Como, Italy saraangela.filippini@uninsubria.it 2 Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy alice.garbagnati@unimi.it URL: http://sites.google.com/site/alicegarbagnati/home Abstract The aim of this paper is to analyze some geometric properties of the rigid Calabi Yau three-fold Z obtained by a quotient of E 3, where E is a specific elliptic curve. We describe the cohomology of Z and give a simple formula for the trilinear form on Pic(Z). We describe some projective models of Z and relate these to its generalized mirror. A smoothing of a singular model is a Calabi Yau three-fold with small Hodge numbers which was not known before. 1 Introduction One of the most exciting mathematical implications of string theory is mirror symmetry, which finds its origin in the papers [13, 28]. A phenomenological verification of the conjecture that Calabi Yau manifolds should appear in pairs was given in [11], and the first nontrivial examples of mirror pairs appeared in [20]. In [20], it was also discovered that mirror symmetry can be used to compute the instanton corrections to the Yukawa couplings (the e-print archive: http://lanl.arxiv.org/abs/1102.1854v2
1746 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI first explicit computations were carried out in [7]), which mathematically corresponds to determine the number of rational curves of given degree embedded in the Calabi Yau manifold. This led to the notion of Gromov Witten invariants and more generally to the one of Gopakumar Vafa invariants [21, 22]. Curiously, in the seminal paper [10], where the relevance of Calabi Yau manifolds in string theory was established, among the few explicit known examples of Calabi Yau manifolds there was the manifold Z, realized as the desingularization of the quotient E 3 /ϕ 3,withϕ 3 = ϕ ϕ ϕ and ϕ the generator of Z 3 which acts on the elliptic curve E. AsZ is a rigid manifold, it cannot admit a Calabi Yau three-fold as mirror partner. This created a puzzle in the general framework of mirror symmetry. However, physically, mirror symmetry arises as a complete equivalence between conformal field theories. In this respect, it should not be surprising that in certain exceptional cases the equivalence could involve more general spaces. Indeed, in [8] it was proposed that the mirror of Z should be a cubic in P 8 quotiented by a suitable finite group. By using the usual mirror methods, the authors were able to reproduce the right Yukawa couplings of Z. The mirror symmetry generalized to rigid Calabi Yau manifolds has been considered also in [32], where the mirror is presented as (embedded in) a higher dimensional Fano variety having the mirror diamond as an embedded sub-diamond and in [1], where it is related to toric geometry. However, a definitive understanding of the question is still open. In this paper, as a preparation to further work on generalized mirror symmetry, we present a very detailed study of the rigid manifold Z. Section 2 is devoted to an explicit description of the cohomology of Z. The Hodge diamond of this three-fold is very well known, but here we identify a set of generators of Pic(Z) made up of surfaces and a set of generators of H 4 (Z, Q) made up of curves. Our goal is to describe the trilinear intersection form on the generators of Pic(Z) relating it with the trilinear intersection form on Pic(E 3 ). Indeed the generators of Pic(Z) are of two types: the ones coming from the generators of Pic(E 3 ) and the ones coming from the resolution of the singularities of E 3 /ϕ 3. The intersection between two divisors of different type is zero, and the trilinear form on the divisors coming from E 3 is, up to a constant, the trilinear form Pic(E 3 ). For this reason it is important to give a good description of the trilinear form on Pic(E 3 ): in (4) the cubic self-intersection form is given for each divisor in Pic(E 3 ). In Section 3, Theorem 3.1, we prove that it can be given in terms of the determinant of a matrix in Mat 3,3 (Q[ζ]), ζ 3 = 1. The locus, where the determinant vanishes is a singular cubic in P 8. We recall that the Yukawa coupling on H 1,1 is strongly related with the cup product on H 1,1 and thus with the intersection form on Pic(E 3 ). Moreover the locus
A RIGID CALABI YAU THREE-FOLD 1747 where the Yukawa coupling vanishes, corresponds to fermion mass generation points. In the second part of the paper, in Section 4, we describe some projective models of Z. Here we will limit ourselves to make some basic observation on mirror symmetry, deferring a systematic analysis to a future paper. We give a detailed description of the images of three maps (called m 0, m 1, m 2 ) defined from Z to projective spaces and we relate these to earlier work. None of the maps m i, i =0, 1, 2, gives an embedding. For this reason, we also prove that a certain divisor on Z is very ample (cf. Proposition 4.12), i.e., it defines a map m such that m(z) Z. The maps m i, i =0, 1, 2, allow us to describe some peculiarities of Z. The map m 0 is 3 : 1 and it gives a model of another rigid Calabi Yau three-fold Y, birational to Z/Z 3. Moreover m 0 (Z) is contained in the Fermat cubic hypersurface in P 8 and this could give a geometrical interpretation of the conjectures on the generalized mirrors of the rigid Calabi Yau three-folds Z and Y presented in [8, 26]. The map m 1 contracts 27 rational curves on Z, and gives a model of Z embedded in P 11. This model will be used in Section 6 to obtain other Calabi Yau three-folds. The map m 2 was already defined by Kimura [25] to show that there exists a birational map between Z and a particular complete intersection of two cubics in P 5, called V 3,3. Several models of the variety V 3,3 were analyzed previously (cf. [24, 37, 29]). We already observed that the Calabi Yau three-fold Z is very well known, but it can be used to construct several other Calabi Yau three-folds, which are not always rigid. In Section 5, we recall constructions which produce Calabi Yau three-folds starting from a given one. In Section 6, we apply one of these constructions (described in [15]) to Z and we obtain non rigid Calabi Yau three-folds. The idea is to contract some curves on Z and then to consider the smoothing of the singular three-fold obtained. One of the Calabi Yau three-folds constructed in this section does not appear in the list of known Calabi Yau three-fold with small Hodge numbers given in [4] and is a new Calabi Yau variety. 2 The three-folds E 3, Ẽ3, Z and their cohomology In this note, we will analyze the properties of the very well known Calabi Yau three-fold Z introduced independently in [3, Example 2] and [34]. In order to describe the trilinear form on Pic(Z) (cf. (8)), which is strongly related to the Yukawa coupling, we will compute the cohomology of Z (Section 2.3) and of the varieties involved in its construction (Sections 2.1 and 2.2).
1748 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI To fix the notation, we recall some definitions and the construction of Z. Definition 2.1. A smooth compact complex variety, X, is called a Calabi Yau variety if it is a Kähler variety, it has a trivial canonical bundle and h i,0 (X) = 0 for 0 <i<dim(x). To give the Hodge diamond of a Calabi Yau three-fold X one has to find h 1,1 (X) andh 2,1 (X). We recall that h 2,1 (X) is the dimension of the family of deformations of X (which are indeed unobstructed by the Tian Todorov theorem), so X has h 2,1 (X) complex moduli. Let E be the Fermat elliptic curve x 3 + y 3 + z 3 = 0, i.e., the elliptic curve admitting a complex multiplication of order 3. We will denote by ϕ : E E the automorphism of E given by (x, y, z) (x, y, ζz), where ζ isaprimitive third root of unity. Let E 3 be the Abelian three-fold E E E and ϕ 3 be the automorphism ϕ ϕ ϕ acting as ϕ on each factor of E 3. The automorphism ϕ has three fixed points on E, which are called p i := ( 1 :ζ i : 0), i =1, 2, 3. Hence ϕ 3 fixes 27 points on E 3, p i,j,k := (p i ; p j ; p k ), i, j, k = 1, 2, 3. Let α : E 3 E 3 /ϕ 3 be the quotient map. The three-fold E 3 /ϕ 3 is singular and its singular locus consists of the 27 points α(p i,j,k ). Let β : Ẽ 3 E 3 be the blow up of E 3 in the 27 points fixed by ϕ 3. The exceptional locus consists of 27 disjoint copies of P 2, and the exceptional divisor over the point p i,j,k will be denoted by B i,j,k. The automorphism ϕ 3 of E 3 induces the automorphism ϕ 3 on Ẽ3. Let Z := Ẽ3 / ϕ 3 and π : Ẽ3 Ẽ3 / ϕ 3 be the quotient map. The following diagram commutes: ϕ 3 E 3 β Ẽ3 ϕ 3 α π E 3 /ϕ 3 γ Z where γ is the contraction of the divisors π( B i,j,k ) to the singular points α(p i,j,k )ofe 3 /ϕ 3. The three-fold Z is smooth (indeed the fixed locus of ϕ 3 on Ẽ3 is of codimension 1) and is a Calabi Yau three-fold. 2.1 The cohomology of E 3 The three-fold E 3 is an Abelian variety. Its canonical bundle is trivial and H p,q (E 3 )= (H a 1,b 1 (E) H a 2,b 2 (E) H a 3,b 3 (E)). a 1 +a 2 +a 3 =p, b 1 +b 2 +b 3 =q
A RIGID CALABI YAU THREE-FOLD 1749 Hence, the Hodge diamond of E 3 is 1 3 3 3 9 3 1 9 9 1 Let z i be the complex local coordinate of the ith copy of E in E 3. Then H 1,0 (E 3 )= dz 1,dz 2,dz 3, H 2,0 = dz 1 dz 2,dz 1 dz 3,dz 2 dz 3 and H 3,0 (E 3 )= dz 1 dz 2 dz 3. The Picard group is generated by: Three classes Φ i, i =1, 2, 3, which are the classes of the fiber of the projection ρ i : E 3 E on the i-th factor, e.g., Φ 1 = q E E for a general point q E; Three classes Δ i, i =1, 2, 3, which are the product of the ith factor of E 3 by the diagonal on the other two factors, e.g., Δ 1 = E Δ= {E q q q E}; Three classes Γ i, i =1, 2, 3, which are the product of the ith factor of E 3 by the graph on the other two factors, i.e., Γ 1 = E Γ={E q ϕ(q) q E}, Γ 2 = {ϕ(q) E q q E}, Γ 3 = {q ϕ(q) E q E}. By the definition of the divisor Φ i it is clear that Φ i = ρ i (q), where q is a general point on E. A similar description can be given for the divisors Δ i and Γ i. Indeed let ρ i : E 3 E, τ i : E 3 E and η i : E 3 E be the maps defined below, then: Φ i = ρ i (P ), ρ i :(q 1,q 2,q 3 ) q i, Δ i = τi (P ), τ i :(q 1,q 2,q 3 ) q j q k, {i, j, k} = {1, 2, 3}, Γ i = ηi (P ), η i :(q 1,q 2,q 3 ) (ϕ(q i+1 ) q i+2 ), {i, i +1,i+2} = {1, 2, 3}, i,i+1,i+2 Z 3. (1) We recall that E R 2 /Λ C/Λ, where Λ is the lattice generated by 1 and ζ. Let(x 2j 1,x 2j ) be the real coordinates of R 2 relative to the jth copy of E and let the isomorphism R 2 C be (x 2j 1,x 2j ) x 2j i + ζx 2j, j =1, 2, 3. Each divisor D on E 3 is a linear combination of surfaces on the three-fold and defines a two-form, c 1 (D), in H 2 (E 3, Z). The classes of the nine divisors defined above can be found by pulling back the class of a point in H 2 (E,Z). The form associated to a point q = y 1 + ζy 2 on E is dy 1 dy 2. We will denote with the same name both the divisor and the corresponding form.
1750 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI So: Φ 1 = ρ 1(dy 1 dy 2 )=dx 1 dx 2, Φ 2 = dx 3 dx 4, Φ 3 = dx 5 dx 6, Δ 1 = τ1 (dy 1 dy 2 )=d(x 3 x 5 ) d(x 4 x 6 ), Δ 2 = d(x 1 x 5 ) d(x 2 x 6 ), Δ 3 = d(x 1 x 3 ) d(x 2 x 4 ), Γ 1 = η1(dy 1 dy 2 )=d(x 5 + x 4 ) d(x 6 x 3 + x 4 ), Γ 2 = d(x 1 + x 6 ) d(x 2 x 5 + x 6 ), Γ 3 = d(x 3 + x 2 ) d(x 4 x 1 + x 2 ). In the last three lines we used: ϕ(q) =ζq, hence ϕ(x 2i 1 + ζx 2i )=ζx 2i 1 + ( ζ 1)x 2i = x 2i + ζ(x 2i 1 x 2i ). Let us now consider the space H 2,2 (E 3 ). We recall that H 2,2 (E 3 ) is the dual of H 1,1 (E 3 ) and in particular H 2,2 (E 3 ) H 4 (E 3, Z) is generated by nine four-forms, which are identified (via Poincaré duality) with nine onecycles. A Q-basis of H 2,2 (E 3 ) H 4 (E 3, Z) generated by classes of curves on E 3, which are the pull back of the class of a general point Q E E along certain maps E 3 E E, is: φ i = ρ j,k (Q), ρ j,k :(q 1,q 2,q 3 ) (q j,q k ), i =1, 2, 3, {i, j, k} = {1, 2, 3} δ i = τj,k (Q), τ j,k :(q 1,q 2,q 3 ) (q i,q j q k ), i =1, 2, 3, {i, j, k} = {1, 2, 3} γ i = ηj,k (Q) η j,k :(q 1,q 2,q 3 ) (q i,ϕ(q j ) q k ), i =1, 2, 3, j = i +1 Z 3,k = i +2 Z 3. One can directly check the following intersection products: φ i =Φ j Φ k, {i, j, k} = {1, 2, 3}, i =1, 2, 3,δ i =Φ i Δ i, i =1, 2, 3, γ i =Φ i Γ i, i =1, 2, 3. (2) As φ i =Φ j Φ k, its class in H 4 (E 3, Z) is the wedge product of the two-forms associated to Φ j and Φ k. The intersection between a divisor in Pic(E 3 )and acurveinh 2,2 (E 3 ) is the wedge product of a four-form and a two-form, hence it is an element in H 6 (E 3, Q) Q, where the isomorphism is given by dx 1 dx 2 dx 3 dx 4 dx 5 dx 6 1. From this one finds the intersection numbers between the divisors generating Pic(E 3 ) and the curves generating
A RIGID CALABI YAU THREE-FOLD 1751 H 2,2 (E 3 ): Φ 1 Φ 2 Φ 3 Δ 1 Δ 2 Δ 3 Γ 1 Γ 2 Γ 3 φ 1 1 0 0 0 1 1 0 1 1 φ 2 0 1 0 1 0 1 1 0 1 φ 3 0 0 1 1 1 0 1 1 0 δ 1 0 1 1 0 1 1 3 1 1 δ 2 1 0 1 1 0 1 1 3 1 δ 3 1 1 0 1 1 0 1 1 3 γ 1 0 1 1 3 1 1 0 1 1 γ 2 1 0 1 1 3 1 1 0 1 γ 3 1 1 0 1 1 3 1 1 0 (3) Now (2) and (3) together give the cubic self-intersection form on Pic(E 3 ): ( 3 ) 3 a i Φ i + b i Δ i + c i Γ i =6 a 1 a 2 a 3 + a i a j (b i + b j + c i + c j ) i=1 i<j + b i b j (a 1 + a 2 + a 3 +3(c 1 + c 2 + c 3 )) i<j + c i c j (a 1 + a 2 + a 3 +3(b 1 + b 2 + b 3 )) i<j +(a 1 + a 2 + a 3 )(b 1 + b 2 + b 3 )(c 1 + c 2 + c 3 )+2 3 i=1 a i b i c i i j a i b j c j. From the cubic self-intersection form, one deduces the trilinear form on Pic(E 3 ). (4) 2.2 The cohomology of Ẽ3 The three-fold Ẽ3 is obtained by blowing up the 27 points p i,j,k := (p i ; p j ; p k ), i, j, k =1, 2, 3, on E 3 and B i,j,k are the exceptional divisors of this blow up. Hence, there is an isomorphism Ẽ3 B i,j,k E 3 p i,j,k. The numbers h i,j with i or j equal to zero are birational invariants, hence h i,j (Ẽ3 )= h i,j (E 3 ), if i or j are zero.
1752 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI Let X be a projective manifold, S be a codimension r submanifold of X and X beablowupofx in S. Then: H k ( X,Z) =H k (X, Z) r 2 i=0 H k 2i 2 (S, Z) ([35, Théorème 7.31]). Applying this result to Ẽ3, the blow up of E 3 in 27 points, we obtain the Hodge diamond of Ẽ3 : 1 3 3 3 36 3 1 9 9 1 Blowing up the 27 points, we introduced 27 exceptional divisors B i,j,k, hence h 1,1 (Ẽ3 )=h 1,1 (E 3 ) + 27 and H 1,1 (Ẽ3 ) is generated by the 36 classes: F h := β (Φ h ), Dh := β (Δ h ), Gh := β (Γ h ), h =1, 2, 3, and by the classes [ B i,j,k ], i,j,k =1, 2, 3. The divisors F i, D i, G i are the classes of the strict transforms of Φ i,δ i,γ i, indeed the Φ i do not pass through the points p i,j,k and Δ i,γ i are cohomologically equivalent to classes, which do not pass through p i,j,k, for example Δ 1 is in the same class of {E q (q + q ) q E} in H 2 (E 3, Z) for any q E. The intersection form on H 1,1 (Ẽ3 )=Pic(Ẽ3 ) is induced by the one on E 3. More precisely: let L be a divisor in Pic(E 3 ), then β (L) B i,j,k = 0, indeed all the divisors in Pic(E 3 ) are linear combinations of Φ i,δ i and Γ i and all these divisors are equivalent to divisors, which do not pass through the points p ijk and thus their strict transforms do not intersect the exceptional divisors; similarly, for L 1, L 2, L 3 Pic(E 3 )wehaveβ (L 1 )β (L 2 )β (L 3 )= L 1 L 2 L 3 ; B i,j,k B h,l,m =0, if (i, j, k) (h, l, m), because they are exceptional divisors over distinct points; 3 B i,j,k = 1 (see (5)). Dually the space H 2,2 (Ẽ3 ) is generated by 36 classes. We give a Q-basis of H 2,2 (Ẽ3 ) H 4 (Ẽ3, Z) made up of curves. Nine of them are the pull-back via β of the classes of the curves generating H 2,2 (E 3 )( f i = β (φ i ), d i = β (δ i ), g i = β (γ i )) and the other 27 are the classes of the lines l i,j,k which generate the Picard group of the exceptional divisors B i,j,k.
A RIGID CALABI YAU THREE-FOLD 1753 By the adjunction formula, the canonical divisor of B i,j,k is ( = K B B i,j,k + KẼ3) B i,j,k i,j,k = B i,j,k + β (K E 3)+2 i,j,k B i,j,k B i,j,k =3 B i,j,k 2. Since B i,j,k P 2, K B i,j,k = 3 l i,j,k, and comparing the two expressions of K B i,j,k,we obtain l i,j,k = B i,j,k 2. Moreover 1=( l i,j,k ) 2 = B i,j,k B i,j,k B i,j,k B i,j,k =( B i,j,k B i,j,k ) B i,j,k = B i,j,k 3. (5) The intersection form between the curves generating H 2,2 (Ẽ3 ) and the divisors generating H 1,1 (Ẽ3 ) is induced by the one on E 3 : if c H 2,2 (E 3 ) and L H 1,1 (E 3 ) are chosen among the classes appearing in table (3), then: c L = β (c) β (L); β (c) B i,j,k = l i,j,k β (L) =0; l 3 i,j,k B i,j,k = B i,j,k = 1; l h,m,n B i,j,k =0if(h, m, n) (i, j, k). 2.3 The cohomology of Z The map ϕ 3 (induced by ϕ 3 ) fixes the divisors B i,j,k and is without fixed points on Ẽ3 B i,j,k=1,2,3 i,j,k. So the fixed locus Fix ϕ (Ẽ3 )= i,j,k=1,2,3 B i,j,k has codimension 1 and hence the three-fold Z, which is the quotient Ẽ3 / ϕ 3, is smooth. Moreover, H p,q (Z) =H p,q (Ẽ3) ϕ 3. We recall that H i,0 (Ẽ3 )=H i,0 (E 3 ) and that the action of ϕ 3 is (z 1,z 2,z 3 ) (ζz 1,ζz 2,ζz 3 ) (where z i are the local complex coordinates of the ith copy of E). Now it is clear that β (dz i )andβ (dz i dz j ), i j, i, j =1, 2, 3, are not invariant under the action of ϕ 3, but that β (dz 1 dz 2 dz 3 ) is invariant under ϕ 3. We conclude that H 1,0 (Z) =H 0,1 (Z) =H 2,0 (Z) =H 0,2 (Z) =0 and H 3,0 (Z) =Cω Z with π ω Z = β (dz 1 dz 2 dz 3 ). Analogously one can compute H 2,1 (Z) =H 2,1 (Ẽ3 ) ϕ 3. Since H 2,1 (Ẽ3 )is generated by β (dz i dz j dz k )for{i, j, k} = {1, 2, 3}, which are not invariant under ϕ 3, we obtain H 2,1 (Z) =H 1,2 (Z) =0. The divisors on Ẽ3 induce divisors on Z. Since the map ( ϕ 3 ) acts as the identity on Pic(Ẽ3 ), the map π :Pic(Ẽ3 ) Q Pic(Z) Q is bijective,
1754 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI and hence, as we will see, the Picard group of Z is generated by the 36 classes π ( F i ), π ( D i ), π ( G i ), π ( B i,j,k ), i, j, k =1, 2, 3, at least over Q and we observe that Pic(Z) Q is H 2 (Z, Q). The divisors F i, D i, G i, B i,j,k generating Pic(Ẽ3 ) correspond to surfaces on Ẽ3. Let us denote by L one of them, then we define L to be L := π( L) as a set, with the reduced scheme structure. Thus, we get the classes F i, D i, G i, B i,j,k which correspond to surfaces on Z. By construction the quotient map π : Ẽ3 Z is a 3:1 cover branched over π( B i,j,k=1,2,3 i,j,k )= i,j,k=1,2,3 B i,j,k. Hence the map π : B i,j,k B i,j,k is 1 : 1. Moreover, also π : F i F i, π : D i D i, π : G i G i are 1 : 1. Thus, π ( B i,j,k )=B i,j,k, π ( F i )=F i, π ( D i )=D i, π ( G i )=G i. The set {F h,d h,g h,b i,j,k } h,i,j,k=1,2,3 is a Q-basis of Pic(Z). However, it is known that this Q-basis is not a Z-basis. Indeed the class of the branch locus of an n : 1 cyclic cover is n-divisible in the Picard group (cf. [5, Lemma 17.1, Chapter I]), in particular there exists a divisor M Pic(Z) such that 3M i,j,k B i,j,k =: B, (6) where C 1 C 2, if the two cycles C 1 and C 2 have the same cohomology class. Of course, M is not a linear combination with integer coefficients of the B i,j,k. We recall that Pic(Z) =H 1,1 (Z) and so the Hodge diamond of Z is 1 0 0 0 36 0 1 0 0 1 The intersection form on Pic(Z) is induced by the one on Pic(Ẽ3 ), but one has to recall that the map π : Ẽ3 Z is a 3 : 1 map away from the ramification locus, where it is a bijection. The map π : H (Z, C) H (Ẽ3, C) is a homomorphism of rings and hence for each D, D,D Pic(Z) H (Z): π (D)π (D )=π (D D ), π (D)π (D )π (D )=π (D D D )=3(D D D ), (7) where the last equality depends on the degree of π (cf. [12, Pag. 9]).
A RIGID CALABI YAU THREE-FOLD 1755 So to compute the intersection form on Pic(Z), it suffices to divide the intersection form on π (Pic(Z)) Pic(Ẽ3 ) by 3, here we sketch this computation: π (B i,j,k )=3 B i,j,k,sinceb i,j,k are in the ramification locus; π (F i )= F i + ϕ 3 ( F i )+( ϕ 3 2 ) ( F i ) 3 F i, in fact ϕ 3 (F i)andf i have the same cohomology class on E 3 (and hence ϕ 3 ( F i )and F i have the same cohomology class on Ẽ3 ); similarly π (D i ) 3 D i, π (G i ) 3 G i. Together with the description of the map π this implies that for every divisor L Pic(Ẽ3 ), π (L) =π (π ( L)) 3 L. By (7), we have (3 L)(3 L )(3 L )=π (L) π (L ) π (L )=3(L L L ) and thus LL L =9 L L L. Hence we obtain that the trilinear form on π (Pic(Ẽ3 )) Pic(Z) is the trilinear form of Pic(Ẽ3 ) multiplied by nine. Since the divisors in π (Pic(Ẽ3 )) define a Q-basis for Pic(Z), this determines the trilinear form on Pic(Z) completely. To recap, we proved that each divisor L Pic(Z) can be written as L = L E + L B, where L E = 3 i=1 (a if i + b i D i + c i G i ), L B = 3 i,j,k=1 α i,j,k B i,j,k and its cubic self-intersection is ( 3 3 3 L 3 = L 3 E + L 3 B =9 (a i Φ i + b i Δ i + c i Γ i )) +. (8) i=1 i,j,k=1 α 3 i,j,k We found a Q-basis of H 2,2 (Ẽ3 ) and this induces, via π, a Q-basis of H 2,2 (Z) (in analogy to what we did for Pic(Z)). Hence a Q-basis for H 2,2 (Z) consists of the curves f i (= π (β (φ i ))), d i (= π (β (δ i ))), g i (= π (β (γ i ))), l i,j,k (= π ( l i,j,k )). The intersection number al between a =: π (ã) H 2,2 (Z) and L, a divisor of the chosen Q-basis of Pic(Z), can be computed by the projection formula (cf. [12, pag. 9]): al = π (ã)l = ãπ L =3ã L (9) For example, choosing a = l i,j,k and L = B i,j,k,wehavel i,j,k B i,j,k = 3. We will need the expression of certain curves and classes in H 2,2 (Z) as linear combinations of the classes generating H 2,2 (Z), so here we compute some of them as examples.
1756 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI Example 2.2. The class M 2. The space H 2,2 (Z) contains all the classes obtained as intersection of two divisors on Z. In particular, the class M 2 = ( 1 2 3 i,j,k i,j,k) B can be written as linear combination of fi, d i, g i, l i,j,k with coefficients in Q, i.e., 1 9 i,j,k (B2 i,j,k )=( 3 h=1 (λ hf h + μ h d h + ν h g h )+ i,j,k α i,j,kl i,j,k ). To find the coefficients of this Q-linear combination, it suffices to compute the intersection of the divisors F i, D i, G i, B i,j,k with M 2. The only non trivial intersections of M 2 with these divisors are M 2 B i,j,k =1. We know that B i,j,k l i,j,k = 3, B i,j,k l a,b,c =0,if(a, b, c) (i, j, k), B i,j,k f h = B i,j,k d h = B i,j,k g h = 0 (cf. (9)). This implies that 1 = M 2 B i,j,k = 3α i,j,k and hence, α i,j,k = 1/3. Using (3) and (9), the intersections of M 2 with F h, D h and G h give λ h = μ h = ν h =0. Example 2.3. The curves C k i,j E 3 and A k i,j Z. Let us consider the curve Ci,j 1 = E {p i} {p j }, where i, j =1, 2, 3andp i are the points fixed by ϕ on E. This curve passes through three fixed points, p a,i,j E 3, a = 1, 2, 3. The curve Ci,j 1 E3 has the same cohomology class as E {q} {r} for two general points q, r E. In particular, the class of the curve Ci,j 1 is the class φ 1 for each i, j. Let C i,j 1 := β 1 (Ci,j 1 ) a=1,2,3{p a,i,j }, it is the strict transform of Ci,j 1. The curve C i,j 1 intersects the exceptional divisors B a,b,c in one point, if and only if (b, c) =(i, j). Hence, C i,j 1 = β (φ 1 ) 3 a=1 l a,i,j. So, π ( C i,j 1 )=π (β (φ 1 ) 3 a=1 l a,i,j )=f 1 3 a=1 l a,i,j. Let us consider A 1 i,j = π( C i,j 1 ) as set with the reduced scheme structure. The map π : C j,k 1 A1 i,j is 3 : 1, hence A1 i,j = 1 3 (π (β (Ci,j 1 ))). More generally, define Ci,j 2 = {p i} E {p j }, Ci,j 3 = {p i} {p j } E, A h i,j = π( C i,j h )asaset with the reduced scheme structure, then: ( A 1 i,j = 1 f 1 3 ( A 3 i,j = 1 f 3 3 ) ( ) 3 l a,i,j, A 2 i,j = 1 3 f 2 l i,a,j, 3 a=1 ) 3 l i,j,a. (10) a=1 a=1 2.4 The Chern classes and the Riemann Roch theorem on Z The ith Chern class of a variety is the ith Chern class of its tangent bundle. For a smooth projective variety X, c i (X) H 2i (X, Z) and, by convention, c 0 (X) =1.
A RIGID CALABI YAU THREE-FOLD 1757 If X is a Calabi Yau variety, then c 1 (X) = 0, indeed c 1 (T X )=c 1 ( 3 T X ) = c 1 ( K X )=0. The third Chern class of a smooth projective variety of dimension three satisfies χ(x) =c 3 (X) (Gauss Bonnet formula, [19, 416]). Here, we compute the Chern classes of Z. From the previous considerations it follows immediately that: c 0 (Z) =1, c 1 (Z)=0, c 2 (Z)= c 3 (Z) =χ(z) =72, 3 (λ h f h + μ h d h + ν h g h )+ h=1 3 α i,j,k l i,j,k, i,j,k=1 where λ h, μ h, ν h, α i,j,k Q. It remains to determine the coefficients of the linear combination defining c 2. To do this we need the following result: Lemma 2.4 ([15, Lemma 4.4]). If X is a complex three-fold with trivial canonical bundle and S is a smooth complex surface in X, then c 2 (X)[S] = c 1 (S) 2 + c 2 (S). We now apply this result to each generator of Pic(Z). As the divisors B i,j,k are isomorphic to P 2, one has c 1 (B i,j,k )=3l i,j,k and c 2 (B i,j,k )=χ(p 2 )= 3. The divisors F i, D i, G i are isomorphic to the Abelian surface E E (indeed the map π is 1 : 1 between the Abelian surface F i E E and F i are isomorphic to E E, similarly D i and G i ). Hence, their first Chern class is zero (since their canonical bundle is trivial) and their second Chern class is zero (since it is equal to their Euler characteristic). Now we compute the coefficients in c 2 (Z) as in Example 2.2. Indeed using (3) and (9) one has: and 6 = c 1 (B x,y,z ) 2 + c 2 (B x,y,z )=c 2 (Z)[B x,y,z ] 3 = h f h + μ h d h + ν h g h h=1(λ )+ α i,j,k l i,j,k [B x,y,z ]= 3α x,y,z, i,j,k 0= c 1 (F 1 ) 2 + c 2 (F 1 )=c 2 (Z)[F 1 ] 3 = h f h + μ h d h + ν h g h h=1(λ )+ α i,j,k l i,j,k [F 1 ] i,j,k =3(λ 1 + μ 2 + μ 3 + ν 2 + ν 3 )
1758 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI Doing this for all divisors, we obtain λ h = μ h = ν h =0,thus: c 2 (Z) =2 3 i,j,k=1 l i,j,k. Remark 2.5. Considering Example 2.2, it is immediate to see that the second Chern class c 2 (Z) is divisible by 6 in H 4 (Z, Z), indeed c 2 (Z)/6 = i,j,k=1,2,3 l i,j,k/3 = M 2 H 4 (Z, Z). The divisibility of this class was already obtained in a different and more involved way by Lee and Oguiso, [27]. The computation of the second Chern class of Z allows also to write down explicitly the Riemann Roch theorem for the divisors on Z. Indeed it is well known (cf. [23]) that the Riemann Roch theorem for a three-fold is: χ(l(d)) = 1 12 D (D K X)(2D K X )+ 1 12 c 2 D +1 p a. In case X = Z, wehavek Z = 0 and p a =1(sinceZ is a Calabi Yau variety), so we obtain: χ(l(d)) = 1 6 D3 + 1 6 i,j,k=1,2,3 l i,j,k D. (11) 3 More on the trilinear form on Pic(E 3 ) To compute the Yukawa coupling on Z it is necessary to describe the trilinear form on Pic(Z). We proved in the previous section that the trilinear form of Pic(Z) depends on the trilinear form on Pic(E 3 ) (cf. (8), (14)). For this reason, we now give a different description of the trilinear form on Pic(E 3 ): we reduce the computation of this trilinear form to the computation of the determinant of a matrix in Mat 3,3 (Q[ζ]) (cf. (13)). More precisely in this section, we give a way to associate to each divisor L on E 3 amatrixω L in Mat 3,3 (Q[ζ]) and we prove the following theorem: Theorem 3.1. There exists a homomorphism of groups μ : Pic(E 3 ) {H Mat 3,3 (Q[ζ]) t H = H} Mat 3,3 (Q[ζ]) such that, for each divisor L Pic(E 3 ), L 3 = 12 1 3det(μ(L)). We already said (Section 2.1) that a divisor D on an Abelian variety A R n /Λ corresponds to a two-form c 1 (D) and hence to a skew-symmetric form E D on the lattice Λ taking values in Z.
A RIGID CALABI YAU THREE-FOLD 1759 The elliptic curve E is obtained as C/Z[ζ]. Since, we are considering the Abelian variety E 3, in this context Λ Z[ζ] 3, and we are saying that each divisor D in Pic(E 3 ) defines a skew-symmetric form E D : Z[ζ] 3 Z[ζ] 3 Z. First of all, we prove that for each L Pic(E 3 ) there exists a matrix Ω L, such that for each x, y Z[ζ] 3, E L (x, y) =Tr( t xω L y), where Tr is the trace of an element in Q[ζ] overq defined as Tr(a + ζb)=(a + ζb)+(a + ζb)= 2a b for a, b Q. SinceE L (x, y) = E L (y, x), t Ω L = Ω L. To compute the matrix Ω L for each of the nine divisors L, which generate the Picard group of E 3, we use the same technique considered in Section 2.1, i.e., we consider divisors which generate Pic(E 3 ) as pull-back of divisors on an elliptic curve E. Indeed, since the map c 1 :Pic(X) H 2 (X, Z) commutes with the pull back, we have that, if L Pic(E 3 )isα (l) for a certain map α : E 3 E and a certain divisor l Pic(E), then E L (x, y) = E l (α(x),α(y)). Let us consider the elliptic curve E [ = C/Z[ζ] ] and a general point P E. 0 1 Then, c 1 (P ) is the skew-form E P = (the unique, up to a constant, 1 0 skew-form on Λ). So E P (a + ζb,c + ζd)=ad bc. The matrix Ω P is a 1 1 matrix with entries in Q[ζ] (i.e., Ω P Q[ζ]) such that [ ) 0 1 c Tr((a + ζb)ω L (c + ζd)) = (a, b) = ad bc. 1 0]( d This gives Ω P = ρ (= t Ω P ) where ρ := (ζ ζ)/3. The matrix Ω L for a certain divisor L Pic(E 3 ) was identified by the property E L (v, w) =Tr( t vω L w), hence to compute it, we consider α : E 3 E, α :(z 1,z 2,z 3 ) 3 i=1 a iz i, where z i are the complex coordinates on the ith copy of E. Let a := (a 1,a 2,a 3 ) and analogously v := (v 1,v 2,v 3 ), w := (w 1,w 2,w 3 ). Let L = α (P ), α(v) = i a iv i = a t v, α(w) = i a iw i = a t w (where t b is the transpose of the vector b). Then E L (v, w) =E P (α(v),α(w)) = E P (a t v, a t w)=tr( t (a t v)ρa t w) =Tr(v t aρa t w), which implies that the matrix Ω L associated to L = α (P )is Ω L := ρ a 1 (a 1, a 2, a 3 ). (12) a 2 a 3
1760 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI Thus to find, for example, Ω Γ1, η 1 :(z 1,z 2,z 3 ) ζz 2 z 3 : it suffices to apply (12) to the map Ω Γ1 = ρ 0 0 0 0 ζ (0, ζ, 1) = ρ 0 1 ζ. 1 0 ζ 1 Similarly, one finds Ω Φi,Ω Δi and Ω Γi (the map associated to each of these divisors is given in (1)). In this way one finds that if L = 3 i=1 (a iφ i + b i Δ i + c i Γ i ), then Ω L = a i Ω Φi + b i Ω Δi + c i Ω Γi is given by a 1 + b 2 + b 3 + c 2 + c 3 b 3 ζc 3 b 2 ζc 2 Ω L = ρ b 3 ζc 3 a 2 + b 1 + b 3 + c 1 + c 3 b 1 ζc 1. b 2 ζc 2 b 1 ζc 1 a 3 + b 1 + b 2 + c 1 + c 2 (13) Now an explicit computation shows that for each divisor L Pic(E 3 ), the determinant of Ω L is, up to a constant, the intersection form computed in (4), ( L 3 = 1 3 ) 3det (a i Ω Φi + b i Ω Δi + c i Ω Γi ) 12 i=1 (14) and this concludes the proof of Theorem 3.1. Remark 3.2. The compatibility between the group structures of Pic(E 3 ) and Mat 3,3 (Q[ζ]) is because of the properties of the skew-symmetric form E L defined by a divisor L and of the trace Tr. Indeed E L M (x) =E L (x)+ E M (x) =Tr( t xω L x)+tr( t xω M x)=tr( t x(ω L +Ω M )x), and so to the line bundle L M, we associate the matrix Ω L +Ω M. We observe that the Picard group of the singular quotient E 3 /ϕ 3 has rank 9 and is induced by the one on Pic(E 3 ). The Picard group of E 3 /ϕ 3 can be identified with the subgroup of Pic(Z) generated by F i, D i, G i. In Section 2.3, we proved that the trilinear intersection form on F i,d i,g i i=1,2,3 Pic(Z) is, up to a constant (multiplication by 9), the trilinear form on Pic(E 3 ). We deduce that the trilinear form on Pic(E 3 /ϕ 3 ) is, up to a constant, the determinant of the matrix as in (13). Since H 2,0 (E 3 /ϕ 3 )=0, Pic(E 3 /ϕ 3 ) C H 1,1 (E 3 /ϕ 3 ) and the cup product on H 1,1 (E 3 /ϕ 3 ) coincides with the C-linear extension of the trilinear form. So the cup product on H 1,1 (E 3 /ϕ 3 ) can be represented as determinant of a matrix in Mat 3,3 (Q[ζ]). This is of a certain interest because of its relation with the Yukawa coupling on H 1,1 (E 3 /ϕ 3 ), obtained as the sum of the cup
A RIGID CALABI YAU THREE-FOLD 1761 product and another addend, involving the Gromov Witten invariants. The values of a i, b i, c i, where the determinant of the matrix (13) is zero correspond to (1, 1) forms, where the cup product is zero. The set of such values is described by the cubic C 3 := V (det( 3 i=1 (a iω Φi + b i Ω Δi + c i Ω Γi ))) in P 8 (projective space with coordinates (a 1 : a 2 : a 3 : b 1 : b 2 : b 3 : c 1 : c 2 : c 3 )). This cubic is singular, where the matrix 3 i=1 (a iω Φi + b i Ω Δi + c i Ω Γi )has rank 1, hence along the intersection of the nine quadrics in P 8 defined by requiring that the nine 2 2 minors of the matrix are zero. The matrices of rank 1 are of type ρ( t a)(a) for a certain vector a =(a 1,a 2,a 3 ). We already showed that the matrix associated to the divisors Φ i,δ i,γ i are of this type and hence, they correspond to singular points of the cubic. We notice that these divisors define a fibration on the three-fold. Remark 3.3. Let Y be a Calabi Yau three-fold. In [36], the cubic hypersurface W in P(Pic(Y ) C), consisting of the points representing divisors L with L 3 = 0, is analyzed. Here, we are considering the cubic C 3 defined in the same way as W, but in the case of the Abelian variety E 3. By the relations between Pic(E 3 )andpic(z) given in Section 2.3, the cubic C 3 is also related to the cubic W in case Y = Z. 4 Projective models of Z The aim of this section is to give explicit relations and equations for Z. To do this we describe some (singular) projective models of the three-fold Z and more in general maps f : Z P N. Each of these maps is associated to a line bundle L := f (O P N (1)), and hence f is given by global sections s 0,...,s N H 0 (Z,L), i.e., f : z (s 0 (z) :...s N (z)). Our strategy will be to construct line bundles L (and maps m L associated to L) one 3 and use these to induce line bundles (and hence maps) on Z. Let L be a line bundle on E 3 such that ϕ 3 L L. Then, ϕ 3 acts on H 0 (E 3,L) and hence the space H 0 (E 3,L) is naturally decomposed in three eigenspaces H 0 (E 3,L) 0, H 0 (E 3,L) 1, H 0 (E 3,L) 2. By construction the maps (m L ) 0 : E 3 P(H 0 (E 3,L) 0 ), (m L ) 1 : E 3 P(H 0 (E 3,L) 1 ), (m L ) 2 : E 3 P(H 0 (E 3,L) 2 ) identify points on E 3 which are in the same orbit for ϕ 3. This implies that these maps (or better the maps induced by these maps on Ẽ3 ) are well defined on Z and thus are associated to line bundles on Z. It is moreover clear that the map (m L ) ɛ, ɛ =0, 1, 2, is the composition of E 3 m L (E 3 )followedbytheprojectionofm L (E 3 ) on the subspace P(H 0 (E 3,L)) ɛ. First of all, we point out the relations between the line bundle and its global sections on E 3 and on Z and then we focus our attention on a specific case.
1762 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI Remark 4.1. We said that the space H 0 (E 3,L) is naturally decomposed in eigenspaces by the action of ϕ 3, and indeed there are three subspaces of H 0 (E 3,L), such that the action of ϕ 3 is the same on all the elements in the same subspace and is different on two elements chosen in two different subspaces. However, the choice of the eigenvalue of each eigenspace is not canonical, but depends on the lift of ϕ 3 on H0 (E 3,L) chosen. Lemma 4.2 ([5, Lemma I.17.2]). Let π : X Y be an n-cyclic covering of Y branched along a smooth divisor C and determined by O Y (L), where L is a divisor such that O Y (nl) O Y (C). Then π (O X )= n 1 k=0 O Y ( kl). Since ϕ 3 acts as a multiplication by ζ on the local equation of each ramification divisor B i,j,k, we can apply the previous lemma to X = Ẽ3, Y = Z, C = B, L = M (cf. (6)), obtaining π (OẼ3) =O Z O Z ( M) O Z ( 2M). (15) Indeed O Z, O Z ( M), O Z ( 2M) correspond to the subbundles of π (OẼ3), which are stable with respect to the action of ϕ 3. In particular, O Z corresponds to the subbundle of the eigenvalue 1. Let L Pic(Ẽ3 ) be such that there exists L Pic(Z) satisfying L = π (L). Then π ( L) =π (π (L) OẼ3) =L π OẼ3 = L L( M) L( 2M), (16) where the last equality follows from (15). This implies that H 0 (Ẽ3, L) =H 0 (Z,π ( L)) = H 0 (Z,L) H 0 (Z,L M) H 0 (Z,L 2M) (17) where in the first equality we used, viewing L as invertible sheaf, H 0 (Z,π ( L)) = (π ( L))(Z) = L(π 1 (Z)) = L(Ẽ3 )=H 0 (Ẽ3, L) and in the last (16). Now, we concentrate on a specific choice of divisors on E 3 and Z: let Φ:=Φ 1 +Φ 2 +Φ 3 Pic(E 3 )andf := F 1 + F 2 + F 3 Pic(Z). Proposition 4.3. The map m 3Φ : E 3 P 26 is an embedding. The automorphism ϕ 3 of E 3 extends to an automorphism, called again ϕ 3,onP 26. Let (P 26 ) ɛ be the eigenspace for the eigenvalue ζ ɛ, ɛ =0, 1, 2, for ϕ 3. The composition of m 3Φ with projection P 26 P 26 0 (resp. P 26 1, P26 2 )isthemap defined on Z, which is associated to the divisor F (resp. F M, F 2M).
A RIGID CALABI YAU THREE-FOLD 1763 Proof. The diagram: E 3 β Ẽ 3 π Z induces H 0 (E 3, 3Φ) β H 0 (Ẽ3, 3β (Φ)) π H 0 (Z,F) The map β is an isomorphism. A section s H 0 (E 3, 3Φ) with divisor D, which has multiplicity α i,j,k in the point p i,j,k pulls back to a section β s with divisor β D = D + i,j,k α i,j,k B i,j,k, where D is the strict transform of D. Since 3β (Φ) = π (F ), using (16) and (17), we have π (3β (Φ)) = F π OẼ3 and Thus H 0 (Ẽ3, 3β (Φ)) = H 0 (Z,F π OẼ3) = H 0 (Z,F) H 0 (Z,F( M)) H 0 (Z,F( 2M)). H 0 (E 3, 3Φ) β H 0 (Ẽ3, 3β Φ) H 0 (Z,F) H 0 (Z,F M) H 0 (Z,F 2M) (18) and the last decomposition is a decomposition in eigenspaces of H 0 (E 3, 3Φ). So H 0 (Z,F am) H 0 (E 3, 3Φ) corresponds to the space of the sections of 3Φ on E 3 with zeros with multiplicity at least a in the points p i,j,k and which are in the same eigenspace for ϕ 3. The map associated to 3Φ is very explicit: Every elliptic curve is embedded in P 2 as a cubic, by the linear system associated to the divisor 3P. In particular, the curve E has the curve x 3 + y 3 + z 3 = 0 as image in P 2 x,y,z. So we can embed E 3 in P 2 x 1,y 1,z 1 P 2 x 2,y 2,z 2 P 2 x 3,y 3,z 3 (embedding each factor of E 3 in the correspondent copy of P 2 ). Now, it is well known that there exists an embedding of P 2 P 2 P 2 in P 26 given by the Segre map s :((x 1 : y 1 : z 1 ), (x 2 : y 2 : z 2 ), (x 3 : y 3 : z 3 )) (x 1 x 2 x 3 : x 1 x 2 y 3 : x 1 x 2 z 3 : x 1 y 2 x 3 :...: z 1 z 2 z 3 ). Hence, there is an embedding of E 3 in P 26, which is the restriction of s to E 3. By construction, this map is associated to the very ample divisor 3Φ on E 3. This map extends to a map defined on Ẽ3, which contracts the exceptional divisors B i,j,k (which are in fact orthogonal to the divisor β (3Φ) defining the map). The action of the automorphism ϕ 3 on E 3 is given by ϕ :((x 1 : y 1 : z 1 ), (x 2 : y 2 : z 2 ), (x 3 : y 3 : z 3 )) ((x 1 : y 1 : ζz 1 ), (x 2 : y 2 : ζz 2 ), (x 3 : y 3 : ζz 3 )) and this
1764 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI automorphism extends to an automorphism on P 26. The eigenspaces with eigenvalue 1, ζ, ζ 2 for ϕ 3 on P 26 are (x 1 x 2 x 3 : x 1 x 2 y 3 : x 1 y 2 x 3 : x 1 y 2 y 3 : y 1 x 2 x 3 : y 1 x 2 y 3 : y 1 y 2 x 3 : y 1 y 2 y 3 : z 1 z 2 z 3 ) (z 1 x 2 x 3 : z 1 x 2 y 3 : z 1 y 2 x 3 : z 1 y 2 y 3 : x 1 z 2 x 3 : x 1 z 2 y 3 : y 1 z 2 x 3 : y 1 z 2 y 3 : x 1 x 2 z 3 : x 1 y 2 z 3 : y 1 x 2 z 3 : y 1 y 2 z 3 ) (x 1 z 2 z 3 : y 1 z 2 z 3 : z 1 x 2 z 3 : z 1 y 2 z 3 : z 1 z 2 x 3 : z 1 z 2 y 3 ) respectively. We observe that the first eigenspace is defined by sections of 3Φ, which are not necessarily zero in the points p i,j,k (for example the monomial x 1 x 2 x 3 is not zero in the points p i,j,k ), the second is defined by sections passing through p i,j,k with multiplicity 1 and the third by sections passing through the points p i,j,k with multiplicity 2. Hence, the first eigenspace is identified (under the isomorphisms (18)) with H 0 (Z,F), the second with H 0 (Z,F M) and the third with H 0 (Z,F 2M). Remark 4.4. From this description of H 0 (Z,F km) we obtain: dim(h 0 (Z,F)) = 9, dim(h 0 (Z,F M)) = 12, dim(h 0 (Z,F 2M)) = 6. If we apply the Riemann Roch theorem (cf. (11)) to the divisor F, F M, F 2M, we find χ(f )=9,χ(F M) = 12, χ(f 2M) = 6. This in particular implies that for a divisor L among F, F M, F 2M, h 2 (Z,L) h 1 (Z,L) = 0, indeed by Serre duality we have h 3 (Z,L) = 0. For the divisors F and F M this is a trivial consequence of the fact that they are big and nef, as we will see in Propositions 4.6 and 4.8, and of the Kawamata Viehweg vanishing theorem. Remark 4.5. Analogously, we can consider the sections of the line bundles hf km, h, k 0, over Z. These correspond (as showed for F km) to sections of 3hΦ overe 3, which vanish at least of degree k in the points p i,j,k. We denote by N h,k the space of such a sections. In case k =0, 1, 2, this gives a decomposition in eigenspaces of H 0 (E 3,hΦ) relative to the eigenvalue ζ k. Let k =0, 1, 2. We denote by (Sym h (E)) k := (Sym h <x,y,z >) k the monomials of degree h in the coordinates of E, which belong to the eigenspace of the eigenvalue ζ k. Now (Sym h (E)) k is generated by the monomials of the form x α y β z γ, such that α + β + γ = h and γ k mod 3. Since z 3 = x 3 y 3 on E, we can assume that γ = k. Thus the eigenspaces have the following dimensions: dim(sym h (E)) k = h +1, h, h 1fork =0, 1, 2, respectively. The sections of 3hΦ one 3 are given by Sym h (E) Sym h (E) Sym h (E) and hence N h,k =Sym h (E) a Sym h (E) b Sym h (E) c with
A RIGID CALABI YAU THREE-FOLD 1765 a, b, c =0, 1, 2anda + b + c k mod 3. After direct computation, we obtain the following dimensions 9h 3 k =0, dim(n h,k )= 9h 3 +3 k =1, 9h 3 3 k =2, whichadduptoh 0 (E 3, 3hΦ) = (3h) 3. Now χ(hf km) =9h 3 + 3 2 k(3 k2 ) by the Riemann Roch theorem (cf. (11)): we notice that dim(n h,k ) equals χ(hf km) for k =0, 1, 2, h>0. This generalizes the result of Remark 4.4 and allows one to describe projective models of Z obtained from the maps associated to the divisors hf km, foreachh>0, k =0, 1, 2. 4.1 The first eigenspace We now analyze the projection to the eigenspace relative to the eigenvalue 1, i.e., the map m 0 on E 3 given by (x 1 x 2 x 3 : x 1 x 2 y 3 : x 1 y 2 x 3 : x 1 y 2 y 3 : y 1 x 2 x 3 : y 1 x 2 y 3 : y 1 y 2 x 3 : y 1 y 2 y 3 : z 1 z 2 z 3 ). Considering the coordinate functions of m 0, we observe that they are invariant not only under the action of ϕ 3, but also under the action of φ : ((x 1 : y 1 : z 1 ), (x 2 : y 2 : z 2 ), (x 3 : y 3 : z 3 )) ((x 1 : y 1 : ζz 1 ), (x 2 : y 2 : ζ 2 z 2 ), (x 3 : y 3 : z 3 )). It is easy to see that the map is 9 : 1 on E 3, and hence the image gives a model of the Calabi Yau variety Y, which desingularizes E 3 / φ, ϕ 3. So Z is a 3 : 1 cover of m 0 (E 3 ). The Calabi Yau Y, of which m 0 (E 3 )is a birational model, is still interesting, so we describe the map m 0 in some details. In this section we prove the following: Proposition 4.6. The map m 0 : E 3 P 8 is well defined on E 3,andisa 9:1map on its image. Its differential fails to be injective only on the curves Cj,k i (cf. Example 2.3). The variety m 0 (E 3 ) is a 3:1 cover of σ(p 1 P 1 P 1 ) where σ : P 1 P 1 P 1 P 7 is the Segre embedding. Moreover, m 0 (E 3 ) is contained in the Fermat cubic hypersurface in P 8. The map m 0 induces the 3:1map m F : Z P 8 associated to the nef and big divisor F. It is immediate to check that the map m 0 is 9 : 1. To analyze its differential, we first consider m 0 as defined on P 2 P 2 P 2 and then we will restrict it to
1766 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI E 3. We recall that P 2 is covered by its open subsets U x := {(x : y : z) x 0}, U y and U z. Since the point (0 : 0 : 1) E, it suffices to consider the open sets U x and U y, but the map is totally symmetric in the x i and y i,soitis enough to study the map on the open set U x U x U x of P 2 P 2 P 2 : U x U x U x U C 8 (y 1,z 1 ) (y 2,z 2 ) (y 3,z 3 ) ( y 3,y 2,y 2 y 3,y 1,y 1 y 3,y 1 y 2, y 1 y 2 y 3,z 1 z 2 z 3 ). The Jacobian is given by 0 0 0 0 1 0 0 0 1 0 0 0 0 0 y 3 0 y 2 0 J m0 = 1 0 0 0 0 0 y 3 0 0 0 y 1 0. y 2 0 y 1 0 0 0 y 2 y 3 0 y 1 y 3 0 y 1 y 2 0 0 z 2 z 3 0 z 1 z 3 0 z 1 z 2 Now we restrict our attention to the tangent space to E 3 : the tangent vectors (u, v) toe in (y, z) satisfyu f y + v f z = 0, where f =1+y3 + z 3 is the equation of E in U x,thus(u, v) =λ( f z, f y )=λ( 3z2, 3y 2 ), λ C. Hence the tangent vectors of E 3 in the point q := (y 1,z 1,y 2,z 2,y 3,z 3 )are (u 1,v 1,u 2,v 2,u 3,v 3 )=( f 1 z 1 λ, f 1 y 1 λ, f 2 z 2 μ, f 2 y 2 μ, f 3 z 3 ρ, f 3 y 3 ρ), where (λ, μ, ρ) C 3 T q E 3. The Jacobian fails to be injective, where J m0 (u 1,v 1,u 2,v 2,u 3,v 3 ) t =0, which gives the following equations: f 3 ρ =0 z 3 f 2 μ =0 z 2 f 1 z 1 λ =0 f 1 f 2 f 3 z 2 z 3 λ + z 1 z 3 μ + z 1 z 2 ρ =0 y 1 y 2 y 3 z 3 =0( y 3 = ζ c 0) or ρ =0 z 2 =0( y 2 = ζ b 0) or μ =0 z 1 =0( y 1 = ζ a 0) or λ =0
A RIGID CALABI YAU THREE-FOLD 1767 Thus, if either z i 0 ior z i =0,z j,z k 0for{i, j, k} = {1, 2, 3}, then λ = μ = ρ =0, which gives no points where (J m0 ) E 3 is not injective. On the other hand, the condition z 1 0,z j =0forj 1 (resp. z i =0 i) implies λ = 0, but does not give conditions on μ, ρ (resp. λ, μ, ρ). We recall that the condition z i = 0 gives exactly the fixed points p i on the ith copy of E in E 3. Therefore the map (J m0 ) E 3,andalsom 0, fails to be injective on the curves Ci,j k (and in particular at the fixed points p i p j p k ). The reason is that these curves are invariant not only under the action of ϕ 3, but also under φ. The map m 0 can also be described in a different way, which exhibits m 0 (E 3 ) as 3 : 1 cover of a subvariety in P 7. Consider the composition γ of the projection α : E 3 P 1 of each elliptic curve E P 2 on the first two coordinates and the Segre embedding σ :(P 1 ) 3 P 7 γ : E 3 α P 1 P 1 P 1 x 3 i + y3 i + z3 i =0 ((x 1 : y 1 ), (x 2 : y 2 ), (x 3 : y 3 )) σ P 7 (x 1 x 2 x 3 : x 1 x 2 y 3 :...: y 1 y 2 y 3 ) The Segre map σ is well known to be an embedding and the map α is clearly 3 3 : 1, hence, γ is 3 3 :1. Let us denote by X 0,...,X 8 the coordinates on the target projective space of the map m 0. The map γ is the composition of projection of m 0 with the projection on the hyperplane P 7 P 8 with coordinates X 0,...,X 7. Thanks to this description one can show that m 0 (E 3 ) is contained in certain quadrics and a cubic hypersurface. Indeed the variety σ(p 1 P 1 P 1 )iscontained in the quadrics X 0 X 3 = X 1 X 2, X 0 X 5 = X 1 X 4, X 0 X 7 = X 2 X 5, X 0 X 7 = X 1 X 6, X 0 X 6 = X 2 X 4, X 0 X 7 = X 3 X 4, X 1 X 7 = X 3 X 5, X 2 X 7 = X 3 X 6, X 4 X 7 = X 5 X 6 and since x 3 i + y3 i + z3 i =0, we have (X 8) 3 = (z 1 z 2 z 3 ) 3 = ( x 3 1 y 3 1 )( x3 2 y3 2 )( x3 3 y3 3 )=X3 0 + X3 1 + X3 2 + X3 3 + X3 4 + X3 5 + X3 6 + X3 7, so m 0 (E 3 ) is contained in the Fermat cubic in P 8, F 8 := V ( 8 i=0 X3 i ). It is now clear that the projection (X 0 :...: X 8 ) (X 0 :...X 7 ) restricted to F 8 and to m 0 (E 3 ) is a cyclic 3 : 1 map with cover transformation (X 0 :...: X 7 : X 8 ) (X 0 :...: X 7 : ζx 8 ).
1768 SARA ANGELA FILIPPINI AND ALICE GARBAGNATI Remark 4.7. The map m F induced by m 0 on Z does not contract curves. This guarantees that F is a big and nef divisor, indeed F 3 > 0 (i.e., F is big) and for each curve C Z, FC =3deg(m F (C)) > 0, and by [12, Theorem 1.26] this suffices to conclude that F is nef. The inclusion m 0 (E 3 ) F 8 is interesting in view of the paper [8], where the authors suggest that a generalized mirror for the Calabi Yau three-fold Z is a quotient of the Fermat cubic in P 8 by an automorphism of order 3. Here, we proved that there exists a 3 : 1 map from Z to a singular model of the Calabi Yau variety Y = Z/Z 3, which is contained in this cubic in P 8. The Hodge numbers of Y are h 1,1 =84andh 2,1 =0. In [8], the authors observe that the middle cohomology of the desingularization F 8 /G of the quotient of F 8 by certain groups G has the following Hodge numbers: H 7 : 0 0 1 β β 1 0 0. (19) For a certain choice of the action of the group G Z 3, the value of β is 36 and hence h 4,3 (F 8 /G) =h 1,1 (Z). The space H 4,3 ( F 8 /G) is the complex moduli space of F 8 /G and has the same dimension of the Kähler moduli space of Z. Requiring that the dimension of the complex moduli space of a variety coincides with the dimension of the Kähler moduli space of another variety is one of the necessary conditions for the two varieties to be mirrors. In [8], deeper relations between the complex moduli of F 8 /G and the Kähler moduli of Z are found using the Yukawa coupling. Because of this, the authors suggest that F 8 /G could be a generalized mirror of Z. Now we observe that, if G is trivial, then β in (19) is 84 and h 4,3 (F 8 )= h 1,1 (Y). Thus, we observe that F 8 has the Hodge numbers of the generalized mirror of Y (a desingularization of Z/Z 3 ). This was already noticed in [26, Section 6.1.3], where the authors analyze a deeper relation between Y and F 8 based on their L-functions (cf. [26, Theorem 2]). We observe that in these two generalized mirrors the desingularization of a quotient by Z 3 is involved: CY Conjectured generalized mirror Z F8 /Z 3 Y = Z/Z 3 F 8
A RIGID CALABI YAU THREE-FOLD 1769 The fact that the Calabi Yau variety Y admits a birational model inside the variety 8 i=0 X3 i = 0 (and Z a 3 : 1 map to a subvariety of 8 i=0 X3 i =0) could be useful to give a geometric explanation of the relations between Z and its generalized mirror and between Y and F 8. 4.2 The second eigenspace We now analyze the projection on the eigenspace of the eigenvalue ζ, i.e the map m 1 on E 3 given by (z 1 x 2 x 3 : z 1 x 2 y 3 : z 1 y 2 x 3 : z 1 y 2 y 3 : x 1 z 2 x 3 : x 1 z 2 y 3 : y 1 z 2 x 3 : y 1 z 2 y 3 : x 1 x 2 z 3 : x 1 y 2 z 3 : y 1 x 2 z 3 : y 1 y 2 z 3 ). We summarize the properties of this map in the following Proposition, which is proved in this section: Proposition 4.8. The base locus of the map m 1 : E 3 P 11 consists of the 27 points p i,j,k. The map m 1 contracts the 27 curves Cj,k i and is 3:1 on E 3 away from these curves. Its differential is injective away from the 27 contracted curves. The image m 1 (E 3 ) has 27 singular points, the images of the curves Cj,k i, which are ordinary double points. The map m 1 induces a well defined map over Ẽ3, which sends the 27 exceptional divisors B i,j,k to 27 copies of P 2 and whose differential is injective away from the contracted curves. The map m 1 induces the map f F M on Z associated to the nef and big divisor F M. It contracts the curves A i j,k and is the isomorphism Z 3 i,j,k=1 Ai j,k f F M (Z ) 3 i,j,k=1 Ai j,k away from the contracted curves. By the definition of m 1, it is clear that the base locus is given by the condition z 1 = z 2 = z 3 = 0 and hence the base locus consists of the 27 points p i,j,k. Let q be one of the following 27 points (0 : :0:1: ζ b : ζ a : ζ a+b ), (0 : :0:1: ζ c : ζ a : ζ a+c :0: : 0) and (1 : ζ c : ζ b : ζ b+c : 0: : 0). Then the inverse image of q is a curve C i j,k (for example C1 1,1 is sent to (1 : ζ : ζ : ζ 2 :0:...: 0)). The inverse image of all the other points in m 1 (E 3 ) consists of 3 points, so the map is generically 3 : 1. To study the Jacobian, we consider the open subset U z U x U x (where the map is surely defined, because the base locus is defined by z 1 = z 2 = z 3 = 0). Since the tangent vectors to E 3 in (x 1,y 1,y 2,z 2,y 3,z 3 )are(u 1,v 1,u 2,v 2, u 3,v 3 )=( f 1 y 1 λ, f 1 x 1 λ, f 2 z 2 μ, f 2 y 2 μ, f 3 z 3 ρ, f 3 y 3 ρ), where (λ, μ, ρ) C 3 T p E 3, the restriction of the differential of m 1 to the tangent space of E 3