Groups Acting Freely on Calabi-Yau Threefolds Embedded in a Product of del Pezzo Surfaces
GGroups Acting Freely on Calabi-Yau ThreefoldsEmbedded in a Product of del Pezzo Surfaces
Gilberto Bini ∗ and Filippo F. Favale † November 17, 2018
Abstract
In this paper, we investigate quotients of Calabi-Yau manifolds Y embed-ded in Fano varieties X which are products of two del Pezzo surfaces -with respect to groups G that act freely on Y . In particular, we revisitsome known examples and we obtain some new Calabi-Yau varieties withsmall Hodge numbers. The groups G are subgroups of the automorphismgroups of X , which is described in terms of the automorphism group ofthe two del Pezzo surfaces. Contents dP × dP with maximal order 12 . . . . . . . . . . . . . . . . . . 94.2 dP × dP with maximal order 3 . . . . . . . . . . . . . . . . . . 11 P × P ) × ( P × P ) with maximal order 16 . . . . . . . . . . . 135.2 dP × dP with maximal order 8 . . . . . . . . . . . . . . . . . . 165.3 P × P with maximal order 9 . . . . . . . . . . . . . . . . . . . . 185.4 dP × dP with maximal order 5 . . . . . . . . . . . . . . . . . . 195.5 P × P × dP with maximal order 4 . . . . . . . . . . . . . . . . 215.6 Other similar examples . . . . . . . . . . . . . . . . . . . . . . . . 22 dP × S , with S ∈ (cid:8) P × P , dP , dP , dP , dP (cid:9) . . . . . . . . . . 236.2 dP × dP with estimated maximal order 7 . . . . . . . . . . . . 246.3 dP × dP with estimated maximal order 9 . . . . . . . . . . . . 26 ∗ Universit`a degli Studi di Milano - Dipartimento di Matematica “F. Enriques” - Via C.Saldini, 50 - 20133 Milano (Italy).
E-mail : [email protected] † Universit`a degli Studi di Pavia (Italy) - Dipartimento di Matematica “F. Casorati” - ViaFerrata, 1 - 27100 Pavia.
E-mail : [email protected] a r X i v : . [ m a t h . AG ] A p r On the Relation between
Aut( S ) × Aut( S ) and Aut( S × S )
298 List of the Threefolds Obtained 36Bibliography 38
In [13] and [14] Tian and Yau discover a new Calabi-Yau manifold with Eulercharacteristic equal to -6. Let us briefly explain their seminal example. To beginwith, they consider the product X of two cubic Fermat surfaces in P C . Next,they pick a smooth hyperplane section Y in X , which is invariant with respectto a group G isomorphic to the cyclic group of order three. By adjunction andby Lefschetz’s Hyperplane Theorem, Y turns out to be a smooth Calabi-Yauthreefold, i.e., a smooth compact K¨ahler threefold with trivial canonical bundleand no holomorphic p -forms for p = 1 ,
2. The Euler characteristic of Y is − h , ( Y ) and h , ( Y ) are 14 and 23,respectively. To reduce to Euler characteristic and the Hodge numbers, Tianand Yau take the quotient of Y with respect to G that turns out to act freelyon it. The quotient manifold Y /G is a Calabi-Yau variety with Hodge numbers h , = 6 and h , = 9.In recent years, physicists have focused on Calabi-Yau manifolds with smallHodge numbers: see, for instance, [2], [4], [3], [6] and [9]. In fact, imagine to plotthe distribution of Calabi-Yau varieties on a diagram with variables the Eulercharacteristic χ ( Y ) (on the horizontal axis) and the height h ( Y ) := h ( Y ) + h ( Y ) (on the vertical axis). Fix a pair ( χ , h ) of positive integers such that χ is even and − h ≤ χ ≤ h . For h ≤
30, it turns out that there arestill a lot of missing examples of Calabi-Yau varieties with Euler characteristic χ and height h . The example in [13] is even more significant because theEuler characteristic is −
6. In general, special attention is given to those Calabi-Yau manifolds that have Euler characteristic 6 in absolute value since theycorrespond to three-generation families (see, for instance, [3]).Remarkably, the example in [13] can be generalized in the following way. Thetwo cubic Fermat surfaces are examples of degree three del Pezzo surfaces, i.e.,smooth surfaces with ample anticanonical divisor which can be obtained as theblow-up of P C at six points in general position. A first generalization in thisdirection was given by Braun, Candelas and Davies in [3]. In that paper, theydiscover a new Calabi-Yau manifold with Euler characteristic − P C by two del Pezzosurfaces of degree six and come up with a group of order twelve that acts freelyon a suitable hyperplane section of the product.In this paper we generalize the examples mentioned above even further and weput them in a more general context. Indeed, let us consider two suitable smoothdel Pezzo surfaces S and S . The product X is a smooth Fano fourfold, i.e., − K X is ample. In X we pick a smooth threefold Y which is in | − K X | . Aspointed out by the example in the Introduction in [11] this requires some work:2n fact, for some choice of the two del Pezzo surfaces it is not even possible.Moreover, we pick a finite group G in Aut( S × S ) that acts freely on Y so thatthe quotient variety is a Calabi-Yau manifold. Since the Euler characteristic χ ( Y ) is negative, it is easy to verify that the height of Y /G is less than theheight of Y for any non-trivial group G . Within this set-up, we obtain the twoexamples mentioned above; further, we find new Calabi-Yau manifolds withsmall Hodge numbers. The smoothness and the free action of G on a suitable Y are proved as follows. We pick a group G that has only finitely many fixedpoints on X . We decompose the representation of G on H ( X, − K X ) as a directsum ⊕ V i of irreducible subrepresentation. We consider a subspace W such thatfor every g ∈ G and every s ∈ W , g ∗ ( s ) = λ g s for some λ g ∈ C ∗ , i.e. forevery g ∈ G , W is an eigenspace for g ∗ .We pick a section s ∈ W , if there aresome, so that the corresponding zero locus does not intersect the fixed locus of G . Next, we look at the base points of the subsystem W ≤ H ( X, − K X ). Incase there are some, we take a generic section and prove that the base pointsare smooth. This is done by direct computation with MAPLE. A Bertini-typeargument yields the existence of a smooth threefold Y in X on which G actsfreely.In Section 5 we present the examples we obtain case by case. Except for thelast subsection of that Section, all the examples have height less than 20. Un-fortunately, we do not obtain any new three-generation manifolds, i.e., a man-ifold with | χ ( Y ) | = 6. Moreover, in Section 8, you may find all the exam-ples of quotients of Calabi-Yau threefolds Y embedded in S × S by groupswhich are of maximal order. In other words, we take the quotient by a group H ≤ Aut( S × S ) such that the restriction to Y yields a free action and H cannot have order greater than the groups used. Finally, we investigate the heightof the quotient variety. In several cases, we are able to say that the height forthe quotient threefold is the least possible within this framework.The following picture represents the tip of the distribution of the Calabi-Yaumanifold with respect to the Hodge numbers. The diagonal axis are h , ( Y ) and h , ( Y ) whereas the horizontal and the vertical axis are χ ( Y ) and h ( Y ), respec-tively. We plot only the known manifolds with height less or equal than 31. Thesolid dots correspond to quotients found in this paper. The blue rings representthe ones known until now (with respect to the data collected in [2], [4], [3] and[6]). The black rings are quotients by groups whose order is maximal. From thepicture below, we can summarize our results as follows. The dots (3 , , (2 , ,
13) represent NEW Calabi-Yau threefolds. There exists a Calabi-Yaumanifold corresponding to the pair (1 ,
5) with non-abelian fundamental group:see [4]. Our example in Section 5.1 has abelian fundamental group isomorphic tothe product of the cyclic group of order two and that of order eight. Moreover,we come up with a Calabi-Yau manifold with Hodge numbers (2 ,
11) (cf. (5.3)),which are the same as those described in [4]. Finally, we construct other vari-eties with greater height (see Section 5.6) but they correspond to existing dotsin the picture below. In all the cases where other Calabi-Yau manifolds alreadyexist, it would be interesting to know whether our examples are isomorphic tothose or not. 3n some cases, it is not possible to consider non-trivial quotients with ourmethod. In fact, we prove, for instance, that there does not exist a Calabi-Yauvariety which is the quotient by a group of order seven of a smooth anticanonicalsection Y in a product of two del Pezzo surfaces of degree two. This type ofresults is collected in Section 6. To prove them, we use the following theoremwhich is proved in Section 7. For this purpose, we first use some Mori theoremof Fano fourfolds which are products of two Fano varieties. Second, we alsorecall that for low degree del Pezzo surfaces are toric varieties. Thus, we applya theorem due to Demazure (later generalized by D. Cox in [5]) on the structureof the automorphism group of toric varieties. More specifically, the followingholds (see Section 7). Theorem.
Let S and S be two del Pezzo surfaces. Then • If S (cid:54) = S , Aut( S × S ) = Aut( S ) × Aut( S ) ; • If S = S (cid:54) = P × P , Aut( S × ) = Aut( S ) × (cid:110) Z ; • If S = S = P × P , Aut(( P ) × ) = Aut( P ) × (cid:110) S , where S is thesymmetric group with elements. Acknowledgments.
During the preparation of this work, we asked some ques-tions and suggestions to various people we kindly acknowledge: Alberto Alzati,Cinzia Casagrande, Philip Candelas, Igor Dolgachev, Bert van Geemen, Grze-gorz Kapustka, Michal Kapustka, Antonio Lanteri and Gian Pietro Pirola. Thiswork was partially supported by MIUR and GNSAGA.4
Preliminaries
We say that a complex surface S is a del Pezzo surface if it is projective, smooth,simply-connected and the anticanonical divisor − K S is ample. Examples of delPezzo surfaces are blow-ups of the projective plane in a finite set ∆ of 0 ≤ n < P × P . As proved in [7], this list is exhaustive.We often write dP d to mean a del Pezzo surface that is obtained by blowing up9 − d points of P that are in general position. Let S = Bl ∆ P . We can identify H ( S, − K S ) with the vector space of the homogeneous polynomials of degree 3with variables { x , x , x } such that f ( P ) = 0 for all P ∈ ∆. It is easy to showthat h ( S, − K S ) = d + 1 if S = dP d . Moreover, if k = 9 − d then − K S = 3 π ∗ H − k (cid:88) i =1 E i , where H is the hyperplane divisor on the projective plane and the E i ’s are theexceptional divisors. Thus, K S = 9 − k = d . For d ≥ − K S isvery ample. For d = 2 the anticanonical system | − K S | gives a 2 : 1 map of S in P branched along a smooth quartic. For d = 1 the anticanonical model of S is a finite cover of degree two of a quadratic cone Q ramified over a curve B inthe linear system |O Q (3) | .Suppose that Y is a Calabi-Yau threefold and that G is a group that acts freelyon Y . Then it is well known that the quotient Y /G has a canonical complexstructure such that the projection on the quotient is holomorphic. Furthermore,the quotient map is a local isomorphism.
Theorem 2.1.
If the action of G is free then Y /G is also a Calabi-Yau threefold.Moreover, the quotient is projective.Proof.
Take g ∈ G \ { Id } . The manifold Y is a Calabi-Yau threefold, so h , ( Y ) = h , ( Y ) = 0 , h , ( Y ) = 1 . There exists ω ∈ H , ( Y ) such that ω P (cid:54)≡ P ∈ Y (this is equivalent to K Y ≡ g ∗ ω = ω . The maps g ∗ : H p, ( Y ) −→ H p, ( Y )are zero for p = 1 , p = 0, g ∗ is the identity. We apply theHolomorphic Lefschetz Fixed Point formula, which in this case reads as follows:0 = 1 − − Tr( g ∗ : H , ( Y ) −→ H , ( Y )) . Since h , ( Y ) = 1 ( Y is a Calabi-Yau manifold) we get g ∗ = Id for p = 3 andfor all g ∈ G . Thus the action of G on H , ( Y ) is trivial. We have the followingisomorphism ([1], p. 198): H p,q ( Y /G ) (cid:39) H p,q ( Y ) G ;hence H , ( Y /G ) (cid:39) H , ( Y ) G = H , ( Y ) and there exists a holomorphic3 − form ˜ ω on Y /G such that π ∗ ˜ ω = ω and, as π is a local isomorphism,˜ ω P (cid:54) = 0 for all P ∈ Y /G . This is equivalent to K Y/G ≡
0. Finally, using h p, ( Y /G ) = h p, ( Y ) G one has h , ( Y /G ) = h , ( Y /G ) = 0 and this concludesthe proof. As for the projectivity of
Y /G , see, for example, [10], p. 127.5e will adopt the following framework. We will take two del Pezzo surfaces S and S , their product X = S × S , which is a Fano fourfold, and a smoothelement Y of | − K X | .First of all, we will define a number M ( S , S ) which bounds the maximumorder of a finite group acting freely on Y and which depends only on the degreeof S and S . Definition 2.2.
Let M ( S , S ) to be the positive greatest common divisor of χ ( Y ) / and χ ( − ι ∗ K X )) , where ι : Y → X is the embedding of Y in X . Notice that if Y ⊂ S × S is a Calabi-Yau threefold and G is a finite groupthat acts freely on Y , then | G | divides M ( S , S ).With the definition of M ( S , S ) in mind, we will search for a group G with thefollowing properties:(a) G is a subgroup of Aut( S × S );(b) | G | = M ( S , S ).Note that if Fix( G ) ⊂ X contains a curve L , by the Nakai-Moishezon criterionof ampleness, − K X · L >
0, and since Y = − K X we will have some fixed pointson Y . Hence it’s necessary to choose groups whose action on X has at most afinite number of fixed points.Finally, Let m ( S , S , Y ) bemax {| G | | g ( Y ) = Y ∀ g ∈ G and satisfies ( a ) and ( b ) } . We anticipate that there are cases in which M ( S , S ) > m ( S , S , Y ) = 1 for all Y ). Assume that S and S are smooth projective surfaces and Y is a Calabi-Yauthreefold embedded in X = S × S . Then the following result holds: Theorem 3.1.
The Euler characteristic of Y is − K S K S . Proof.
By the exact sequence of vector bundles0 → T Y → T X → N Y/X → Y is a Calabi-Yau manifold ( which implies c ( Y ) = 0), we have:(1 + c ( Y ) + c ( Y )) · (1 + c ( N Y/X )) = ι ∗ (1 + c ( X ) + c ( X ) + c ( X ) + c ( X ))and, in particular, c ( N Y/X ) = ι ∗ c ( X ) , c ( Y ) = ι ∗ c ( X ) and c ( Y ) = ι ∗ c ( X ) − c ( Y ) c ( N Y/X ) . X is a product of surfaces we have c ( X ) = c ( S ) + c ( S ) , c ( X ) = c ( S ) + c ( S ) + c ( S ) c ( S )and c ( X ) = c ( S ) c ( S ) + c ( S ) c ( S ) . Hence, by the identification H ( Y, Z ) (cid:39) Z , we have c ( Y ) = ι ∗ ( c ( X ) − c ( X ) c ( X )) = c ( X ) c ( X ) − c ( X ) c ( X ) = c ( S ) c ( S ) + c ( S ) c ( S ) − c ( S ) c ( S ) − c ( S ) c ( S ) − c ( S ) c ( S ) == − c ( S ) c ( S ) = − K S K S . Now, assume Y is asmooth ample divisor in X . Thus, the following isomor-phisms hold: H ( Y, Z ) (cid:39) H ( X, Z ) (cid:39) H ( S , Z ) ⊕ H ( S , Z ) . For any divisor class D ∈ H ( Y /G, Z ) denote by D and D divisors classessuch that π ∗ ( D ) = D + D , where π is the projection of Y onto the quotient.Finally, we denote by K i the divisor classes such that K X = K + K . Thenthe following holds. Theorem 3.2.
Let G be a group that acts freely on Y . Then for any D ∈ H ( Y /G, Z ) and D , D as above, we have χ ( D ) = − D D ( D K + D K )2 | G | − χ ( O S ) K D + χ ( O S ) K D | G | . Proof.
We recall that the Riemann-Roch formula for the Calabi-Yau threefold
Y /G is χ ( D ) = D c ( Y /G ) D . The action of G is free, hence | G | D = π ∗ ( D ) and | G | c ( Y /G ) D = c ( Y ) π ∗ ( D ) . This yields π ∗ ( D ) = ( D + D ) ( c ( S ) + c ( S )) == 3 D D c ( S ) + 3 D D c ( S ) = − D D ( D K + D K ) . In a similar way, we obtain c ( Y ) π ∗ ( D ) = − ( χ ( S ) + K ) K D − ( χ ( S ) + K ) K D . Merging these results and using N¨other formula , we complete the proof. χ ( O S ) = K S + χ ( S )12 .
7e focus our attention on a particular divisor on the quotient: a divisor D such that π ∗ D = − ι ∗ K X =. Such a divisor always exists because the canonicaldivisor is G -invariiant for any gruoup of automorphisms G . We can specializethe previous formula for nD obtaining χ ( nD ) = n K K | G | + n χ ( O S ) K + χ ( O S ) K | G | = χ ( − nι ∗ K X ) | G | . Hence | G | has to divide χ ( − nι ∗ K X ) for all n . We can obtain a similar conditionusing Theorem 3.1: the Euler characteristic of the quotient Y /G of Y by a finitegroup G that acts freely is the Euler characteristic of Y divided by the order ofthe group. Moreover, it is known that a Calabi-Yau threefold has even Eulernumber so we obtain that | G | must divide χ ( Y ) /
2. This gives a motivation toDefinition 2.2.The following table gives the values of M ( S , S ) for every distinct values ofdegrees of S and S , with S and S del Pezzo surfaces - distinguishing thecase dP and P × P . M ( S , S ) P P × P dP dP dP dP dP dP dP dP P P × P dP dP dP dP dP dP dP dP X = dP × dP ( M ( dP , dP ) = 1) it isn’t possible to find a pair( Y, G ) with Y embedded in X and Id (cid:54) = G ≤ Aut( Y ) that acts freely on Y . Ifwe choose X = dP × dP ( M ( dP , dP ) = 5) a pair ( Y, Z ) with Z withoutfixed points might exist.The self-intersection of − K S , where S is a del Pezzo surface, is positive and isequal to its degree and this, using Theorem 3.1, means that χ ( Y ) < G is free, we have that the height h := h , + h , of Y and that of One could easily check that | G | divides χ ( − nι ∗ K X ) ∀ n ∈ Z ⇐⇒ | G | divides χ ( − ι ∗ K X ) . /G satisfy the following inequality: h ( Y /G ) = h , ( Y /G ) + h , ( Y /G ) = 2 h , ( Y /G ) − χ ( Y )2 | G | = 2 h , ( Y ) G + | χ ( Y ) | | G | < h , ( Y ) + | χ ( Y ) | h ( Y ) . By finding a group whose order is maximal - and such that the dimension h , ( Y ) G of the invariant part of H , ( Y ) is the smallest possible - we obtainthe least possible height for the quotient.In the following sections we give some examples (both known and new) andsome results of non-existence. With the following examples we revisit some known examples in the frameworkpresented. The first one is due to Braun, Candelas and Davies and can be foundin [3]. The second one is due to Tian and Yau and is presented in [14] and [13]. dP × dP with maximal order There is a unique del Pezzo surface of degree 6 and this surface can be obtainedas the complete intersection of two global sections of O P × P (1 , S to be the surface in P × P given by the equations f = x x − x x and g = x x − x x , where x ij is the j -th coordinate on the i -th copy of P . In this way, S is thesurface obtained by blowing up the points P = (1 : 0 : 0) , P = (0 : 1 : 0) and P = (0 : 0 : 1) of P and the exceptional divisors E i are given by E := V ( x , x , x ) , E := V ( x , x , x ) and E := V ( x , x , x ) . We define S = S = S and embed X = S × S in ( P ) using x i , x i and x i asprojective coordinates of the i − th P for i = 1 , , ,
4. Let P be the point P := (( x , x , x ) , ( x , x , x ) , ( x , x , x ) , ( x , x , x )) . Consider the automorphism of X defined by g ( P ) = (( x : x : x ) , ( x : x : x ) , ( x : x : x ) , ( x : x : x )) g ( x , x , x , x ) = ( x , x , x , x ) . It is easy to check that g = g = Id and g g = g g hence G = < g , g > (cid:39) Z (cid:110) Z :=: Dic , G is given by the union ofFix( g ) = (cid:8) ( Q , Q ) | Q , Q ∈ (cid:8) (1 : a : a ) × (1 : a : a ) | a = 1 (cid:9)(cid:9) and Fix( g ) = { ( T × T × Q × Q | T, Q ∈ { (1 : ± ± }} so we have a total of 25 fixed points.We are looking for a global section s of O X ( − K X ) that is G − invariant andwhose zero-locus V ( s ) is smooth and doesn’t intersect Fix. We have an exactsequence 0 → < f, g >(cid:44) → H ( P × P , O (1 , (cid:16) H ( S, − K S ) → ι : S → P × P . Hence, we have asurjection H (( P ) , O (1 , , , (cid:16) H ( X, − K X )with kernel given by < f , g > · H (( P ) , O (0 , , , < f , g > · H (( P ) , O (1 , , , . The representation of Dic in H ( X, − K X ) (cid:39) C has an invariant space H ( X, − K X ) G of dimension 5. By direct inspection, we have checked that thegeneric invariant section s doesn’t intersect Fix and is smooth. Then Y = V ( s )is a Calabi-Yau threefold with a free action of Dic .If we call R the representation of Dic in H ( Y, C ) (cid:39) H ( X, C ) given by g i (cid:55)→ g ∗ i ∈ GL( H ( X, C )) (cid:39) GL( H ( S, C ) ⊕ H ( S, C )) (cid:39) GL( C ) we have R ( g ) (cid:33) A := and R ( g ) (cid:33) A := − − − − − − − − − , where we used the base { H , E , E , E , H , E , E , E } π ∗ i ( E j ) = E ij and π ∗ i ( π ∗ H ) = H i . Hence dim H ( Y, C ) G = 1, so wehave h , ( Y /G ) = 1. By Theorem 3.1 we know that χ ( Y ) = −
72 and then χ ( Y /G ) = − Y /G
10 00 1 01 4 4 1 , h ( Y /G ) = 5. Note that, because h , ( Y /G ) = 1, this example achievesthe minimum of the height for the quotient
Y /G , where G is isomorphic to the Dic and Y is as above. It is interesting to note that taking g (cid:48) ( P ) = (( x : x : x ) , ( x : x : x ) , ( x : x : x ) , ( x : x : x ))the group G (cid:48) spanned by g and g (cid:48) is cyclic of order 12 and a generator is g (cid:48) g := g . Following the same argument as the previous case it can be shownthat exist a Calabi-Yau Y such that G (cid:48) acts on Y freely. The quotient Y /G (cid:48) ishence again a Calabi-Yau and has the same Hodge diamond as
Y /G . Howeverthese two manifolds aren’t even diffeomorphic because Π ( Y /G ) (cid:39) G (cid:39) Dic (cid:54)(cid:39) Z (cid:39) G (cid:48) (cid:39) Π ( Y /G (cid:48) ). dP × dP with maximal order Suppose S and S del Pezzo surfaces of degree 3. Then − K S i is very ample andgives an embedding in P . The surface obtained is a cubic (is called anticanonicalmodel of S i ) and all smooth cubic surfaces in P can be obtained in this way.Set f := x + x + x + x and f := y + y + y + y and consider the Fermatsurfaces S i := V ( f i ). Denote, as usual, X = S × S ⊂ P × P and considerthe automorphism given by ϕ ( x, y ) = (( x : x : x : ωx ) , ( y : y : y : ω y ))where ω (cid:54) = 1 is a fixed root of z −
1. The group G = < ϕ > is cyclic of order 3;hence we haveFix( ϕ ) = Fix( G ) = (cid:8) ((1 : ω : ω : c ) , (1 : ω : ω : d )) | d = c = − (cid:9) . There is an isomorphism H ( P × P , O (1 , (cid:39) H ( X, − K X )11o we have to study the polynomial of bidegree (1 , Z on X gives a representation of Z in H ( X, − K X ) and a basis for the invariant spaceis { G , G , G , G , G , G } where G = ωx y + ω x y + x y , G = ω x y + ωx y + x y ,G = x y + x y + x y , G = x y + x y + x y ,G = x y + x y + x y and G = x y . By direct computation, one can check that the generic section s doesn’t intersectFix( Z ) hence the action of G restricted to V ( s ) is free. For example, taking s to be G + G = x y + x y + x y + x y gives a section whose zero locus Y is smooth and Y ∩ Fix( Z ) is empty.Assume ϕ ∈ Aut( S ) × Aut( S ) with o ( ϕ ) = 3. By the Lefschetz fixed-pointtheorem, one can show that h , ( Y ) G = 2 + 23 ( χ (Fix( π ◦ ϕ )) + χ (Fix( π ◦ ϕ ))) , where π i : X → S i is the projection onto the i − th factor of the product X .In fact, by Lefschet’s Hyperplane Theorem, the group H , ( Y ) is isomorphic to H ( X ). The dimension of the space of invariants with respect to G is equalto the traces of the homomorphisms induced on the second cohomology groupof X = S × S by the elements of G . By linear algebra and the K¨unnethformula, the traces on the cohomology groups of the product X is the sum ofthe traces on the cohomology on the factors H ( S i ) for i = 1 ,
2. These tracescan be computed via the Lefschetz fixed-point Theorem. In this case we obtain h , ( Y /G ) = 6. The same number could be obtained by studying the invariantspace of H ( X, C ) with respect to the representation of Z given by ϕ (cid:55)→ ϕ ∗ (cid:33) (cid:20) A A (cid:21) where A and A are respectively − − − − − −
20 0 0 0 − − − − − −
10 0 − − − − − −
11 1 1 0 1 2 3 and − − −
10 0 0 0 0 1 0 − − − − − − − − − − − −
10 0 − − −
11 1 1 1 2 0 3 . By Theorem 3.1 we have χ ( Y / Z ) = − / −
6; so the Hodge diamond of the12uotient is the foolowing one 10 00 6 01 9 9 10 6 00 01In particular the height is h ( Y / Z ) = 15.As shown in [7], up to isomorphism of P , there are 3 possible pairs ( f, G ) where f is a homogeneous polinomial of degree 3 and G is a group fixing f of order 3.One can show that Fix( f ) is either one of the following: 3 points or 6 points,or one line. Thus, the least value that can be assumed by χ (Fix f ) is 3 if weexclude the case with one line of fixed points. Hence, the example presentedhere achieves the minumum for h ( Y /G ). We present some new examples. ( P × P ) × ( P × P ) with maximal order Take S = S = P × P and define X to be S × S . We begin to search for agroup H ≤ (Aut( P )) (cid:110) S ≤ Aut( X ) such that | H | = 8 and | Fix( H ) | < ∞ .Moreover, we want a section s that is an eigenvector for the action of H on H ( X, − K X ) and does not intersect Fix( H ). After that, we try to extend H toa group of order 16 with the same properties.Let g ∈ (Aut( P )) (cid:110) S be an element of finite order. Without loss of generality,we can take g of the form˜ g ◦ σ := (cid:18)(cid:18) a (cid:19) , (cid:18) a (cid:19) , (cid:18) a (cid:19) , (cid:18) a (cid:19)(cid:19) ◦ σ where σ ∈ S and a i ∈ C ∗ for i = 1 , , , o ( g ) of g is 2, we can choose σ ∈ { Id , (12) , (12)(34) } . An easy checkshows that (( x : y ) , ( x : a y ) , (1 : 0) , (1 : 0))is a line of fixed points if σ = (12) or σ = (12)(34); so we must take σ = Id.The only possible case is a j = − g = g := (cid:18)(cid:18) − (cid:19) , (cid:18) − (cid:19) , (cid:18) − (cid:19) , (cid:18) − (cid:19)(cid:19) g ) = { ( P , P , P , P ) | P i ∈ { (1 : 0) , (0 : 1) }} . If o ( g ) = 4, we can take σ ∈ { Id , (12) , (12)(34) , (1234) } . The automorphism σ cannot be a permutation of order 4. In fact, in this case g would have afixed line, as previously showed. Then, we have Fix( g ) ⊂ Fix( g ) = Fix( g ).Suppose σ = Id or σ = (12) and consider an eigenvector s ∈ H ( X, − K X ) = O X (2 , , , X ). The condition o ( g ) = 4 is then equivalent to a j = 1 for σ = Id and a a = a = a = 1 for σ = (12). Necessarily g satisfies g = g and this implies respectively a j = − a a = a = a = −
1. One can seethat for all P ∈ Fix( g ) there exists a unique element e i of the usual basis of O X (2 , , , X ) such that e i ( P ) (cid:54) = 0. For example, we have x x x x | (1:0) = 1 and x x x y | ((1:0) , (0:1)) = 1 . Then s has to be an element of the eigenspace of both x x x x and x x x y ,but these have different eigenvalues ( 1 and a = − s = 0.Suppose that σ = (12)(34). The conditions o ( g ) = 4 and g = g show that g has to be of the form (cid:18)(cid:18) a (cid:19) , (cid:18) − a − (cid:19) , (cid:18) − a (cid:19) , (cid:18) − a − (cid:19)(cid:19) ◦ (12)(34)for some a , a ∈ C ∗ .Finally, take g to be an automorphism of order 8. Then σ has to be a per-mutation of order
4. For example, pick σ = (1324) (that gives the followingconditions on the a i ’s: a a a a = −
1) and let a = a = a = − a = 1. Abasis for H ( X, − K X ) is given by { e , . . . e } , where e = x x y y x y + x y x x y y − x y y x x y + y x y x y x ,e = x x y x y y − x y x y x y + x y y x y x + y x y x x y ,e = x y x y + x y y x + y x x y + y x y x ,e = − x y x y x y + x y x y x y − x y x y y x + y x x y x y ,e = x x x y + x x y x + x y x x + y x x x ,e = x x y x x y + x x y x y x − x y x x x y + x y x x y x ,e = x x x x ,e = y y y y ,e = x y y y + y x y y + y y x y + y y y x ,e = x x y y + y y x x ,e = x y y x y y − x y y y x y + y x y x y y + y x y y x y . By this result, one can show that an element of order 16 cannot exist in (Aut( P )) (cid:110) S with the request we made. In fact, if such g existed, g would have order 8 and g =( A , A , A , A ) ◦ σ with σ permutation of order 4. This is not possible for an element of S . H . Define h to be the involution of ( P ) suchthat ( x i : y i ) (cid:55)−→ ( y i : x i ) . An easy check shows that gh = hg and that the following hold:Fix( h ) = { ((1 : ± , (1 : ± , (1 : ± , (1 : ± } and Fix( g h ) = { ((1 : ± i ) , (1 : ± i ) , (1 : ± i ) , (1 : ± i )) } . For every k (cid:54) = 0 , g k h ) = g k h = g k so Fix( g k h ) ⊂ Fix( g ) =Fix( g ) . This means that, defining G to be the group generated by g and h ,Fix( G ) is a finite set composed of 48 points and G (cid:39) Z × Z .If we take s = (cid:88) i =1 C i e i and impose both s ( P ) = 1 for all P ∈ Fix( g ) and h ∗ ( s ) = s , we have thefollowing conditions on the C i ’s: C = C = C = C = C = C = 1 , C = C , C = C , C = 0 . By evaluating at the other fixed points, we obtain 4 different non identically-zero linear-combinations of the C i ’s; so the generic invariant section does notintersect Fix( G ). For example, the section obtained by taking C = 1 and C = 2 fulfills all our requests. Moreover, it is smooth, so there exists a groupof order 16 = M ( P × P , P × P ) that acts freely on a Calabi-Yau threefoldembedded in ( P ) .The representation of G on H ( Y, C ) is given by g (cid:55)→ g ∗ (cid:33) and h (cid:55)→ h ∗ (cid:33) so both h and g are trivial on H ( Y, C ) = H ( P , C ) ⊕ .This action has then a unique fixed class in H ( Y, C ) (the sum of the four P ’s).By Theorem 3.1, we have χ ( Y /G ) = − /
16 = −
8, so the Hodge diamond of15he quotient
Y /G is the following one: 10 00 1 01 5 5 10 1 00 01In particular, the height is 6 and it’s the least possible for a quotient of aCalabi-Yau in ( P ) because h , ( Y /G ) = 1. dP × dP with maximal order As proved, for instance, in [7], every del Pezzo surface of degree 4 can be obtainedas a complete intersection of two quadrics of P . Moreover, one can choose theequations to be of the form f = x + x + x + x + x and g = a x + a x + a x + a x + a x where a i (cid:54) = a j ∈ C for i (cid:54) = j . We choose g = x − ix − x + ix and S (cid:39) S (cid:39) S = V ( f, g ) ⊂ P . Let r be the automorphism which sends( x, y ) to the point(( x : x : − x : x : − x ) , ( y : y : − y : y : − y )) . Denote by t the automorphism which sends ( x, y ) to(( y : y : − y : − y : y ) , ( x : x : x : x : x )) . Consider the groups H = < r, t > (cid:39) Z × Z and G = < r, t > (cid:39) Z × Z .By adjunction − K S × S := − K X (cid:39) O X (5 , ⊗ O X ( − , −
4) = O X (1 , ι : S × S −→ P × P induces an isomorphism ι ∗ : H ( P × P , O (1 , −→ H ( S × S, O X (1 , { x i y j } ≤ i,j ≤ as a basis of the space of sections of the anticanonical bundle. It is easy to seethat the vector space V spanned by { x y , x y , x y , x y , x y , x y , x y }
16s such that for all h ∈ H and for all s ∈ V , h ∗ ( s ) = λs for some λ ∈ C ∗ . Bytaking the generic section s ∈ V and imposing r ∗ s = t ∗ s = s (so that for everyautomorphism g of G , V is an eigenspace with respect to g ∗ ), we obtain s = A x y + A y x + A x y + A x y + A x y , where A i ∈ C . Let a and b be fixed roots of 2 z + 1 + i and 2 z + 1 − i ,respectively. Then F ix ( r ) = { ( P, Q ) | P, Q ∈ { (1 : ± a : 0 : ± b : 0) }} F ix ( t ) = { ( P, Q ) | P, Q ∈ { ( ± a : ± b : 0 : 0 : 1) }} and F ix ( rt ) = { ( P, Q ) | P, Q ∈ { ( ± b : 1 : ± a : 0 : 0) }} . To look for the fixed points of G it suffices to know the fixed points of r, t and rt . In fact, the following holds:Fix( t ) = Fix( t ) ⊆ Fix( t ) = Fix(( rt ) ) ⊇ Fix( rt ) = Fix( rt ) . An easy check shows that for generic values of A , A , A and A , the section s does not intersect Fix( G ).We can check directly that the section corresponding to A = 1 , A = − , A =3 and A = 1 is smooth and doesn’t intersect Fix( G ); so there exists a Calabi-Yau threefold Y embedded in S × S with Z × Z acting freely on Y .We don’t have an explicit description of a basis for Pic( Y ) = Pic( S ) ⊕ Pic( S ) (cid:39) Z , but we can use the Lefschetz Fixed Point fomula to get the traces we needto compute h , ( Y ) G . For example, notice that r = r × r with r i ∈ Aut( S i );so the trace of r ∗ : H ( S × S , C ) → H ( S × S , C ) is equal to the sum of thetraces of r ∗ i : H ( S i , C ) → H ( S i , C ) . By recalling that16 = χ (Fix( r )) = χ (Fix( r × r )) = χ (Fix( r i )) and by Lefschetz Fixed Point formula, we haveTr( r ∗ ) = Tr( r ∗ ) + Tr( r ∗ ) = χ (Fix( r ∗ )) − χ (Fix( r ∗ )) − −
2) = 4 . With the same method we obtain Tr(( t ∗ ) ) = Tr( r ∗ ( t ∗ ) ) = 4. We can write t as( t × t ) ◦ σ where σ is the the permutation of the two copies of S . Hence t ∗ willswap H ( S ) and H ( S ) in the sum H ( S ) ⊕ H ( S ) and this means that itstrace is zero. In the same way we obtain Tr(( t ∗ ) ) = Tr( r ∗ t ∗ ) = Tr( r ∗ ( t ∗ ) ) = 0.Merging these results and recalling that χ ( Y ) = −
32, we obtain h , ( Y /G ) = 12 + 4 + 4 + 4 + 0 + 0 + 0 + 08 = 3 and h , ( Y /G ) = 517o the quotient has the following Hodge diamond10 00 3 01 5 5 10 3 00 01In particular, the height is 8. P × P with maximal order Let ( x : x : x ) and ( y : y : y ) be the projective coordinates on the twocopies of P and set a = e πi/ . Consider the automorphism of P × P := X defined by g := ( x : ax : a x ) × ( y : ay : a y ) := g × g and h := ( x : x : x ) × ( y : y : y ) := h × h . It is easy to show that the group G generated by g and h is isomorphic to Z × Z .Moreover, it is easy to see thatFix( G ) =(Fix( g ) × Fix( g )) ∪ (Fix( h ) × Fix( h )) ∪ (Fix( g h ) × Fix( g h )) ∪ (Fix( g h ) × Fix( g h ))where Fix( g i ) = { (1 : 0 : 0) , (0 : 1 : 0) , (0 : 0 : 1) } Fix( h i ) = (cid:8) (1 : 1 : 1) , (1 : a : a ) , (1 : a : a ) (cid:9) Fix( g i h i ) = (cid:8) (1 : 1 : a ) , (1 : a : 1) , ( a : 1 : 1) (cid:9) Fix( g i h i ) = { (1 : 1 : a ) , (1 : a : 1) , ( a : 1 : 1) } . Consider the following global sections of O P (3) = − K P : e i, = x + a i x + a i x e i, = x x + a i x x + a i x x e i, = x x + a i x x + a i x x e = x x x Then g ∗ ( e i,j ) = a j e i,j , h ∗ ( e i,j ) = a i e i,j , g ∗ ( e ) = h ∗ ( e ) = e ; hence { e , e i,j } ≤ i,j ≤ is a basis of H ( P , O P (3)) composed of eigenvectors of both g ∗ and h ∗ . Since H ( X, − K X ) (cid:39) H ( P , − K P ) ⊗ H ( P , − K P ) ,
18 basis for the space of invariant sections is given by { e i ,j ⊗ e i ,j } i + i ≡ ,j + j ≡ ∪ { e ⊗ e , , e , ⊗ e , e ⊗ e } . By direct computation, we can show that the generic invariant section doesn’tintersect Fix( G ). Moreover, the system | H ( X, − K X ) G | is base-point free. ByBertini’s Theorem, the generic section is smooth. Hence there exists a Calabi-Yau threefold Y embedded in P × P equipped with a free action of G .The space H ( X, Z ) is free of rank two and is generated by π ∗ H and π ∗ H where < H > = H ( P , Z ). Every automorphism of P fixes H , so H ( X, C ) G = H ( X, C ). This implies that the following is the Hodge diamond of Y /G :10 00 2 01 11 11 10 2 00 01 . Its height is 13. An element g ∈ Aut( P × P ) = (Aut( P ) × Aut( P )) (cid:110) Z of order 3 has to be of the form g = g × g with g i ∈ Aut( P ). This meansthat H ( X, Z )
Y /G with Euler characteristic −
10. As in section 5.1 we compute h , ( Y /G ) using the Lefschetz Formula andwe obtain h , ( Y /G ) = 2. Then h , ( Y /G ) = 7 and the Hodge diamond is thefollowing one: 10 00 2 01 7 7 10 2 00 01Note that
Y /G realize the minimum for the height. P × P × dP with maximal order Let us consider again the del Pezzo surface S of degree 4 embedded in P usedin section 5.2. If we denote with g and h the automorphism of S = P × P such that g (( x : x ) , ( x : x )) = (( x : − x ) , ( x : − x ))and h (( x : x ) , ( x : x )) = (( x : x ) , ( x : x )) ,
21e obtain the relation g = h = g h g − h − = Id that is < g , h > (cid:39) Z ⊕ Z .The same holds for the automorphism g and h of S such that g (( y : y : y : y : y )) = ( y : y : − y : y : − y )and h (( y : y : y : y : y )) = ( y : y : − y : − y : y ) . Denote by g = g × g and h = h × h ; hence we have G := < g, h > (cid:39) Z ⊕ Z .We recall (see Section 5.2) that if a and b are fixed roots of 2 z + 1 + i and2 z + 1 − i then Fix( g ) = { (1 : ± a : 0 : ± b : 0) } Fix( h ) = { ( ± a : ± b : 0 : 0 : 1) } and Fix( g h ) = { ( ± b : 1 : ± a : 0 : 0) } . It is easy to see that | Fix( α ) | = 4 for each α ∈ < g , h > \ { Id } and, conse-quently, that | Fix( G ) | = 48.Analogously to the previous cases, we can conclude that there exists a smoothCalabi-Yau threefold Y ⊂ X and a group G (cid:39) Z ⊕ Z acting freely on it. Thequotient has the following Hodge diamond10 00 5 01 13 13 10 5 00 01 . Hence the height of the quotient is 18.
For brevity we don’t treat explicitly some examples. These are some threefoldsin P × dP , P × dP , ( P × P ) × dP and dP × dP . The threefolds in P × dP and in P × dP admit a free action of Z (in both cases M ( S , S ) = 3). The22uotients have Hodge diamonds respectively:10 00 3 01 21 21 1 and0 3 00 01 10 00 4 01 13 13 10 4 00 01These are threefolds with minimal height. The threefolds in ( P × P ) × dP andin dP × dP admit a free action of Z (again this hits the maximum because M ( S , S ) = 2 for these two cases). The Hodge diamonds are10 00 5 01 29 29 1 and0 5 00 01 10 00 7 01 19 19 1 . In this section we present some results of non-existence. In particular, we showthat there are cases for which M ( S , S ) > G that fulfills ourrequests doesn’t exist. dP × S , with S ∈ { P × P , dP , dP , dP , dP } We will show that in these cases m ( S , S , Y ) = 1 for all Y . The key points areCorollary 6.2 and some structural results on Aut( dP ). Lemma 6.1. If S is a del Pezzo surface and g ∈ Aut( S ) is such that o ( g ) = p is prime, then g has a fixed point.Proof. Every del Pezzo surface S is a rational surface. Suppose that the fixedlocus of g is empty. Recall that p is prime. Let G := < g > be the groupgenerated by g . Then Fix( G ) is empty. In fact, for every n (cid:54)≡ p m such that nm ≡ p
1; this impliesFix( g n ) ⊂ Fix(( g n ) m ) = Fix( g ) . R := S/G is a smooth surface and R is rational. In particular Π ( R ) = { Id } . But this is not possible because S is simply connected, so Π ( R ) (cid:39) G (cid:54)(cid:39){ Id } . Hence, g must have at least one fixed point. Corollary 6.2.
For every finite subgroup G of Aut( S ) , | Fix( G ) | > . By [7], every automorphism of a del Pezzo surface S of degree 8 comes from anautomorphism of P that fixes the point R such that S = Bl { R } P . Suppose S (cid:54) = dP . Then we search for a group G ≤ Aut( dP ) × Aut( S ). We are interestedin the cases S ∈ (cid:8) P × P , dP , dP , dP (cid:9) for which M ( dP , S ) is respectively16 , , Y . Let g = ( g , g ) be an involution. By Corollary6.2 there exists a fixed point P of g . The automorphism g comes from aninvolution of P , hence it has a line L of fixed points, therefore L × { P } is a lineof fixed points for g .If S = dP , then Aut( dP × ) = Aut( dP ) × (cid:110) Z . Let G = < g > where g = ( g , g ). Using the same result as above, we will have a surface of fixedpoints. Then, it suffices to analyze the case g = ( g , g ) ◦ τ , where τ is theinvolution that switches the two copies of dP . Then, by changing projectivecoordinates, we can assume that( g , g ) = a b
00 0 1 , a − b −
00 0 1 for some a, b ∈ C ∗ . It is easy to see that (( ax : by : 0) , ( x : y : 0)) is a line offixed points.In conclusion, we have shown that m ( dP , S, Y ) = 1 for a del Pezzo surface S (here we have checked all the cases for which M ( dP , S ) (cid:54) = 1) and for all Y Calabi-Yau embedded in dP × S . dP × dP with estimated maximal order There is only one del Pezzo surface S of degree 7. It is given as the blow-up of P in P = (1 : 0 : 0) and P = (0 : 1 : 0). We will show that there does not exista section s of − K S × S such that g ∗ s = cs for some c ∈ C ∗ and g ∈ Aut( S × S )of order 7 which doesn’t intersect the fixed locus of < g > .By [7], every automorphism of a del Pezzo surface of degree 7 comes from anelement of P GL (3) fixing the set { P , P } . Thus, we haveAut( S ) (cid:39) (cid:42) b a c d , (cid:43) . Recall that Aut( S × S ) = Aut( S ) × (cid:110) Z . Since we need g of order 7, we haveto choose an element of the form g = ( g , g ), where g i ∈ Aut( S ) and g i = b i a i c i d i . P and P , we mayassume b i = c i = 0 so that g i is in diagonal form. The condition o ( g ) = 7 gives a i = d i = 1. Since we need a finite number of fixed points, we must impose a i (cid:54) = 1 (cid:54) = d i and a i (cid:54) = d i .In conclusion, we can take g of the form λ m
00 0 λ n × λ m
00 0 λ n where λ = e πi/ and 0 (cid:54) = n i , m i and n i (cid:54) = m i .The fixed points of g i as an automorphism of P are P , P and P , whereas thefixed points of g i as an automorphism of S are { ( P , Q ) , ( P , Q ) , P | Q ∈ { (1 : 0) , (0 : 1) }} . Here, for example, with ((0 : 1 : 0) , (1 : 0)) we mean the point (1 : 0) on theexceptional divisor E = π − ( P ), where we use the standard local descriptionof S in a neighbourhood of E as the surface of C × P such that um = vl with { ((0 , , ( l : m )) } = E . Hence, in total, G := < g > has 25 fixed points.We blow up P in P and P . Then, the following isomorphism holds: H ( S, − K ) (cid:39) < x , x x , x x , x x , x x , x x , x x , x x x > . The correspondence is given by taking the strict transform of a polynomial seeas a global section of O P × P (3 , e i the elements of the base on thefirst del Pezzo surface and f i the elements of the base on the second one so that,by the K¨unneth formula, we obtain H ( S × S, − K S × S ) (cid:39) < e i ⊗ f j > . Suppose that s is an eigenvector of H ( S × S, − K S × S ) and that s ( P ) (cid:54) = 0 forall P fixed points of G . Then, for example, s (((1 : 0 : 0) , (1 : 0)) , ((1 : 0 : 0) , (1 : 0))) (cid:54) = 0if and only if s belongs to the eigenspace of x x y y and s (((1 : 0 : 0) , (1 : 0)) , ((1 : 0 : 0) , (0 : 1))) (cid:54) = 0if and only if s is in the eigenspace of x x y y . But these two eigenvectors havecorresponding eigenvalues λ m + m and λ m + n and these numbers are differentif and only if m (cid:54) = n , which it is true by hypotesis. This means that s mustbe zero and we have a contradiction.Albeit M ( dP , dP ) = 7, this shows that an automorphism of S × S with finiteorder cannot act freely on a smooth section of − K S × S .25 .3 dP × dP with estimated maximal order In this case recall that M ( dP , dP ) = 9. Nonetheless, the maximum order of G to have a free action on a Calabi-Yau threefold Y embedded in X is 3. Wewill also give an example for which m ( dP , dP , Y ) = 3.Suppose that G ≤ Aut( dP ) × Aut( dP ) has order 9. Then either G (cid:39) Z or G (cid:39) Z × Z . First, we will show that if G (cid:39) Z then G must have a fixedcurve and so it can’t satisfy our assumption on G . Next, we will deal withthe case G (cid:39) Z × Z . We’ll first find all the groups whose fixed locus is finite.Essentially, this will be done by projecting G on Aut( dP ) and Aut( dP ) so thatthe projections G and G satisfy G (cid:39) G (cid:39) G (cid:39) Z × Z . There is only oneuseful choice for G = < g , h > whereas there are infinitely many possibilitiesfor G , which are parametrized by ( C ∗ ) . Once we fix G := < u, v > , we willconsider all the possible G (cid:48) s such that the projection of G on Aut( dP ) andAut( dP ) are G and G , respectively. This will be done by choosing all thepossible pairs ( g , h ), not necessarily equal to ( u, v ), that generate G . We thusconsider the group G := < g, h > , where g = g × g and h = h × h . For everycase we have checked that all the sections of H ( X, − K X ) that are eigenvectorsof both g ∗ and h ∗ are zero on a fixed point of the group G (we will show anexplicit calculation for one of the cases).Suppose that G (cid:39) Z and consider its projection G on Aut( dP ). Necessarily, G (cid:39) G . On the contrary, if G = < g × g > with g = Id, G would haveinfinitely many fixed points. Hence G has to be a group isomorphic to Z inAut( dP ). If S is a smooth cubic surface in P and if g ∈ Aut( S ) has order 9then, by [7], there exist a projective automorphism of P such that( S, g ) = V ( x + x x + x x + x x ) , a a
00 0 0 a where a satisfies a (cid:54) = 1 = a . On the other hand, we have g = a a
00 0 0 a . Hence Fix( < g > ) contains a curve C . This means that, by Corollary 6.2, wehave a fixed curve in Fix( G ), which contradicts our assumptions.Suppose, now, that G (cid:39) Z × Z ≤ Aut( dP ) × Aut( dP ) and consider theprojection G on Aut( dP ) so that G (cid:39) G . Fix two generators g , h of G and consider dP = V ( f ) ⊂ P . By [7], if V ( f ) is a smooth cubic and ˜ G (cid:39) Z × Z ≤ Aut( V ( f )) , we can change coordinates to obtain f = (cid:80) y i . In thiscase Aut( V ( f )) (cid:39) Z (cid:110) S , where each Z acts as multiplication of a variableby a k (we write the elements in Z as (1 , a k , a k , a k )) and S = Sym(0 , , , | Fix( G ) | < ∞ we obtain G ≤ Z . There is only one group isomorphic to G in Z that has a26nite number of fixed points on V ( f ) and it is < g , h > where g = (1 , , a, a )and h = (1 , a, a , a ). We call V (2) i,j the maximal subspace of H ( dP , − K dP )such that g ∗ ( s ) = a i s and h ∗ ( s ) = a j s for every s ∈ V (2) i,j . This vector space isthe intersection of the eigenspaces Λ a i of g and Λ (cid:48) a j of h relative to a j . Thefollowing table summarizes the situation providing generators for these spaces. g \ h Λ (cid:48) Λ (cid:48) a Λ (cid:48) a Λ x Λ a x Λ a x x Now, consider the projection G of G on Aut( dP ) = ( S × Z ) (cid:110) ( C ∗ ) . Anyelement of order 3 can be written in the form diag(1 , b, c ) ◦ (123) k for somefixed b, c ∈ C ∗ and k = 0 , ,
2. Easy arguments show that G cannot satisfy G ≤ ( C ∗ ) (if it happens, one has | Fix( G ) | = ∞ ) and that G has exactly twonon-trivial elements in ( C ∗ ) . These are diag(1 , a, a ) and its inverse. Moreover,these two elements commute with every element of the form (1 , b, c ) ◦ (123) k ,thus every subgroup of Aut( dP ) isomorphic to Z × Z and with a finite numberof fixed points can be written in the form < u, v > where u = diag(1 : a : a ) and v = diag(1 : b : c ) ◦ (123)for some fixed b, c ∈ C ∗ . We define d to be a fixed third root of bc . Set F = x x ,F = x x + 1 b x x + 1 c x x ,F = x x + 1 c x x + bc x x ,F = x x + a b x x + ac x x ,F = x x + a c x x + abc x x ,F = x x + ab x x + a c x x ,F = x x + ac x x + a bc x x . Then F j is an eigenvector of both u and v and the corresponding eigenvaluesare the ones in the following table: u \ v Λ Λ a Λ a Λ F F F Λ a F F Λ a F F This shows that { F j } form a base for H ( dP , − K dP ). The following are the27xed points of the elements of G and G :Element Fixed points ( k = 0 , , , (0 : 1 : 0)) , ((1 : 0 : 0) , (0 : 0 : 1)) ,u, u ((0 : 1 : 0) , (1 : 0 : 0)) , ((0 : 1 : 0) , (0 : 0 : 1)) , ((0 : 0 : 1) , (1 : 0 : 0)) , ((0 : 0 : 1) , (0 : 1 : 0)) v, v ((1 : da k : ( da k ) b , (1 : da k : b ( da k ) ) uv, u v ((1 : da k : ( da k ) ba , (1 : da k : ba ( da k ) ) u v, uv ((1 : da k : ( da k ) ba , (1 : da k : ba ( da k ) )Element Fixed points ( k = 0 , , g , g (1 : − a k : 0 : 0) h , h (0 : 0 : 1 : − a k ) g h , g h (1 : 0 : − a k : 0) , (0 : 1 : 0 : − a k ) g h , g h (1 : 0 : 0 : − a k ) , (0 : 1 : − a k : 0)Suppose g = u . Let h be any element of G such that G = < g , h > anddenote Q = ((1 : 0 : 0) , (0 : 1 : 0)) and Q = ((1 : 0 : 0) , (0 : 0 : 1)). Then P := ((1 : 0 : 0) , (0 : 1 : 0) , (1 : − P := ((1 : 0 : 0) , (0 : 0 : 1) , (1 : − g = g × g . Suppose that s = (cid:88) i,j a i,j F i y j is a section such that g ∗ ( s ) = a k s and that s ( P j ) (cid:54) = 0. Then s ( P ) = (cid:88) i =2 , , ( a i, − a i, ) F i ( Q ) (cid:54) = 0and s ( P ) = (cid:88) i =1 , , ( a i, − a i, ) F i ( Q ) (cid:54) = 0 . This means that at least one between x i F j with i = 0 , j = 2 , , x i F j with i = 0 , j = 1 , ,
5. But,if i = 0 , g ∗ ( x i F j ) = a x i F j if j = 2 , , g ∗ ( x i F j ) = ax i F j if j = 1 , , g is zero if evaluated in P or in P .The same result is true for every other case: we have checked that, for every b, c ∈ ( C ∗ ), for every choice of g , h generators of G = < u, v > , every sectionof H ( X, − K X ) that is an eigenvector of both g and h where g = g × g and h = h × h is zero on at least one fixed point of G = < g, h > . Inconclusion the restriction of the action of a group G ≤ Aut( dP ) × Aut( dP )of order 9 to a Calabi-Yau threefold Y ⊂ dP × dP cannot be free. Hence m ( dP , dP , Y ) < M ( S , S ) = 9 for every Y .28e have obtained m ( dP , dP , Y ) ≤ Y . We now give an example suchthat m ( dP , dP , Y ) = 3. Take dP to be the Fermat surface in P . Call g theautomorphism of dP such that x i,j (cid:55)→ x i,j +1 and g the authomorphism ω
00 0 0 ω of dP . Notice that the minimum for the number of fixed points for an auto-morphism of order 3 in Aut( dP ) × Aut( dP ) is achieved by g = g × g . Thedimension of H ( X, − K X ) G , where G = < g > , is 10. It can be shown that thebase locus for | H ( X, − K X ) G | has only 9 points and that these are((1 : ω i : ω i ) , (1 : ω i : ω i ) , (0 : 0 : − ω j : 1))with 0 ≤ i, j ≤
2. By direct inspection, the generic invariant section s is smoothat these points and does not intersect the fixed locus, so, by Bertini’s Theorem,there exists a Calabi-Yau Y embedded in dP × dP and a group G (cid:39) Z actingfreely on Y . The Hodge diamond for Y /G is10 00 5 01 11 11 10 5 00 01and it’s height is 16, that is the minimum for the height.
Aut( S ) × Aut( S ) and Aut( S × S ) Let X be a projective complex manifold. We will denote by NE( X ) the cone ofeffective curves of X . An extremal subcone V of NE( X ) is a closed convex conesuch that for every v, w ∈ NE( X ) if v + w ∈ V then v, w ∈ V . An extremalray is an extremal subcone of dimension 1. For every D divisor on X a subcone V ⊂ NE( X ) is said to be D − negative if for every v ∈ V one has v · D < K X -negative subcone V of NE( X ) the contraction c V of V is well defined, that is to say, a morphism c V : X → W with connected fibers such that W is a normal variety. Moreover,a curve in X is contracted if and only if is numerically equivalent to a curve in V and the Picard number ρ ( W ) is equal to ρ ( X ) − dim( < V > ). For a morphism f we recall that NE( f ) is given by the intersection NE( X ) ∩ ker( f ∗ ), where f ∗ is the map induced by f on the vector space spanned by NE( X ).29f φ ∈ Aut( S × S ) we will write φ ( x, y ) = ( φ ( x, y ) , φ ( x, y )) where φ i = π i ◦ φ where π i is the projection of S × S on S i . Lemma 7.1.
Let S and S be two del Pezzo surfaces and let φ ∈ Aut( S × S ) Let π i be the projection from S × S onto the i -th factor S i for i = 1 , .If φ ∗ (NE( π i )) = NE( π i ) , then φ ( x, y ) = ( φ ( x ) , φ ( y )) where φ i ∈ Aut( S i ) .If φ ∗ switches the cones NE( π ) and NE( π ) , then S = S and φ ( x, y ) =( φ ( y ) , φ ( x )) with φ ∈ Bihol( S , S ) and φ ∈ Bihol( S , S ) .Proof. Assume φ ∗ (NE( π i )) = NE( π i ). Fix x , x ∈ S and take two distinctirreducible curves C and C on S whose intersection is non empty and suchthat x i ∈ C i . We have φ ( C i × y ) = D i × y i because the image of C i × y is a curve that is numerically equivalent to a curvein NE( π ). But C × y and C × y are two curves with nonempty intersection sotheir images have nonempty intersection. In particular y = y and this impliesthat φ ( x, y ) = φ ( y ). The same argument works with the first component( φ ( x, y ) = φ ( x )) and with φ − meaning that φ i is an automorphism of S i .With the same method, if φ ∗ switches the two cones, one has φ ( x, y ) = ( φ ( y ) , φ ( x ))and that φ i are biholomorphism thus S = S . Lemma 7.2.
Let S and S be two del Pezzo surfaces such that ρ ( S ) , ρ ( S ) ≥ .If ρ ( S ) (cid:54) = ρ ( S ) then Aut( S × S ) = Aut( S ) × Aut( S ) . The same holds if ρ ( S ) = ( S ) and S (cid:54) = S . Instead, if S = S one has Aut( S × S ) = (Aut( S ) × Aut( S )) (cid:110) Z . Proof.
Call X the product S × S . Then X is a Fano fourfold andNE( X ) = NE( X ) ∩ NE( π , ∗ ) + NE( X ) ∩ NE( π , ∗ ) . In particular, every extremal ray of X is generated by a curve of the type P × E or E × P , where E i is a ( − − curve on S i . Observe that the image V (cid:48) of anextremal subcone V by an automorphism φ is again an extremal subcone. Infact, if v + w ∈ V (cid:48) for some v, w ∈ NE( X ) then φ − ∗ ( v ) and φ − ∗ ( w ) are effectivecurves such that φ − ∗ ( v ) + φ − ∗ ( w ) = φ − ∗ ( v + w ) ∈ V . But if V is extremal both φ − ∗ ( v ) and φ − ∗ ( w ) are in V . This implies that v and w are in V (cid:48) , so V (cid:48) alsois extremal. This implies that φ induces a permutation of the extremal rays of X .Suppose that there exists an extremal curve E × P such that φ ∗ ( E × P ) = P × E . Then φ ∗ maps the extremal ray V := [ E × P ] to the extremalray V (cid:48) := [ P × E ]. The contractions c V and c V (cid:48) associated to the extremalsubcones V and V (cid:48) are respectively p × Id and Id × p , where p i : S i → ˆ S i are the blow up with exceptional divisor E i . Observe that ˆ S i is smooth and30hat the fibers of c V and c V (cid:48) have dimension 0 or 1 and are connected. Byconstruction a curve C is contracted by c V if and only φ ∗ C is contracted by c V (cid:48) . These two facts imply that the map f : ˆ S × S → S × ˆ S such that f ( P ) = ( c V (cid:48) ◦ φ )( c − V ( P )) is well defined. S × S (cid:9) c V (cid:15) (cid:15) φ (cid:47) (cid:47) S × S c V (cid:48) (cid:15) (cid:15) ˆ S × S f (cid:47) (cid:47) S × ˆ S Let’s see that the map f is injective. Call Q i the point of ˆ S i such that p − i ( Q i ) = E i . If f ( Q × R ) = f ( Q × R ) with R (cid:54) = R then, to calculate the imageof Q × R i we obtain first two disjoint curves in S × S of the form E × R i .Then these two are sent to two disjoint curves of the form T i × E by φ and,at last, contracted to the same point by c V (cid:48) . This implies that the fiber of thispoint with respect to c V (cid:48) contains two disjoint curves and, being connected, hasto be at least of dimension 2. But we have seen that every fiber has dimensionat most 1, so necessarily R = R . By construction f is also surjective and soit is a bijective map.The map f is a morphism because it is everywhere well defined and it is holo-morphic outside Q × S that has codimension 2 in ˆ S × S . Hence, by Hartogs’Theorem, it is holomorphic on ˆ S × S . This is enought to conclude that f isan isomorphism. This implies χ ( ˆ S × S ) = χ ( S × ˆ S );but χ ( ˆ S i ) = χ ( S i ) − b ( ˆ S i ) = b ( S ) − χ , we have ( χ ( S ) − χ ( S ) = χ ( S )( χ ( S ) − χ ( S ) = χ ( S ). But this contraddicts the hypothesis ρ ( S ) (cid:54) = ρ ( S ); hencethe image of E × P by φ ∗ has to be of the same type. This implies that φ ∗ NE( π j ) = NE( π j ) and this is sufficient to conclude that φ can be written asa product of two automorphisms by Lemma 7.1.Suppose, now, that ρ ( S ) = ρ ( S ) ≥
3. Fix a blow-up model for S i . Then the( − − curves on S i are either E ij , and are contracted to points by the model,or are sent to curves (lines, conics ( ρ ( S i ) ≥
5) and cubics ( ρ ( S i ) ≥ j , the image of E j × P belongs to [ Q × E ] for some ( − − curve E thatdepends on j , then the same holds true for the other exceptional curves of thesame type: φ ( E × P ) ∈ [ Q × E ] for some E depending on E . Thus, sayingthat there exist two exceptional curves E i × P such that φ ( E × P ) ∈ [ Q × E ]and φ ( E × P ) ∈ [ E (cid:48) × Q ] is equivalent to requiring that there are two indices(for examples j = 1 and j = 2) such that φ ( E × P ) ∈ [ Q × E ] and φ ( E × P ) ∈ [ E × Q ] . Suppose, then, that this could happen. Then, as in the previous case, we can31onstruct a commutative diagram S × S (cid:9) c V (cid:15) (cid:15) φ (cid:47) (cid:47) S × S c V (cid:48) (cid:15) (cid:15) ˜ S × S f (cid:47) (cid:47) ˆ S × ˆ S where c V = r × Id and c V (cid:48) = p × p where r : S → ˜ S is the contractionof two E = r − ( R ) and E = r − ( R ) whereas p and p are the blow-upwith exceptional divisor respectively E and E . Note that the cone V spannedby E × P and E × P is an extremal subcone because for a >> L := O (( aH − E − E ) × S ) is a nef line bundle such that V = NE( S × S ) ∩ L ⊥ .This implies that its image V (cid:48) is extremal. Again, the construction of f makesense because c V (cid:48) contracts a curve if and only if c V contracts its preimage andbecause all the fibers of c V are connected and have at most dimension one.Assume f ( R × Q ) = f ( R × Q ). The fibers E × Q i are mapped to twodisjoint curves of the form ˜ Q i × E and then contracted to the same point. Thenthe fiber S of this point has dimension at least 2 (exactly 2 by construction) andcontains ˜ Q i × E . Recall that − K X | S := D (cid:48) is ample so it intersects ˜ Q i × E . D (cid:48) is then an effective curve that is contracted to a point by c V (cid:48) so its preimage D intersects E × Q i and is contracted by c V . Hence Q = Q . In a similar waywe dealt with the other cases and prove that f is injective. By construction, f is also surjective and hence bijective.Again f is a map that is holomorphic outisde two disjoint smooth subvariety of˜ S × S whose codimension is 2. Thus, by Hartogs’ Theorem, f is everywhereholomorphic. Then f is an isomorphism but checking the equality of the Eulernumbers one obtain2 + ρ ( S ) = 2 + ρ ( S ) = χ ( S ) = χ ( S ) + 1 = 3 + ρ ( S )and then again a contradiction. Hence the two types of extremal rays cannotbe mixed by φ . There are two cases: the first corresponding to the case forwhich ∀ φ ∈ Aut( S ) , φ ∗ NE( π i ) = NE( π i ) and the second where there exists φ ∈ Aut( X ) that switches the two cones. By Lemma 7.1, in the first caseAut( S × S ) = Aut( S ) × Aut( S ) and S (cid:54) = S whereas, in the second, wehave S = S and Aut( S × S ) = Aut( S ) × (cid:110) Z . Lemma 7.3.
Let S and S be two del Pezzo surfaces with ρ ( S ) ≤ and ρ ( S ) ≥ . Then Aut( S × S ) = Aut( S ) × Aut( S ) .Proof. There are three cases: ρ ( S ) = 1 with S = P and ρ ( S ) = 2 with S = P × P or S = dP .If S = P and φ ∈ Aut( X ), fix a point s ∈ S and consider the map obtained ascomposition of the inclusion P (cid:39) P ×{ s } ⊂ P × S , φ and the projection on S .The resulting map β s cannot be a dominant morphism because, in this case, P would have divisors with negative self-intersection . Moreover its image cannot The pullback D of a ( − − line E for example. P → C inducesa surjective map P → P but this cannot exist. Hence β s ( P ) is a point, orequivalently, doesn’t depend on P . Hence φ ( P, s ) = ( α ( P, s ) , β ( s ))and the same holds true for φ − so β ∈ Aut( S ) and, by a composition withId × β − , we can restrict to the case β = Id. Consider now for a fixed s ∈ S the morphism α s : P → P . As before, its image cannot have dimension 1. Ifdim( α s ( P )) = 0 then φ ( P ×{ s } ) ⊂ P t × S , and because φ is an automorphism,we would obtain an isomorphism between P and a del Pezzo surface of Picardnumber strictly greater than 1 which is impossible. Hence α s is a dominantmap. Suppose α s ( P ) = α s ( Q ). Then φ ( P, s ) = ( α s ( P ) , s ) = ( α s ( Q ) , s ) = φ ( Q, s )but φ is injective so P = Q and α s is also injective. This shows that α s is anautomorphism for every s and in particular we have a map f : s ∈ S (cid:55)→ α s ∈ P GL (3) = SL (3) / Z . Then f lifts to a map from S to SL (3) that is affine andthen f doesn’t depend on s . So Aut( P × S ) = Aut( P ) × Aut( S ).If S = P × P then the extremal rays of X = S × S are of the form [( P × P ) × E ] , [( P × P ) × Q ] or [( P × P ) × Q ] where E is a ( − − curve on S . Inparticular (( P × P ) × E ) · ( K X ) = − P × P ) × Q ) · K X = (( P × P ) × Q ) · K X = − . In particular, because extremal rays are permuted by every automorphism andbecause the intersection numbers are preserved, we have φ ∗ (NE( π i )) = NE( π i )and then Aut( S × S ) = Aut( S ) × Aut( S ).If S = dP and ρ ( S ) ≥ E × P ] , [( H − E ) × P ] and [ P × E ] where E is the only ( − − curve on S and E is a( − − curve on S . In particular − K X · (( H − E ) × P ) = 2 whereas for allthe other extremal curves the intersection with − K X is 1; hence φ ∗ fixes thisextremal ray. Assume that φ ∗ ([ E × P ]) = ([ P × E i ]). Then, denoting V = R + [ E × P ] and V (cid:48) = R + [ P × E i ], we obtain the following commutative diagram dP × S (cid:9) c V (cid:15) (cid:15) φ (cid:47) (cid:47) dP × S c V (cid:48) (cid:15) (cid:15) P × S f (cid:47) (cid:47) dP × ˆ S where f is again an isomorphism. This gives χ ( S ) = 4 but ρ ( S ) ≥ χ ( S ) ≥ S i ) = φ ∗ (NE( S i )) and thenAut( S × S ) = Aut( S ) × Aut( S ). Lemma 7.4.
Let S and S be two del Pezzo surfaces such that ρ ( S ) , ρ ( S ) ≤ .Then: • If S (cid:54) = S , Aut( S × S ) = Aut( S ) × Aut( S ) ; If S = S (cid:54) = P × P , Aut( S × S ) = (Aut( S ) × Aut( S )) (cid:110) Z ; • If S = S = P × P , Aut( S × S ) = (Aut( P ) × ) (cid:110) S . Proof. If ρ ( S i ) ≤ S i is a smooth toric variety. For a complete simplicial toricvariety the sequence1 → Aut ( X ) → Aut( X ) → Aut( N, ∆)Π S ∆ i → X ) can be seen as a semidirect productof Aut ( X ) and Aut( N, ∆)Π S ∆ i . The proof will be completed analysing the structureof these two groups.We call ∆ S i ⊂ Z =: N i the fan of S i and denote with ∆ S i (1) = { e , . . . , e r i } the set of the rays of the fan. The following table summarizes the rays of thefans we need. S e e e e e e P [1,0] [0,1] [-1,-1] P × P [1,0] [0,1] [-1,0] [0,-1] dP [1,0] [0,1] [-1,0] [-1,-1] dP [1,0] [0,1] [-1,0] [0,-1] [-1,-1] dP [1,0] [0,1] [-1,0] [0,-1] [-1,-1] [1,1]If ∆ ⊂ Z = N is the fan of X , then ∆(1) = (∆ S × { [0 , } ) ∪ ( { [0 , } × ∆ S ).Aut( N, ∆) will denote the group of the automorphisms of the lattice N thatfixes the fan ∆. By direct computation, we show that • If S (cid:54) = S , Aut( N, ∆) = Aut( N , ∆ S ) × Aut( N , ∆ S ); • If S = S (cid:54) = P × P , Aut( N, ∆) = (Aut( N , ∆ S ) × Aut( N , ∆ S )) (cid:110) Z ; • If S = S = P × P , Aut( N, ∆) = S (cid:110) Z .It is possible to associate a divisor D i to each e i ∈ ∆(1) and we say than e i ∼ e j iff D i and D j are linearly equivalent. Call { ∆ i } the partition of ∆(1) obtainedby taking the quotient with respect to ∼ . Call S ∆ i the pemutation group over∆ i . It is easy to see that this partition doesn’t mix rays coming from differentfactors of the product so we can write S i or S i to mean a permutation groupthat acts on the first or on the second factor. Call H the quotient of Aut( N, ∆)with respect to Π S ∆ i = Π S i × Π S i . Then • If S (cid:54) = S , H = Aut( N , ∆ S )Π S i × Aut( N , ∆ S )Π S i ; • If S = S (cid:54) = P × P , H = (cid:18) Aut( N , ∆ S )Π S i × Aut( N , ∆ S )Π S i (cid:19) (cid:110) Z ; • If S = S = P × P , H = S (cid:110) Z Z (cid:39) S .Here a small summary of these groups.34 Aut( N S , ∆ S ) (cid:81) S ∆ i Aut( N S , ∆ S ) / (cid:81) S ∆ i P Sym( e , e , e ) Sym( e , e , e ) Id P × P < (13) , (1234) > < (13) , (24) > Z dP < (24) > < (24) > Id dP < (12)(34) > Id Z dP Sym( e , e , e ) × < − Id > Id S × Z To see that the sequence splits, consider, for example, the case X = dP × dP for which H = Id × Z = < σ > . This group is generated by the automorphism ofthe fan of dP that switches the rays associated to the two exceptional divisorsof dP , thus a section of Aut( X ) → H is given by σ (cid:55)→ A where A is anautomorphism of P that switches the two points that are blown-up to obtain dP . All the other cases can be described in a similar way.Aut ( X ) is the connected component of the identity in Aut( X ) and now we willshow that Aut ( X ) = Aut ( S ) × Aut ( S ). By a result of Cox (see again [5])Aut ( X ) (cid:39) Aut g ( S ) Hom Z (Pic( X ) , C ∗ )where Aut g ( S ) is the group of the automorphisms of the homogeneous coor-dinate ring S of X , regarded as graded C − algebra. This group is spannedby ( C ∗ ) | ∆(1) | = ( C ∗ ) | ∆ S (1) | + | ∆ S (1) | and by the elements y m ( λ ) where λ ∈ C and m ∈ R ( N, ∆) (the elements of R ( N, ∆) are the roots of Aut( X )).We show that each y m ( λ ) can be written in a unique way as the productof f i ∈ Aut g ( R i ) where R i is the coordinate ring of S i . This shows thatAut g ( S ) (cid:39) Aut g ( R ) × Aut g ( R ). The group Hom Z (Pic( X ) , C ∗ ) splits as Hom Z (Pic( S ) , C ∗ ) × Hom Z (Pic( S ) , C ∗ ) because Pic( X ) = Pic( S ) ⊕ Pic( S ).Then, the quotient can be viewed as a product of the quotient givingAut ( X ) = Aut ( S ) × Aut ( S ) . The claim follows from the combination of the facts above. For example, con-sider again the case X = dP × dP . Since Aut( dP ) is connected, we haveAut ( X ) = Aut( dP ) × K , where K (cid:39) (cid:42) ∗ ∗ ∗ ∗ (cid:43) . Since H = Id × Z , we obtainAut( X ) (cid:39) (Aut( dP ) × K ) (cid:110) (Id × Z ) =Aut( dP ) × ( K (cid:110) Z ) = Aut( dP ) × Aut( dP ) . Combining all these results, we obtain
Theorem 7.5.
Let S and S be two del Pezzo surfaces. Then If S (cid:54) = S , Aut( S × S ) = Aut( S ) × Aut( S ) ; • If S = S (cid:54) = P × P , Aut( S × ) = Aut( S ) × (cid:110) Z ; • If S = S = P × P , Aut(( P ) × ) = Aut( P ) × (cid:110) S . In the previous sections we constructed examples of quotients of Calabi-Yauthreefolds Y embedded in S × S by groups that are of maximal order in thesense that a group H ≤ Aut( S × S ) such that the restriction to Y gives afree action, cannot have greater order than the ones used. If Y is a Calabi-Yau threefold and G is a group acting freely on Y the same holds true each H ≤ G . Moreover Y /H → Y /G is an ´etale covering. In the following tablewe summarize all the quotients analyzed and all the ´etale coverings obtained bytaking quotient with respect to subgroups. Also the known examples are shown.The column m ( | G | ) /M represents the ratio of the maximal order of the existinggroup action freely on Y and the estimated ( M = M ( S , S )). In the columnΠ ( Y /G ) the fundamental group of the quotient is written. When for twoisomorphic subgroups H and H of G we obtain h ( Y /H ) = h ( Y /H ) and h ( Y /H ) = h ( Y /H ) we represent them in the table in one row indicatingthat multiple subgroups give the same result by their number between roundbrackets. For example, taking S = S = P and G (cid:39) Z ⊕ Z there are 4subgroups of order 3 and each of them gives a manifold with Hodge numbers(2 , Z (4) in the column ofΠ ( Y /H ). In the last column a ” Y ” means that the height obtained for thequotient threefold is the least possible, a ” N ” means the opposite and a ”?”means that we don’t know if this is the case or not. The pairs ( S , S ) for which M ( S , S ) = 1 are omitted. 36 S max( | G | ) /M | G | Π ( Y /H ) h h h min? P P / Z ⊕ Z Z (4) 2 29 31 N1 { Id } P dP / Z { Id } P dP / Z { Id } P × P P × P /
16 16 Z ⊕ Z Z ⊕ Z Z (2) 1 9 10 N4 Z ⊕ Z Z (2) 2 18 20 N2 Z (3) 4 36 40 N1 { Id } P × P dP / Z { Id } dP dP /
12 12 Z Z Z Z Z { Id } dP dP /
12 12 Dic Z Z (3) 3 12 15 N3 Z Z { Id } dP dP / Z { Id }
10 34 44 N dP dP / Z { Id }
11 29 40 N dP dP / Z { Id }
10 35 45 N dP dP / Z ⊕ Z Z ⊕ Z Z (2) 4 8 12 N2 Z (3) 8 16 24 N1 { Id }
12 28 40 N dP dP / Z { Id }
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