AATTRACTORS ARE NOT ALGEBRAIC
YEUK HAY JOSHUA LAM AND ARNAV TRIPATHY
Abstract.
The Attractor Conjecture for Calabi-Yau moduli spaces predicts the alge-braicity of the moduli values of certain isolated points picked out by Hodge-theoreticconditions. We provide a family of counterexamples to the Attractor Conjecture in allsuitably high, odd dimensions conditional on the Zilber-Pink conjecture.
Contents
1. Introduction 12. The Attractor Conjecture 43. Dolgachev Calabi-Yau varieties 54. Reduction to Shimura theory 125. Unlikely intersection 21References 231.
Introduction
Statement of results.
In this paper, we study the following remarkable conjecturedue to string theorists:
Conjecture 1.1.1 (Moore) . If X is an attractor Calabi-Yau 3-fold, then it is defined over Q . We recall the definition of an attractor Calabi-Yau variety:
Definition 1.1.2.
For X a Calabi-Yau d -fold, we say that it is an attractor variety if thereis a nonzero integral cohomology class γ ∈ H d ( X, Z ) satisfying γ ⊥ H d − , , where H d − , ⊂ H d ( X, C ) denotes the ( d − ,
1) piece of the Hodge decomposition.These varieties were originally introduced and studied by Ferrara-Kallosh-Stromingerfor Calabi-Yau threefolds as the case most directly of interest in string theory; Calabi-Yaufourfolds were also considered shortly thereafter [Moo07, Section 3.8]. The above definitionin general dimension was then given in [BR11], as we discuss somewhat further in § h d − , conditions, where wenote h d − , = dim H ( X, T X ) as the dimension of Calabi-Yau moduli space; as such, onetypically expects attractor Calabi-Yaus (for some fixed γ ) to be isolated in moduli space,which is indeed the case for the examples we consider below. It is hence certainly of interest, Date : September 30, 2020. a r X i v : . [ m a t h . N T ] S e p YEUK HAY JOSHUA LAM AND ARNAV TRIPATHY irrespective of the physical genesis of the question, to investigate the arithmetic structureof the points picked out by this natural Hodge-theoretic condition.Our main result is then the following:
Theorem 1.1.3.
Under the Zilber-Pink conjecture, the analogue of Conjecture 1.1.1 forCalabi-Yau varieties of arbitrary dimension is false. More precisely, there exist attractorCalabi-Yau varieties in all odd dimensions except 1, 3, 5 and 9 which are not defined over Q . Indeed, for the family of counterexamples we consider, we show the much strongerstatement that the set of attractor points defined over Q must be non-Zariski dense in theCalabi-Yau moduli space. As we will check that the set of attractor points is indeed dense(even in the analytic topology), these families give extremely strong counterexamples in thesense that almost all attractor points fail to be defined over Q .Amusingly, the Calabi-Yau examples we consider are decidedly not counterexamples indimensions 1, 3, 5 or 9. These examples have already been well-understood in the context offlat surfaces (in the theory of Teichm¨uller dynamics). Indeed, in these cases the attractorsare indeed algebraic and are moreover examples of CM points on Shimura varieties.While we give the specific counterexamples above due to the particular techniques webring to bear, we expect a much more general transcendence property for these attractorpoints. Conjecture 1.1.4.
The algebraic attractor points in the moduli space of a Calabi-Yau X are Zariski dense if and only if said moduli space is a Shimura variety. We pause to explain the nomenclature and history of our examples, as well as to pointto some related examples. The Calabi-Yau construction we use is that of a crepant res-olution of an n -fold cyclic cover of P n − branched at a suitable hyperplane collection,following [SXZ13]. We follow these authors in citing Dolgachev for his study of the mod-uli spaces thereof (as attempting to answer the famous question of B. Gross on realizingball quotients as geometric moduli spaces) as in [DGK05; DK07], terming these DolgachevCalabi-Yaus .In particular, one could certainly consider variants of the construction we investigatehere, such as a family of double covers of projective space now branched at some other suit-able hyperplane arrangement. This latter family contains a Calabi-Yau threefold examplewith non-Shimura moduli [SXZ15]. Following our conjecture above, we hence suggest thefollowing case as a particularly attractive next area for investigation:
Question . Does the Attractor Conjecture hold for the family of double cover DolgachevCalabi-Yau threefolds?1.2.
History of the problem and related works.
Attractor varieties in the context ofCalabi-Yau threefolds were originally discovered by Ferrara-Kallosh-Strominger in the con-text of Calabi-Yau threefold compactifications of string theory. They have been the subjectof focused study since; mathematically, for example, they are conjectured to govern thebehavior of the enumerative geometry of Calabi-Yau threefolds [KS14]. Moore in [Moo98]performed an in-depth study and made various conjectures about their possible arithmeticproperties, including the Conjecture 1.1.1 above. In particular, Moore investigated variousexamples such as S × E for S a K3 surface and E an elliptic curve, a quotient thereofknown as the FHSV model, and abelian threefolds. In all these cases, the attractor pointsare defined over Q ; however, note that all these examples have Shimura moduli (and indeed,the attractor points are special points in said Shimura variety). TTRACTORS ARE NOT ALGEBRAIC 3
These attractor points have many other becoming properties analogous to those ofspecial points of Shimura varieties. Douglas and his coauthors studied the distribution ofthese points in their moduli space in [Dou03; DD04; DSZ04; DSZ06a; DSZ06b]; the lastseries of papers by Douglas-Shiffman-Zelditch in particular developed strong heuristics tosuggest that attractor points equidistribute in moduli space with its natural Weil-Peterssonmetric (together with strong numerical evidence in myriad cases to support said claim). Wein fact use such distributional results in our proof of Theorem 1.1.3, although we only needthe much weaker statement that attractor points are Zariski dense, which we can verifydirectly in § Outline of proof of Theorem 1.1.3.
We sketch the proof of Theorem 1.1.3.Weproceed by contradiction, so we assume that all attractor Calabi-Yau varieties are in factdefined over Q . We consider the Dolgachev Calabi-Yau varieties as constructed in § X areconstructed from an associated curve C , and the middle Hodge structures of X and C are closely related as reviewed in Section 3.1. Next, it is not difficult to check that theattractors are Zariski (in fact, even analytically) dense in the moduli space M . Secondlywe show that, for X a Dolgachev Calabi-Yau variety, if it is attractor and defined over Q ,then its Jacobian splits in the isogeny category as A × A where the abelian variety A hascomplex multiplication (CM) by a fixed cyclotomic field. The crucial ingredient here is atheorem of Shiga-Wolfart (following W¨ustholtz) in transcendence theory, which, informally,implies that an abelian variety defined over Q is CM as soon as it has sufficiently manyalgebraic period ratios, which in our case follows from the attractor condition and priorHodge-theoretic analysis.This splitting of Jac ( C ) up to isogeny then may now be thought of as a problem inthe intersection theory of Shimura varieties: a priori , Jac ( C ) naturally defines a point ofan ambient Shimura variety Sh and the isogeny splitting condition above implies that M intersects the Hecke translates of a sub-Shimura variety Sh A of Sh in a dense set of points.The attractor condition has hence reduced to a problem in unlikely intersection theory . Inparticular, when the codimensions of M and Sh A in Sh sum to less than the dimensionof Sh, the Zilber-Pink conjecture implies that M is contained in some proper Shimurasubvariety. This is the point where the argument fails for small values of the dimensionof the Calabi-Yau varieties; in fact, in these cases the moduli space M turns out to be aShimura variety, the attractor points are CM points, and therefore the Attractor Conjectureholds. In the general case, we instead use a result of Deligne-Mostow on the monodromygroups of these varieties to show that, for almost all dimensions, M not contained in anyproper Shimura subvariety of Sh, and hence we have a contradiction as desired.1.4. Outline of the rest of the paper.
In Section 2 we introduce the attractor conditionon a Calabi-Yau variety as well as the Attractor Conjecture, along with providing somewhatmore context for the interested reader. Section 3 continues with the main thrust of theproof above by defining the Dolgachev Calabi-Yaus X along with their associated curves C before establishing the relation between the Hodge structures thereof. Section 4 appliesthe theorem of Shiga-Wolfart to reduce to a problem in Shimura theory before setting upthe formalism of the ambient Shimura variety Sh and its special Shimura subvariety Sh A .We finally conclude in Section 5 with a discussion on the unlikely intersection theory of thisShimura variety problem. YEUK HAY JOSHUA LAM AND ARNAV TRIPATHY
Notations and conventions.
We set a few conventions now. We work throughoutwith the Hermitian intersection pairing on the middle-degree complex cohomology of amanifold; under this pairing, distinct Hodge summands are orthogonal. For a vector space V defined over some field K and a field extension K ⊂ L , we often denote the extensionof scalars V ⊗ K L as V L . For an algebraic variety X over C , we will sometimes say that X is algebraic to mean that it is defined over Q , since the latter is tautologically equivalentto the statement that the coordinates of the point corresponding to X in its moduli spacebeing algebraic numbers.Much of the motivation for this work arose from discussions with Shamit Kachru andAkshay Venkatesh, and it is a pleasure to thank them. We are also grateful to Matt Emerton,Phil Engel, Mark Kisin, Barry Mazur, Greg Moore, Curt McMullen, Minhyong Kim, AnanthShankar, and Max Zimet for illuminating comments, questions, and discussions. AT issupported under NSF MSPRF grant 1705008.2. The Attractor Conjecture
We recall the attractor condition for (higher-dimensional) Calabi-Yaus with slightlymore precision now. Note that in general, we take a Calabi-Yau variety to be a smooth,projective variety X with trivial canonical bundle and ( a priori ) defined over the complexnumbers C . In practice, however, we will work with a specific family defined shortly in § Definition 2.0.1.
Given a Calabi-Yau d -fold X , then for each nonzero class γ ∈ H d ( X, Z ), X is said to be an attractor for the class γ if the following condition γ ⊥ H ,d − ( X )holds. If in addition we have γ d, (cid:54) = 0, we will refer to X as an attractor point . Remark . Note the condition γ d, (cid:54) = 0 is vacuous in the original case of threefolds,as γ , = 0 and the attractor condition would simply imply γ = 0 by Hodge theory. Ourreason for emphasizing this condition is that in general, for higher-dimensional Calabi-Yaus,omitting this condition would allow the possibility for non-isolated attractors (i.e. positive-dimensional families); it is indeed straightforward to construct examples where this happens,such as the very families of Dolgachev Calabi-Yaus considered in this paper when n is notprime. As such, the higher-dimensional Attractor Conjecture would be trivially false.In fact, this attractor condition in high dimensions is essentially due to [BR11] in their §
4, although they instead consider critical points of the central charge function Z γ : = (cid:104) γ, Ω (cid:105) (cid:113) (cid:104) Ω , Ω (cid:105) . One may easily verify that this condition is equivalent to the attractor condition; it may beamusing to note that in fact there is a natural flow on (the universal cover) of the Calabi-Yaumoduli space induced by gradient flow for log | Z γ | which naturally dynamically producesthese special points as fixed points of said flow. These considerations will play no role inour analysis, however. Conjecture 2.0.3 ([Moo98, Conjecture 8.2.2] for the case of threefolds) . If X is an at-tractor variety for a non-zero class γ ∈ H d ( X, Q ) such that γ d, (cid:54) = 0 , then it has a modelover Q . TTRACTORS ARE NOT ALGEBRAIC 5
The above conjecture would suggest that these points picked out by the Hodge-theoreticattractor condition are a class of special points analogous in myriad aspects to special pointsof Shimura varieties. (Indeed, Moore also makes a counterpart conjecture that the periodsof these points are algebraic.) And while Moore originally makes the conjecture for Calabi-Yau threefolds, Brunner-Roggenkamp exhibit that the same considerations apply in allrespects to higher-dimensional Calabi-Yaus, and so we find it equally interesting to studythe veracity of this conjecture in higher dimensions. Note that in our phrasing above, weassume that γ d, (cid:54) = 0 to avoid the possibility of non-isolated attractor points.We briefly give some more context to the above conjecture for the interested reader;this discussion will play no role in the proof and may be skipped without consequence.First, we note that the failure of the Attractor Conjecture should quite reasonably havebeen expected. Let us return for a moment to the case of Calabi-Yau threefolds, whereagain the attractor condition is that the two-dimensional vector space H , ( X ) ⊕ H , ( X )contains a vector in the integral lattice. If instead we impose the stronger condition thatit contains a rank-two sublattice, known as the rank-two attractor condition, then thestandard conjectures tell us that we expect H , ⊕ H , to split off as a motive, a conditionwhich moreover would take place at algebraically-defined points. So, rank-two attractorpoints should certainly be algebraic; by contrast, the attractor condition itself is too weakto suggest any motivic splitting, and one should expect no particular algebraicity in general.Second, as remarked above, Moore also conjectures in [Moo98] not just algebraicity ofthe attractor points but also algebraicity of their periods. In fact, the paragraph abovealready suggests distrust of this conjecture as well: when we have a motive that splits off,for example an H , ( X ) ⊕ H , ( X ) motive of a (Tate-twisted) CM elliptic curve in the casewhen b ( X ) = 4, there is no reason to suspect that the period of said CM elliptic curveneed always be algebraic. And indeed, in examples with such motivic splitting such asKlemm-Scheidegger-Zagier’s study of the conifold point of the mirror quintic or the exam-ples of [Can+19], said periods have (numerically) been found to agree with the expectedspecial values of the appropriate L -function from the motivic splitting. Hence, morally, ourmain theorem should be thought of as analogous (or even mirror symmetric ) to provingtranscendence of special values of L -functions.3. Dolgachev Calabi-Yau varieties
Defining the Calabi-Yaus.
We will consider a family of Calabi-Yau varieties con-structed as crepant resolutions of n -fold cyclic covers of projective space. Most of the dis-cussion holds for any n ≥
2, although the case n = 2 is completely classical and returns theLegendre family of elliptic curves; we hence restrict to n ≥ n = 3 , , M CY is Shimura); all three of these hence display qualitatively different behavior and wenote at the appropriate point where the condition n (cid:54) = 3 , , n -fold cyclic coversof P n − branched along 2 n hyperplanes; this construction is due to Dolgachev, and we firstcollect here some basic properties about the Hodge structure of such a Calabi-Yau. So,consider 2 n points x , · · · , x n in P . Then, we may certainly consider the curve C givenby the n -fold cyclic cover of P branched at those 2 n points; more precisely, we mean thecover determined by the same n -cycle monodromy about each branch point in the base, or YEUK HAY JOSHUA LAM AND ARNAV TRIPATHY simply the smooth, projective curve C whose affine model is given by(1) C ◦ = { y n = n (cid:89) i =1 ( x − x i ) } . Then, by construction, H ( C, Q ) has both a Hodge splitting and a Z /n -action; in fact,these two structures are compatible. More precisely, if we let ζ = e πi/n , we know that H ( C, Q ( ζ )) splits into eigenspaces for the Z /n -action. Let µ ∈ Z /n be the generator whichacts on C by y (cid:55)→ ζy. Definition 3.1.1.
Let H ( C )[ i ] ⊂ H ( C ; Q ( ζ ))be the sub Q ( ζ )-vector space such that µ acts by ζ i .Then H ( C )[ i ] is a Q ( ζ )-sub-Hodge structure of H ( C, Q ( ζ )) in the sense that we havethe decomposition H ( C )[ i ] ⊗ Q ( ζ ) C (cid:39) H , ( C )[ i ] ⊕ H , ( C )[ i ]given by the Hodge splitting of H ( C ; C ). Proposition 3.1.2.
The Hodge numbers of H ( C )[ i ] are given by (2 i − , n − i ) − , for ≤ i ≤ n − ; that is dim C H , ( C )[ i ] = 2 i − , dim C H , ( C )[ i ] = 2( n − i ) − . We refer the reader to [Loo07, Lemma 4.2] for this computation. Note that the i = 0eigenspace is trivial as any cohomology class in this space comes from H ( P ) (cid:39) Construction 3.1.3.
It remains to introduce the Calabi-Yau (2 n − X . Herewe follow the treatment in [SXZ13, Sections 2, 3]. For a collection of 2 n hyperplanes( H , · · · , H n ) of P n − , we say that they are in general position if no 2 n − H i ’s.Then we may define an n -fold cyclic cover X (cid:48) of P n − branched along these hyperplanes.We give the rigorous construction here. For a line bundle L on an arbitrary variety Y and a positive integer n , consider the rank n vector bundle E := O ⊕ L ∨ ⊕ · · · ⊕ ( L ∨ ) ⊗ n − on Y ; here L ∨ denotes the dual of L . Now given a section of L ⊗ n , or equivalently a map( L ∨ ) ⊗ n → O , we may define an algebra structure on E in the obvious way, and therefore we may form thevariety X := Spec( E );and by construction X admits a map to Y ; in fact, this is a cyclic n -fold covering. In otherwords, a section σ ∈ Γ( L ⊗ n ) defines a cyclic n -fold cover X → Y . Definition 3.1.4.
For a collection of 2 n points p , · · · p n on P , we may consider 2 n hyperplanes on P n − as follows: recall that there is an isomorphism(2) Sym n − P ∼ = P n − , TTRACTORS ARE NOT ALGEBRAIC 7 and also that, for each i = 1 , · · · , n , the set of points of the form { p i } × P × · · · × P onthe left hand side of (2) gives a hyperplane on the right hand side. Therefore we obtain 2 n hyperplanes H , · · · , H n on P n − , in general position.We now apply the above construction to Y = P n − , L = O (2), and σ ∈ Γ( O (2 n )) suchthat the zero locus of σ is precisely D := n (cid:88) i =1 H i ⊂ P n − to obtain a cyclic n -fold covering of P n − , which we denote by X (cid:48) .Note that a pleasant computation shows that the canonical bundle of the cyclic cover X (cid:48) defined above is trivial: indeed, using the formula K X = π ∗ K Y + R for a covering map π : X → Y , where R ⊂ X is the ramification divisor, we deduce that K X (cid:48) = π ∗ ((2 − n ) H ) + (2 n )( n − H (cid:48) = ( n (2 − n ) + 2 n ( n − H (cid:48) = 0 , and so the canonical class of X (cid:48) is trivial; here H denotes a hyperplane class in P n − and H (cid:48) is the class of one of the n components of the pullback of H .Denote the moduli space of collections ( H , · · · , H n ) of hyperplanes of P n − by H n .The following was proved by Sheng-Xu-Zuo [SXZ13, Corollary 2.6]: Theorem 3.1.5.
Denote by f (cid:48) : X (cid:48) → H n the family of n − -folds over the moduli ofhyperplane arrangements constructed above. Then (1) there is a family of smooth Calabi-Yau (2 n − -folds f : X → H n , as well as a commutative diagram (3) X X (cid:48) H nσf f (cid:48) where σ is a simulatneous crepant resolution; (2) the middle degree Hodge structures of the families X (cid:48) and X agree: R n − f (cid:48)∗ Q ∼ = R n − f ∗ Q ;(3) furthermore, the family f is maximal in the sense that the Kodaira-Spencer map isan isomorphism at each point p ∈ H n . Definition 3.1.6.
We refer to the varieties constructed in Theorem 3.1.5 as the DolgachevCalabi-Yau varieties. For X (cid:48) = f (cid:48)− ( p ) the (2 n − p ∈ H n , wedenote by X := f − ( p ) the corresponding crepant resolution. Remark . This is a slight abuse of terminology since the crepant resolution σ : X → X is not unique; on the other hand any two resolutions are of course birational to each other.
YEUK HAY JOSHUA LAM AND ARNAV TRIPATHY
Hodge structures of Dolgachev Calabi-Yaus.
Now we relate the curves con-structed earlier as branched covers of P to the Calabi-Yau varieties constructed as coversof P n − . Recall from Theorem 3.1.5 that the cyclic cover X (cid:48) has a crepant resolution X ,which is Calabi-Yau and we have an isomorphism H n − ( X ; Q ) (cid:39) H n − ( X (cid:48) ; Q ) . The right hand side has a natural Z /n -action since it arises as a cyclic cover, and thereforethe left hand side does as well. We may therefore decompose H n − ( X ) into eigenspacesas follows. As in Section 3.1 we fix a generator µ ∈ Z /n . Definition 3.2.1.
We define H n − ( X )[ i ] ⊂ H n − ( X ; Q ( ζ ))as the sub-vector space over Q ( ζ ) on which µ acts by ζ i .Then we have the crucial relationship between the Hodge structures of C and X : Lemma 3.2.2 ([SXZ13, Proposition 2.2, Remark 2.2]) . We have the following isomorphismof Hodge structures: H n − ( X )[ i ] (cid:39) (cid:94) n − H ( C )[ i ] . Remark . Perhaps a more intrinsic way of phrasing the above lemma is that there isan isomorphism of Q -Hodge structures with A := Q ( X ) / ( X n − H n − ( X, Q ) ∼ = (cid:94) A n − H ( C, Q ) . In other words, we view H ( C, Q ) as an A -module, where the generator X ∈ A acts through µ ∈ Z /n , and then we take its (2 n − rd wedge power over A .We have the following simple Corollary 3.2.4.
The Hodge structure H n − ( X ) ⊗ Q ( ζ ) decomposes as a sum H n − ( X ) ⊗ Q ( ζ ) ∼ = n − (cid:77) i =1 V i where V i is a Q ( ζ ) -Hodge structure, concentrated only in Hodge degrees ( p, q ) = (2 i − , n − i − and (2 i − , n − i − furthermore the dimensions of these pieces of the Hodge decomposition are i − and n − i − , respectively.Proof. Indeed, we define the Hodge structures V i to be (cid:86) n − H ( C )[ i ] from Lemma 3.2.2.By Proposition 3.1.2 we may write the Hodge decomposition of H ( C )[ i ] as H ( C )[ i ] ⊗ C ∼ = H , ⊕ H , , where we have omitted the dependence on i on the right hand side, anddim H , = 2 i − , dim H , = 2( n − i ) − . For convenience let us pick a basis { e i } (respectively { f j } ) for H , (respectively H , ).Since the dimension of H ( C )[ i ] is 2 n −
2, upon taking the (2 n − TTRACTORS ARE NOT ALGEBRAIC 9 only non-zero elements obtained by wedging together e i ’s and f j ’s must omit precisely one e i or one f j . Therefore the Hodge degrees of such an element are either( p, q ) = (2 i − , n − −
1) or (2 i − , n − i ) − i − n − i ) −
1) choices of an e i (respectively f j ) toomit, and therefore the Hodge numbers are 2 i − n − i − (cid:3) Notation 3.2.5.
We will sometimes use the following piece of notation for bookkeepingwhen dealing with these Hodge numbers. We record the dimensions in a ( n − × dim H , ( C )[1] dim H , ( C )[1]dim H , ( C )[2] dim H , ( C )[2]... ...dim H , ( C )[ n −
1] dim H , ( C )[ n − . For example, in the case n = 5, it follows from Proposition 3.1.2 that the above matrix is . Remark . We therefore verify from the case of i = n − h n − , = 1, as expected fora Calabi-Yau variety of dimension 2 n −
3; moreover, we find dim M CY = h n − , = 2 n − M , n , the moduli of 2 n points in P .Indeed, this construction exactly accounts for the full moduli space of Calabi-Yau varietiesso constructed, i.e. M , n ∼ → M CY . As a consistency check let us see that the dimensions of M , n and H n agree: indeed,each moduli space parametrizes hyperplanes inside a projective space modulo the action ofa projective linear group, and the coincidence of the dimensions is the equality(2 n ) − (3) = (2 n − n ) − ((2 n − − Remark . Note that in the case when n is not a prime, for any Dolgachev Calabi-Yauvariety X there exists classes γ ∈ H n − ( X, Q ) with no component in H n − ( X )[1]: indeed,just take any element in H n − ( X, Q ( ζ ))[ i ] for some i not coprime to n , and take the sumof all of its Galois conjugates. In fact, when we take the parallel transport of such a class γ , it will continue to have no component in H n − ( X )[1] for any X ; this shows that thecondition γ n − , (cid:54) = 0 condition is necessary in Conjecture 2.0.3.3.3. The attractor condition for Dolgachev Calabi-Yau varieties.
We are now fi-nally in the position to study the attractor condition for Calabi-Yau varieties X constructedas above. We first show that the attractor condition is equivalent to a condition on theperiods of the associated curve C as follows: Lemma 3.3.1.
The variety X satisfies the attractor condition if and only if there exists ω ∈ H , ( C )[1] ∩ H ( C )[1] . Indeed, recall that H , ( C )[1] is one-dimensional, so the above condition is that theabove subspace of H ( C )[1], a priori only defined once one tensors up to C , is in factdefined over Q ( ζ ). Proof.
Suppose X satisfies the attractor condition. Then there exists γ ∈ H n − ( X ; Z )orthogonal to H n − , ( X ); equivalently, γ is orthogonal to H , n − ( X ). Recall from thediscussion above that H , n − ( X ) is contained within the H n − ( X )[1] eigenspace. Thedistinct µ -eigenspaces are certainly orthogonal under the intersection pairing, and if γ i ∈ H n − ( X )[ i ] denotes the summand of γ under the decomposition of H n − ( X ; Q ( ζ )), theabove condition is equivalent to γ orthogonal to H , n − ( X ). As the Hermitian pairing isperfect on H , n − ( X ), γ cannot have any support within said Hodge summand of H n − ( X )[1] C (cid:39) H , n − ( X ) ⊕ H , n − ( X ) , and so we must have γ ∈ H , n − ( X ). But γ was defined as an element of the vector space H n − ( X )[1], a vector space defined over Q ( ζ ), and so both H , n − ( X ) and H , n − ( X ),as the orthogonal complement of H , n − ( X ) under the intersection pairing restricted to H n − ( X )[1], are defined over Q ( ζ ). But then H , n − ( X ) (cid:39) ∧ n − H , ( C )[1] ⊗ H , ( C )[1] (cid:39) (cid:16) H , ( C )[1] (cid:17) ∨ ⊗ (cid:16) det H , ( C )[1] (cid:17) ⊗ H , ( C )[1] (cid:39) (cid:16) H , ( C )[1] (cid:17) ∨ ⊗ det H ( C )[1]as a subspace of H n − ( X )[1] (cid:39) ∧ n − H ( C )[1] (cid:39) (cid:16) H ( C )[1] (cid:17) ∨ ⊗ det H ( C )[1] . Above, we use the notation det V = ∧ dim V V and the isomorphism ∧ dim V − V (cid:39) V ∨ ⊗ det V. In any case, we have that the decomposition H n − ( X )[1] C (cid:39) H , n − ( X ) ⊕ H , n − ( X )is isomorphic to the decomposition (cid:16)(cid:16) H ( C )[1] (cid:17) ∨ ⊗ det H ( C )[1] (cid:17) C (cid:39) (cid:16)(cid:16) H , ( C )[1] (cid:17) ∨ ⊗ det H ( C )[1] (cid:17) ⊕ (cid:16)(cid:16) H , ( C )[1] (cid:17) ∨ ⊗ det H ( C )[1] (cid:17) induced from the Hodge splitting of H ( C )[1] C . As H ( C )[1] and hence det H ( C )[1] aredefined over Q ( ζ ), however, the condition that the first decomposition be defined over Q ( ζ )is equivalent to the condition that the second decomposition be defined over Q ( ζ ), which inparticular implies that there exists some ω ∈ H , ( C )[1] ∩ H ( C )[1].Conversely, given such an ω , we have that the subspace H , ( C )[1] ⊂ H ( C )[1] C is infact defined over Q ( ζ ) and hence so is H , ( C )[1] as its orthogonal complement; as above,the decomposition H n − ( X )[1] C (cid:39) H , n − ( X ) ⊕ H , n − ( X ) is then also defined over Q ( ζ ). Then take some γ ∈ H , n − ( X ) defined over Q ( ζ ) so that by construction, γ is orthogonal to H , n − ( X ), and consider the Galois conjugates γ i under the action ofGal( Q ( ζ ) / Q ) on H n − ( X ; Q ( ζ )). These Galois conjugates will lie within the H n − ( X )[ i ]eigenspaces for values of i coprime to n and thereby be concentrated in Hodge summandsaway from the (1 , n −
4) and (0 , n −
3) summands, so that if we now define γ = (cid:80) i γ i ,the Galois-theoretic construction will give us γ ∈ H n − ( X ; Q ) while its summand in the H n − ( X )[1] eigenspace is still the original γ we started with. As such, scaling γ as TTRACTORS ARE NOT ALGEBRAIC 11 necessary so it in fact lies in H n − ( X, Z ), we have produced some integral cohomologyclass orthogonal to H , n − ( X ), or equivalently H n − , ( X ), as desired. (cid:3) Algebraicity of the associated curve.
In this section we show that the algebraicityof the Dolagchev Calabi-Yau variety implies that of the curve associated to it. Thereforeto show that Dolgachev Calabi-Yau varieties provide counterexamples to the AttractorConjecture it suffices to show that for the attractor varieties, the assocaited curves are notdefined over Q . Proposition 3.4.1. X is defined over Q if and only if C is defined over Q .Proof. We first do the easier direction, so suppose C is defined over Q . Let µ : C → C denote a generator of the Z /n action, which must also be defined over Q : we spell out thisargument here as we will use its basic idea (“spreading out”) frequently. So, consider µ as apoint of the quasiprojective Q -scheme Aut( C ). If the field of definition K of µ is larger than Q , and in particular contains some pure transcendental extension thereof, we may freelyspecialize that transcendental variable to produce a family of automorphisms of C , but anycurve has only finitely many automorphisms. Hence µ must have been defined over Q . Butnow the morphism C → C/µ (cid:39) P is defined over Q , and so the 2 n points x , · · · , x n ∈ P of ramification are defined over Q (after an appropriate automorphism of P ). But then itis clear that the cover X (cid:48) and all the blow-up centers within X (cid:48) are defined over Q , andhence so is X .The more interesting direction is the reverse argument, where we begin by supposingthat X is defined over Q . The morphism X → P n − corresponds to some line bundle L ∈
Pic X given by the pullback of O (1), but note that Pic X (cid:39) H ( X ; Z ) is simply adiscrete set of points as a scheme over Q , and hence all points must be defined over Q . Claim 3.4.2.
The complete linear system of L defines precisely the morphism X → P n − .Proof. It suffices to show that the pullback map induces an isomorphismΓ( X, L ) ∼ = Γ( P n − , O (1)) . Recall that we have the factorization X σ −→ X (cid:48) α −→ P n − , where σ is the crepant resolution from Theorem 3.1.5, and α : X (cid:48) → P n − denotes the n -fold covering of P n − from Definition 3.1.4. We first show thatΓ( X (cid:48) , L (cid:48) ) ∼ = Γ( P n − , O (1)) , where L (cid:48) := α ∗ O (1). By construction of X (cid:48) (see Definition 3.1.4 and the paragraph preced-ing it), α ∗ O X (cid:48) ∼ = O P n − ⊕ L ∨ ⊕ · · · ⊕ ( L ∨ ) ⊗ n − , where L ∨ ∼ = O ( − α ∗ L ∼ = O P n − (1) ⊕ O P n − ( − ⊕ · · · ⊕ O P n − ( − n + 3) , and hence Γ( X (cid:48) , L (cid:48) ) = Γ( P n − , O (1)) as required. On the other hand, X is obtained from X (cid:48) by blowing up along subvarieties of codimension at least two, and we claimΓ( X, L ) ∼ = Γ( X (cid:48) , L (cid:48) ) as well. Indeed, as X and X (cid:48) fail to be isomorphic only in codimension two, this statementwould follow from (algebraic) Hartogs’ Lemma provided X and X (cid:48) are both normal. That X is normal follows from its smoothness, while X (cid:48) is normal as it is both R and S . Indeed,its singular set has codimension two, while it is S given its construction as a hypersurfacein a smooth ambient variety (the total space of a line bundle over P n − ). (cid:3) Hence L , and its linear system X → P n − is defined over Q , and so we learn thatthe ramification locus with irreducible components the 2 n hyperplanes H , · · · , H n maybe taken to be defined over Q – i.e. are defined over Q after, possibly, an application ofsome P GL n − projective transformation to their original definition as corresponding to thepoints x i . However, this condition is precisely the same as that the original 2 n points maybe taken to be defined over Q , i.e. possibly after some P GL transform, or equivalently thatall their cross-ratios are in Q , and so it is then easy to reconstruct C over Q . Indeed, themap from the 2 n points to the 2 n hyperplanes (or 2 n points in a dual projective space) maybe regarded as a morphism between (open loci of) Sym n − P /S → P n − /S n − (by usingthe simple 3- or 2 n − P GL - and P GL n − -actions, respectively) whichis explicitly defined over Q . Indeed, one may write down this map in explicit coordinates:we refer the reader to [SXZ13, Claim 3.6]. (cid:3) Attractors are dense.
In this section we show that the attractor points are Zariskidense in moduli space. This fact will be used when we apply the Zilber-Pink conjecture.We define the auxiliary space M (cid:48) , n := { ( s, ω ) | s ∈ M , n , ω ∈ H , ( C s )[1] , ω (cid:54) = 0 } where we have denoted by C s the n -fold cover of P branched at the configuration of 2 n points given by s ∈ M , n . Recall that H , ( C s )[1] is a one dimensional and hence M (cid:48) , n is a G m -bundle over moduli space. Also let (cid:102) M (cid:48) , n denote the universal cover of M (cid:48) , n ; onthis universal cover we have a well defined basis of the cohomology group H ( C, Q ( ζ n ))[ − γ , · · · , γ n − .We may now consider the so-called Schwarz map defined as follows: π : (cid:102) M , n → C n − ( s, ω ) (cid:55)→ (cid:16) (cid:90) γ ω, · · · , (cid:90) γ n − ω (cid:17) . Now by Lemma 3.3.1 we have that a point ( s, ω ) ∈ (cid:102) M , n is an attractor point (moreprecisely the point s gives rise to an attractor CY and ω witnesses this) if and only if π (( s, ω ))has coordinates in Q ( ζ ) ⊂ C . Now note that π is a holomorphic local homeomorphism, andso the image π ( (cid:102) M , n ) contains some open ball inside C n − . Since Q ( ζ n ) ⊂ C is dense,we have that the attractors are topologically dense, and hence Zariski dense as well. Tosummarize we have the following: Proposition 3.5.1.
The attractor points are Zariski dense in the moduli space M CY . Reduction to Shimura theory
Algebraic attractors split off CM abelian varieties.
In this subsection we showthat if an attractor is algebraic, then the Jacobian of the corresponding curve C must splitoff CM factors. TTRACTORS ARE NOT ALGEBRAIC 13
We make use of the following theorem of Shiga-Wolfart [ref][SW95, Proposition 3], aconsequence of the analytic subgroup theorem of W¨ustholz [ref]:
Theorem 4.1.1 (Shiga-Wolfart) . Suppose A is an abelian variety over Q endowed with ω ∈ Γ( A, Ω A ) Q such that for any two classes γ , γ ∈ H ( A ; Z ) , we have that the periodratios are algebraic: (cid:104) ω, γ (cid:105)(cid:104) ω, γ (cid:105) ∈ Q . Then A has complex multiplication. Moreover, if K is the number field generated by theperiod ratios above then the CM field of A is precisely K . The last sentence above is not part of the proposition cited but follows from theirproof, which uses the analytic subgroup theorem to directly construct endomorphisms of A from the hypothesized period relations. More generally, it follows that if A splits in theisogeny category as some product of abelian varieties A i , then for all A i on which the ω above is supported (i.e. restricts to nontrivially), A i has complex multiplication by some(possibly varying with i ) subfield of K . Applied to the case above, we have the immediateconsequence: Proposition 4.1.2. If X as above satisfies the attractor condition, then Jac C has a sum-mand A in the isogeny category such that the following conditions hold: (1) ω restricts non-trivially to A ; (2) A has complex multiplication (in the isogeny category) by Q ( ζ ) . Proof.
Let A denote the simple abelian variety, which is a summand of A in the isogenycategory, on which ω restricts non-trivially. Then the Hodge structure of A is a Q -sub-Hodge structure of H ( A, Q ), and hence must contain all the Galois conjuagtes of ω . Since ω lives in H ( A )[1] on which µ acts by a primitive root of unity, we havedim A ≥ φ ( n ) / . On the other hand, since X satisfies the attractor condition, all the periods (cid:104) ω, γ (cid:105) for γ ∈ H ( A , Q )are contained in Q ( ζ ), and applying Theorem 4.1.1 to A and ω we deduce that A hascomplexmultiplication by a subfield of K = Q ( ζ ), and by the inequality on the dimensionabove we conclude that A has complex multiplication by K , as required. (cid:3) In fact, it is possible to be more precise still in this case: the endomorphisms constructedfrom the W¨ustholz analytic subgroup theorem commute with the Z /n cyclic action and soeach CM abelian variety A i produced from the Shiga-Wolfart argument continues to respectthe Z /n -equivariant structure. But ω is the unique holomorphic form in its eigenspace, upto scaling, and so there can only be one A i upon which ω is supported.The theory of complex multiplication now tells us that there is some finite list of abelianvarieties A , up to isogeny, with CM related by Q ( ζ ) as above. The dimension of A is dim[ Q ( ζ ) : Q ] = φ ( n ), and so we find that the following characterization of the attractorpoints in M CY : Theorem 4.1.3.
The attractor points for the Dolgachev family of CYs considered hereare exactly the intersection of M CY (cid:39) M , n with the Hecke translates of the sub-Shimuravarieties above of A ( n − under M , n → M ( n − → A ( n − . Prym varieties.
In fact, while M , n does naturally map to the Shimura varietyparametrizing ( n − -dimensional (principally polarized) abelian varieties as above, forthe application of the Zilber-Pink conjecture it is necessary to refine this map slightly,especially when n is not a prime. The end result will be a map to a PEL type Shimuravariety instead of simply A ( n − . Therefore in this section we study the construction ofPrym varieties, which is a certain quotient of the Jacobian.Recall that, for n ≥ C → P denotes the cyclic n -fold covering of P branched at 2 n points, whose affine model is given in (1); moreover there is an action of Z /n on C . Nowsuppose we have a divisor n (cid:48) of n , and let Z /n (cid:48) ⊂ Z /n denote the unique order n (cid:48) subgroupof Z /n ; we fix a generator µ ∈ Z /n as before, and further denote by µ (cid:48) := ( n/n (cid:48) ) µ , whichis a generator of this Z /n (cid:48) subgroup. Definition 4.2.1.
For each n (cid:48) dividing n , define C (cid:48) := C/ ( Z /n (cid:48) ) , where Z /n (cid:48) acts on C via the inclusion Z /n (cid:48) ⊂ Z /n . Also let π n (cid:48) : C → C (cid:48) denote the natural quotient map. By further quotienting by a Z / ( n/n (cid:48) ), we also have amap C (cid:48) → P , which is a cyclic n/n (cid:48) -fold covering. Proposition 4.2.2.
Let J n (cid:48) denote the cokernel of the pullback map π ∗ n (cid:48) : Jac ( C (cid:48) ) → Jac ( C ) . Then its cohomology is given by H ( J n (cid:48) , Q ( ζ )) ∼ = (cid:77) i H ( C, Q ( ζ ))[ in (cid:48) ]; equivalently, the above is the sum of all H ( C, Q ( ζ ))[ j ] where j satisfies ζ jn/n (cid:48) = 1 . Remark . As a consistency check, we see that the above sum is over i = n (cid:48) , · · · , ( nn (cid:48) − n (cid:48) , and so there are ( n/n (cid:48) −
1) non-trivial summands, as expected, since C (cid:48) → P is now a n/n (cid:48) -fold cyclic cover. Proof.
Applying the Riemann-Hurwitz formula to the covering map C (cid:48) → P , we have2 − g ( C (cid:48) ) = nn (cid:48) (2) − n ( nn (cid:48) − , and hence g ( C (cid:48) ) = (1 + n ) (cid:0) − nn (cid:48) (cid:1) . Here g ( C (cid:48) ) denotes the genus of C (cid:48) . On the other hand, recall that the points of the Jacobianof a curve are the degree zero divisors modulo rational equivalence, and therefore the imageof the pullback map π ∗ n (cid:48) : Jac ( C (cid:48) ) → Jac ( C )is invariant under the Z /n (cid:48) -action. On the other hand π ∗ n (cid:48) is injective, since if D is a degreezero divisor on C (cid:48) such that π ∗ n (cid:48) ( D ) = ( f ) TTRACTORS ARE NOT ALGEBRAIC 15 for some rational function f on C , then f is invariant under the Galois group Z /n (cid:48) of thecovering map C → C (cid:48) , and therefore f descends to C . Therefore the map on homology in-duced by π ∗ n (cid:48) is also injective, and lands inside the invariant subspace H (Jac ( C ) , Q ( ζ )) Z /n (cid:48) (using Q ( ζ )-coefficients). By the genus computation above, this gives an isomorphism H (Jac ( C (cid:48) ) , Q ( ζ )) ∼ = H (Jac ( C ) , Q ( ζ )) Z /n (cid:48) . Dualizing, we have H (Jac ( C (cid:48) ) , Q ( ζ )) being the coinvariants of the Z /n (cid:48) -action on H (Jac ( C ) , Q ( ζ )),which gives the desired result: indeed, since the action of a generator µ (cid:48) := ( n/n (cid:48) ) µ ∈ Z /n (cid:48) on H (Jac ( C ) , Q ( ζ )[ j ] is given by ζ nj/n (cid:48) , the coinvariants are given by n − (cid:77) i =1 H (Jac ( C ) , Q ( ζ ))[ j ] / ( ζ nj/n (cid:48) − H (Jac ( C ) , Q ( ζ ))[ j ] , whose non-trivial summands are indexed by i such that ζ jn/n (cid:48) = 1, as claimed. (cid:3) We immediately deduce the following simple
Corollary 4.2.4.
We denote by π ∗ n (cid:48) the map on cohomologies induced by π ∗ n (cid:48) . Then thequotient of H ( C, Q ( ζ )) by the images of π ∗ n (cid:48) for all proper divisors n (cid:48) (i.e. n (cid:48) (cid:54) = 1 , n ) isprecisely the sum (cid:77) i ∈ ( Z /n ) × H ( C, Q ( ζ ))[ i ] . The above can be refined integrally, or equivalently as a statement aboue abelian vari-eties.
Definition 4.2.5.
We now define the abelian varietyPrym := Jac ( C ) / (cid:88) n (cid:48) Im( π ∗ n (cid:48) ) , and refer to it as the Prym variety. Here the sum is over proper divisors n (cid:48) as above. Corollary 4.2.6.
The abelian variety
Prym has endomorphisms by Q ( ζ ) , and its Q ( ζ ) -Hodge structure is given by (cid:77) i ∈ ( Z /n ) × H ( C, Q ( ζ ))[ i ] . As such, it has dimension ( n − φ ( n ) . PEL Shimura varieties.
Now that we have the neccesary statements on the Prymconstruction from Section 4.2, we can define the refined period, whose image is a certainPEL type Shimura variety. Denote by V the Q subspace of H ( C, Q ) such that V ⊗ Q ( ζ ) = (cid:77) r ∈ ( Z /n ) × V [ r ] , where V [ r ] ⊂ H ( C, Q ( ζ )) denotes the ζ r eigenspace for the action of µ ∈ Z /n ; as before V has an action of Q ( ζ ). Definition 4.3.1.
The abelian variety Prym from Section 4.2 furnishes us with an integrallattice V Z := H (Prym , Z ) ⊂ H (Prym , Q ) = V, equipped with a symplectic form Ψ. Let S := Res C / R G m denote the Deligne torus, and let H denote the space of homomorphisms h : S → GSp( V R , Ψ)which define Hodge structures of type ( − ,
0) + (0 , −
1) on V Z . The space H is isomorphicto the Siegel upper half space of dimension( n − φ ( n )(( n − φ ( n ) + 1)2 , since it parametrizes abelian varieties of dimension ( n − φ ( n ).Now the Shimura datum (GSp( V, Ψ) , H ) certainly defines a Shimura variety to which M , n maps; we can describe its C -points as follows. The integral structure on V defines amaximal compact subgroup K ⊂ GSp( V, Ψ)( A ), and then we haveSh(GSp( V, Ψ) , H )( C ) = GSp( V, Ψ)( Q ) \ H × GSp( V, Ψ)( A f ) /K ;here A (respectively A f ) denotes the ring of (respectively finite) adeles. However, as weshall see presently M , n lands inside a smaller Shimura subvariety. Definition 4.3.2. (1) For a Q -algebraic subgroup H ⊂ GSp( V, Ψ), define H H := { h ∈ S → GSp( V R , Ψ) | h factors through H R } . (2) Recall that there is a Q ( ζ )-action on V . Define the algebraic group G := GL Q ( ζ ) ( V ) ∩ GSp( V, Ψ);here GL Q ( ζ ) ( V ) denotes the elements of GL( V ) commuting with the action of Q ( ζ )on V .(3) Let Sh denote the Shimura variety associated to the Shimura datum ( G, Y G ); byconstruction this is a subvariety of the Shimura variety associated to the Shimuradatum (GSp( V, Ψ) , H ). Proposition 4.3.3. (1)
The real points of the group H are given by G R ∼ = (cid:89) r ∈ ( Z /n ) × U ( V [ r ] ⊗ Q ( ζ ) C ) , where, on the right hand side, in the r th factor of the product the embedding Q ( ζ ) (cid:44) → C is the one sending ζ to ζ r . (2) The dimension of Sh is (4) 12 (cid:88) r ∈ ( Z /n ) × (2 r − n − − r ) . Here the sum is over representatives between and n of the elements of ( Z /n ) × .Proof. The first part follows from [Moo10, Remark 4.6]. For the second part, it suffices tofind the signature of the pairing on each of the subspaces V [ r ], since the hermitian symmetricdomain for the unitary group U ( a, b ) has dimension ab . The signatures, or equivalently theHodge numbers, are given by Proposition 3.1.2, and (4) follows immediately. (cid:3) TTRACTORS ARE NOT ALGEBRAIC 17
The Prym construction therefore gives us a map P : M , n → Sh . We have the following result, which is analogous to the classical result that the Torelli mapis an embedding, although we only require an infinitesimal version of this.
Lemma 4.3.4.
The derivative of P is injective.Proof. We will show equivalently that, for each x ∈ M , n , the codifferential map P ∗ : T ∗ P ( x ) Sh → T ∗ x M , n is surjective. Here for a variety X and a point x ∈ X we denote by T ∗ x X the cotangentspace to X at x .First we identify the source and target of P ∗ in terms of the geometric structures athand. Claim 4.3.5.
We have the following identifications of the cotangent spaces: T ∗ P ( x ) Sh ∼ = (cid:77) r ∈ ( Z /n ) × r 3, since it has to be the dimension of M , n .We will now show that the restriction (which we continue to denote by P ∗ )(7) P ∗ : H ( C, Ω C )[1] ⊗ H ( C, Ω C )[ n − → H ( C, (Ω C ) ⊗ ) inv is in fact an isomorphism; certainly the dimensions of the source and target agree, and so itsuffices to show this map is injective. But this is clear since the space H ( C, Ω C )[1] is one-dimensional and spanned by ω , say, and so anything in the kernel of the map (7) takes theform ω ⊗ η for some η ∈ H ( C, Ω C )[ n − ω and η are non-zero 1-formson C , and the quadratic differential obtained by multiplying them together is certainlynon-zero. This shows that the map (7) is injective, and therefore it is an isomorphism, asrequired. Therefore it suffices to prove Claim 4.3.5: Proof of Claim. By construction, we have an embeddingSh → Sh(GSp( V, Ψ) , H ) , where the right hand side denotes the Shimura variety attached to the Shimura datum(GSp( V, Ψ) , H ). The latter is the moduli space of abelian varieties of dimension ( n − φ ( n )equipped with a polarization of the fixed type specified by the polarization on the Prymvariety. Therefore the tangent space to Sh(GSp( V, Ψ) , H ) at P ( x ) is given by(8) Sym ( t Prym ) ⊂ t Prym ⊗ t Prym ∨ , where for an abelian variety A we denote by t A the tangent space at the origin, and A ∨ itsdual abelian variety. On the left hand side of the above we have also made the identification t Prym ∼ = t Prym ∨ using the polarization on Prym. Note that in (8) the right hand side is the deformationspace of Prym with no reference to polarizations. By Corollary 4.2.6, we may make theidentification t Prym ∼ = (cid:77) i ∈ ( Z /n ) × H ( C, O C )[ i ];for convenience we denote the right hand side of this identification by H ( C, O C ) prim ; sim-ilarly we define H ( C, Ω C ) prim to be the sum of the eigenspaces of H ( C, Ω C ) with eigen-values primitive n th roots of unity.On the other hand, as mentioned above, the right hand side of (8) is the deformationspace of the abelian variety Prym, and hence there is a Kodaira-Spencer map(9) KS : t Prym ⊗ t Prym ∨ → Hom( H ( C, Ω C ) prim , H ( C, O C ) prim ) , which is the natural isomorphism once we make the identifications t Prym ⊗ t Prym ∨ ∼ = H ( C, O C ) ⊗ , and H ( C, O ) prim ∼ = H ( C, Ω C ) ∨ prim , the latter of which is induced by Serre duality.By definition, Sh is contained in the locus of Sh(GSp( V, Ψ) , H ) where the Hodge struc-ture H admits a Q ( ζ )-action and a splitting H ⊗ Q Q ( ζ ) ∼ = (cid:77) r ∈ ( Z /n ) × H [ r ]with prescribed Hodge numbers. Therefore the Kodaira-Spencer map (refeqn:ks) restrictedto Sh must preserve the different eigenspaces. In other words, we have(10) KS | Sh : T P ( x ) Sh → (cid:77) r ∈ ( Z /n ) × Hom( H ( C, Ω C )[ r ] , H ( C, O C )[ r ]);now since KS itself is an isomorphism, KS | Sh is injective at least; on the other hand,deformations in Sh are also required to preserve the polarization, which means further that(11) KS | Sh : T P ( x ) Sh → (cid:77) r ∈ ( Z /n ) × r Corollary 4.3.6. The dimension of the image P ( M , n ) inside Sh is n − . Special subvarieties. Now that we have defined the relevant PEL type Shimuravariety Sh, we can rephrase Theorem 4.1.3 in terms of special subvarieties of Sh.For A one of the finite number of isogeny representatives of abelian varieties as inProposition 4.1.2, we denote by Sh A the sub-Shimura variety of Sh classifying abelianvarieties parametrized by Sh that split, as a product, of A and some other abelian variety.Then Sh A is a Shimura variety associated to group the Weil restriction, from Q ( ζ n ) + to Q ,of a unitary group G A . We now describe this in detail.We now introduce some notation to describe the signatures of G A under the variousembeddings Q ( ζ n ) + (cid:44) → R . Indeed, A is also an abelian variety with compatible Z /n -actionand Hodge structure; in other words, each eigenspace H ( A )[ i ] over Q ( ζ n ) again splits as H ( A )[ i ] ⊗ Q ( ζ n ) C (cid:39) H , ( A )[ i ] ⊕ H , ( A )[ i ] . (We will almost immediately argue that this splitting, once again, is defined over Q ( ζ n ).)But the Galois conjugates of ω are in distinct eigenspaces H ( A )[ i ] and already account for φ ( n ) dimensions’ worth of cohomology – which is the total dimension of H ( A ) as a vectorspace. Hence, for each i , the total dimension of H , ( A )[ i ] ⊕ H , ( A )[ i ] is one, and so onespace has dimension one while the other has dimension zero. Definition 4.4.1. Define n A ( i ) := dim H , ( A )[ i ] . From the remark above the n A ( i )’s take values either zero or one. We will sometimes alsorefer to the n A ( i )s as the CM type of A , since they encode information equivalent to thestandard notion of CM types.In particular, note that as ω itself is in the first eigenspace and is holomorphic, we know n A (1) = 1.Recall that V Z denotes the integral lattice given by the Prym variety and that G denotesthe Q -algebraic group associated to the Shimura variety Sh. Now for each CM type { n A ( i ) } ,we fix a splitting of the integral Hodge structure of the form(14) V Z ∼ = V A ⊕ V (cid:48) , where both V A and V (cid:48) have actions by Z [ ζ ], and such that V A has the CM type givenby { n A ( i ) } ; more precisely, the Z [ ζ ]-action on V A allows us to define eigenspaces V A [ i ] asbefore, and we require dim C V , A [ i ] = n A ( i )for each i = 1 , · · · , n . Definition 4.4.2. Let G A denote the Q -algebraic group which is the subgroup of G pre-serving the subspaces V A ⊗ Z Q and V (cid:48) ⊗ Z Q . Definition 4.4.3. (1) For a Q -subgroup H ⊂ G such that the subspace H H (see Definition 4.3.2) is non-empty, let H + H denote a connected component of H H . A special subvariety (for thesubgroup H ) is the image under the uniformization map H × G ( A f ) /K → G ( Q ) \ H × G ( A f ) /K = Sh( C )of H + H × ηK , for some element η ∈ G ( A f ).(2) Let Sh A denote the special subvariety associated to the inclusion of a connectedcomponent of H G A , and the element η = 1.From Theorem 4.1.3 we immediately deduce the following slight refinement: Proposition 4.4.4. For each attractor point x in M , n there exists a CM type { n A ( i ) } such that x lies in a special subvariety of Sh associated to the group G A .Remark . The element η ∈ G ( A f ) is measuring the isogeny to the reference integralHodge splitting (14).We may now describe the signatures the unitary form defining G A in terms of thesenumbers n A ( i ): under the i th embedding of Q ( ζ n ) + (cid:44) → R induced from ζ n (cid:55)→ ζ in , we havethat the unitary form now has signature (2 i − − n A ( i ) , n − i ) − n A ( i )) anddim Sh A = (cid:98) ( n − / (cid:99) (cid:88) i =1 ,i ⊥ n (2 i − − n A ( i ))(2( n − i ) − n A ( i )) . More interesting is the codimension of Sh A within Sh, namelycodim X Sh A = (cid:98) ( n − / (cid:99) (cid:88) i =1 ,i ⊥ n (cid:16) n A ( i )(2( n − i ) − 1) + (1 − n A ( i ))(2 i − (cid:17) . TTRACTORS ARE NOT ALGEBRAIC 21 In particular, as n ( A ) = 1, we have(15) codim Sh Sh A ≥ n − , with equality if and only if there is only one value of i in the sum, i.e. φ ( n ) = 2. On theother hand, dim M , n = 2 n − 3, so unless equality holds in (15), one generically expects M , n and (Hecke translates of) Sh A to not intersect within Sh purely for dimensionalreasons. The Zilber-Pink conjecture of transcendental number theory makes precise thisexpectation and we recall it shortly – but first we allow a brief digression on the particularcases of φ ( n ) = 2 where this dimensional expectation does not hold. Example 4.4.6. We give an example to illustrate the numerology that is at play here,for n = 5, which is the smallest value for which the Dolgachev Calabi-Yau varieties givecounterexamples to the Attractor Conjecture. In this case, since 5 is a prime number, thePrym variety is simply the Jacobian of C . Let us also fix a CM type, say n A (1) = n A (2) = 1;then we write schematically the splitting of the Hodge structure as(16) = + . In the equation above we follow Notation 3.2.5, and on the right hand side we have writtenthe dimension matrices of the summands of this splitting.4.5. A brief digression: the arithmetic cases of n = 3 , , and connections totilings of the sphere. In this subsection we observe that the Attractor Conjecture worksremarkably well in the cases when the Calabi-Yau moduli space does happen to be a Shimuravariety, and point out a connection to the tilings of the sphere by polygons due to Engel-Smillie [ES18]. Proposition 4.5.1. For n = 3 , , , there is a bijection between attractor points and tilingsof the sphere by triangles, squares and hexagons, respectively.Proof. After quotienting by the appropriate arithmetic group, the Schwarz map π consideredin Section 3.5 coincides precisely with the map denoted by D in [ES18, Proof of Proposition2.5, p.7]. Furthermore, the integral points in the image of D correspond to tilings of thesphere, as required. (cid:3) In fact, we mention an intriguing question for the interested reader: Engel-Smillie inthe above paper study a very precise generating function of the attractor points in thearithmetic n = 3 , , Unlikely intersection We now recall the Zilber-Pink conjecture: Conjecture 5.0.1 ([Pin05, Conjecture 1.3]) . Given a subvariety Y ⊂ X of a Shimuravariety and a countable collection of special Shimura subvarieties { X α } of codimensiongreater than dim Y , if (cid:91) α Y ∩ X α ⊂ Y is Zariski-dense, then Y ⊂ X (cid:48) is contained within some proper special Shimura subvariety X (cid:48) ⊂ X . We will now argue that for M , n is not contained in any special Shimura subvarietyof Sh. Once again, these arguments hold for all n ≥ 2; the only place where the φ ( n ) > G = GL Q ( ζ ) ( V ) ∩ GSp( V, Ψ) from Definition 4.3.2. Let G (cid:48) denote the Q -algebraic group, obtained via restriction of scalars from Q ( ζ ) + , whose Q ( ζ )-points are given by G (cid:48) ( Q ( ζ )) = (cid:89) r ∈ ( Z /n Z ) × SU ( V [ r ]) . Proposition 5.0.2. Let G denote the Q -Zariski closure of the fundamental group of M , n acting on H ( C ) . Then G contains the group G (cid:48) above. Before turning to the argument for this proposition, we note that this requirement isexactly the hypothesis necessary to apply Zilber-Pink to conclude Theorem 1.1.3 in the φ ( n ) > X cor-respond to subgroups of G and being contained within some special Shimura subvarietywould imply a corresponding restriction on the Zariski-closure of the monodromy group. Ithence remains to establish the above proposition. Proof. This is essentially due to Deligne-Mostow [DM86], but we will use the version statedby Looijenga in his review of Deligne-Mostow, which we now describe. For brevity, in thefollowing, for r ∈ ( Z /n ) × we denote by V [ r ] the Q ( ζ )-vector space H ( C, Q ( ζ ))[ r ], and by V the Q -subspace of H ( C, Q ) such that V ⊗ Q ( ζ ) = (cid:77) r ∈ ( Z /n ) × V [ r ] . As in [Loo07, Proof of Theorem 4.3], we have a decomposition G ( C ) = (cid:89) r ∈ ( Z /n ) × G r ( C )with G r ( C ) ⊂ GL( V [ r ])( C ). Since this was stated without proof in loc.cit., we provide anexplanation here, although no doubt it is well known to experts. The reason is that theaction of the fundamental group Γ preserves each direct summand V [ r ], and therefore the Q -algebraic group G commutes with the action of Q ( ζ ) on H ( C, Q ). Therefore the basechange G ⊗ Q ( ζ ) commutes with the action of Q ( ζ ) ⊗ Q Q ( ζ ); the latter can be written as (cid:77) r ∈ ( Z /n ) × Q ( ζ ) , and in particular contains idempotents (cid:15) r , the projection onto the r th factor, for each r ∈ ( Z /n ) × . Therefore we have a decomposition G ⊗ Q ( ζ ) = (cid:89) r ∈ ( Z /n ) × G r , as claimed.Furthermore, again according to [Loo07, Proof of Theorem 4.3], G r ( C ) contains thespecial unitary group of V [ r ]. Therefore the Zariski closure contains G (cid:48) , as required. (cid:3) We now put all the ingredients together to conclude the proof of our main result. EFERENCES 23 Proof of Theorem 1.1.3. Suppose that n (cid:54) = 3 , , 6. We assume for the sake of contradictionthat attractor points are defined over Q . By Section 3.5 the attractor points are Zariskidense in moduli space. Now recall that we have the Prym map P : M CY → Sh , where Sh denotes the Shimura variety from Definition 4.3.2, and consider its image P ( M CY )inside the Shimura variety Sh. That this subvariety has dimension 2 n − 3, i.e. the samedimension as M CY , is precisely Corollary 4.3.6.On the other hand, by Proposition 4.4.4, each attractor point lies in a sub-Shimura va-riety, whose codimension inside Sh is strictly greater than 2 n − P ( M CY ) must be contained insome proper special subvariety of Sh, which is impossible by the monodromy computation inProposition 5.0.2. Therefore the attractor points cannot be defined over Q , as required. (cid:3) References [BR11] Ilka Brunner and Daniel Roggenkamp. “Attractor flows from defect lines”. In: J.Phys. A issn : 1751-8113. doi : . url : https://doi.org/10.1088/1751-8113/44/7/075402 .[Can+19] Philip Candelas, Xenia de la Ossa, Mohamed Elmi, and Duco van Straten. “AOne Parameter Family of Calabi-Yau Manifolds with Attractor Points of RankTwo”. In: (2019). url : https://arxiv.org/abs/1912.06146 .[DM86] P. Deligne and G. D. Mostow. “Monodromy of hypergeometric functions andnonlattice integral monodromy”. In: Inst. 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