Disproof of modularity of moduli space of CY 3-folds of double covers of P3 ramified along eight planes in general positions
aa r X i v : . [ m a t h . AG ] S e p DISPROOF OF MODULARITY OF MODULI SPACE OFCY 3-FOLDS OF DOUBLE COVERS OF P RAMIFIEDALONG EIGHT PLANES IN GENERAL POSITIONS
RALF GERKMANN, MAO SHENG † , AND KANG ZUO Abstract.
We prove that the moduli space of Calabi-Yau 3-foldscoming from eight planes of P in general positions is not modular.In fact we show the stronger statement that the Zariski closure ofthe monodromy group is actually the whole Sp(20 , R ). We con-struct an interesting submoduli, which we call hyperelliptic locus ,over which the weight 3 Q -Hodge structure is the third wedge prod-uct of the weight 1 Q -Hodge structure on the corresponding hy-perelliptic curve. The non-extendibility of the hyperelliptic locusinside the moduli space of a genuine Shimura subvariety is proved.1. Introduction
In the study of geometry of moduli space, it is important to characterize thosemoduli spaces which are locally Hermitian symmetric varieties. We refer the readerto [20], [21], [12] for such a theory based on the Arakelov equality. On the otherhand, in order to prove a negative result it is also important to find some necessaryconditions, which can be checked quite easily for explicitly given moduli spaces.In this paper, we will work with an interesting moduli space of CY 3-folds, whichcomes from the hyperplane arrangements in P consisting of eight planes in generalpositions. The aim of our present work is to disprove the modularity of this modulispace by two different methods. Before stating the main theorem, we shall makethe meaning of modularity precise, since it could be ambiguous in certain cases. Forexample, the moduli space of six lines of P in general positions, which is identicalto the moduli space of six points of P in general positions, can be openly embeddedeither into an arithmetic quotient of type four bounded symmetric domain [11] orinto an arithmetic ball quotient [1], [5] by different period mappings.Let M be the coarse moduli scheme representing a moduli functor M of polarizedalgebraic manifolds of dimension n . After a finite base change of M , one obtainsa universal family f : X → S . The rational primitive middle cohomologies of the † The second named author is supported by a postdoctoral fellowship in the EastChina Normal University. fibers of f constitute a Q -polarized variation of Hodge structures V . It induces theperiod mapping φ : S ֒ → Γ \ G ′ /K ′ where G ′ /K ′ is the classfying space of polarized Hodge structures. If there existsa locally Hermitian symmetric variety Γ \ G/K and a locally homogenous PVHS W over it such that φ factors through the period mapping ψ : Γ \ G/K ֒ → Γ ′ \ G ′ /K ′ defined by W and such that the induced map φ : S → Γ \ G/K is an open embedding, then we say that f is a modular family with respect to( G/K, W ). In the case that the reference locally homogenous PVHS W and theHermitian symmetric space G/K are clear from the context, we simply say f ismodular family. The moduli space M is said to be modular if a certain universalfamily of M is a modular family. Under this definition, it is clear that if f : X → S is a modular family with respect to ( G/K, W ), then one has a factorization of themonodromy representation ρ of V : π ( S ) G ′ G ........................................................................................................................................................................................................ ............ ρ ................................................................................... ............ ..................................................................................................... ̺ where ̺ : G → G ′ is the group homomorphism determined by W .Now let M CY be the moduli space of CY 3-folds from eight planes of P ingeneral positions. The classfying space of the polarized Hodge structure on themiddle cohomology of such a CY 3-fold is D ′ = Sp(20 , R )U(1) × U(9) . The natural Hermitian symmetric space in this case is D = SU(3 , × U(3)) , and the locally homogenous PVHS W is the Calabi-Yau like PVHS over Γ \ D (cf. [19]), which is induced from the group homomorphism ^ : SU(3 , → Sp(20 , R ) . The main theorem of this paper is the following
Theorem 1.1.
Let M CY and ( D, W ) be as above. Then M CY is not modular withrespect to ( D, W ) . ISPROOF OF MODULARITY 3
To keep our theorem in perspective, we would like to point out that the analogousmoduli spaces of CY n -folds for n ≤ n ≥ , R ) and an application of the plethysm methodwe can deduce a stronger result about the Zariski closure of the monodromy group. Theorem 1.2.
Let f : X → S be a universal family of M CY , and V = R f ∗ Q X be the interesting weight 3 VHS. Then the image of the monodromy representation ρ : π ( S ) → Sp(20 , R ) of V is Zariski dense. In the work of [11], a special submoduli, which is isomorphic to an arithmeticquotient of Sp(4 , R ) / U(2), was constructed. We shall generalize their constructionto our case. Since this submoduli arises from the moduli of hyperelliptic curvesof genus 3, we simply call it the hyperelliptic locus . We show that over the fivedimensional hyperelliptic locus the weight 3 VHS of CY 3-folds is isomorphic tothe wedge product of weight 1 VHS (cf. Prop. 2.4). It is then natural to ask if onecan extend the hyperelliptic locus in M CY to a six dimensional submoduli whichis isomorphic to an arithmetic quotient of Sp(6 , R ) / U(3). Using the concept ofcharacteristic subvarieties we arrive at a negative answer of this extension problem.
Theorem 1.3.
Let H CY be the hyperelliptic locus of M CY . Then there exists noextension of H CY inside M CY such that it is isomorphic to a Zariski open subset ofan arithmetic quotient of Sp(6 , R ) / U(3) which is embedded into M CY via the wedgeproduct of the weight 1 VHS of a universal family of abelian 3-folds. The paper is organized as follows. In § § § Acknowledgements:
The major part of this work was done during an academic visitof the second named author to the department of mathematics of the universityof Mainz in 2006. He would like to express his hearty thanks to the hospitality of
RALF GERKMANN, MAO SHENG, AND KANG ZUO the faculty, especially Stefan M¨uller-Stach. We also thank Eckart Viehweg for hisinterests and several helpful conversations about this work.2.
Calabi-Yau manifolds from eight planes of P in general positions Let H , · · · , H denote eight planes of P in general position. They sum up to asimple normal crossing divisor B = H + · · · + H on P . Since B is even, we can form the double cover X of P with branch locus B .Obviously, X is only a singular variety because B is singular. Fixing an ordering ofthe irreducible components H ij = H i ∩ H j of singularities of B , we use the canonicalresolution of double covers to obtain a smooth model ˜ X of X . Namely, we havethe following commutative diagram X ˜ X P ˜ P .................................................................................................... τ .................................................................................................... σ .................................................................................................... π .................................................................................................... ˜ π where σ : ˜ P → P is the composition of the sequence of blows-up with smoothcenters (the strict transform of) H ij . The variety ˜ X is a smooth projective CY3-fold with h , ( ˜ X ) = 9 and h , ( ˜ X ) = 29 . This construction can actually be extended to all 2 n +2 hyperplane arrangements of P n in general positions, and the Hodge numbers of the primitive middle cohomologyof the resulting smooth CY n -fold ˜ X are h p,n − ppr ( ˜ X ) = (cid:18) np (cid:19) . For the details, we refer to chapter 3 of [14].It is easy to see that the moduli space of ordered eight hyperplane arrangementsof P in general positions is of dimension 9. Hence, by fixing an ordering of theindex set I = { ( i, j ) ∈ N | ≤ i < j ≤ } , the above constructions gives rise to a complete moduli scheme M CY of smooth CY3-folds. We note that a different ordering of I yields a different birational minimalmodel of the singular CY X . ISPROOF OF MODULARITY 5
Now we consider the embedding determined by the starting hyperplane arrange-ment ( H , ..., H ), namely j : P ֒ → P , x ( ℓ ( x ) : · · · : ℓ ( x ))where ℓ i : C → C denotes a linear form such that H i = { x ∈ P | ℓ i = 0 } for1 ≤ i ≤
8. The defining equations of j ( P ) ⊂ P are the four linearly independentrelations among ℓ , · · · , ℓ . Written out explicitly, they are a y + · · · + a y = 0... ... a y + · · · + a y = 0where ( y : · · · : y ) denote homogenous coordinates of P . We can define a newCY 3-fold Y which is the complete intersection of four quadrics in P defined by(2.1) a y + · · · + a y = 0... ... a y + · · · + a y = 0 . The variety Y is smooth since any 4 × A := ( a ij ) is nonzero.The two Hodge numbers of Y are computed to be h , ( Y ) = 65 and h , ( Y ) = 1 . In particular, the moduli space of complex structures on Y is of dimension 65. Thecovering map P −→ P , ( y : · · · : y ) ( y : · · · : y )restricts to p : Y → j ( P ). The composite map, denoted again by p , p : Y −→ j ( P ) ∼ = P exhibits Y as the Kummer covering of P , branched along B with degree 2 . Clearly,the Galois group Aut( Y | P ) of Y over P is isomorphic to ( Z / Z ) . There is acanonical surjection G := ( Z / Z ) −→ Aut( Y | P ) , a = ( a , ..., a ) σ a with σ a ( y i ) = ( − a i y i for 1 ≤ i ≤
8. Its kernel of order 2 is generated by (1 , ..., N ⊳ G given by N := ker (cid:18) G ∼ = ( Z / Z ) P −→ Z / Z (cid:19) . The following proposition reveals the geometric relation between two CY manifoldscoming from the same hyperplane arrangement.
RALF GERKMANN, MAO SHENG, AND KANG ZUO
Proposition 2.1.
Let ( H , ..., H ) be a hyperplane arrangement of P in generalposition. Then one can construct two smooth CY 3-folds ˜ X and Y as above. Onehas a natural isomorphism H ( ˜ X, Q ) ∼ = H ( Y, Q ) N . Proof.
The quotient map p factors as Y α −→ Y /N −→ P . The degree of
Y /N over P is 2 and one can directly check that Y /N branchesexactly along B . Since P has no torsion element in the second integral coho-mology, we have the identification X = Y /N . Now let us examine the followingcommutative diagram Y ˜ Y P ˜ P .................................................................................................... ˜ τ .................................................................................................... σ .................................................................................................... p .................................................................................................... ˜ p where ˜ Y is the normalization of the fiber product of Y and ˜ P over P . Obviously,˜ τ is a contraction map. Since Y is smooth, ˜ τ induces the isomorphismH ( Y, Q ) ∼ = H ( ˜ Y , Q ) . We put ˜ B to be the strict transform of B under σ . Then the projection ˜ p is theKummer covering map of degree 2 with branch locus ˜ B . Argued as previously, ˜ p factors as ˜ Y ˜ α −→ ˜ Y /N = ˜ X ˜ p −→ ˜ P . Since ˜ τ is G -equivariant, we have H ( Y, Q ) N ∼ = H ( ˜ Y , Q ) N , and as both ˜ X and˜ Y are smooth,(2.1) H ( ˜ X, Q ) ∼ = H ( ˜ Y , Q ) N . Therefore, combining the last two isomorphisms, we obtain the isomorphism statedin the proposition. (cid:3)
We proceed to construct the hyperelliptic locus H CY inside our moduli space M CY , generalizing the construction in [11]. We first recall that there is a naturalGalois covering γ : ( P ) −→ P with Galois group S , the symmetric group of three letters. Explicitly, let( x i : y i ) , ≤ i ≤ i -th factor of ( P ) , such that the componentsof the quotient map γ are given by the t -coefficients of the polynomial f ( t ) = ISPROOF OF MODULARITY 7 Q i =1 ( x i t + y i ). Now we take arbitrary eight distinct points p , p , ..., p ∈ P andconstruct a hyperplane arrangement from it. Lemma 2.2.
For ≤ i ≤ let H i denote the image of { p i } × P × P under γ .Then H i is a hyperplane in P , and the hyperplane arrangement ( H , H , ..., H ) isin general position.Proof. Let ( z : z : z : z ) be the homogenous coordinates of P , and p = ( a : b )be a point of P . Then using the expression of γ , the defining equation of the imageset γ ( { p } × P × P ) is easily seen to be b z − ab z + a bz − a z = 0 . Therefore, H i is obviously a hyperplane in P . We can choose an appropriatesystem of coordinates on P such that the eight points have coordinates ( − a i : 1)for 1 ≤ i ≤
8. Then the columns of the following matrix give the defining equationsof the arrangement ( H , H , · · · , H ): · · · a a · · · a a a · · · a a a · · · a Now the property of the hyperplane arrangement to be in general position is equiv-alent to that all 4 × a , a , ..., a aredistinct from each other by our assumption, all 4 × (cid:3) Now let C be the hyperelliptic curve over P branched at p , · · · , p , and let q denote the corresponding covering map. The Galois group G of the compositionof morphisms C q −→ ( P ) γ −→ P is isomorphic to the semi-direct product N ⋊ S , where N = h ι , ι , ι i is thegroup generated by the hyperelliptic involutions on each factor of C × C × C . Oneobserves that there is a distinguished index two subgroup G ′ = N ′ ⋊ S of G ,where N ′ is the kernel N ′ := ker (cid:18) N ≃ ( Z / Z ) P −→ Z / Z (cid:19) RALF GERKMANN, MAO SHENG, AND KANG ZUO of the multiplication map. This gives the following commutative diagram of Galoiscoverings: C X ( P ) P ........................................................................................ ............ δ .................................................................................................... q π .............................................................................. ............ γ Lemma 2.3.
The double cover π : X → P branches along the union of the hyper-plane arrangement ( H , H , · · · , H ) .Proof. The Galois group of π is generated by ι in G /G ′ . By the commutativityof the above diagram, the branch locus of π is the image of the fixed locus of ι under the morphism γ ◦ q . By Lemma 2.2, it is clear that the image is S i =1 H i . (cid:3) By this lemma, the moduli of hyperelliptic curves of genus 3 are embedded intothe moduli space M CY . We call the image H CY the hyperelliptic locus , which isfive-dimensional. In [11], the analogous submoduli were also characterized as thosesix lines in general positions tangential to a smooth conic of P , and it was shownthat this submoduli gives the family of Kummer surfaces. The Hodge structure ofCY threefold over the hyperelliptic locus is also special in our case. Proposition 2.4.
Let ˜ X be the canonical resolution of X . We have an isomor-phism of rational polarized Hodge structures H ( ˜ X, Q ) ∼ = ^ H ( C, Q ) . Proof.
As a consequence of Proposition 2.1, we know thatH ( ˜ X, Q ) ∼ = H ( X, Q ) . So it suffices to prove the isomorphism for X . For this purpose we consider thefollowing commutative diagram C S ( C ) Jac( C ) X ......................................................................... ............ δ ϕ ........................................................................................................................................ ............ δ .................................................................................................... δ where ϕ is the Abel-Jacobi map, δ is the quotient map by the subgroup S ≤ G ′ and δ is the projection map. One notes that, since S is not normal in G ′ , themap δ is only a finite morphism. However, δ induces the embedding δ ∗ : H ( X, Q ) ∼ = H ( C , Q ) G ′ ֒ → H ( C , Q ) . ISPROOF OF MODULARITY 9
Since δ = δ ◦ δ , the pullback δ ∗ gives the embedding H ( X, Q ) δ ∗ −→ H ( S ( C ) , Q ) . By the Abel-Jacobi theorem, ϕ is a birational morphism and thus induces an iso-morphism of Hodge structures on the middle cohomology: ϕ ∗ : H (Jac( C ) , Q ) ∼ = −→ H ( S ( C ) , Q ) . In particular, dim Q H ( S ( C ) , Q ) = 20. Since we computed before that the dimen-sion of H ( ˜ X, Q ) is also 20, the map δ ∗ is in fact an isomorphism. The compositionmap H ( X, Q ) δ ∗ −→ H ( S ( C ) , Q ) ( ϕ ∗ ) − −→ H (Jac( C ) , Q ) ∼ = ^ H ( C, Q )gives the isomorphism required in the proposition. (cid:3) Remark 2.5.
It is worthwhile to remark that the same construction and argu-ments generalize to n ≥ n − n -dimensional moduli of CY manifolds, over which the primitive mid-dle dimensional rational Hodge structures are wedge products of weight 1 Hodgestructures. 3. Characteristic Subvariety and Plethysm
In this section, we will present two different methods to disprove the modularityof M CY . Our first method is to study a series of invariants of IVHS, introduced in[19], which we call characteristic subvarieties . These invariants exploit the geometryof the kernels of iterated Higgs fields of the associated system of Hodge bundles withthe given IVHS. In the case of Calabi-Yau like PVHS over bounded symmetricdomain, these invariants are proved to be the characteristic bundles introducedin [10] by N. Mok, which played a pivotal role in the proof of the metric rigiditytheorem of compact quotient of bounded symmetric domains of rank ≥
2. Thesecond method uses the idea of plethysm in representation theory (cf. [6]). For afixed simple complex Lie algebra g the plethysm describes the decompositions ofrepresentations derived from a given irreducible representation of g .3.1. Characteristic Subvariety.
We first recall some results in [19]. The boundedsymmetric domain D = SU(3 , U (3) × U(3))is of rank 3. Let W be the Calabi-Yau like PVHS over Γ \ D and ( F, η ) be theassociated system of Hodge bundles. By Theorem 3.3 in [19] we have the following
Lemma 3.1.
For k = 1 , the k -th characteristic subvariety S k of ( F, η ) coincideswith k -th characteristic bundle. In particular, for every point x ∈ Γ \ D , ( S ) x ∼ = P × P , and ( S ) x is isomorphic to the determinantal hypersurface in P . Now we take a universal family f : X → S of M CY . Let V := R f ∗ Q X and E = M p + q =3 E p,q , θ = M p + q =3 θ p,q ! be the corresponding system of Hodge bundles. Since V is of weight 3, we havealso two characteristic subvarieties of ( F, η ), which are denoted by R k for k = 1 , f is a modular family, then the period mapping φ : S ֒ → Γ \ D will induce anisomorphism φ ∗ W ∼ = V , hence an isomorphism φ ∗ ( E, θ ) ∼ = ( F, η ). This implies the isomorphisms φ ∗ S k ∼ = R k for k = 1 , . Using Lemma 3.1, we then have the following
Proposition 3.2.
If there exists a point x ∈ S such that ( R ) x = P × P or ( R ) x is not isomorphic to the determinantal hypersurface in P , then f is nota modular family. Remark 3.3.
It was first pointed out by E. Viehweg that the iterated Higgs fieldsfor (
E, θ ) are surjective. Namely, the maps θ : S k ( T S ) −→ Hom( E , , E − k,k )are surjective for 1 ≤ k ≤
3, where T S denotes the tangent bundle over S . If oneof these maps were not surjective, then the disproof of modularity of M CY wouldhave been obtained at this stage already. This phenomenon (or difficulty) actuallymotivated the two latter authors to study the characteristic subvariety in [19]. Itturned out that the present work gives a non-trivial application of the theory ofcharacteristic subvarieties.3.2. Plethysm.
The simple real Lie group SU(3 ,
3) is a real form of SL(6 , C ). ByWeyl’s unitary trick, one has an equivalence of categories of finite dimensional com-plex representations of SU(3 ,
3) and finite dimensional complex representations of g := sl (6 , C ). So the plethysm problem for SU(3 ,
3) is transformed into the plethysmproblem for g .Let V := C be the standard representation of g . We shall study the plethysmfor the fundamental representation W := V ( V ). In other words, we shall study thedecomposition of S λ ( W ) for a Schur functor S λ . The two simplest Schur functorsare S and V . By Exercise 15.32 in [6] we have the following decompositions:(3.1) S ( W ) = Γ ⊕ Γ , ^ W = Γ ⊕ Γ ISPROOF OF MODULARITY 11
By formula (15.17) in [6] it is easy to compute thatdim C Γ = 1 , dim C Γ = 35 , dim C Γ = 189and dim C Γ = 175. However, V W will be of no use for us. That is because,considering W as sp (20 , C )-representation, one also has a decomposition ^ W = C ⊕ W ′ where C is the trivial representation of sp (20 , C ) spanned by the symplectic form.On the other hand, S ( W ) is an irreducible representation of sp (20 , C ). It is actuallythe adjoint representation. Proposition 3.4.
Let f : X → S be a universal family of M CY . If S ( E, θ ) isnot decomposed according to the following pattern, then f is not a modular family.Explicitly, S ( E, θ ) = ( E , θ ) ⊕ ( E , θ ) where E = E , ⊕ E , ⊕ E , E = E , ⊕ E , ⊕ E , ⊕ E , ⊕ E , ⊕ E , ⊕ E , . Furthermore, the dimensions of Hodge bundles of E are respectively , , , , , , and those of E are , , , , , , .Proof. The modularity of f will imply a factorization of the monodromy represen-tation ρ : π ( S ) −→ SU(3 , V −→ Sp(20 , R ) . Thus for any Schur functor S λ the derived PVHS S λ ( V ) will decompose into irre-ducible SU(3 , S ( V ) = V ⊕ V . The system of Hodge bundles S ( E, θ ) decomposes into a direct sum of system ofHodge bundles accordingly, S ( E, θ ) = ( E , θ ) ⊕ ( E , θ ) . Since W is of weight 3, S ( W ) is of weight 6. One can compute the Hodge numbersof ( E i , θ i ) for i = 1 , ,
3) tothe center U(1) of its maximal compact subgroup S(U(3) × U(3)). If denotes the3 × ( C z := z z − ! ∈ GL ( C ) | z ∈ U(1) ) is the center of S(U(3) × U(3)). We choose the standard basis ( e , ..., e ) of V = C such that C z ( e i ) = ze i for 1 ≤ i ≤ C z ( e i ) = z − e i for 4 ≤ i ≤ . One notes that Γ is the unique nontrivial component in Γ ⊗ Γ . Itis easy to compute that C z acts on Γ with three characters z , z , z − , andthe dimensions of their eigenspaces are respectively 9 , ,
9. Then the characters of C z on the other direct component Γ are z , z , z , z , z − , z − , z − , and theirdimensions of eigenspaces are computed to be 1 , , , , , ,
1, respectively. Theproof of the proposition is complete. (cid:3) The Jacobian Ring
In the subsequent part we will carry out the strategies described in section 3to the special family of CY 3-folds constructed in section 2. For this purpose welet S denote the moduli space of eight planes in P in general positions. Everypoint s ∈ S can be determined by a matrix A ∈ C × with the property that all(4 × A are non-zero. Furthermore, we let f : ˜ X −→ S denote the universal family of M CY such that for every every fiber ˜ X := ˜ X s isobtained by resolution of singularities from the ramified double cover X → P as-sociated to a certain matrix A as described in section 2. For our purposes it willbe necessary to give an explicit description of the PVHS V := R f ∗ C X and theassociated system ( E, θ ) of Higgs fields in every fiber.First we give a description of V as a local system of graded C -vector spaces.Let O S denote the sheaf of holomorphic functions on S and a ij ∈ Γ( S, O S ) thecoordinate functions for 1 ≤ i ≤ ≤ j ≤
8. Furthermore, we let R := O S [ x , ..., x , y , ..., y ]denote the free O S -algebra in 12 indeterminates. For p ∈ N we define R p to bethe O S -submodule of elements which have total degree deg X = 2 p in the variables x j and total degree deg Y = p in the variables y i . We define a global sections f i , f ∈ Γ( S, R ) by f i := P j =1 a ij x j for 1 ≤ i ≤ F := P i =1 y i f i . The twelvepartial derivatives ∂F∂x j for 1 ≤ j ≤ ∂F∂y i for 1 ≤ i ≤ R which we denote by I . Finally, we let the group G from section 2 act on the sheaf R by sending a = ( a , ..., a ) σ a with σ a ( x i ) = ( − a i x i for 1 ≤ i ≤ σ a ( y j ) = y j for 1 ≤ j ≤ . ISPROOF OF MODULARITY 13
Then obviously σ a ( I ) ⊆ I holds for all a ∈ G . Now we obtain the followingexplicit description of our PVHS V . Proposition 4.1.
There is a canonical isomorphism of local systems V ⊗ C O S ∼ = ( R / I ) N which maps V − p,p ⊗ C O S onto the submodule generated by R p for ≤ p ≤ .Proof. Let g : Y → S denote the family of intersections of four quadrics in P as constructed in section 2, i.e. for every s = A = ( a ij ) ∈ S the fiber Y s inthe intersection of quadrics given by the equations (2.1). Furthermore, by W := R g ∗ C Y we denote the associated PVHS. According to Proposition 2.1 we have acanonical isomorphism V ∼ = W N , so that it remains to establish the isomorphism(4.1) W N ⊗ C O S ∼ = ( R / I ) N . First we show that W ⊗ C O S ∼ = R / I . This is a special case of Proposition 2.2.10in [13], and although it is stated only for individual varieties, the result carries overto algebraic families. Here we just sketch the essential steps. Let P S denote theprojective 7-space over S on which the coherent sheaf E := O P S (2) ⊕ is defined, and let P := P ( E ) denote the associated projective bundle. Then P contains a toric hypersurface ˆ Y given by the equation F = P i =1 y i f i from above.Let π : P → P S denote the canonical projection, extend g to a map g : P S → S andlet h := g ◦ π . Then the embedding π − ( Y ) ֒ → ˆ Y induces a natural isomorphism R h ∗ C ˆ Y ∼ = W ⊗ H ( P , C )of PVHS on S , the right part of the tensor product being constant of rank one.Now let V := P \ Y denote the open complement of Y . Then the Gysin sequencerelating the PVHS’s of P , ˆ Y and V gives rise to an isomorphism R h ∗ C ˆ Y ∼ = R h ∗ C V of PVHS. In order to compute the latter, we make use of de Rham’s theorem whichenables us to describe the cohomology R h ∗ C V ⊗ C O S ∼ = R h ∗ Ω ·V| S ∼ = R h ∗ Ω · P | S ( ∗ ˆ Y )in terms of the sheaf Ω · P | S ( ∗ ˆ Y ) of relative differentials on P with poles along ˆ Y ,where the functor R h ∗ denotes hypercohomology. Since the sheaves Ω iP | S ( m ˆ Y )are acyclic for i, m >
0, it can be computed by taking global sections. That is, if Z P | S ⊆ Ω P | S ( ∗ ˆ Y ) denotes the subsheaf of closed differentials and B P | S the subsheafof exact ones, then simply R h ∗ Ω · P | S ( ∗ ˆ Y ) ∼ = ( h ∗ Z P | S ) / ( h ∗ B P | S ) . The sections of h ∗ Ω P | S ( ∗ ˆ Y ) can be described in terms of the O S -algebra R . Namely,let ω denote the homogeneous differential formˆΩ := dx ∧ · · · ∧ dx ∧ dy ∧ · · · ∧ dy and define the vector fields ϑ i := ∂/∂x i and λ j := ∂/∂y j for 1 ≤ i ≤ ≤ j ≤
4. If we put θ := X j =1 y j λ j , θ := X i =1 x i ϑ i − X j =1 y j λ j and Ω := θ θ ( ˆΩ), then every section ω of h ∗ Ω · P | S ( p ˆ Y ) can be written in the form(4.2) ω = H Ω F p +4 where H is a section of R p . In degree 9, any section ψ of h ∗ Ω P | S ( ∗ ˆ Y ) can be written as ψ = P i =1 G i Ω i − P j =1 H j Ω ′ j F p +4 where Ω i := θ θ ϑ i , Ω ′ j := θ θ λ j and G i , H j are sections of R such that deg X ( G i ) =2 p + 1 , deg Y ( G i ) = p and deg X ( H j ) = 2 p , deg Y ( H j ) = p + 1 for 1 ≤ i ≤ ≤ j ≤
4. Its exterior derivative is dψ = H Ω /F p +5 where H = 2 X i =1 ∂F∂x i G i + 2 X j =1 ∂F∂y j h j − F X i =1 ∂G i ∂x i + X j =1 ∂H j ∂y j . We see that ω can be reduced to lower pole order if and only if the section h is asection of the ideal sheaf I . This shows that( h ∗ Z P | S ) / ( h ∗ B P | S ) ∼ = R / I . Combing all isomorphisms, the desired assertion W ⊗ C O S ∼ = R / I follows. Ob-serving that the action of G on W ⊗ C O S is compatible with the action definedabove on R / I , we obtain W N ⊗ C O S ∼ = ( R / I ) N .In order to prove the refined statement on the grading, notice that by the aboveconstruction W − p,p ∼ = R − p, p h ∗ C ˆ Y ∼ = R − p, p h ∗ C V . By the comparison of Hodge and pole filtration, the part ( F p h ∗ C V ) ⊗ C O S coin-cides with the subsheaf of R h ∗ Ω · P | S ( ∗ ˆ Y ) generated by differentials of pole order ≥ p + 4. This shows that W − p,p ⊗ C O S corresponds to the subsheaf of R / I generated by R p . (cid:3) The description of the local system V in terms of the Jacobian ring R / I admitsan explicit computation the Gauss-Manin connection and the Higgs field in one-parameter families. Let h : ˆ Y → S denote the family of toric hypersurfaces thatwe used in the proof of Proposition 4.1. Furthermore, let T denote an open subsetof A and h : ˆ Y T → T be the family obtained by restriction. Over T the defining ISPROOF OF MODULARITY 15 equation of ˆ Y inside the toric variety P is given by an equation F = 0 with F ∈ C ( t )[ x , ..., y ], and the Gauss-Manin connection ∇ : R h ∗ C ˆ Y T −→ R h ∗ C ˆ Y T ⊗ O T Ω T acts on the de Rham cohomology of the complement by(4.3) ω = H Ω F p +4
7→ − ( p + 4)( ∂ t F ) H Ω F p +5 ⊗ dt provided that the section H of R T is chosen such that ∂ t H = 0. Thus if one fixesa local basis of ( R T / I T ) N given by polynomials over the function field C ( t ), onecan compute a representation matrix of ∇ by applying the map (4.3) to all basiselements and reducing them with respect to the basis. A representation matrix forthe Higgs field θ : R h ∗ C ˆ Y T −→ R h ∗ C ˆ Y T ⊗ O T Ω T is obtained by projecting the images of the basis elements inside the R pT -part ontothe subspace generated by R p +1 T , for 0 ≤ p ≤
3. By the canonical isomorphism ofProposition 4.1 this also yields a local representation matrix of θ : V T → V T ⊗ O T Ω T ,or equivalently, of(4.4) θ : T T ⊗ O T V T −→ V T . For our purposes it will be sufficient to compute the map (4.4) in the infinitesimalneighborhood of a point x ∈ T , which turns out to be much easier. Let ˜ X := ˜ X x denote the fiber at x . We have an exact sequence of vector bundles over ˜ X givenby(4.5) 0 −→ T ˜ X −→ T ˜ X | ˜ X −→ f ∗ ( T S ) | ˜ X −→ T S,x , the tangent space of S at x . Since ˜ X is compact, all sectionsof the trivial bundle are constant, so that T S,x = H ( X, f ∗ ( T S ) | ˜ X )holds. Now the long exact cohomology sequence associated to the short exactsequence (4.5) yields a map ρ : T S,x −→ H ( ˜ X, T ˜ X ) , the Kodaira-Spencer map . It is know to be an isomorphism.Let R denote the stalk of the local system ( R / I ) N at x , and by R p the stalksof the images of R p , for 0 ≤ p ≤
3. Then R = L p =0 R p is a finite-dimensional C -algebra. Lemma 4.2.
There is a canonical isomorphism R ∼ = H ( ˜ X, T ˜ X ) . Proof.
Since ˜ X is a Calabi-Yau manifold, the canonical bundle K ˜ X = Ω X is trivial,which gives rise to a natural identification T ˜ X = (Ω X ) ∗ ∼ = Ω X . It implies that H ( ˜ X, T ˜ X ) is isomorphic to H , ( ˜ X ) = H ( ˜ X, Ω X ). On the otherhand, if we specialize the isomorphism from Proposition 4.1 to the stalks at x , weobtain H , ( ˜ X ) ∼ = R . (cid:3) Proposition 4.3.
For ≤ p ≤ there is a commutative diagram T S,x ⊗ H − p,p ( ˜ X ) H − p,p +1 ( ˜ X ) R ⊗ R p R p +1 ................................................................................................................. ............ θ .................................................................................................... ∼ = .................................................................................................... ∼ = .............................................................................................................................................................................................................................................. ............ µ where the vertical arrows are induced by the Kodaira-Spencer map and Proposition4.1, and where the lower horizontal arrow denotes multiplication on the graded C -algebra R .Proof. It is known that the derivation of a cohomology class in H − p,p ( ˜ X ) withrespect to a tangent direction v ∈ T S,x is given by the cup productH ( ˜ X, T ˜ X ) ⊗ H q ( ˜ X, Ω p ˜ X ) ∪ −→ H q +1 ( ˜ X, Ω p − X )with the Kodaira-Spencer class ρ ( v ) (see e.g. [2], Lemma 5.3.3). In the de Rham co-homology of the toric hypersurface ˆ Y , the cup product between cohomology classescorresponds to the wedge product between differential forms. Furthermore, wehave seen in (4.2) that every differential is defined by a polynomial in R . It canbe checked easily that the multiplication of polynomials corresponds to the wedgeproduct of the corresponding differential forms. (cid:3) For later use we need an explicit, fiberwise description of the characteristicsubvarieties R k introduced in section 3 associated to our special universal family f : ˜ X → S . To this end we introduce the symmetric algebra S · ( R ∗ ) over the dual of R , which is the homogeneous coordinate ring of P ( R ∗ ). Taking the multiplicationmap to its dual, we obtain a linear map µ ∗ : R ∗ −→ S ( R ∗ ) , and we let a denote the ideal generated by the image of µ ∗ . Similiarly, we let a denote the ideal generated by the image of the dualized multiplication map µ ∗ : R ∗ → S ( R ∗ ). Then the fibers of the characteristc varieties are obtained inthe following way. Lemma 4.4.
For a point x ∈ S as above and k = 1 , , the fiber of the k -thcharacteristic subvariety ( R k ) x is isomorphic to the projective subvariety Z k :=Proj( A k ) of P ( R ∗ ) , where A k denotes the graded quotient ring S · ( R ∗ ) / a k . ISPROOF OF MODULARITY 17
Proof.
We recall the definition of the k -th characteristic subvariety as given in [19].For our system ( E, θ ) of Hodge bundles, the ( k + 1)-st iterated Higgs field definesa map θ k +1 : S k +1 ( T S ) −→ Hom( E , , E − k,k +1 )whose kernel we denote by I k . Then R k = Proj( I k ) as a subvariety of P ( T S ). For k = 1 the stalk ( I ) x at x ∈ S is the kernel of θ x : S ( T S,x ) ∼ = S ( R ) −→ Hom( R , R ) ∼ = R , the first isomorphism coming from Lemma 4.2 and the Kodaira-Spencer map, thesecond being a consequence of the fact that R is one-dimensional. If we dualize thismap, up to a non-zero constant we obtain µ ∗ , and the kernel of θ x is isomorphicto S ( R ∗ ) / a , the cokernel of µ ∗ . Since this quotient generates A , we obtain Z ∼ = ( R ) x . The proof for k = 2 is similar. (cid:3) Proofs of the Main Theorems
We recall some basic notions from computational commutative algebra. Let K be a field and R := K [ x , ..., x n ] the polynomial ring in n indeterminates. A monomial ordering is a total ordering ≺ on the set of monomials in R such that f ≺ g implies f h ≺ gh for monomials f, g, h ∈ R . In our computations we will usethe graded lexicographical ordering , which is defined as follows: First one fixes anordering on the set of indeterminates by requiring x ≻ x ≻ · · · ≻ x n . Now let f = y y · · · y r and g = z z · · · z s with y i , z i ∈ { x , ..., x n } for all i such that y i ≻ y j or y i = y j for i ≤ j , and similary for the factors of g . Then bydefinition f ≻ g if either r > s or r = s and there is an m ∈ N such that y i = z i for 1 ≤ i ≤ m and y m +1 ≻ z m +1 .The total ordering on the monomials extends to a partial ordering on R by defin-ing f ≺ g iff the maximal monomial of f is smaller than the maximal monomial of g . Furthermore, zero is defined to be the least element in R . If a ⊆ R is an ideal,then we say that an element f ∈ R is in normal form with respect to a and write f = NF( f ) if f is minimal inside the coset f + a . It can be shown that the normalform is unique; in particular, NF( f ) = 0 if and only if f ∈ a .Let f : ˜ X −→ S denote the family of CYs defined at the beginning of section 4.In order to prove the theorems from section 1, it suffices to consider one particularfiber of this family. Let λ j := j for 1 ≤ j ≤ A ∈ C , by a ij := λ ij for 1 ≤ i ≤ ≤ j ≤
8. We define x ∈ S to be the pointcorresponding to the matrix A and let ˜ X := ˜ X x denote its fiber. For p = 0 , ..., R p be the ring defined before Lemma 4.2. By Proposition 4.1, R N p is isomorphicto H − p,p ( ˜ X ) for 0 ≤ p ≤ Lemma 5.1.
The following elements constitute a basis of R N p for p = 0 , ..., . p = 0 1 p = 1 x y , x y , x y , x y , x y , x y , x y , x y , x y p = 2 x y , x y y , x y , x x y , x x y y , x x y , x y , x y y , x y p = 3 x y Proof.
For each p we list all monomials with deg X = 2 p and deg Y = p . If we let e , ..., e denote the canonical basis of ( Z / Z ) , then N is generated by the set B := { e i + e i +1 | ≤ i ≤ } ∪ { e + e } . We remove all elements g from the list with σ a ( g ) = g for some a ∈ B or withNF( g ) = g . By uniqueness and linearity of the normal form, the remaining ele-ments are linearly independent in R N p . Since the Betti numbers of ˜ X are 1 , , , (cid:3) Proposition 5.2.
The fiber ( R ) x of the first characteristic subvariety at x istwo-dimensional.Proof. By Lemma 4.4 we have to compute the ideal a ⊂ S · ( R ∗ ) which is generatedby the image of the dual multiplication map µ ∗ : R ∗ → S ( R ∗ ). Let v , ..., v denotethe basis of R and w , ..., w the basis of R as defined in 5.1. Furthermore, we fixa bijection ϕ : { ( i, j ) ∈ N | ≤ i ≤ j ≤ } ∼ −→ { , ..., } and put u ϕ ( i,j ) := v i v j . The first step is to compute a representation matrix of themultiplication map µ : S ( R ) −→ R with respect to the basis u , ..., u and w , ..., w . By computing the normal formsof elements with respect to the Jacobian ideal I x ⊆ R , we determine c ϕ ( i,j ) k ∈ Q such that NF( v i v j ) = X k =1 c ϕ ( i,j ) k w k for 1 ≤ i, j ≤ . Then C := ( c ℓk ) ∈ C , is the desired representation matrix. Its transpose repre-sents µ ∗ with respect to the dual basis w ∗ , ..., w ∗ and u ∗ , ..., u ∗ .Notice that ( v i v j ) ∗ = 2 v ∗ i v ∗ j for 1 ≤ i, j ≤
9. Thus if we define˜ c kℓ := c kℓ k = ϕ ( i, j ) , i = j c kℓ k = ϕ ( i, j ) , i = j then t ˜ C ∈ C , is a representation matrix of µ ∗ with respect to w ∗ , ..., w ∗ and˜ u , ..., ˜ u , where ˜ u ϕ ( i,j ) := v ∗ i v ∗ j . Each row corresponds to one generator of a in S ( R ∗ ). Furthermore, the choice of a basis v ∗ , ..., v ∗ admits a natural identificationProj( S · ( R ∗ )) ∼ = P . ISPROOF OF MODULARITY 19
Let z , ..., z denote a new set of indeterminates. If we define f , ..., f by f ℓ := X i =1 9 X j = i ˜ c ϕ ( i,j ) ℓ z i z j then the variety in P defined by f = · · · = f = 0 is isomorphic to ( R ) x . Sincethe matrix ˜ C is known in explicit term, we can use computer algebra to computeits dimension. We obtain dim( R ) x = 2. (cid:3) In order to prove the non-modularity of f , by Proposition 3.2 it is sufficient todetermine a single points x ∈ S such that the fiber ( R ) x is not isomorphic to P × P . By Proposition 4.4, the fiber ( R ) x is only two-dimensional. Thus both f and M CY cannot be modular, and Theorem 1.1 is proved. Proof of Theorem 1.3:
The proof will be achieved by contradiction. Let H CY ⊂ H ′ CY ⊂ M CY be an extension as described in the theorem, and let f : X → S denote a universal family. Let Z be the sublocus in S mapping to H ′ CY , and g = f | Z : X | f − ( Z ) → Z be the corresponding subfamily. Then g is a modularfamily with respect to (Sp(6 , R ) / U(3) , W ) where W is the Calabi-Yau like PVHSover Sp(6 , R ) / U(3)(cf. [19]). Let (
F, η ) be the corresponding Higgs bundle of thesubfamily g . By Theorem 3.3 in [19], the Higgs bundle ( F, η ) has two characteristicsubvarieties and the fibers of the first characteristic subvariety are all isomorphicto P . Take one point x ∈ Z , and denote by ( R ′ ) x be the fiber over x of the firstcharacteristic subvariety of ( F, η ). By the geometric description of the characteristicsubvariety in Lemma 3.2 [19], we know that in P ( T S,x ) the equality( R ′ ) x = ( R ) x ∩ P ( T Z,x )holds. Now if dim( R ) x = 2, then we neccessarily have an isomorphism( R ) x = ( R ′ ) x ≃ P . Since our computation is local, we simply take the point x to be the same point asused in the above proof of Theorem 1.1. The arithmetic genus of ( R ) x is calculatedto be -41, whereas the arithmetic genus of P is 0. So ( R ) x is non-isomorphic to P . Therefore such an extension does not exist. (cid:3) Now we give a second proof of Theorem 1.1 which is based on the plethysmmethod described in subsection 3.2. As before by (
E, θ ) we denote the Hodgebundle associated to the family f : ˜ X → S . The Higgs field θ x : T S,x ⊗ E , x −→ E , x induces in a natural way a linear map S ( θ x ) : T S,x ⊗ S ( E x ) , −→ S ( E x ) , on the symmetric 2-space. By threefold iteration we obtain S ( θ x ) : S ( T S,x ) ⊗ S ( E x ) , −→ S ( E x ) , . Proposition 5.3.
The image of S ( θ x ) is -dimensional.Proof. By Proposition 4.3 it is sufficient to compute the image of the linear map S ( µ ) : S ( R ) ⊗ S ( R ) −→ S ( R )induced by the multiplication map µ : R ⊗ R → R . Let v , ..., v denote thebasis of R = ⊕ p =0 R p specified in Lemma 5.1; in particular, we assume that theelements w k := v k +1 span the subspace R , where 1 ≤ k ≤
9. For each k wedetermine the coefficients c ( k ) ij ∈ C such that v i w k = X j =1 c ( k ) ij v j for 1 ≤ i ≤
20 and 1 ≤ j ≤ I x as in the proof of Theorem 5.2. Then C k := c ( k ) ij is a representation matrix of the linear map µ w k : R −→ R, v vw k with respect to v , ..., v . With these matrices it now an easy task to compute theinduced action of µ w k on S ( R ). Fix a bijection ϕ : { ( i, j ) ∈ N | ≤ i ≤ j ≤ } ∼ −→ { , ..., } and define a basis u , ..., u of S ( R ) by u ϕ ( i,j ) := v i v j for 1 ≤ i ≤ j ≤
20. Then S ( µ w k ) acts on this basis by(5.1) S ( µ w k )( u ϕ ( i,j ) ) := X ℓ =1 c ( k ) iℓ v j v ℓ + X ℓ =1 c ( k ) jℓ v i v ℓ . The subspace U , of S ( R ) of degree zero is one-dimensional and generated by u ϕ (0 , . Now the space S ( µ )( U , ) is generated by the images of all maps S ( µ w k )( k = 1 , ...,
9) applied to u ϕ (0 , . By (5.1) and computational linear algebra it turnsout to be 9-dimensional and thus all of U , . Applying all maps S ( µ w k ) to U , we obtain a subspace of S ( R ) of dimension 45 contained in U , , and a thirdapplication yields a 78-dimensional subspace of U , . (cid:3) Now we explain how Proposition 5.3 implies Theorem 1.1. If f : ˜ X → S weremodular, by Proposition 3.4 there would be a decomposition of Hodge bundles S ( E, θ ) = ( E , θ ) ⊕ ( E , θ ) , such that each graded piece E − p,pi has a specific dimension. This decompositionwould exist in any fiber. In particular, the image of S ( E ,x ) , = S ( E x ) , under the iterated Higgs field S ( θ x ) : S ( T S,x ) ⊗ S ( E x ) , −→ S ( E x ) , ISPROOF OF MODULARITY 21 would be contained in S ( E ,x ) , and thus be at most 65-dimensional. But sincethe image of S ( θ x ) has dimension 78, the decomposition cannot exist. Proof of Theorem 1.2:
Let ρ : π ( S ) → Sp(20 , R ) be the monodromy rep-resentation. We know that ρ is irreducible since the VHS V is irreducible. Let G be the Zariski closure of the monodromy group in Sp(20 , R ). Then we have afactorization: ρ : π ( S ) −→ G ̺ −→ Sp(20 , R ) , and ̺ : G → Sp(20 , R ) is irreducible. If G is not the whole group, G must be aproper Lie subgroup of Sp(20 , R ). We will now derive a contradiction by a sequenceof steps. Step 1.
Differentiating ̺ we pass to the real Lie algebra monomorphism χ : g −→ sp (20 , R )where g = Lie( G ). By Deligne [3] Cor. 4.2.9 we know that g is semi-simple. Wethen complexify χ to obtain χ C : g C → sp (20 , C ), which is irreducible in the sensethat after composition with the natural representation sp (20 , C ) −→ gl (20 , C ) χ C is an irreducible representation of the semi-simple complex Lie algebra g C . Step 2.
In this step we classify all possible complex Lie algebra monomorphism χ C : g C → sp (20 , C ) where g C is semi-simple and χ C is irreducible in the sensedescribed above. In order to classify ( g C , χ C ), we observe that it suffices to considerall 20-dimensional irreducible representations of complex semi-simple Lie algebras g C . Actually, an irreducible representation g C → gl ( V C ) with dim( V C ) >
20 admitsno factorization g C −→ sp (20 , C ) −→ gl ( V C ) . The reason is that, since g C is mapped onto a proper subspace of sp (20 , C ), thecomposition must decompose and hence is reducible. We can list all such possibili-ties. Our method is first to find all 20-dimensional representation of a semi-simpleLie algebra, and then exclude those whose images do not lie in sp (20 , C ). Case 1. g C has only one simple factor:(a) ( A , [19]),(b) ( A , [0 , , , , C , [3 , Case 2. g C has two simple factors:(a) ( A ⊕ C , [1] ⊗ [2 , A ⊕ C , [4] ⊗ [1 , A ⊕ D , [1] ⊗ [1 , , , , (d) ( C ⊕ C , [1 , ⊗ [0 , sp (20 , C ) if and only if there exists anone-dimensional component in the irreducible decomposition of the second wedgepower. Here is an example. The pair ( A ⊕ A ⊕ A , [1] ⊗ [1] ⊗ [1 , , , A ⊕ A ⊕ A a 20-dimensional representation withthe highest weight [1] ⊗ [1] ⊗ [1 , , , ^ ([1] ⊗ [1] ⊗ [1 , , , ≃ [0] ⊗ [2] ⊗ [2 , , , ⊕ [2] ⊗ [0] ⊗ [2 , , , ⊕ [0] ⊗ [0] ⊗ [0 , , , ⊕ [2] ⊗ [2] ⊗ [0 , , , , there is no one dimensional component. Step 3 . All possible simple real groups of Hodge types are listed in § C -PVHS structures on a tensor product will simplify our argument toa large extent. Since this result is of interest in itself, we would like to include aproof in this paper.Let V be an irreducible C -PVHS over a quasi-projective manifold X \ S and withunipotent local monodromy around S . Let ρ : π ( X \ S ) → GL( V )be the corresponding representation of the fundamental group and G be the Zariskiclosure of ρ . Assume G = G × G with G i simple. Then according to Schur’slemma V is decomposed into V ≃ V ⊗ V , where V i corresponds to a representation ρ i : π ( X \ S ) → G i . Proposition 5.4.
The C -PVHS on V factors into PVHS’s on each factor V i , i.e.each V i admits a C -PVHS structure such that their tensor product on V i coincideswith the C -PVHS on V .Proof of Proposition 5.4: We write dim V i = n i for i = 1 , n ≥ n without lose of generality. We first need the following lemma. Lemma 5.5.
Each factor ρ i has quasi-unipotent local monodromy around S .Proof. By choosing a base point in S , the tensor product decomposition of V givesthe tensor product decomposition of the vector space V ≃ V ⊗ V with group action,and since G i is simple, G i ⊂ SL( V i ) for i = 1 ,
2. Now we apply V n on the above ISPROOF OF MODULARITY 23 isomorphism. Ex. 6.11(b) in [6] tells us that, for V considered as a representationspace of SL( V ) × SL( V ), there exists an irreducible component S n ( V ) ⊂ n ^ ( V ) . Since V is of unipotent local monodromy, each direct component of V n ( V ) isof unipotent local monodromy, too. In particular, S n ( V ) is of unipotent localmonodromy. Let T be one of local monodromy operators of ρ , and λ be one ofeigenvalues of T . Then clearly, λ n is one of eigenvalues of T on S n ( V ), henceis equal to one. This proves that ρ is of quasi-unipotent local monodromy. Andby the unipotency of ρ , ρ is of quasi-unipotent local monodromy as well. Thiscompletes the proof of Lemma 5.5. (cid:3) Since ρ i : π ( X \ S ) → G i is a Zariski dense representation into the simplealgebraic group G i and with quasi-unipotent local monodromy around S , by Jost-Zuo [8] there exists a pluri-harmonic metric on the flat bundle V i with finite energy,which makes V i into a Higgs bundle ( E, θ ) i over X \ S . Furthermore, T. Mochizuki[9] has analyzed the singularity of this harmonic metric in detail and has shownthat ( E, θ ) i admits a logarithmic extension ( ¯ E, ¯ θ ) i over X, i.e. ¯ E i is an extensionof E i , ¯ θ i is an extension of θ i and such that¯ θ : ¯ E i → ¯ E i ⊗ Ω X (log S ) . Such a pluri-harmonic metric is called tame . In this case the residue of ¯ θ along S is nilpotent.From the proof of Lemma 5.5, we know that, by applying the Schur functor V n ,one finds a direct factor of V n ( V ⊗ V ) of the form S n ( V ) ⊗ det( V ) , and S n is non-trivial. Since G is simple, det( V ) is the trivial representation.We consider G as a simple algebraic subgroup of GL( V ) . Since the Schur functor S n is non-trivial and G is a simple algebraic group, the representation S n : G → GL( S n ( V ))is faithful. Since ρ is Zariski dense in G , S n ( ρ ) is irreducible. Since V n ( V ⊗ V )is semi-simple, there exists a decomposition n ^ ( V ⊗ V ) = M i S i ⊗ W i , where S i are irreducible and W i are trivial. By Deligne’s Prop. 1.13 in [4], thereexists uniquely C -PVHS on S i and C -HS on W i such that the direct sum of the tensor products of them coincides with the C -PVHS on V n ( V ⊗ V ). So, in par-ticular, there exists a C -PVHS on S n ( V ) . By the uniqueness of such pluri-harmonic metric, S n ( ¯ E, ¯ θ ) coincides with the C -PVHS on S n ( V ). Hence S n ( ¯ E, ¯ θ ) is a fixed point of the C × -action. Therepresentation G → GL( S n ( V )) induces a morphism φ S n : M ( π ( X \ S ) , G ) s.s → M ( π ( X \ S ) , GL( S n ( V ))) s.s between the corresponding moduli spaces of semi-simple representations. By Simp-son’s Cor. 9.16 in[17], φ S n is finite.If S = ∅ , then C × acts on both moduli spaces continuously via Hermitian-Yang-Mills metrics on poly-stable Higgs bundles ( E, tθ ). And this action is compatiblewith φ S n . Since S n ( ρ ) is a fixed point of C × -action, ρ is a fixed point of C × -action. This just means that ( E, θ ) is a fixed point of C × -action. Hence ( E, θ ) is a C -PVHS on V . In general S = ∅ . We take a curve C \ S ⊂ X \ S , which is acomplete intersection of ample hypersurfaces. Taking the restrictions ρ | C \ S ∈ M ( π ( C \ S ) , G ) s.s , we have S n ( ρ ) | C \ S ∈ M ( π ( C \ S ) , GL( S n ( V ))) s.s . We consider the map φ S n : M ( π ( C \ S ) , G ) s.s → M ( π ( C \ S ) , GL( S n ( V ))) s.s . By Simpson’s main theorem in [15], there exist Hermitian-Yang-Mills metrics onpoly-stable Higgs bundles on C with logarithmic pole of the Higgs field on S . Andthe C × -action can be defined on both spaces of semi-simple representations on C \ S via Hermitian-Yang-Mills metric on ( ¯ E, t ¯ θ ) . Applying the same argument as aboveto the compact case, we show that the pulled back Higgs bundle ( ¯ E, ¯ θ ) to C \ S isa fixed point of the C × -action. If we choose C \ S sufficiently ample, then ( ¯ E, ¯ θ ) is also a fixed point of the C × -action. (The isomorphism ( ¯ E, ¯ θ ) | C ≃ ( ¯ E, t ¯ θ ) | C extends to an isomorphism ( ¯ E, ¯ θ ) ≃ ( ¯ E, t ¯ θ ) if C is sufficiently ample.) Again bySimpson, ( ¯ E, ¯ θ ) is a C -PVHS on V .Similarly, we also show that V admits a C -PVHS. The tensor product of C -PVHS on V and on V is a C -PVHS on V ⊗ V . By Deligne’s uniqueness theoremon C -PVHS on irreducible local systems, this tensor product coincides with theoriginal C -PVHS on V ⊗ V . Proposition 5.4 is completed. (cid:3)
Now we start with the analysis of case 2. By the above proposition, we knowthat in this case we have an isomorphism(
E, θ ) ≃ ( E , θ ) ⊗ ( E , θ ) , ISPROOF OF MODULARITY 25 where each ( E i , θ i ) is a system of Hodge bundles. Because the Hodge numbersof E are 1,9,9,1, it is not difficult to see that, up to permutation of factors, theHodge numbers of ( E , θ ) are 1,1, and those of ( E , θ ) are 1,8,1. Since ( E , θ )comes from a R -PVHS, ( E , θ ) also comes from a R -PVHS. This implies that, if g = g ⊕ g , then up to permutation of factors, g = su (1 ,
1) and g ⊂ so (2 , § sp (4 , C ) are sp (1 ,
1) and sp (4 , R ), and those of so (10 , C ) are so (2 ,
8) and so (4 , • case (2c) ( su (1 , ⊕ so (2 , , id ⊗ id).Obviously, case (1a) is impossible since it is of weight 19. For those real forms ofHermitian types in case 1 we can again compare the Hodge numbers. The onlypossibilities are • case (1b) ( su (3 , , V ); • case (1c) ( sp (1 , , S ).Note that case (1c) is of non-Hermitian type. Step 4.
We have already excluded case (1b) using the method of plethysm. In thelast step, we apply the method further in order to exclude the left two cases. Theargument for case (2c) is similar. We give the analogous statement of Proposition3.4 as follows: S ( E, θ ) = ( E , θ ) ⊕ ( E , θ ) ⊕ ( E , θ )where E = E , ⊕ E , ⊕ E , E = E , ⊕ E , ⊕ E , E = E , ⊕ E , ⊕ E , ⊕ E , ⊕ E , ⊕ E , ⊕ E , . The Hodge numbers of E are 0 , , , , , ,
0, respectively, those of E are 0 , , , , , E are 1 , , , , , ,
1. So by the computational resultin Prop. 5.3, we see that case (2c) is impossible. For case (1c), the correspondingresult is the following: S ( E, θ ) = ( E , θ ) ⊕ ( E , θ ) ⊕ ( E , θ ) ⊕ ( E , θ )where the respective dimensions of E i are 10 , , ,
84. But we are unable toobtain further information on the Hodge numbers of E i , because Sp(1 ,
1) is of non-Hermitian type. But fortunately we can still get a contradiction by the actualcomputation. The argument works as follows. The first Hodge bundle of S ( E, θ )is of dimension 1, it must lie in one of ( E i , θ i ), and hence the rank of the Higgssubsheaf generated by the first Hodge bundle will not exceed 84. Over the point used in Prop. 5.3, the rank of the stalk of the Higgs subsheaf generated by the firstHodge bundle is not less than1 + 9 + 45 + 78 = 133 . This gives the desired contradiction for case (1c). The proof is complete. (cid:3)
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Universit¨at Mainz, Fachbereich 17, Mathematik, 55099 Mainz, Ger-many
E-mail address : [email protected] East China Normal University, Dep. of Mathematics, 200062 Shang-hai, P.R. China
E-mail address : [email protected] Universit¨at Mainz, Fachbereich 17, Mathematik, 55099 Mainz, Ger-many
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