aa r X i v : . [ m a t h . AG ] A p r Rigidity of Spreadings and Fields ofDefinition
Chris PetersTechnical University Eindhovenand Universit ´e Grenoble Alpesemail: [email protected]
November
Abstract
Varieties without deformations are defined over a number field.Several old and new examples of this phenomenon are discussedsuch as Bely˘ı curves and Shimura varieties. Rigidity is related tomaximal Higgs fields which come from variations of Hodge struc-ture. Basic properties for these due to P. Gri ffi ths, W. Schmid, C.Simpson and, on the arithmetic side, to Y. Andr´e and I. Satake allplay a role. This note tries to give a largely self-contained expositionof these manifold ideas and techniques, presenting, where possible,short new proofs for key results.AMS Classification G , C , D , G Introduction
Results stating that certain types of algebraic varieties are definable over anumber field are scattered in the literature. Arguably, those most studiedform the class of Shimura varieties [Sh, F, Mi]. Another famous exampleis Bely˘ı’s theorem [Bel] which characterizes curves over Q as those whichhave a Bely˘ı representation, i.e., a branched cover of the line branched inexactly three points. In dimension two, the fake projective planes [Pr-Y, r-Y- ] and the Beauville surfaces [Be, Exercise X. ( )], [Ba-C-G] areknown to have models over Q .Such examples can be uniformly explained by constructing a suitablespread of the varieties concerned as demonstrated in Sections . and .Of a totally di ff erent flavor are the applications to special subvarietiesof Shimura varieties in Section . on the one hand, and to splittings ofHiggs bundles as given in Section . on the other hand. As has beenknown since the work of Viehweg and Zuo [Vie-Z], the last two are justfacets of the same phenomenon: Higgs bundles of a very special kind,those that they called ”maximal” are directly related to special subvari-eties of certain Shimura varieties. One of the goals of this note is to showthat rigidity plays a central role in this; exploiting this, simplifies severalof the arguments.The overall goal of this note is to show how a few relatively simpleideas plus some standard techniques from deformation theory and Hodgetheory explain a wide range of phenomena of the above kind. It brings to-gether various known results from very di ff erent subfields of mathematics.This is the reason why I thought to explain some of the basic notions andtechniques from these fields, and also to search for new simpler proofs.This note has been inspired by discussions with Stefan M ¨uller-Stach.Equally influentual has been [G-G, Ch. ] as well as the last chapter of[Mu-O]. Finally I would like to thank Christopher Deninger for pointingout the references [F, Sh]. ontents Spreads of varieties and rigidity . Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformations and rigidity . . . . . . . . . . . . . . . . . . . . Rigidity and fields of definition . . . . . . . . . . . . . . . . Further examples of models over number fields . Locally symmetric spaces . . . . . . . . . . . . . . . . . . . . . Holomorphic maps into locally symmetric spaces . . . . . Applications to variations of Hodge structure . Hodge theory revisited . . . . . . . . . . . . . . . . . . . . . . Application to variations of weight 1 and 2 . . . . . . . . . . . Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . Monodromy and rigidity . . . . . . . . . . . . . . . . . . . . . Special subvarieties of Shimura varieties . . . . . . . . . . . Applications to Higgs bundles . Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic variant . . . . . . . . . . . . . . . . . . . . . . . . Rigid maximal Higgs subsytems . . . . . . . . . . . . . . . . Spreads of varieties and rigidity . Spreads
The ”spread philosophy” roughly states that a complex algebraic varietycan be seen as a family over a base variety determined by specifying sometranscendence basis of the field of definition of the variety. Spreads are byno means unique but all share the crucial property that, by construction,the total family is always defined over a number field.Although this construction can be phrased in varying generality [G-G,§ . ], the following somewhat restricted version su ffi ces for this note. Proposition . . Let X be smooth complex quasi projective variety. Thereexists a smooth family f : X → S defined over Q such that . X , S are smooth quasi-projective; . there is a canonical point o ∈ S such that f − o = X ; . if s ∈ S is a Q –rational point, the fiber X s = f − s is defined over Q .Proof. Suppose for simplicity that the variety X is projective and is givenby a finite set of polynomials. The coe ffi cients of these polynomials gener-ate a field k of finite transcendence degree r over Q , say k = k ( α , . . . , α r )where { α , . . . , α r } is a transcendence basis for k and where k is a numberfield, say of the form Q [ x ] /P with P some monic irreducible polynomial.Then k is the function field of some complex algebraic variety S ′ . Thedeformation will be constructed over a Zariski open subset S of S ′ .The basic idea is to replace the coe ffi cients α j of each of the polynomi-als defining the variety X , by variables x j . A point s ∈ S ′ corresponds to afield k ( s ) isomorphic to k . If one replaces the coe ffi cients in k of a definingset of homogeneous equations for X by the corresponding coe ffi cients in k ( s ) one gets a variety X s . The X s glue to a variety X fibered over V ( P ). In-deed, it is given by the same equations as X except that the coe ffi cients forthese equations are not considered as numbers but as Q –polynomials inthe supplementary variables x j tied by the extra equation P ( x , . . . , x r ) = 0.Substituting x j = α j gives a canonical k –valued point o ∈ S ′ and byconstruction X o = X . Since k/k is separable, this point is a non-singularpoint. Now replace S ′ by a suitable Zariski open neighborhood S of o such that the variety S is smooth. Again by separability, this variety issmooth along X o . But it might still be singular or reducible. To remedythis, first take the component of X which contains f − o . Then replace S bya smaller neighborhood of o such that not only the total space is smooth,but also all of the fibers of the fibration are smooth. The resulting family,still denoted f : X → S , is a smooth deformation of X . By construction,the Zariski-open subsets figuring in the construction are complements ofequations over Q , and so the resulting family is defined over Q .Finally, since Q is algebraically closed, S contains points s defined over Q . This amounts to replacing the variables x j figuring in the coe ffi cientsfor the equations of X by suitable algebraic numbers and hence X s is de-fined over Q . Remark . . There are several variants of this result: one can spread pairs(
X, Z ) with Z a closed subvariety of X . Similarly, one can spread a givenmorphism f : X → Y between varieties. . Deformations and rigidity
Let me first recall some basic definitions and facts. More details andproofs can be found for example in [Sern].
Kodaira-Spencer classes
A complex variety X is said to be infinitesimally, respectively locally rigid if any infinitesimal deformation of X , resp. any local deformation p : X → S of X with S su ffi ciently small, is trivial, i.e. isomorphic to the productdeformation. This can be rephrased by saying that if, say o ∈ S is such thatthe fiber of p over it is isomorphic to X , say ι : X o = f − o ≃ −→ X , then there isa morphism S → Aut( X ) , s g s , g o = id X inducing a product structureon the family X → S : X × S ≃ −→ X , ( x, s ) ( ι ◦ g s ( x ) , s ) . As is well known, a variety X is indeed infinitesimally or locally rigid if H ( X, Θ X ) = 0. If such a variety appears in a deformation p : X → S of X ≃ X o , o ∈ S , finer information is present by looking at the the Kodaira-Spencer map . Recall that it is given as the extension class of the exactsequence 0 → T o ( S ) ⊗ O X → Θ X | X → Θ X → )of O X –modules. In other words, it gives a characteristic map κ p : T o ( S ) → H ( X, Θ X ) . For a given deformation, it measures deviation of triviality of the defor-mation:
Theorem . ([K-S, Thm. . ]) . Suppose that a family p : X → S is regularin the sense that dim H ( X s , Θ X s ) is constant for s ∈ S . Then it is trivial if andonly if κ p = 0 . Observe that this theorem gives back the criterion that X is rigid if andonly if H ( X, Θ X ) = 0. Indeed, if this is the case, by semi-continuity, anysu ffi cently small deformation of X is regular and the theorem applies toshow rigidity. ariants . Infinitesimal deformations of pairs ( X, Z ) with Z a closed subscheme of a smooth variety X . Any such deformation p with base ( S, o ) (i.e. with fiberover o isomorphic to ( X, Z )) is classified by its Kodaira-Spencer map κ p : T o S → H ( X, Θ X ( Z )) , ( )where Θ X ( Z ) is the sheaf of germs of vector fields on X tangent to Z . Thisdeformation is rigid precisely when κ p = 0 as before. . Deformations of morphisms f : X → Y . These are given by a commutativediagram X f / / YX oι ≃ O O (cid:127) _ (cid:15) (cid:15) F | X o / / Y oι ′ ≃ O O (cid:127) _ (cid:15) (cid:15) X F / / p ❆❆❆❆❆❆❆❆ Y p ~ ~ ⑦⑦⑦⑦⑦⑦⑦ S. A deformation of morphism as above is a deformation keeping the source, re-spectively target fixed if p resp. p are a trivial deformations. A morphism f is rigid, if all infinitesimal deformations of f are trivial in the sense thatthere are morphisms S → Aut( X ) s g s , g o = id X S → Aut( Y ) s g ′ s , g ′ o = id Y . which trivialize the deformation: for all s ∈ S there is a commutative dia-gram X ≃ ι ◦ g s (cid:15) (cid:15) f / / Y ≃ ι ′◦ g ′ s (cid:15) (cid:15) X s F | X s / / Y s . Two special cases will be used in this note: This is Sernesi’s notation; if Z is a normal crossing divisor it is dual to Ω X (log Z ) andother auhors use Θ X ( − log Z ) in this case. ) Deformations of a morphism f : X → Y between non-singular vari-eties keeping source and target fixed. Such morphisms are classifiedby the vector space H ( X, f ∗ Θ Y ).b) Deformations of closed embeddings f : Z ֒ → X between smooth va-rieties with target fixed. Here the characteristic morphism is κ F : T o S → H ( Z , N Z | X ) , where N Z | X is the normal bundle of Z in the ambient manifold X .Note that automorphisms of X yield non-trivial deformations of f but these are trivial as deformations of Z itself. Indeed, there is anexact sequence0 → H ( Z , Θ Z ) i ∗ −→ H ( Z , Θ X | Z ) → H ( Z , N Z | X ) δ −→ H ( Z , Θ Z ) . The quotient H ( Θ X | Z ) /i ∗ H ( Θ Z ) is the space of isomorphisms classesof infinitesimal deformations of f keeping Z and X fixed; the nextterm in the sequence, H ( X, N Z | X ), is the space of infinitesimal defor-mations of f keeping only X fixed and δ maps such a deformationto the corresponding deformation of Z , i.e., it is the forgetful map.The embedding f is rigid in this case precisely if κ F = 0. If Z itself isrigid, this would follow if H ( Z , Θ Z ) → H ( Z , Θ X | Z ) is surjective. Incase Z admits no vector fields, this means H ( Z , Θ X | Z ) = 0. For lateruse, consider the following special case, that of a totally geodesicsubmanifold Z of X : Proposition . ([Ca-MS-P, Sect. . ] ) . Let X be a manifold equippedwith a hermitian metric, and let Z ⊂ X be a totally geodesic submanifoldfor which H ( Z , Θ Z ) = 0 . Then the tangent bundle sequence for Z ⊂ X splits. Hence H ( Z , Θ X | Z ) = 0 ⇐⇒ H ( Z , N Z | X ) = 0 . It follows that Z is rigidly embedded (keeping the target fixed) if and only if the embeddingis rigid keeping source and target fixed. In particular, since δ is the zero map in this case, it is irrelevantwhether Z itself is rigid or not. Kodaira-Spencer classes and spreading
The Kodaira-Spencer class of the spread family f : X → S from Prop. . incorporates arithmetic information, since the dual of T o ( S ) is the complex ector space Ω k/ Q ⊗ Q C . Also, Ω X | X = Ω X/k , the sheaf of K¨ahler di ff eren-tials on the k -variety X . The dual of the exact sequence ( ) then reads0 → Ω X → Ω X/k → Ω k/ Q ⊗ C O X → . The extension class of the dual of the above sequence is the Kodaira-Spencer class for the spread family f : X → S at o . It depends on thechoice of the field k : κ X/k ∈ H (Hom O X ( Ω X , Ω k/ Q ⊗ C O X )) ≃ Hom C ( T o S, H ( X, Θ X )) . ( ) Corollary . . The spread family from Prop. . is regular. It is a trivialdeformation if and only if the Kodaira-Spencer class ( ) vanishes.Proof. To see regularity, first observe that dim H ( X, Θ X ) depends on theisomorphism class of X as an abstract algebraic variety. Secondly, since all X s , s ∈ S with the property that s corresponds to a transcendental numberare mutually isomorphic as abstract algebraic varieties, dim H ( X s , Θ X s ) isthe same for all such s ∈ S . This set corresponds to points in S not lying onany proper subvariety of S and hence is dense in S . Upper semicontinuityof dim H ( X s , Θ X s ) then implies that this dimension is locally constant, i.e.,the family is regular. The result follows from Theorem . . . Rigidity and fields of definition
Proposition . . ) Let X be a smooth complex quasi-projective variety. As-sume that the Kodaira-Spencer class ( ) of some spread family of X vanishes(e.g. in case X is rigid). Then X has a model over a number field , i.e., X ≃ X ′ ⊗ Q C , X ′ is defined over Q ,and where the isomorphism is defined over C . This model is unique if H ( Θ X ) =0 . ) Let ( X, Z ) be a pair of varieties, where X is smooth and Z ⊂ X a closedembedding. Assume that the Kodaira-Spencer class ( ) of a spread family for ( X, Z ) vanishes (e.g. in case ( X, Z ) is rigid). Then the pair ( X, Z ) has a modelover a number field, The model is unique if H ( Θ X ( Z )) = 0 . ) In the relative situation of a morphism f : X → Y between complex quasi-projective varieties suppose that f is rigid. Then X, Y and f have a model over . ) In the relative situation, suppose that Y is defined over Q and that f is rigidfixing the target. Then the same conclusion as in ) holds.Proof. ) Rigidity implies that the fibers of any su ffi ciently small defor-mation of X are isomorphic to X . This holds in particular for the spread f : X → S from Prop. . . So, if s ∈ S ( Q ), one has an isomorphism X s ≃ X o = X and since X s is defined over Q , X has a model over Q . If, moreover, H ( Θ X ) = 0 there is no non-trivial deformation of id X and the isomor-phism X s ≃ X o is unique (compare with the definition above). ) The argument is as for ), using an obvious variant of Prop. . for pairs.See remark . .Note that ) and ) can be reduced to embeddings, since f is rigid ifand only the embedding of graph of f in X × Y is a rigid morphism, and thegraph is defined over Q precisely when f is. For embeddings i : X ֒ → Y , tofind a variety over which to spread, start with equations for Y and let k be the field extension of Q obtained by adjoining the coe ffi cients. The em-bedding is then specified by supplementary equations whose coe ffi cientsare adjoined to k . The resulting field k = Q ( S ) is the function field of thebase variety S . Observe that if the variety Y is defined over a number field, k is also a number field and then S parametrizes a deformation of X inthe fixed variety Y . Rigidity in both cases ensures that the embedding hasa model over a number field. Examples . . . Fake projective planes are compact complex surfaces ofgeneral type with p g = q = 0 and with K = 9. They are known to bequotients of the complex unit 2–ball by an arithmetic subgroup, and arealso known to be rigid. See [Pr-Y, Pr-Y- ]. . Let S be a Beauville surface [Be, Exercise X. .( )] and [Ba-C-G]. Theseare certain minimal surfaces of general type with K = 8, p g = q = 0. Sucha surface is rigid [Cat] and so, by Proposition . , it has a model over Q .Its complex conjugate cousin, also a Beauville surface, is rigid as well.By [Ba-C-G], there are a two more types of surfaces similar to Beauville’sexamples in that they are all quotients of a product of two curves of gen-era > G and having moduli spaces of di-mension 0. Here G is one of two non-abelian groups of order . Thefirst gives an example whose moduli space consists of three 0–dimensionalcomponents, the second group leads to a unique example. he next result gives an application in the relative setting. It leads upto Belyˇı curves: Proposition . . Suppose
X, Y are smooth projective of the same dimension, p : X → Y is a surjective finite morphism with smooth branch locus B ⊂ Y .Assume that Y is rigid and that B is rigidly embedded in Y . Then X has amodel over a number field.Proof. One constructs a spread of the morphism p : X → Y as in the proofof Cor. . . . Call it ˜ p : X → Y × S . We do not now that p is rigid. Butthe induced deformation of f , the family X → Y × S → S , is di ff erentiablylocally trivial over S and so the topological structure of the fibers p s of themap ˜ p does not vary. Away from the branch locus, the map p s is a finite´etale cover and so the complex structure on X s := p − s ( Y − B s ) ⊂ X s is locally determined by the complex structure on Y − B s , which by rigidityof the embedding of B in Y is independent of s . The manifold structure of X s is fixed and so it only has to be checked that the complex structure onit is completely determined by the complex structure on the Zariski opensubset X s .To show this, note that holomorphic functions on Y are bounded nearthe branch locus and so, by Riemann’s extension theorem, their lifts to X s can be extended uniquely to X s . So indeed, up to isomorphism, thecomplex structure on X s does not depend on s . As before, pick any s ∈ S defined over Q (which exists since S is by construction defined over Q ).Then, not only Y s is defined over Q , but also X s is, and hence, by rigidity,so is the variety X o = X . Remark.
By [Mu-O, p. – ], a variant of the above proof is apparentlydue to Carlos Simpson. Examples . . . Recall that a Bely˘ı curve [Bel] is a complex projectivecurve admitting a cover to P ramified only in the three points 0 , , ∞ .Three distincts points in P define a rigid divisor since three distinct pointscan always be mapped to three given distinct points by a projective trans-formation of P . Bely˘ı showed (loc. cit.) that a complex projective curvecan be defined over Q if and only if it is isomorphic to a Bely˘ı curve. Theabove Proposition shows that the fact that Bely˘ı curves are defined over a umber field is an example of a quite general phenomenon. The conversestatement however requires an explicit construction which is very partic-ular to curves. See [Mu-O, Sect. . ] for a proof in the style of this paper. . For higher dimensional examples, including branched covers of P branched in 4 or less lines, see [Pa]. Further examples of models over number fields . Locally symmetric spaces
Let D = G ( R ) /K be a hermitian symmetric domain, Γ a torsion free arith-metic subgroup of G ( R ) and let X = Γ \ D be the corresponding locallysymmetric space. Such X give examples of Shimura varieties for whichit is known that they can be defined over a number field. See e.g. [Mi] forbackground. Shimura varieties will be investigated more in detail belowin Section . .Here I want to present another approach, due to Faltings which is morein the spirit of this note. Proposition . ([F]) . The pair ( X, ∂X ) has a unique model over Q .Proof. I give a sketch of Faltings’ proof. The specific Kodaira-Spencerclass κ ( X,∂X ) coming from the derivations of C / Q given by ( ) lands in thevector space H ( X, Θ X ( ∂X )) measuring infinitesimal deformations of thepair ( X, ∂X ). Using harmonic theory, Faltings shows that each of these canbe represented by a unique vector valued harmonic form H κ ( X,∂X ) on D oftype (0 , X, ∂X ) H κ ( X,∂X ) is functorial andequivariant with respect to group actions.Using this property for the various Hecke correspondences, one showsthat such a harmonic form is Γ –invariant for all possible arithmetic sub-groups Γ ⊂ G . This form lifts to D as a G ( Q )–invariant harmonic 1-formwith values in the tangent bundle. By density it is then G ( R )–invariant on D . But such a form must vanish. One sees this as follows. By [Helg, Ch.VIII, § ] the complex structure on the tangent space T o D at a any point o of the hermitian symmetric domain D is induced from the action of the For more details and a generalization of the results of [Cal-V] to the non-compactsituation see [Pe ]. enter Z ≃ U of the isotropy group of o on D : z ∈ Z induces multiplicationwith z . Hence, if α is a global (0 , D with values in the tangentbundle point o the action is given by z ∗ ( α ) = ( ¯ z − · z ) · α . So Z -invariance,implies α ( o ) = 0. Since o is arbitrary, α = 0.Next, one observes that the spread family for the pair ( X, ∂X ) is regularin the Kodaira-Spencer sense. The proof is similar to the proof of Cor. . .Hence one may apply (a relative variant of) Theorem . : ( X, ∂X ) is rigid,and hence this pair has model over Q .Uniqueness then follows from H ( X, Θ X ( ∂X )) = 0 (no vectorfields canbe tangent along the boundary divisor). Faltings gives an explicit argu-ment reducing the proof to the assertion that there exists no G ( R )–invariantholomorphic vector fields on D . For the last assertion in loc. cit. no proofis given, but the argument is similar to what we did before: The element z ∈ Z = { center of the isotropy group of G ( R ) at o } acts as multiplicationwith z on tangent vectors at o and so, invariance implies that any globaltangent vector field on D invariant under the action of G ( R ) vanishes at o and hence everywhere. . Holomorphic maps into locally symmetric spaces
As before, let X = Γ \ G/K be a locally symmetric space of hermitian type.To D = G/K and a parabolic subgroup P ⊂ G one associates a boundarycomponent D ( P ) which is also a bounded symmetric domain. Introducerank of D = ℓ ( D ) = min P dim D ( P ) . The numbers ℓ ( D ) for D irreducible are collected in Table . The rigidityresult I use here is due to Sunada: Proposition . ([Su]) . With the above notation, let M be projective, f : M ֒ → X = Γ \ D with X compact, is rigid keeping source and target fixed, whenever dim M ≥ ℓ ( D ) + 1 . From Prop. . , Cor. . . , together with the fact that X is defined over Q whenever Γ is arithmetic, we deduce: Corollary . . If moreover, Γ ⊂ G is a neat congruence subgroup, and M isembedded in D as a totally geodesic submanifold, then M has a model over anumber field. able : Hermitian symmetric domainsDomain dim D ℓ ( D ) I p,q = SU( p, q ) / S(U( p ) × U( q )) pq ( p − q − I I g = SO ∗ (2 g ) / U( g ) g ( g − ( g − g − I I I g = Sp( g ) / U( g ) g ( g + 1) g ( g − I V n = SO o (2 , n ) / SO(2) × SO( n ) n V = E / SO(10) · SO(2) 16 1
V I = E /E · SO(2) 27 8
Examples . . . Since the unit ball B n in C n can be represented as thedomain I ,n and since ℓ ( I ,n ) = 0, all (positive dimensional) geodesicallyembedded subvarieties of a compact arithmetic quotient of the unit ballhave models over a number field. . A domain of type I V n with n ≤
18 is a parameter space for lattice polar-ized K surfaces, and since ℓ ( I V n ) = 1, using local Torelli, we deduce thatif we have a family of K surfaces over a compact base B of dimension ≥ B has a model over a number field. Applications to variations of Hodge structure . Hodge theory revisited
As a preliminary to the topic of Shimura varieties, it is useful to view aHodge structure as a representation space for a certain algebraic torus, asobserved by Deligne. See e.g. [Del-M-O-S, Chap. I], [Ca-MS-P, Chap. ].To explain this briefly, giving a Hodge structure on a real vector space V is the same as giving a morphism h : S → GL( V ) , S = Res C / R G m , where I recall that the Weil restriction Res C / R G m is just the group C × con-sidered as a real group. In other words, a real Hodge structure is just arational (or ”algebraic”) representation of the torus group S . One sees thisby observing that on the complexified vector space V C = V ⊗ R C the action f S diagonalizes and the Hodge subspace V p,q ⊂ V C by definition is thesubspace where h ( z ) acts as multiplication with z p ¯ z q .If the Hodge structure has pure weight k this shows up as follows: viathe natural inclusion w : R × → S , the action of t ∈ R × is multiplication by t k . This motivates introducing w h = h ◦ w : G m → GL( V ) , the weight morphism .If, moreover, V has a rational structure, say V = V Q ⊗ R , this weight mor-phism is obviously defined over Q . When this is the case, one definesthe Mumford-Tate group of h as the smallest closed subgroup M = M ( h ) ofGL( V Q ) such that h factors through the real algebraic group M R .Hodge structures coming from geometry carry a polarization, whereI recall that a polarization consists of a Q -valued bilinear form b on V Q satisfying the two Riemann relations . b C ( x, y ) = 0 if x is in V p,q and y is in V r,s for ( r, s ) , ( k − p, k − q ), where k is the weight of the Hodge structure; . i p − q b ( x, x ) > x is a nonzero vector in V p,q .A Hodge structure is polarizable if such a b exists and then M is knownto be reductive. See [Del-M-O-S, Prop. I. . ], [Ca-MS-P, Prop. . . ].Using this language, one singles out a CM-Hodge structure as one whoseMumford-Tate group is abelian and hence, by reductivity, an algebraictorus.Let me next discuss the notion of a variation of Hodge structure . It con-sists of a local system V on a smooth quasi-projective variety S of finite di-mensional Q -vector spaces, such that all fibers admit a polarizable Hodgestructure. More precisely, V should come from a representation of thefundamental group of S in a finite dimensional vector space V equippedwith a non-degenerate bilinear form b such that . the locally free sheaf V = V ⊗ O S carries a descending filtration F • by holomorphic subbundles; . the natural flat connection ∇ on V lowers degrees of this filtrationby at most 1 (Gri ffi ths’ transversality); . b and F • induce a polarized Hodge structure in each stalk. iven such a variation of Hodge structure, the Hodge structure over x ∈ S corresponds to h x : S → GL( V ) and its Mumford-Tate group may vary,However, outside a countable union of proper subvarieties, M = M ( h x ) isthe same, the generic Mumford-Tate group , and a point with this Mumford-Tate group is called Hodge generic .The group G = Aut( V , b )is a Q –algebraic group. The representation of π ( S, x ) in V defining thelocal system V preserves the polarization b and the image Γ of π ( S, x ) in G ( R ) is discrete. It is called the monodromy group of the variation. Definition . . The connected component of the Q –Zariski closure of themonodromy group in G is called the algebraic monodromy group .The group G ( R ) acts transitively on a so called period domain D , whichclassifies the Hodge structures on V with a fixed set of Hodge numbers po-larized by b . The obvious map p : S → Γ \ D is holomorphic; it is called the period map . The Gri ffi ths’ transversality condition is in general a furtherconstraint. It is vacuous for weight one variations and also for variationsof K -type. . Application to variations of weight and For a weight two variation with Hodge numbers h , = p, h , = q , theperiod domain has shape D = SO(2 p, q ) / U( p ) × SO( q ), the K -case corre-sponding to p = 1 , q = 19. For weight two domains one further introducesthe rank ℓ ( D ) of D which generalizes the concept for hermitian symmetricspaces from Table : ℓ ( D ) = p = 1 ,q − p = 2 , ( p − t + ǫ if p ≥ , t = ⌊ ( q − ⌋ , ǫ = q odd1 if q even.One has the following rigidity result: Theorem . ([Pe , Theorem . ]) . Let D be a period domain for polarizedweight or Hodge structures. An immersive period map f : S → Γ \ D with quasi-projective is rigid keeping source and target fixed as soon as dim S ≥ ℓ ( D ) + 1 . Using Prop. . , as a corollary, we get: Corollary . . Let D be a period domain for polarized weight or Hodgestructure. Let S be quasi-projective and f : S → Γ \ D an immersive period mapof rank ≥ ℓ ( D ) + 1 . Suppose moreover, that S is geodesically embedded, then S has a model over Q . . Shimura varieties
One needs a Hodge theoretic interpretation of Shimura varieties, i.e., va-rieties the form X = Γ \ D for which D = G ( R ) /K is a Hermitian symmetricdomain of non-compact type and G is a connected Q –algebraic group. Fordetails of the discussion that follows see e.g. [Ca-MS-P, Chap. , ],[Mi].A point x ∈ D turns out to correspond to a unique h x : S → G R andso a given representation of G in V defines a real Hodge structure. Ifthe representation comes from a Q –representation ρ : G → GL( V Q ) onemight not get a rational Hodge structure. However, we do get a direct sumof such structures (possibly of di ff erent weights) if the weight morphism ρ ◦ h x ◦ w : R × → GL( V ) is defined over Q . Such representations exist: takethe adjoint representation, with H = Lie G and ρ = ad : G → GL(V): itsweight is zero and hence the weight morphism is automatically definedover Q .The group G ( R ) acts by conjugation on h x . Let h ( g ) x denote the conjugateof h x by g ∈ G ( R ). Then one has the basic equality h gx = h ( g ) x and hence, since G ( R ) acts transitively on D , one may view D as an entireconjugacy class of maps h : S → G R . Each point in D can be identified withsuch a map since h = h ( g ) precisely if g belongs to the isotropy group of thecorresponding Hodge structure. For clarity, let me write [ h ] for the pointin D corresponding to h ∈ Mor( S , G ( R )). Not any G ( R )–conjugacy class ofa morphism S → G R underlies a Hermitian symmetric domain. For this tobe true, such a morphism has to verify certain axioms, as given in [Del].If this is the case, the corresponding pair ( G, D ) is called a
Shimura datum nd D is called a Shimura domain . By the previous remarks about theadjoint representation, all Hermitian symmetric domain thus arise with G the group of holomorphic automorphisms of D , which is known to be Q -algebraic and of adjoint type.It makes sense to define the Mumford-Tate group of a point [ h ] ∈ D asthe smallest closed subgroup M ( h ) of G such that h factors through thereal algebraic group M ( h ) R . Then ρ ( M ( h )) is the Mumford-Tate group ofthe Hodge structure ρ ◦ h . The orbit of h ∈ D under its Mumford-Tate group M ( h ) is a holomorphic submanifold of D which turns out to be a Shimuradomain for M ( h ). It is called the submanifold of Hodge type passing through[ h ]. Its image in X is called a special subvariety .As recalled above, for a point [ h ] ∈ D outside a countable union ofproper closed subvarieties in D , the Mumford-Tate group is precisely G .Call such a point Hodge-generic . For such points, D is the submanifold ofHodge type through [ h ]. At the other end of the spectrum one has the CM-points in D , by definition those points [ h ] for which M ( h ) is abelian (i.e.an algebraic torus). In this case it is its own submanifold of Hodge type.Concerning these points, one has: Lemma . ([Ca-MS-P, Corr. . . ]) . A Shimura subdomain contains adense set of CM-points. . Monodromy and rigidity
The geometry of the variation is reflected in the algebraic monodromy , whichas I recall, is the connected component M mon of the Q –Zariski closure inGL( V ) of the monodromy group of the variation. Any reductive groupsuch as M has a canonical almost direct product decomposition M = M der · (center of M ) , where M der is the derived subgroup of M , its maximal semi-simple sub-group. There are two important results concerning the relation of the twogroups: Theorem . . )[An, Thm] The algebraic monodromy group is a normal sub-group of the generic Mumford-Tate group. In fact, one has M mon ⊳ M der . )[An, Prop. ] If there are CM-points in the variation, this is an equality M mon = M der . et me now consider a more general situation of a homomorphic map p : S → Γ \ D to a Shimura variety, i.e. D = G ( R ) /K is a bounded Her-mitian symmetric domain. This defines a polarizable variation of Hodgestructures on S where Gri ffi ths’ transversality is automatic. Here Γ is themonodromy group of the variation. The group that determines the defor-mations of p is the centralizer of the algebraic monodromy group insidethe group G : G ′ := Z G ( M mon ) . Indeed, one has:
Proposition . . Under the assumption that X = Γ \ D is a Shimura variety,the ”period map” p : S → Γ \ D is rigid if and only if G ′ ( R ) is compact.Proof. The Lie algebra g of G ( R ) consists of the endomorphisms of V thatare skew with respect to b . The Cartan involution induces a direct sumdecomposition g = k ⊕ p where k is the Lie algebra of the maximal compactsubgroup K ( R ) ⊂ G ( R ). The Lie algebra has a natural structure of a weightzero Hodge structure inherited from the one on End( V ). Indeed g C = g − , ⊕ g , ⊕ g , − , g , = k C . The Lie algebra g ′ ⊂ g of G ′ ( R ) consists of those endomorphisms in g thatcommute with the action of the monodromy group. This subalgebra inher-its a weight zero Hodge structure and by [Pe , Theorem . ], the tangentspace to infinitesimal deformations of p is isomorphic to ( g ′ C ) − , and inthis case, as a real space it is isomorphic to p ∩ g ′ . Hence p ∩ g ′ = 0 if andonly if g ′ = k ∩ g ′ if and only if G ′ ( R ) is compact.Observe next that G ′ is also a reductive group of Hermitian type: D := G ′ ( R ) /K ∩ G ′ ( R ) is a bounded subdomain of D and if ˜ S is a universal coverof S with lifting ˜ p : ˜ S → D , there is an induced holomorphic map F : S × D → D extending ˜ f . This maps parametrizes the deformations of f keeping S and D fixed. If f embeds ˜ S as a subdomain D ⊂ D , i.e. if f isa geodesic embedding, then one has a product situation˜ F : D × D ֒ → D. In other words, the deformations of the embedding Γ \ D ֒ → Γ \ D betweentwo Shimura varieties are parametrized by a Shimura variety of the form Γ \ D . By Prop. . one then concludes: orollary . ([Ab, § ]) . Let G be a Q -algebraic group of Hermitian type, G ⊂ G a reductive subgroup, and let D = G ( R ) /K , D = G ( R ) /K ∩ G ( R ) thecorresponding domains. Put G = Z G G , D = G ( R ) /G ( R ) ∩ K . Let Γ be aneat arithmetic subgroup of G ( Q ) such that Γ i = Γ ∩ G i ( Q ) i = 1 , is also neat.The embedding Γ \ D ֒ → Γ \ D between the corresponding Shimura varieties.is rigid with fixed target precisely when G ′ ( R ) is compact. In particular, theembedding Γ \ D × Γ \ D ֒ → Γ \ D is rigid. Let me next consider the algebraic monodromy group M mon ⊂ G froman arithmetic perspective. First recall that for any Q -simple algebraicgroup G there is a totally real number field F and an absolutely simple F –group ˜ G such that G = Res F/ Q ˜ G. Here Res F/ Q is the Weil-restriction whereby an F –group is viewed in afunctorial way as a Q –group. For a real embedding σ : F ֒ → R let ˜ G σ bethe corresponding conjugate of ˜ G . It is called a factor of G . Then G R = Y σ ∈ S ˜ G σ R , S the set of embeddings F ֒ → R . Hence, assuming for simplicity that the algebraic monodromy group issimple over Q , one may write: M mon = Res F/ Q ˜ M mon = ⇒ G ′ = Res F/ Q Z ˜ G ˜ M mon . In particular, for every factor ( ˜ M mon ) σ there is a corresponding factor ˜ G ′ σ .This can be used in the weight one case as follows: Corollary . . Let there be a weight one variation over a quasi-projective va-riety with Q -simple algebraic monodromy group. Assume that M mon has nocompact factor. Then the variation (and the period map) is rigid.Proof. In the weight one case, by [Sat , Prop. IV. . ], M mon ( R ) and G ′ ( R )are in a sense ”dual”: every non-compact factor ( ˜ M mon ) σ corresponds to acompact factor ˜ G ′ σ . The assumption implies that all factors of G ′ must becompact and so the deformation is rigid.This result implies a quite curious result that states that non-trivialmonodromy at the boundary implies rigidity: roposition . ([Sa, Th. . ]) . A weight one variation over a quasi-projectivevariety S with a non-trivial unipotent element in the monodromy is rigid.This holds in particular if S is not compact and there is at least one non-finite local monodromy operator at infinity.In these instances, if moreover S is geodesically embedded, it has a modelover Q .Proof. First I need a result about ranks of simple groups. Recall that areductive k –algebraic group G has k –rank zero if it has no k –split tori. By[Bo, § . ] this is the case if and only if G has no non-trivial characters over k and no unipotent elements g ∈ G ( k ), g ,
1. For k = R , the R –rank is zeroprecisely when G R is compact. Lemma . . If G is a Q –simple group such that G R has at least one compactfactor, then the Q –rank of G is zero. In particular, G contains no unipotentelements g , . To show this, as before, write G = Res F/ Q ˜ G with ˜ G an absolutely simplegroup defined over a totally real number field F .A character χ for G induces a character χ σ for ˜ G σ and any unipotent g ∈ G gives a unipotent element g σ in ˜ G σ . Suppose ˜ G σ R is compact. Then χ σ = 1 and g σ = 1 and also χ = 1, g = 1. This finishes the proof of theLemma.The Lemma implies that the algebraic monodromy group has no com-pact factors. Hence, by Cor. . the deformation is rigid.A similar result can be shown for variations of K -type: Proposition . ([Sa-Zu, Cor. . . ]) . Suppose we have a non-isotrivialK –variation over a quasi-projective variety S with a non-trivial unipotentelement in the monodromy. Assume that the variation is not isotrivial. Supposemoreover, that its rank is not . Then the variation (and the period map) isrigid, and if S is also geodesically embedded, then S has a model over Q .Proof. Here Lemma . is used in a di ff erent manner.: For a non-isotrivialisotypical variation which is non-rigid, M mon ( R ) has one conjugate iso-morphic to SL(2 , R ) with representation space R ⊗ R and the remainingconjugates are ≃ SU(2) with representation space C . It follows from theLemma that the only possibility to accommodate a non-trivial unipotentelement T is when no compact conjugates are present and then the localsystem has rank 4. . Special subvarieties of Shimura varieties
Recall (§ . ) that a special subvariety of a Shimura variety X = Γ \ G/K , ora subvariety of Hodge type, comes from the orbit of a point in D = G/K under its own Mumford-Tate group. In this subsection we study them inmore detail.A morphism of Shimura varieties X = Γ \ G ( R ) /K | {z } D → X = Γ \ G ( R ) /K | {z } D is by definition induced by an equivariant morphism of Shimura domains.Such a morphism is given by a morphism ϕ : G → G of Q –algebraicgroups. It then induces a holomorphic maps of Shimura domains f : D → D by stipulating that f ([ h ] = [( ϕ ◦ h )] for one hence all points[ h ] ∈ D . The Mumford-Tate group of [ h ] maps under ϕ to the Mumford-Tate group of f ([ h ]). It follows that the subvariety f ( D ) is special in D :if [ h ] is Hodge generic, then f ([ h ) has Mumford-Tate group ϕ ( G ) and f ( D ) is the orbit of this group acting on f ([ h ]).Suppose next that f is an embedding. One then may assume that ϕ is also an embedding. It is not hard to see that f is a totally geodesicembedding. See e.g. [Ca-MS-P, Chap . ]. Then the conjugate map f ( g ′ ) , g ′ ∈ G ( R ) is also a totally geodesic embedding. It may or may not arisefrom a morphism of Shimura domains. Indeed, its image may not haveCM-points at all.The variation of Hodge structure on D restricts to one on f ( g ′ ) D andthis descends to give one on its image in X . The monodromy of this vari-ation is Γ ( g ′ )1 . The connected component M mon of its Zariski closure in G acts transitively on D = f ( g ′ ) D ⊂ D and so, if it would be the genericMumford-Tate group M or, which su ffi ces, its derived group, the subdo-main D would be of Hodge type. This is the case if D contains a CM-point.Indeed, by Theorem . one then has M mon = M der . Concluding, I haveshown: Lemma . . Let D , D two Shimura domains and let be i = f ( g ′ ) : D ֒ → D as above. Then i is a morphism of Shimura domains if and only if g ′ ∈ G ( Q ) if and only if i ( D ) contains a CM-point of D . This can be used to give another proof of Abdulali’s criterion [Ab, Thm. . ]: roposition . . Let i : X ֒ → X be a totally geodesic embedding of Shimuravarieties. If the embedding is rigid, i ( X ) is a special subvariety.Proof. Since Shimura varieties are defined over a number field (cf. [Mi]),one may apply Cor. . . . So, if the embedding is rigid, the image is de-fined over a number field. To show that the image is a special subvariety,by the previous Lemma, it su ffi ces to find a CM-point in the image. But, if x ∈ X is a CM-point, then i ( x ) is also a CM-point since the Mumford-Tategroup of the Hodge structure corresponding to x is an algebraic torus andhence, so is the one associated to i ( x ) since i is defined over Q .Corollary . then yields examples for weight one Hodge structures: Examples . . . The group G = GL(2) can be embedded in Sp( g ) as fol-lows. Set V k = ( R k , J k ), J k = k − k ! . The direct sum ⊕ k V is isomorphicto the symplectic space V k . Whence a faithful representation ρ k of 2: a bc d ! ρ k −−→ a k b k c k d k ! . For any k = 1 , . . . , g the direct sum representation ρ k ⊕ (rank ( g − k ) trivialrepresentation) induces a holomorphic embedding h ֒ → h g . It gives thenon-compact embedded Shimura curves starting from the Shimura da-tum (SL(2) , h ). There is no locally constant factor if and only if k = g andthen the embedding is rigid. This follows from Corollary . . These non-compact rigid curves are often called rigid curves of Satake type . . There are also examples where G has compact factors. Here I use againthe Satake ”duality” mentioned before, but in its precise form as explainedin [Sa, § ]. It applies to G and G ′ := Z Sp( g ) G and gives:( G ) R ≃ SL(2) × SU(2) × · · · ×
SU(2) | {z } m − = ⇒ ( G ′ ) R ≃ SO(2) × SU(1) × · · · ×
SU(1) | {z } m − The latter group is compact and hence the deformation is rigid. There areindeed examples of such embeddings, the
Mumford type curves . See [Ad,§ . ]. Applications to Higgs bundles . Basic notions A Higgs bundle over a complex manifold B is a pair ( V , τ ) of a holomor-phic bundle together with an End( V )–valued 1-form τ such that τ ◦ τ = 0.The form τ can also be viewed as a Higgs field , a homomorphism τ : V → V ⊗ Ω B . A graded Higgs bundle is a Higgs bundle such that V = ⊕ r V r ,with V r locally free and such that τ | V r : V r → V r − ⊗ Ω B .The standard example comes from polarized complex variations ofHodge structures on B . Recall [Simp, § ] that such a system consist of• a local system of C –vector spaces V equipped with a flat non-degeneratebilinear form. In other words, if π is the fundamental group of B based at o ∈ S , V comes from a representation ρ : π → O( V , b ), V thefiber of V at o ;• a direct sum decomposition V ⊗ C ∞ B = ⊕ r V r ∞ into locally free C ∞ B –modules such that – the hermitian form h ( x, y ) = ( − r b ( x, ¯ y ) on V r ∞ is positive defi-nite and the above decomposition is h -orthogonal; – the natural flat connection ∇ on V ⊗ C ∞ B obeys V r ∇ −→ A , B ( V r − ∞ | {z } ↓ ⊕ A B ( V r ∞ ) | {z } ↓ ⊕ A , B ( V r +1 ∞ ) | {z } ↓ τ + d + τ ∗ , where τ ∗ is the h -adjoint of τ .These demands imply that F p = ⊕ r ≥ p V r ∞ is a holomorphic subbundle of V ⊗ O B and that Gri ffi ths’ transversality holds. This filtration is the Hodgefiltration . It also follows that the holomorphic bundle V = ⊕ p F p / F p +1 , C ∞ B ( F p / F p +1 ) = V p ∞ with the underlying local system V admits the structure of a graded Higgsbundle with τ the Higgs field. Flatness (i.e., ∇ ◦ ∇ = 0) implies the Higgs For more details on Higgs bundles see e.g. [Ca-MS-P, Chapter ]. ondition τ ◦ τ = 0. Moreover, the Chern connection, that is, the uniqueholomorphic connection on this Higgs bundle which is metric with respectto the hermitian metric h turns out to be ¯ ∂ + τ . So on any subbundle onwhich τ = τ ∗ = 0, the flat connection ∇ induces the Chern connection andso the metric h coincides with the flat metric. Moreover, such a subbundlecomes from a local subsystem of V since it is preserved by ∇ . Also, it is unitary since it admits the flat unitary metric h . This holds in particularfor the largest subbundle for which τ = τ ∗ = 0: U = U ⊗ O B : the maximal unitary Higgs subbundle . There is an h -orthogonal splitting V = U ⊕ W , W = U ⊥ . ( ) . Logarithmic variant If B is quasi-projective, one usually considers Higgs bundle with logarith-mic growth near the boundary. To explain this, assume for simplicity thatdim B = 1 and that B get compactified to a a smooth projective curve B .Then the boundary Σ = B − B consists of finitely many points. A gradedlogarithmic Higgs bundle V = ⊕ p V p on B , with V p locally free, by defini-tion admits a Higgs field with components τ : V p → V p − ⊗ Ω B (log Σ ) . For a variation of Hodge structure on B with unipotent monodromy at thepunctures, one lets V be the associated graded of the Deligne extendedHodge filtration. Then the Gauss-Manin connection induces a Higgs fieldas above.Even more is true. Choose a coordinate patch ( ∆ , t ) around a punc-ture and let T be the (unipotent) local monodromy operator around thepuncture. For v a local holomorphic section of V on the disc, write ∇ v = R dtt , R ∈ End( V | ∆ ) . Then N := R (0) = log( T ) ∈ End( V ) nd the Higgs field at the puncture is given by τ (0) : V → V ⊗ dtt , v p (Gr p N ) v p ⊗ dtt . ( )Suppose k is the first index in the grading for which V k , k + w +1the last. Then the number w is called the width.In this general setting, one says that for a Higgs bundle of width w , theHiggs field is generically maximal if for all p ∈ [ k, k + w + 1] one has V p , τ | V p generically an is an isomorphism for p = k, . . . , p + w . . Rigid maximal Higgs subsytems
The following rigidity result [Vie-Z, Lemma . ], stated without proof,can be formulated in a slightly di ff erent way which fits better within thegeneral framework set up so far: Proposition . . Let B be a smooth quasi-projective variety, V be a local sys-tem on B of finite dimensional Q -vector spaces and let W C a subsystem of V ⊗ C .Suppose W C is rigid as a subsystem of V ⊗ C . Then W C is defined over Q in thesense that W C = W ⊗ C , where W is a local system of F -vector spaces for somenumber field F .Proof. Let π be the fundamental group of B ( C ) based at o ∈ B ( C ) and let V be the fiber at o of V C , considered as a π -representation space. The group π acts on the Grassmannian G ( r, V ) of r -dimensional subspaces W ⊂ V ,where r = rank W C . A fixed point [ W ] of this action corresponds to a com-plex subsystem of V ⊗ C . More precisely, the corresponding π -invariantsubspace is the fiber U [ W ] at [ W ] of the tautological bundle U → G ( r, V ).The spread of the point [ W ] is a subvariety Y ⊂ G ( r, V ) contained in thelocus of fixed points of the π -action, because this action is defined over Q . The tautological subbundle over Y gives a deformation of W C ⊂ V C ,and so rigidity implies that Y ( Q ) = [ W ], an isolated point. Hence the localsystem W C , which corresponds to U [ W ] , is defined over Q .As a direct application, one has: Proposition . ([Vie-Z, Lemma . ] ) . Let B = B − Σ as above. Suppose that V underlies a polarizable Q -variation of Hodge structure and let ( V , σ ) the cor-responding graded logarithmic Higgsbundle over B with unipotent monodromy round points of Σ . Let V C = U ⊕ W the splitting ( ) . Suppose that the loga-rithmic Higgs subbundle W corresponding to W is a generically maximal Higgssubbundle. Then the splitting is defined over Q .Proof. To show how this result is implied by Proposition . , it is enoughto show that W is rigidly embedded in V . Again, with V a typical fiberof V , small deformations of W are parametrized by the tangent space tothe fixed locus under the π -action on the Grassmannian G ( r, V ) at a π -invariant point [ W ]. A tangent vector is therefore represented by a homo-morphism of local systems q : W → V / W = U which is compatible with the structure as a complex system of Hodge bun-dles: a small deformation of W within V inherits this structure from theone on V and the map q is the embedding of the deformed V followedby restriction to U . But the Higgs field for the left hand is generically anisomorphism while on the right hand it is zero. This is impossible unless q = 0. Remark.
A variant of this (loc. cit.) is when W is a direct sum of complexsystems of Hodge bundles of di ff erent widths, all with generically max-imal Higgs field. Then almost the same argument shows that also thissplitting is defined over Q . There is one subtlety here: one has to com-pare projections between complex systems of di ff erent widths and thenone needs semi-simplicity for variations of Hodge structures. This prop-erty is a highly non-trivial consequence of another rigidity property dueto Schmid [Sch]. See [Pe-St, § ] for details.One can say more, if there are punctures with infinite local monodromy: Proposition . . The situation is as in Prop. . . In particular, all local mon-odromy operators at the boundary are unipotent. Assume that at least one localmonodromy operator has infinite order. Then the splitting V = U ⊕ W from ( ) is defined over Q and U extends as a local system to B . The monodromy of thislast system is finite.Proof. The property that a Higgs field is an isomorphism on B extends to B . If all graded fields τ p are isomorphisms, at a puncture ( ) implies thatthe Gr p N are isomorphisms and hence that N is an isomorphism. Thisholds for W , while the fact that the Higgs field for U remains zero at a uncture implies that the local monodromy for U is the identity and thusthat this local system extends to B .Suppose that we have a splitting as above, valid over a Galois extension F/ Q . The property that N is an isomorphism or zero is preserved by theaction of the Galois group G . It follows that for σ ∈ G the natural inclusionfollowed by projection U σ → V → V / U = W sends the fiber U σs at a puncture s ∈ Σ to zero. In other words U s = U σs for all σ ∈ G and so this fiber is defined over Q . Since U extends to B ,and since the entire monodromy action is defined over Q , the local system U which is built from the monodromy representation on some fiber U s is then defined over Q . The polarization is defined over Q as well andhence W = U ⊥ is defined over Q . The finiteness of the monodromy followssince the system is defined over Q and the polarization h on it is a positivedefinite Q -valued form preserved by the monodromy. Example . (An interesting Shimura curve in the Torelli locus) . The aboveresult definitely fails when B = B : the global monodromy of U may be in-finite. The simplest example from [M-O, Example . ] is a Shimura curveand can be described as follows. Consider the family of projective curveswith a ffi ne equation y = x ( x − x − t ) . This gives a family C t over P of genus 4 curves. The fibers are smoothfor t , , , ∞ . Note however that local monodromy operators are quasi-unipotent in this case, but this does not really matter since this could betaken care of by a finite branched cover of P . For simplicity this will notbe done since the above analysis still works after some minor modification.Let ζ be primitive root of unity. Then the cyclic group Z / Z gener-ated by ( x, y ) ( ζ y, x ) preserves C t and the Hodge structure H ( C t ) ofweight one admit an action of Z / Z . Let F = Q ( ζ ). The Galois group G of F/ Q is generated by the element σ which sends ζ to ζ . It permutesthe eigenspaces of Z / Z acting on V t = H ( C t , C ) as in the following table.Next, consider the splitting of the corresponding Higgs bundle. The Higgsbundle splits also in eigenspace subbundles; the Higgs field is zero for thesubbundles corresponding to the first two rows and an isomorphism for able : Eigenspaces for Z / Z on V t Eigenvalue h , h , ζ σ ( ζ ) = ζ σ ( ζ ) = ζ σ ( ζ ) = ζ U = V ζ ⊕ V ζ , V = V ζ ⊕ V ζ . If M mon is the algebraic monodromy group of this family, one has a corre-sponding splitting M mon ( R ) = SU(2) × SU(1 , . The actual monodromy group Γ in this case is dense in both factors andhence cannot be finite for the local system U . Proposition . then im-plies that the local monodromy around the punctures cannot be of infiniteorder. Indeed, one can show that the local monodromy operators are allof order 5 in this case. Hence the period map extends over the puncturesand the period map embeds the base curve as a compact curve in the pe-riod domain. This is consistent with the fact that fibers over the puncturesare of so called compact type: their generalized Jacobians are products ofprincipally polarized Abelian varieties whose dimensions sum up to 4.Note also that this is an elementary example giving a negative answerto the following question of Fujita [Ue]:”for a family f : X → B of complex algebraic manifolds over a curve B ,is the sheaf f ∗ ω X/B is semi-ample?”In the above example the latter sheaf is just the graded part H , = U , ⊕ V , of the Higgs bundle and while the second bundle is ample, thefirst is flat and would be semi-ample if and only if its global monodromywould be finite, but this group is dense in M mon ( R ). Compare also themuch more elaborate examples in [Cat-D]. eferences [Ab] Abdulali, S : Conjugates of strongly equivariant maps, Pac.J. Math. – ( )[Ad] Addington, S. : Equivariant holomorphic maps of symmet-ric domains, Duke Math. Journal ( ), - [An] Andr ´e, Y .: Mumford-Tate groups of mixed Hodge struc-tures and the theorem of the fixed part. Compositio Math. ( ) – [Ba-C-G] Bauer, I., F. Catanese and F. Grunewald : Surfaces with p g = q = 0 isogenous to a product. ArXiv:math/ [Be]
Beauville, A. : Complex algebraic surfaces , Cambridge Univ.Press ( ) (translation of
Surfaces alg´ebriques complexes ,Ast´erisque Soc. Math. France, Paris ( )).[Bel]
Bely˘ı, G. V .: Galois extensions of a maximal cyclotomicfield. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. ( ), – [Bo] Borel, A. : Linear algebraic groups, in
Proc. Symp. PureMath. ( ) – [Cal-V] Calabi, E. and E. Vesentini : On compact, locally symmet-ric K¨ahler manifolds, Ann. of Math. ( ) – .[Ca-MS-P] Carlson, J., S. M ¨uller-Stach and C. Peters : Periodmaps and period domains, Second edition , C.U.P., Cambridge( ).[Cat]
Catanese, F.:
Fibred Surfaces, Varieties Isogenous to aProduct and Related Moduli Spaces, American Journal ofMathematics, ( ), – [Cat-D] Catanese, F. and M. Dettweiler : Answer to a question byFujita on Variation of Hodge Structures, arXiv: . [math.AG] Del]
Deligne, P.:
Vari´et´es de Shimura: interpr´etation modu-laire, et techniques de construction de mod`eles canoniquesin
Automorphic forms, representations and L -functions (Proc.Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., ), Part – , Proc. Sympos. Pure Math., XXXIII ,Amer. Math. Soc., Providence, R.I., ( )[Del-M-O-S]
Deligne, P., J. S. Milne, A. Ogus and K. Shih : Hodge cycles,motives and Shimura varieties , Lecture Notes in Mathemat-ics, Springer Verlag, Berlin, etc. , ( )[F]
Faltings, G.:
Arithmetic varieties and rigidity. in
Semi-nar on number theory, (Paris, / ), Progr. Math., ,Birkh¨auser Boston, Boston, MA, ( ) – [G-G] Green, M. and P. Griffiths : On the tangent space to the spaceof algebraic cycles on a smooth algebraic variety , Annals ofMath. Studies
Princeton Univ. Press ( )[Helg]
Helgason, S. : Di ff erential Geometry, Lie Groups and Sym-metric Spaces , Academic Press, New York/London ( )[K-S] Kodaira, K. and D. Spencer : On deformations of complexstructure I,II, Ann. Math. ( ) – .[Mi] Milne, J.:
Introduction to Shimura varieties , available onlineas ( )[M-O] Moonen, B. and F. Oort : The Torelli locus and special sub-varieties , Handbook of moduli. Vol. II, – , Adv. Lect.Math. (ALM), , Int. Press, Somerville, MA, .[Mu] Mumford, M. : Hirzebruch’s proportionality theorem in thenon-compact case. Invent. Math. – ( )[Mu-O] Mumford, M. and T. Oda : Algebraic Geometry II , Text andreadings in mathematics , Hindustan book agency (In-dia) ( )[Pa] Paranjape, K. H. : A geometric characterization of arith-metic varieties. Proc. Indian Acad. Sci. Math. Sci. ( ) – . Pe ] Peters, C. : Rigidity for variations of Hodge structure andArakelov-type finiteness theorems, Comp. Math. ( ) – .[Pe ] Peters, C. : On the rank of non-rigid period maps in theweight one and two case, in
Complex algebraic varieties, Pro-ceedings, Bayreuth – , Lect. Notes in Math. Springer-Verla, Berlin etc. ( )[Pe ] Peters, C.:
On rigidity of locally symmetric spaces.Preprint .[Pe-St]
Peters, C. and J. Steenbrink : Monodromy of variations ofHodge structure, Acta Appl. Math. – ( )[Pr-Y] Prasad, G. and S. K. Yeung : Fake projective planes, InventMath , – ( ), arXiv:math// [Pr-Y- ] Prasad, G. and S.K. Yeung : Addendum to ‘Fake ProjectivePlanes’, http://arXiv.org/abs/ . [Sa] Saito, M.-H. : Classification of non-rigid families of abelianvarieties, Tohoku Math. J. ( ) – .[Sa-Zu] Saito, M.-H. and S. Zucker : Classification of non-rigid families of K -surfaces and a finiteness theorem ofArakelov type, Math. Ann. ( ) – .[Sat ] Satake, I. : Symplectic representations of algebraic groupssatisfying a certain analyticity condition,
Acta Math. ( ) - [Sat ] Satake, I.:
Algebraic Structures of Symmetric Domains , Publ.Math. Soc. Japan , Iwanami Shoten, Japan, and PrincetonUN. Press ( )[Sch] Schmid, W.:
Variation of Hodge structure: the singularitiesof the period mapping, Invent. Math. , – ( )[Sern] Sernesi, E. : Deformations of Algebraic Schemes , Springer-Verlag Berlin, Heidelberg ( ) Simp]
Simpson, C. : Higgs bundles and local systems, Publ. Math.IHES ( ) – [Sh] Shimura, G .: Algebraic varieties without deformations andChow variety, J. Math. Soc. Japan ( ) – [Su] Sunada, T. : Holomorphic mappings into a compact quo-tient of symmetric bounded domain. Nagoya Math. J. ( ), – .[Ue] Ueno, K. (editor): Open problems: Classification of al-gebraic and analytic manifolds, in
Classification of alge-braic and analytic manifolds, Proc. Symp. Katata/Jap. ,Progress in Mathematics, , Birkh¨auser, Boston, Mass.( ), – .[Vie-Z] Viehweg, E. and K. Zuo : A characterization of certainShimura curves in the moduli stack of abelian varieties. J.Di ff erential Geom. – ( ).).