aa r X i v : . [ m a t h . AG ] A p r On rigidity of locally symmetric spaces
Chris PetersJuly , Introduction
A classical result due to Calabi and Vesentini [Cal-V] states that a com-pact locally symmetric space is rigid, provided all of its irreducible factorshave dimension at least . This implies that such varieties (known to bealgebraic) can be defined over a numberfield. This was first remarked byShimura in [Sh]. For a modern variant of the proof see [Pe].Faltings [F] remarked that one can show that the Kodaira-Spencer classfor any ”spread family” of the given variety is zero which su ffi ces for rigid-ity. This is true without any restriction on the type of irreducible factors,and even for non-compact locally symmetric spaces. The proof uses first ofall Mumford’s theory of toroidal compactifications [A-Mu-R-T] of locallysymmetric varieties together with the existence of ”good” extensions ofmetric homogeneous vector bundles to these compactifications as shownin [Mu]. The second ingredient is a careful analysis of the extension ofclassical harmonic theory to a suitable L version.I show in this note that the same techniques can be used to extend theresults of Calabi and Vesentini to the non-compact case. This is stated asTheorem . .Mumford’s ideas are sketched in Sect. and in Sect. I have explainedthe basic L –techniques used by Faltings. This is done in some detail sincethe arguments in [F] are rather sketchy.Thanks to Christopher Deninger for pointing out to the reference [F]. Poincar ´e growth and good metrics
In this section I recall some concepts and results from [Mu]. Let X be asmooth quasi-projective complex variety and let X be a ”good” compacti-fication: X is non-singular, projective and ∂X := X − X a normal crossingdivisor. Hence, locally at a point of the boundary, coordinates ( z , . . . , z n )can be chosen such that the boundary is given by the equation z · · · z r = 0and ∂X can be covered by a collection of polydisks ∆ n on which X cuts out( ∆ ∗ ) r × ∆ n − r . Let k k P be the Poincar´e norm on such a product. Any smooth p form, say η on X is said to have Poincar´e growth near the boundary , iffor all tangent vectors { t , . . . , t p } at a point of ∆ n ∩ X , one has the esti-mate | η ( t , · · · , t p ) | ≤ Const. k t k P · · · k t p k P . This notion does not depend onchoices. By [Mu, Prop. . ] such a form defines a current on X . Mumfordcalls a smooth form ω on X a good form if ω as well as dω have Poincar´egrowth near the boundary.Let ( E, h ) be a hermitian holomorphic vector bundle on X . Recall thefollowing definition: Definition . . The
Chern connection for (
E, h ) is the unique metric con-nection ∇ E on E whose (0 , ∂ : A X ( E ) → A , X ( E ) com-ing from the complex structure on E .Assume that E = E | X where E is a holomorphic vector bundle on X . Definition . . The metric h is good relative to E , if locally near the bound-ary for every frame of E the following holds: . the matrix entries h ij of h , respectively h − ij of h − , with respect to theframe grow at most logarithmically: in local coordinates z , . . . , z n asabove, | h ij | , | h − ij | ≤ Const. · (log | z · · · z k | ) N for some integer N . . the entries of the connection matrix ω h = ∂h · h − for the Chern con-nection are good forms.By [Mu, Prop. . ] there is at most one extension E of E such that h isgood relative to that extension. Note also that the dual E ∗ carries a naturalmetric and this metric is good relative ( E ) ∗ .If h is a good metric on a vector bundle E relative to an extension E ,then, by definition any Chern form calculated from the Chern connectionis good and by [Mu, Thm. . ], the class it represents, is the correspondingChern class of E . Relevant L harmonic theory Let me continue with the set-up of the previous section. So (
E, h ) is a her-mitian holomorphic vector bundle on X such that E is the restriction to X of a holomorphic vector bundle E on X with the property that h is goodrelative to E . In addition, make the following, admittedly strong assump-tions: Assumption . . . X carries a complete K¨ahler metric h X whose (1 , ∂X (and hence its volume form has Poincar´egrowth). . Smooth sections of the bundle A kX ( E ) of complex k –forms with values in E are bounded in the metric induced from h and h X .Let me recall how to introduces metrics on the spaces A k ( E ) of global complex k -forms with values in E . On a fibre A kX,x ( E ) at x ∈ X of the vectorbundle A kX ( E ), one has a fiberwise metric induced by the metrics h and h X : h x ( α ⊗ s, β ⊗ t ) = h X ( α, β ) h ( s, t ) , α, β ∈ A kX,x , s, t ∈ E x . ( )Assumption means that for any two sections ω i ∈ A k ( E ), i = 1 , { x h x ( ω , ω ) } is bounded on X . Since by assumption , thevolume form for h X has Poincar´e growth near ∂X it follows that the globalinner product h ω , ω i = Z X h x ( ω , ω ) · vol. form w.r. to h X , ω , ω ∈ A k ¯ X ( E )exists; in other words, one has an inclusion A k ( E ) ֒ → L ( X, A k ( E )) = { square integrable E -valued k forms } and one can do harmonic theory for certain di ff erential operators on thesespaces. The particular operators here are those that are induced from theChern connection ∇ = ∇ E (see Defn. . ), namely ∇ : A kX ( E ) → A k +1 X ( E ) , ∇ , = ¯ ∂,α ⊗ s dα ⊗ s + ( − k α ⊗ ∇ s. he operator ¯ ∂ , extends in the distributional sense to an operator¯ ∂ : L ( X, A ,q ( E )) → L ( X, A ,q +1 ( E ))and since the metric on X is complete and ¯ ∂ = 0, one can apply a result ofVan Neumann (cf. [De, Sect. ]) which says that there is a formal adjointoperator ¯ ∂ ∗ : L ( X, A ,q +1 ( E )) → L ( X, A ,q ( E )) in the sense of distributions.Moreover, the formal adjoint of ¯ ∂ ∗ exists and equals ¯ ∂ . These adjoints,viewed as operators on the bundles A , ∗ X ( E ) coincide with the classical ones: Lemma . . Let ∗ E : A p,qX ( E ) → A n − q,n − pX ( E ) be the fiber wise defined operatorinduced by the Hodge star-operator. ) The formal adjoint ¯ ∂ ∗ is induced by − ∗ E ∇ , ∗ E : A ,q +1 X ( E ) → A ,qX ( E ) . ) The formal adjoint of ∇ , equals ( ∇ , ) ∗ = − ∗ E ¯ ∂ ∗ E .Proof. Since ¯ ∂ = − ( ∗ E ∇ , ∗ E ) ∗ = − ∗ E ( ∇ , ) ∗ ∗ E , the second assertion followsfrom the first. The meaning of the first assertion is that for ω ∈ A ,q ( E )and ω ∈ A ,q +1 ( E ) one has h ¯ ∂ω , ω i = −h ω , ( ∗ E ∇ , ∗ E ) ω i . ( )To show this, let me go through the classical calculation. First, using themetric contraction h E : A k ( E ) ⊗ A ℓ ( E ) → A k + ℓ ( α ⊗ s, β ⊗ t ) h E ( s, t ) α ∧ ¯ β one observes the fundamental equaton h E ( ϕ , ∗ E ϕ ) = h x ( ϕ , ϕ ) · vol. form dV , x ∈ X, ϕ , ϕ ∈ A k ( E ) . ( )Next, the Chern connection being metric implies that for the forms re-stricted to X (denoted by the same symbols) one has h E ( ∇ ω , ∗ E ω ) + ( − k h E ( ω , ∇ ( ∗ E ω )) = dh E ( ω , ∗ E ω ) , and hence, using ( ) and the relation ∗ E · ∗ E = ( − k , one finds¯ ∂h E ( ω , ∗ E ω ) = h h x ( ¯ ∂ω , ω ) + h x ( ω , ( ∗ E ∇ , ∗ E ) ω ) i · dV . ( ) claim that ¯ ∂h E ( ω , ∗ E ω ) is bounded near ∂X and that it integrates over X to zero. Assume this for a moment. Since the first term on the right isbounded, the other is too. Hence after integration one obtains0 = h ¯ ∂ω , ω i + h ω , ( ∗ E ∇ , ∗ E ) ω i and the result follows.It remains to show the assertion about ¯ ∂h E ( ω , ∗ E ω ). Let U δ be a tubu-lar neighborhood of ∂X with radius δ . By Stokes’ theorem, Z X ¯ ∂h E ( ω , ∗ E ω ) = lim δ → Z ∂U δ h E ( ω , ∗ E ω ) = 0 . ( )The last equality follows since by ( ) the integrand has Poincar´e growthnear the boundary and hence the integral tends to zero (compare the proofof [Mu, Prop . ].The Laplacian ∆ E := ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ preserves L ( A ,q ( X )) and the forms ω with ∆ E ω = 0 are by definition the harmonic forms. Reasoning as in theclassical situation (cf. [De, Sect. ]) one shows: Corollary . . . For all ω ∈ A ,q ¯ X ( E ) one has h ∆ E ω, ω i = h ¯ ∂ω, ¯ ∂ω i + h ¯ ∂ ∗ ω, ¯ ∂ ∗ ω i . Hence, in the distributional sense, one has ∆ E ω = 0 ⇐⇒ ¯ ∂ω = 0 = ¯ ∂ ∗ ω . . There is an orthogonal decomposition L ( X, A ,qX ( E )) = [ ¯ ∂A ,q − X ( E )] cl ⊕ [ ¯ ∂ ∗ A ,q +1 X ( E )] cl ⊕ H ,q (2) ( E ) , ( ) where the symbol cl stands for ”topological closure” and the symbol H (2) standsfor the harmonic L -forms, i.e. L -forms ω with ∆ E ω = 0 in the sense of distri-butions. To apply this, recall that by Dolbeault’s theorem the cohomology group H k ( X, E ) can be calculated as the k -th cohomology of the complex A , ∗ X E ). Proposition . ([F, Lemma ]) . Assume that E is a holomorphic vector bun-dle on X and that ( E = E | X , h ) is a hermitian bundle on X such that h is good elative E . If assumption . holds, then there is natural injective homomor-phism j ∗ L : H k ( X, E ) = H k ( A , ∗ X ( E )) → H ,k (2) ( X, E ) , with target the space of E -valued harmonic square integrable (0 , k ) –forms.Proof. The map j ∗ L is induced from orthogonal projection to H kL ( E ). Theprocedure is as follows. Pick α ∈ A ,kX ( E ) for which ¯ ∂α = 0 representing agiven cohomology class [ α ] ∈ H k ( X, E ). By assumption . , β = α | X is an E - valued L -form whose orthogonal projection to the harmonic forms is j ∗ L α . One needs to verify independence of choices: since ¯ ∂α = 0, one has¯ ∂β = 0 in the sense of currents and so, another representative for α leadsto a form which di ff ers from β by a current of the form ¯ ∂γ . Hence theharmonic projection is independent of choices.To see that it is injective, suppose that the harmonic part of β vanishes.By ( ) one has h β, ¯ ∂ ∗ ϕ i = h ¯ ∂β, ϕ i = 0 and hence β belongs to the first sum-mand of ( ) so that β = lim j →∞ ¯ ∂γ j , γ j ∈ A ,k − X ( E ) . To test that this gives the zero class in H k ( X, E ), one uses the Serre dualitypairing: H k ( X, E ) ⊗ H n − k ( X, Ω nX ⊗ E ∗ ) → H n,n ( X ) = C as induced by the pairing A ,kX ( E ) ⊗ A ,n − kX ( Ω nX ⊗ E ∗ ) → A n,nX . To this end, consider for a closed β ′ ∈ A ,n − kX ( Ω nX ⊗ E ∗ ). I claim that near ∂X it is bounded in norm. To see this let s ∈ Γ ( X, Ω nX ( E ∗ )), then, with f alocal equation for ∂X , the product f · s is a section in the unique extension Ω n ( X )(log ∂X ) ⊗ E ∗ on X of the bundle Ω nX ⊗ E ∗ on X for which h = h X ⊗ h E ∗ is good. That this is the case will be shown later (Examples . . ). Inparticular, since h ( f · s, f · s ) = | f | h ( s, s ) has logarithmic growth near ∂X it follows that h ( s, s ) and hence also h ( β ′ , β ′ ) must vanish near ∂X . Hence β ′ ∈ L ( A ,n − kX ( Ω nX ⊗ E ∗ )). The Serre pairing therefore is given by( β, β ′ ) := lim j →∞ Z X ¯ ∂γ j ∧ β ′ = lim j →∞ lim δ → Z ∂U δ γ j ∧ β ′ , here U δ is a tubular neighborhood of ∂X whose radius is δ (the last equa-tion follows from Stokes’ theorem). Since β ′ tends to zero near ∂X , this in-tegral vanishes. Consequently, the cohomology class of β is zero by Serreduality.I want to finish this section by showing that the Nakano inequality [Na]still holds for E -values harmonic (0 , q )-forms on X . To explain this, oneneeds some more notation. The Lefschetz operator L - which is wedgingwith the fundamental (1 , h X - preserves L –formssince the fundamental form has Poincar´e growth near ∂X . Moreover, since L is real, h x ( Lα, β ) dV = h E ( Lα, β ) = Lα ∧ ∗ β = α ∧ ∗ ( ∗ − L ∗ β )and so Λ = ∗ − L ∗ is the formal adjoint of L . Since ∗ is an isometry, oneconcludes that also Λ preserves the L –forms. Lemma . (Nakano Inequality [Na]) . Let ω ∈ H ,k (2) ( X, E ) . With F h the curva-ture of the metric connection on ( E, h ) and Λ the formal adjoint of the Lefschetzoperator, one has the inequality i h Λ F h ω, ω i ≥ . Proof.
For simplicity, write ∇ , = ∂ E with adjoint ∂ ∗ E . One has the K¨ahleridentity (see e.g. [De, Sect. ]) Λ ¯ ∂ − ¯ ∂ Λ = − i ∂ ∗ E , which is derived in the L -setting as in the classical setting. Using thisrelation, ¯ ∂ω = 0 = ¯ ∂ ∗ ω , as well as F h ( ω ) = ¯ ∂∂ω , one calculates0 ≤ h ∂ E ω, ∂ E ω i = h ∂ ∗ E ∂ E ω, ω i = i h Λ ¯ ∂∂ E ω − ¯ ∂ Λ ∂ E ω, ω i = i h Λ F h ω, ω i − i h Λ ∂ E , ¯ ∂ ∗ ω i = i h Λ F h ω, ω i . The Calabi-Vesentini method in the L –setting In this section I shall indicate how the method used in [Cal-V, Sect. , ] toshow vanishing of the groups H q ( T X ) for X compact can be adapted stepby step to the non-compact setting. et ( X, h ) be a K¨ahler manifold and let T X be the holomorphic tangentbundle. Suppose that the assumptions . hold. The metric h induceshermitian metrics on the bundles A p,qX = ∧ p T ∗ X ⊗∧ q ¯ T ∗ X of forms on X of type( p, q ). The Chern connection on T X is the standard Levi-Civita connectionand its curvature is a global T X –valued (1 , F h ∈ A , X (End( T X )) . Using the metric one has an identification ¯ T ∗ X ≃ T X and hence F h inducesan endomorphism of T X ⊗ T X : F h ∈ T ∗ X ⊗ ¯ T ∗ X ⊗ T ∗ X ⊗ T X ≃ T ∗ X ⊗ T ∗ X ⊗ T X ⊗ T X ≃ End( T X ⊗ T X ) . One can show, using the Bianchi identity, that the resulting endomor-phism vanishes on skew-symmetric tensors and hence induces Q : S T X → S T X , R = 2 Tr( Q ) , ( )where the function R is the scalar curvature of the metric. The operator Q is self-adjoint and hence at each x ∈ X it has real eigenvalues. Let λ x bethe smallest eigenvalue at x and suppose that − ∞ < λ := Z x ∈ X λ x < , λ x smallest eigenvalue of Q x . ( )The operator Q together with the metric h induces a Hermitian form h Q on the bundles A ,q ( T X ), q > h Q : ( ∧ q ¯ T ∗ X ⊗ T X ) ⊗ ( ∧ q ¯ T ∗ X ⊗ T X ) ≃ T X ⊗ T X ⊗ ( ∧ q ¯ T ∗ X ⊗ ∧ q ¯ T ∗ X ) Q −−→ T X ⊗ T X ⊗ ( ∧ q ¯ T ∗ X ⊗ ∧ q ¯ T ∗ X ) → C , where the last map is induced from the hermitian metric h . If h is K¨ahler-Einstein, one has [Cal-V, Sect. ]:i h x ( Λ Fω, ω ) = R n k ω k − h Q ( ω, ω ) ( )On the other hand, by [Cal-V, Lemma ] one has the inequality h Q ( ω, ω ) ≥
12 ( q + 1) λ x { ω k . ( ) n (loc. cit.) it is shown that first of all R < λ <
0, and hence,combining ( ) and ( ) thati h x ( Λ Fω, ω ) ≤ (cid:18) R n −
12 ( q + 1) λ x (cid:19) k ω k . ( )The above function is ≤ R n − ( q + 1) λ < ω = 0. Now contrast this with the version . of Nakano’s Lemma which holds under the assumptions of Sect. . Theconclusion is: Proposition . . Suppose that the assumptions . hold for a quasi projectiveK¨ahler-Einstein manifold ( X, h ) and its holomorphic tangent bundle ( T X , h ) .Suppose also that R < , where R is the scalar curvature.Then for all integers q for which q < Rnλ − , one has H ,q (2) ( X, T X ) = 0 .Remark . . The above proof has to be modified slightly for q = 0. In thatcase the term h Q ( ω, ω ) in ( ) vanishes and since R < H ( X, T X ) = 0. This implies that ¯ X admits no vector-fields tangent to ∂X . Application to locally symmetric varieties ofhermitian type
Let G be a reductive Q –algebraic group of hermitian type, i.e. for K ⊂ G ( R )maximal compact, D = G ( R ) /K is a bounded symmetric domain. Fix someneat arithmetic subgroup Γ ⊂ G ( Q ) and let X = Γ \ D be the correspondinglocally symmetric manifold. It is quasi-projective and by [A-Mu-R-T] ad-mits a smooth toroidal compactification X with boundary a normal cross-ing divisor ∂X .Let ρ : G → GL( E ) be a finite dimensional complex algebraic represen-tation with ˜ E ρ the corresponding holomorphic vector bundle on D and E ρ the bundle it defines on X . Fix also a G –equivariant hermitian metric ˜ h on ˜ E ρ (which exists since the isotropy group of the G ( R )–action on D isthe compact group K ) and write h for the induced metric on E ρ . By [Mu,Thm. . .], there is a unique extension of E ρ to an algebraic vectorbundle E ρ on X with the property that the metric h is a called good metric for thebundle E ρ relative to E ρ . or what follows it is important to observe: Lemma . . The metric (1 , -form ω h X of a K¨ahler-Einstein metric h X hasPoincar´e growth near ∂X .Proof. The K¨ahler-Einstein condition means that ω h X = − k · i ∂ ¯ ∂ log(det h X ) , for some positive real constant k . Up to some positive constant, the righthand side can be identified with the first Chern form for the canonical linebundle Ω nX with respect to the metric induced by h X . Since this metric is G ( R )-equivariant, it is good in Mumford’s sense and so ω h X is also good.Clearly, if this is to be useful in applications, given a bundle (with some G ( R )–equivariant hermitian metric), one needs to get hold of the extensionmaking the metric good. Examples . . . Let E = Ω pX . Then E = Ω pX (log ∂X ), the bundle of p -forms with at most log-poles along ∂X . This is not trivial. See [Mu, Prop. . .] where this is shown for p = 1. Since Ω pX (log ∂X ) = V p Ω X (log ∂X )this implies the result for all p . In particular, smooth sections of Ω arebounded near ∂X . Indeed, if f = 0 is a local equation for ∂X and ω asmooth section of Ω X , then f · ω is a smooth section of Ω X (log ∂X ). Then k f · ω k = k f k k ω k and since k ω k ≤ C (log k f k ) N , k f · ω k is bounded. Asimilar argument holds for smooth sections of Ω pX and hence for sectionsof A p,qX . . One has T X = T X ( − log ∂X ), the bundle of holomorphic vector fields on X which are tangent to the boundary ∂X , since this is the dual of the bun-dle Ω X (log ∂X ). Any smooth section of this bundle is bounded near theboundary: its normal component tends to zero and the Poincar´e growthof the metric implies (by compactness of ∂X ) that tangential componentremains bounded. . These two remarks show that the holomorphic tangent bundle T X sat-isfies assumption . .I can finally state the main result: heorem . . Let ( X, ∂X ) as before, e.g. X = Γ \ D , D = G ( R ) /K hermitiansymmetric, Γ a neat arithmetic subgroup of G ( Q ) and X a good toroidal com-pactification with boundary ∂D . Let R be the scalar curvature of the G ( R ) –equivariant (K¨ahler-Einstein) metric and let λ be as before (cf. ( ) ). Set γ ( D ) := R/nλ . This is a positive integer and H ,q (2) ( X, T X ) = 0 , for all q for which q < γ ( D ) − . If no irreducible factor of D has dimension , one has γ ( D ) ≥ . In particular,the resulting pairs ( X, ∂X ) are infinitesimally rigid.Proof. Since X admits a K¨ahler-Einstein metic h X , by Lemma . its fun-damental (1 , . is fulfilled. By example . . the second condition isalso fulfilled.In order to apply Prop. . , one observes that the K¨ahler manifold X ishomogeneous and that therefore λ = λ x , x ∈ X , a constant. Since the scalarcurvature of D is known to be negative, this proves the result, except that γ ( D ) is an integer ≥
2. The calculation of γ ( D ) is local and has been donein [Bo, Cal-V] and it implies that it is an integer ≥
2. Also, it is shown therethat γ ( D ) ≥ D has no irreducible factor of dimension 1. Fordetails, see [Cal-V, Sect. ] and [Bo, Sect. ]. See also Remark . below.I apply this to infinitesimal deformations of ( X, ∂X ) as follows. As iswell known, these correspond bijectively to elements of H ( X, T X ( − log ∂X )).See e.g. [Sern, Prop. . . ].Now assume that α ∈ A , X ( T X ( − log ∂X )) represents a given cohomologyclass [ α ] ∈ H ( X, T X ( − log ∂X )). By Prop. . , the class β = α | X is an L -harmonic form and it su ffi ces to show that β = 0 which follows from thevanishing of H , ( X, T X ). Remark . . For irreducible D there is a table for the values of γ ( D ) in[Cal-V] and [Bo]. I copy their result:type I p,q I I m , m ≥ I I I m , m ≥ I V m , m ≥ V V Iγ ( D ) p + q m − m + 1 m
12 18dim C D pq m ( m − m ( m + 1) m
16 27 f D = D × · · · × D N is the decomposition into irreducible factors, one has γ ( D ) = min j γ ( D j ). One sees from this that γ ( D ) ≥ D contains a factor of type I , ≃ I I ≃ I I I . One also sees thatthe best vanishing result is for the unit ball I p, where all groups vanish. Corollary . . Under the assumptions of Theorem . , the pair ( X, ∂X ) has aunique model over a number field.Proof. This follows using spreads. For details see [Pe, Sh].
Remark.
The above theorem is false for Shimura curves (one dimensionallocally homogeneous algebraic manifolds). However, the corollary is truesince all Shimura curves have models over Q . A proof which is a variantof the above method was given in [F] which motivated in fact this note. References [A-Mu-R-T]
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