aa r X i v : . [ m a t h . AG ] O c t TWISTOR DEFORMATION OF RANK ONE LOCAL SYSTEMS
MORIHIKO SAITO
Abstract.
We determine the twistor deformation of rank one local systems on compactKaehler manifolds which correspond to smooth twistor modules of rank one in the sense ofC. Sabbah. Our proof is rather elementary, and uses a natural description of the modulispace of rank one local systems together with the canonical morphism to the Picard variety.The corresponding assertion for smooth twistor modules of rank one follows from the theoryof C. Simpson and has been known to the specialists, according to T. Mochizuki.
Introduction
Let X be a compact K¨ahler manifold of dimension d . Set n := dim H ( X, O X ). There is acanonical morphism(0 . ρ : Hom (cid:0) H ( X, Z ) fr , C ∗ (cid:1) ∼ = ( C ∗ ) n → Pic ( X ) , where the first isomorphism is given by choosing free generators { γ i } of H ( X, Z ) fr := H ( X, Z ) /H ( X, Z ) tor . Note that the source of ρ is the identity component of the moduli space of rank 1 localsystems H ( X, C ∗ ) = Hom (cid:0) H ( X, Z ) , C ∗ (cid:1) , and the latter is the product of this identity component with a finite group by choosing asplitting of the torsion subgroup H ( X, Z ) tor . This finite group is identified with H ( X, Z ) tor ,and can be neglected for the study of the kernel of the canonical morphism to the modulispace of line bundles. The C -local systems of rank 1 on X are uniquely determined upto an isomorphism by the monodromies λ i ∈ C ∗ along the free generators γ i , where themonodromies along torsion elements of H ( X, Z ) are assumed trivial.The target of ρ is defined by(0 .
2) Pic ( X ) := H ( X, C ) / (cid:0) Γ( X, Ω X ) + H ( X, Z (1)) (cid:1) . This is the moduli space of topologically trivial holomorphic line bundles on X by usingthe exponential sequence as is well-known, where Z (1) := 2 π √− Z , see [De2]. (Note that H ( X, Z ) is torsion-free.) It is easy to see that(0 .
3) The morphism ρ is a surjective morphism of complex analytic Lie groups.(See Propositions (1.1-2) below.)Consider the morphism(0 . σ : Γ( X, Ω X ) ∋ θ (cid:0) exp (cid:0) − R γ θ (cid:1) , . . . , exp (cid:0) − R γ n θ (cid:1)(cid:1) ∈ ( C ∗ ) n . In this paper, we show the following.
Theorem 1.
We have (0 .
5) Ker ρ = Im σ. Actually the proof of Theorem 1 turns out almost trivial (up to the verification of thecompatibility of some canonical morphisms) although the minus sign in the definition of σ in (0.4) does not appear here, see (2.2). The first proof of Theorem 1 used the reduction to theabelian variety case via the Albanese map, and solved a certain differential equation, wherewe get the minus sign as in (0.4), see (2.1). By Theorem 1 and (0.3), we get the following. Corollary 1.
For any ξ ∈ Pic ( X ) , let λ = ( λ i ) be an element of ρ − ( ξ ) ⊂ ( C ∗ ) n . Then (0 . ρ − ( ξ ) = (Im σ ) λ. It is rather easy to show that there is a unique rank 1 unitary local system η in ρ − ( ξ )(see Proposition (1.3) below), and we may assume that λ is this element. Corollary 1 isclosely related to the theory of C. Simpson, see Example after Prop. 1.5 in [Si1], where heassumes dim X = 1. His C ∗ -action on Higgs line bundles ( L , θ ) seems to be defined by( L , tθ ) ( t ∈ C ∗ ) where θ ∈ Γ( X, Ω X ) is the Higgs 1-form. In terms of the moduli space ofrank 1 local systems, it is then given by(0 . η (cid:0) exp (cid:0) − R γ ( tθ + tθ ) (cid:1) , . . . , exp (cid:0) − R γ n ( tθ + tθ ) (cid:1)(cid:1) ( t ∈ C ∗ ) , where η ∈ ( C ∗ ) n is the rank 1 unitary local system corresponding to the line bundle L . Notethat Re( θ ) := ( θ + θ ) / θ , i.e., the image of θ by the canonical morphismΓ( X, Ω X ) → H ( X, C ) = H ( X, R ) ⊕ H ( X, R (1)) pr → H ( X, R ) , and the exponential map is equivalent to the passage to the quotient by H ( X, Z (1)).On the other hand, we have the twistor deformation of a rank 1 local system, i.e., aholomorphic family of rank 1 local systems on X parametrized by z ∈ C ∗ , which correspondsto a smooth twistor module of rank 1 in the sense of C. Sabbah [Sa] (where the local systemat z = 1 is the given local system), see also [Si2], [Si3]. The following has been informedfrom T. Mochizuki [Mo]: Theorem 2.
The smooth twistor module of rank associated with a Higgs line bundle ( L , θ ) can be expressed by using the holomorphic family of connections ∇ h + z − θ + z θ ( z ∈ C ∗ ) , where ∇ h is the unitary connection on the underlying C ∞ line bundle of L associated with apluri-harmonic metric h . This follows from the theory of Simpson ([Si2], Thm. 4.3) and has been known to thespecialists, according to Mochizuki. By translating Theorem 2 to an assertion on localsystems (see Remark (1.5) below), we get the following.
Theorem 3.
The twistor deformation, i.e. the family of rank local systems associatedwith the twistor module corresponding to a Higgs line bundle ( L , θ ) is written as (0 . η (cid:0) exp (cid:0) − R γ ( z − θ + z θ ) (cid:1) , . . . , exp (cid:0) − R γ n ( z − θ + z θ ) (cid:1)(cid:1) ( z ∈ C ∗ ) . where η is as in (0 . . We give a proof of Theorem 3 using the moduli space of rank 1 local systems togetherwith the morphism to the Picard variety, see (2.3) below. This is rather elementary, andseems to be relatively easy to follow even for non-experts of harmonic bundles.By Theorem 1, considering the images of (0.7) and (0.8) in the moduli space of line bundle(i.e. in the Picard variety) is equivalent to forgetting the holomorphic terms tθ and z − θ .Hence these images coincide up to the complex conjugation of the parameters (i.e. by setting z = t ). Note that z − θ + z θ in (0.8) cannot be replaced with z θ + z − θ as someone mightthink, since z − θ does not converge at z = 0 if θ = 0 (by considering its image in the Picardvariety by using Theorem 1, see the proof of Theorem 3 in (2.3)). WISTOR DEFORMATION 3
We would like to thank those who are interested in this manuscript, especially ProfessorsT. Mochizuki and C. Sabbah who informed us of the relation with the theory of Simpson([Si1], [Si2], [Si3]) and also with that of twistor modules.In Section 1 we review some well-known facts from the theory of rank one local systems.In Section 2 we prove the main theorems.
1. Preliminaries
In this section we review some well-known facts from the theory of rank one local systems.
The morphism ρ in (0 . is a morphism of complex analytic Lie groups.Proof. For a contractible Stein manifold S , the exponential sequence induces a long exactsequence(1 . . H ( X, Z (1)) → H ( S, R ( pr ) ∗ O X × S ) → H ( X × S, O ∗ X × S ) → H ( X, Z (1)) , by applying the Leray spectral sequence to pr : X × S → S . So a topologically trivialholomorphic line bundle on X × S gives a section of a locally free sheaf R ( pr ) ∗ O X × S . This implies that ρ is complex analytic since the universal family of rank 1 local systems canbe constructed by dividing the trivial line bundle on the product of the universal covering of X with ( C ∗ ) n , and this gives a holomorphic family of line bundles. The compatibility withthe multiplicative structures is trivial since these are defined by using the tensor product. The morphism ρ is surjective.Proof. We have the commutative diagram(1 . .
1) 0 −−−→ H ( X, Z (1)) −−−→ H ( X, C ) −−−→ H ( X, C ∗ ) −−−→ H ( X, Z (1)) (cid:13)(cid:13)(cid:13) y α y β (cid:13)(cid:13)(cid:13) −−−→ H ( X, Z (1)) −−−→ H ( X, O X ) −−−→ H ( X, O ∗ X ) −−−→ H ( X, Z (1))where α is surjective by Hodge theory. Moreover, by the universal coefficient theorem wehave a canonical isomorphism(1 . .
2) Hom( H ( X, Z ) , C ∗ ) = H ( X, C ∗ ) , (since C ∗ is an injective Z -module), and ρ can be identified with a restriction of β . So theassertion follows.The above argument can be modified to get the following (which implies that there is aunique unitary local system of rank 1 in each fiber of ρ ). Set C ∗ := { λ ∈ C ∗ | | λ | = 1 } . We have a canonical isomorphism H ( X, C ∗ ) ∼ −→ H ( X, O ∗ X ) . Proof.
This follows by replacing the first exact sequence in (1.2.1) with the long exactsequence associated with the exponential sequence(1 . .
1) 0 → Z (1) → R (1) → C ∗ → , since Hodge theory implies the canonical isomorphism(1 . . H ( X, R (1)) ∼ −→ H ( X, O X ) . M. SAITO
For a compact K¨ahler manifold X of dimension d , its Albanesevariety Alb( X ) is defined by(1 . .
1) Alb( X ) := H d − ( X, C ) / (cid:0) F d H d − ( X, C ) + H d − ( X, Z ( d )) fr (cid:1) , where H d − ( X, Z ) fr := H d − ( X, Z ) / torsion. Choosing a point x of X , the Albanese map α X : X → Alb( X ) can be defined by using the integrals(1 . . H ( X, Ω X ) ∋ ω Z xx ω ∈ C mod h H ( X, Z ) , ω i , where H d − ( X, C ) /F d H d − ( X, C ) is identified with the dual of H ( X, Ω X ) and we haveby Poincar´e duality H d − ( X, Z ( d )) = H ( X, Z ) . Setting Y := Alb( X ), this construction implies an isomorphism(1 . .
3) ( α X ) ∗ : H ( X, Z ) fr ∼ −→ H ( Y, Z ) , and hence(1 . . α ∗ X : H ( Y, Z ) ∼ −→ H ( X, Z ) ,α ∗ X : Γ( Y, Ω Y ) ∼ −→ Γ( X, Ω X ) . The translation between Theorems 2 and 3 consists of the calculation of theglobal monodromies of the C ∞ connection ∇ h + z − θ + z θ. By using the tensor product of connections, this can be reduced to the case L = O X and ∇ h = d . Then the monodromies are calculated by using the pull-back of the connection bysmooth paths representing generators of H ( X, Z ) fr .
2. Proof of the main theorems
In this section we prove the main theorems.
By (1.4), the assertion is reduced to the case where X is a complex torus of dimension n . We have the universal covering π : e X → X , and e X has a natural structure of a C -vector space. Choosing a basis we have e X = C n . Wealso choose generators γ i ( i ∈ [1 , n ]) of H ( X, Z ) which correspond to generators ω i ofKer π ∼ = Z n ⊂ C n .Then the problem is about an integrable holomorphic connections on a trivial line bundleon X , see [De1]. Let v be a nonzero global section of a trivial line bundle over X . This isunique up to a nonzero constant multiple. Let ξ j ( j ∈ [1 , n ]) be the invariant vector field on X with π ∗ ξ j = ∂/∂x j where the x j are the natural coordinates of C n . Then a holomorphicconnection ∇ is uniquely determined by c j ∈ Γ( X, O X ) = C satisfying(2 . . ∇ ξ j v = c j v ( j ∈ [1 , n ]) , where the integrability trivially holds since c j ∈ C . This c j is independent of the choice of v . The connection ∇ can be written as(2 . . ∇ = d + θ ∧ , where θ is the Higgs one-form with π ∗ θ = P i c i dx i in this case, see [Si1] for the general caseof Higgs fields. WISTOR DEFORMATION 5
The monodromy of the corresponding local system is given by solutions of the differentialequation on X (2 . . ξ j f = − c j f ( j ∈ [1 , n ]) , which comes from ∇ ξ j f v = ( ξ j f ) v + c j f v = 0 . The pull-back of the differential equation to e X = C n is given by(2 . . ∂f /∂x j = − c j f. This has nontrivial solutions on e X = C n of the form:(2 . . f ( x ) = ae −h c,x i with h c, x i := P nj =1 c j x j and a ∈ C ∗ . Setting a = 1 so that f (0) = 1, the corresponding point of ( C ∗ ) n is then given by(2 . .
6) ( e −h c, ω i i ) ∈ ( C ∗ ) n , where the ω i are generators of Ker π ∼ = Z n ⊂ C n corresponding to γ i ∈ H ( X, Z ). We thusget a family of rank 1 local systems L c on X whose associated line bundles L c := O X ⊗ C L c are trivial. This is parametrized by c = ( c , . . . , c n ) ∈ C n . So Theorem 1 follows from(2.1.6). (This argument can be extended to the case of any topologically trivial line bundlesby replacing the differential d with the connection ∇ associated to a unitary local system ofrank 1.) The morphisms σ and ρ can be identified respectivelywith the compositions of canonical morphisms(2 . .
1) Γ( X, Ω X ) → H ( X, C ) → H ( X, C ) /H ( X, Z (1)) , (2 . . H ( X, C ) /H ( X, Z (1)) → H ( X, C ) / (cid:0) Γ( X, Ω X ) + H ( X, Z (1)) (cid:1) . So the assertion follows.
Associated with a smooth twistor module of rank 1 on X ,we have a twistor deformation, i.e., a holomorphic family of rank 1 local systems on X parametrized by C ∗ . (Note that a smooth twistor module means that it corresponds to alocal system.) Set V := H ( X, C ) , Γ := H ( X, Z (1)) . Any twistor deformation is expressed by using a multi-valued holomorphic function on C ∗ with values in V :(2 . .
1) Ψ( z ) = (2 πi ) − (log z ) η + X k g k ( z ) θ k + X k h k ( z ) θ k ( z ∈ C ∗ ) , where η ∈ Γ, { θ k } is a basis of Γ( X, Ω X ), and g k ( z ), h k ( z ) are holomorphic functions on C ∗ . In fact, taking the exponential map is equivalent to the passage to the quotient by Γ,see the remark after (0.7). Hence a twistor deformation is expressed by some multi-valuedholomorphic function Ψ( z ) on C ∗ with values in V , and Ψ( z ) − (2 πi ) − (log z ) η is univaluedfor some η ∈ Γ by considering its monodromy, since Ψ( z ) is well-defined modulo Γ. So(2.3.1) follows.By the definition of twistor modules (see [Sa]) the underlying holomorphic family of linebundles is extended over C . Hence η = 0 and the h k ( z ) are holomorphic at z = 0 byconsidering the image by the morphism to the Picard variety where the g k ( z ) θ k are neglected.(Indeed, Ψ( z ) mod Γ( X, Ω X ) + H ( X, Z (1)) is holomorphic at z = 0 if and only if Ψ( z ) modΓ( X, Ω X ) is. Moreover, the latter condition is equivalent to that η = 0 and the h k are M. SAITO holomorphic at z = 0.) As for the g k , we see that the zg k ( z ) are holomorphic at z = 0 by thedefinition of twistor modules in loc. cit. (considering the corresponding holomorphic familyof connections). We thus get η = 0 , g k = X i ≥− g k,i z i , h k = X i ≥ h k,i z i . From the polarizability of twistor module which is a kind of Hermitian self-duality, we candeduce(2 . . X k g k ( z ) θ k + X k h k ( z ) θ k = − X k g k ( − /z ) θ k − X k h k ( − /z ) θ k mod Γ . Here the right-hand side is obtained by calculating the Hermitian dual (which produces thecomplex conjugation on the values together with the minus sign) and applying the “complexconjugation” defined by z
7→ − /z , see loc. cit.Set ζ := X k g k, θ k , ζ ′ := X k h k, θ k , ξ := X k g k, − θ k , ξ ′ := X k h k, θ k . Then (2.3.2) implies g k, − = h k, , g k,i = 0 ( i ≥ , h k,i = 0 ( i ≥ , and ζ ′ = − ζ mod Γ , ξ ′ = ξ, since mod Γ affects only the constant term. We thus get(2 . . X k g k ( z ) θ k + X k h k ( z ) θ k = ζ − ζ + z − ξ + z ξ mod Γ . Here ζ − ζ (mod Γ) is identified with a rank 1 unitary local system η . This corresponds to theline bundle L by considering the limit for z → ξ coincides with the Higgs 1-form θ of the Higgsline bundle ( L , θ ) by comparing (0.7) and (0.8) (with θ replaced by ξ ) at z = 1. So theassertion follows from (2.3.3). For a local system L c in (2.1) (which is identified witha local system on the given K¨ahler manifold X by (1.4.3)), we have the Hodge spectralsequence(2 . . E p,q = H q ( X, Ω pX ⊗ O X L c ) ⇒ H p + q ( X, L c ) , where L c := O X ⊗ C L c . This does not degenerate at E , for instance, if c is nonzero andbelongs to the kernel of ρ in the notation of Theorem 1, since H j ( X, L c ) = 0 for any j . Indeed, H j ( X, L c ) is calculated by using the Koszul complex associated with the multiplication by λ i − i ∈ [1 , n ]) on C where λ i = e −h c, ω i i is the monodromy along the path γ i . (For theproof of the E -non-degeneration, it is actually enough to compare H ( X, L c ) and H ( X, L c )in this case.) The E -differential d is given by the Higgs one-form θ , and we have π ∗ θ = P i c i dx i . Hence the E -term vanishes in this case, and in particular, the spectral sequence degeneratesat E . WISTOR DEFORMATION 7
It seems that this E -degeneration of (2.4.1) holds for any rank 1 local systems L onsmooth projective varieties (see [Ar1], [Ar2]) by comparing the spectral sequence (2.4.1)(with L c , L c replaced by L and L := O X ⊗ C L ) with the following spectral sequence:(2 . . E p,q = H q ( X, Ω pX ⊗ O X L ) ⇒ H p + q ( X, (Ω • X ⊗ O X L , θ ∧ )) . These two spectral sequences are induced by the truncations σ ≥ p on Ω • X (see [De2]), and areassociated with the complexes(2 . .
3) (Ω • X ⊗ O X L , ∇ + θ ∧ ) , (Ω • X ⊗ O X L , θ ∧ ) , which have the same components but different differentials ∇ ′ := ∇ + θ ∧ and θ ∧ , where ∇ is the unitary connection on L := L ⊗O X . Recall that the first complex is the deRham complex associated with ( L , ∇ ′ ), where ∇ ′ = ∇ + θ ∧ is a non-unitary connection on aline bundle L (assuming θ = 0). This complex is quasi-isomorphic to a rank 1 local systemwhich will be denoted by L θ .The two spectral sequences have the same E -terms since the morphism induced by ∇ in the E -differential d vanishes by applying Hodge theory to the unitary connection ∇ .Moreover, the second spectral sequence for the differential θ ∧ degenerates at E , see Prop. 3.7and Remark 1 after its proof in [GL]. So the problem is whether they have the same totaldimensions of the E ∞ -terms. (Here Lemma 2.2 in [Si1] does not immediately imply this bythe same reason as in Remark (2.5)(i) below.) Note that the E -degeneration implies that H j ( X, L αθ ) is independent of α ∈ C ∗ since this holds for the complex defined by αθ ∧ . (i) The connection ∇ = d + θ ∧ in (2.1.2) associated with a Higgs one-form θ is a holomorphic connection on a trivial holomorphic line bundle, and the C ∞ connection on the trivial C ∞ line bundle which is associated to θ in [Si1] does not necessarily have thesame solution local system as that of the above holomorphic connection ∇ . This is informedfrom Professor T. Mochizuki, and we would like to thank him. As is seen from Example afterProp. 1.5 in loc. cit., the C ∞ connection on the trivial line bundle which is associated to aHiggs one-form θ in loc. cit. seems to be d + ( θ + ¯ θ ) ∧ , where the latter d is a C ∞ connection ,and is the sum of ∂ and ¯ ∂ .(ii) By Proposition (1.3) there is a unique unitary local system of rank 1 in each fiber of ρ .This can be shown also by using the fact that the local systems L c in (2.1) are never unitarylocal systems, i.e. they do not belong to ( C ∗ ) n . (Indeed, e −h c, ω i i belongs to C ∗ if and onlyif the real part of h c, ω i i vanishes.) This uniqueness implies that we have a section of ρ , butthis is never complex analytic.(iii) An argument similar to the proof of Theorem 3 in (2.3) seems to apply to the case X = C ∗ , where the local systems of rank 1 on X are parametrized by C ∗ via the monodromyeigenvalue. The twistor deformation in this case seems to be expressed as z m exp( iα + zβ + z − β ) ( z ∈ C ∗ ) for m ∈ Z , α ∈ R , β ∈ C . Here it is unclear whether m = 0 and β is a real number unless we assume that the twistormodule is extendable over a partial compactification of C ∗ so that the nearby cycle functorat this point can be considered. (Indeed, the condition on the roots of the b -function oftwistor deformation implies that m = 0 and β is a real number.)(iv) In twistor theory, it seems that the twistor module M X corresponding to the constantsheaf on a smooth complex manifold X has always weight 0 independently of the dimension of X , see [Sa]. However, this does not seem to be compatible, for instance, with the calculation M. SAITO of the nearby cycles in the normal crossing case in loc. cit., Lemma 3.7.9. In fact, it seemsto assert that the primitive part of the ℓ -th graded piece of the monodromy filtration of thenearby cycles of M X for the function f = x x is given by P Gr Wℓ ψ f M X = (L a =1 , ( i Y a ) + M Y a if ℓ = 0 , ( i Z ) + M Z if ℓ = 1 , where Y a := { x a = 0 } , Z = Y ∩ Y . (Here the monodromy is unipotent, and ψ f, − is denotedby ψ f to simplify the notation.) Indeed, we get the above formula from the one in loc. cit.,if we set M Y a = M X /x a M X , M Z = M X / ( x , x ) M X , see Lemma 3.7.8 in loc. cit., where x x M seems to mean rather ( x , x ) M . However, itis quite unclear why we get naturally the shift of weight for ℓ = 1. Notice that the weightof a twistor module can be changed arbitrarily since the Tate twist ( i/
2) for i ∈ Z exists intwistor theory. However, the Tate twist ( − /
2) should be noted explicitly after ( i Z ) + M Z inthe above formula for ℓ = 1, and it should be clarified where this Tate twist comes from.There is also a problem about the shift of the monodromy filtration on the nearby andvanishing cycle functors. In the minimal extension case, the vanishing cycle functor ϕ f isidentified with the image of N on the nearby cycle functor ψ f (restricted to the unipotentmonodromy part). Then the monodromy filtration on the nearby cycles must be shifted by 1in order that the morphism can : ψ f → ϕ f induce isomorphisms between the primitive partsof the graded pieces of the monodromy filtration (except for the lowest degree). In fact, thisis clearly impossible without shifting the center of the symmetry for one of them since thecanonical morphism “can” must preserve the weights . But the filtration on the vanishingcycles cannot be shifted by considering the case where the module is supported inside thedivisor. So we have to shift the monodromy filtration on the nearby cycle functor by 1.The above problems can be solved rather naturally if one thinks, for instance, that theweights decrease by the codimension under the direct images by closed immersions. In theabove case it is natural to put the Tate twist ( −
1) after M Z , since we get M Z for theco-primitive part naturally. This is closely related with the inclusion of twistor modules M Y ֒ → ψ f M X , which appears in the exact sequence0 → M Y → ψ f M X → ϕ f M → , or 0 → M Y → ψ f M X N → ψ f M ( − , where M Y is the twistor module on X corresponding to the constant sheaf on Y := Y ∪ Y .(It may coincide with i + i ∗ M X up to a shift of complex if i ∗ can be defined appropriately,where i : Y ֒ → X is the inclusion.)It is expected that the weight filtration W on M Y satisfies:Gr Wk M Y = (L a =1 , ( i Y a ) + M Y a if k = − , ( i Z ) + M Z if k = − . In fact, this should be closely related withGr Wk j ! M U = M X if k = 0 , L a =1 , ( i Y a ) + M Y a if k = − , ( i Z ) + M Z if k = − , WISTOR DEFORMATION 9 where U := X \ Y with the inclusion j : U ֒ → X . Indeed, there should be a short exactsequence of twistor modules on X → M Y → j ! M U → M X → . Note, however, that there is no nontrivial morphism of twistor modules on X : M X → M Y , and we have only an element in the extension class, i.e. M X → M Y [1] . This comes from the difference in t -structure for constructible sheaves and D -modules. References [Ar1] Arapura, D., Higgs line bundles, Green-Lazarsfeld sets and maps of K¨ahler manifolds to curves, Bull.AMS 26 (1992), 310–314.[Ar2] Arapura, D., Geometry of cohomology support loci for local systems, I, J. Alg. Geom. 6 (1997),563–597.[De1] Deligne, P., Equations diff´erentielles `a points singuliers r´eguliers, Lect. Notes in Math. 163, Springer,Berlin, 1970.[De2] Deligne, P., Th´eorie de Hodge II, Publ. Math. IHES, 40 (1971), 5–58.[GL] Green, M., Lazarsfeld, R., Deformation theory, generic vanishing theorems and some conjectures ofEnriques, Catanese and Beauville, Inv. Math. 90 (1987), 389–407.[Mo] Mochizuki, T., letters (in Japanese).[Sa] Sabbah, C., Polarizable twistor D -modules, math.0503038v2.[Si1] Simpson, C., Higgs bundles and local systems, Publ. IHES 75 (1992), 5–95.[Si2] Simpson, C., The Hodge filtration on nonabelian cohomology, Algebraic geometry–Santa Cruz 1995,Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, 217–281.[Si3] Simpson, C., Mixed twistor structures, arXiv:alg-geom/9705006.-modules, math.0503038v2.[Si1] Simpson, C., Higgs bundles and local systems, Publ. IHES 75 (1992), 5–95.[Si2] Simpson, C., The Hodge filtration on nonabelian cohomology, Algebraic geometry–Santa Cruz 1995,Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, 217–281.[Si3] Simpson, C., Mixed twistor structures, arXiv:alg-geom/9705006.