CHAPTER 4

VARIATIONS OF ON ACOMPLEXMANIFOLD

Summary. The notion of a variation of Hodge structure on a complex mani- fold is the first possible generalization of a Hodge structure. It naturally occurs when considering holomorphic families of smooth projective varieties. Later, we will identify this notion with the notion of a smooth Hodge module.Weconsider global properties of polarizable variations of Hodge structure on a smooth pro- jective variety. On the one hand, the Hodge theorem asserts that the de Rham of a polarizable variation of Hodge structure on a smooth is itself a polarizable sl2-Hodge structure. On the other hand, we show that the local system underlying a polarizable variation of Hodge structure on a smooth projective variety is semi-simple, and we classify all such variations with agivenunderlyingsemi-simplelocalsystem.

4.1. Variations of Hodge structure

4.1.a. Variations of C-Hodge structure. The definition of a variation of C-Hodge structure is modeled on the behaviour of the cohomology of a family of smooth projec- tive varieties parametrized by a smooth algebraic variety, that is, a smooth projective morphism f : Y X,thatwecallbelowthe“geometricsetting”. ! Let us first motivate the definition. Let X be a connected (possibly non compact) . In such a setting, the generalization of a Ho is a locally constant of vector spaces H on X.LetuschooseauniversalcoveringX X of ! X and let us denote by ⇧ its group of deck-transformations, which is isomorphic to ⇡ (X,?) for any choice of a base-point ? X.LetusdenotebyH the spacee of global 1 2 sections of the pullback H of H to X. Then, giving H is equivalent to giving the monodromy representation ⇧ GL(H).However,itisknownthat,inthegeometrice ! setting, the Hodge decompositione ine each fiber of the family does not give rise to locally constant sheaves, but to C1-bundles.e k In the geometric setting, to the locally constant sheaf R f CX (k N)isassocia- ⇤ 2 ted the Gauss-Manin connection, which is a holomorphic on Y endo- wed with a holomorphic flat connection. In such a case, the Hodge filtration can be 72 CHAPTER 4. VHS ON A COMPLEX MANIFOLD naturally defined and it is known to produce holomorphic bundles. Therefore, in the general setting of a variation of C-Hodge structure that we intend to define, a better o analogue of the complex vector space H is a holomorphic vector bundle H0 equipped 1 with a flat holomorphic connection : H0 ⌦ O H0,sothatthelocallyconstant r ! X ⌦ X sheaf H0 =Ker ,thatwealsodenotebyH0r,isthedesiredlocalsystem.Notethat r we can recover (H0, ) from H0 since the natural morphism of flat bundles r (O H0, d Id) (H0, ) X ⌦C ⌦ ! r is an isomorphism. A filtration is then a finite (exhaustive) decreasing filtration by p sub-bundles F •H0 (recall that a sub-bundle F H0 of H0 is a locally free OX -submodule p of H0 such that H0/F H0 is also a locally free OX -module ; F •H0 is a filtration by p p+1 sub-bundles if each F H0/F H0 is a locally free OX -module). The main property, known as Griffiths transversality property is that the filtration should satisfy p 1 p 1 (4.1.1) (F H0) ⌦ OX F H0 p Z. r ⇢ X ⌦ 8 2 However, the analogue of a bi-filtered vector space is not a bi-filtered holomorphic flat bundle, since one knows in the geometric setting that one of the filtrations should behave holomorphically, while the other one should behave anti-holomorphically. This leads to a presentation by C1-bundles. Let H = C1 O H0 be the associated C1 bundle and let D be the connection X ⌦ X on H defined, for any C1 function ' and any local holomorphic section v of H0,by D(' v)=d' v + ' v (this is a flat connection which decomposes with respect ⌦ ⌦ ⌦r to types as D = D0 + D00 and D00 =d00 Id). Then D00 is a holomorphic structure ⌦ on H,i.e.,Ker D00 is a holomorphic bundle with connection induced by D0 :thisis p r p (H0, ) by construction. Each bundle F H0 gives rise similarly to a C1-bundle F 0 H r p 0,1 p 2 which is holomorphic in the sense that D00F 0 H EX F 0 H (and thus (D00) =0 p ⇢ ⌦ on F 0 H). On the other hand, D0 defines an anti-holomorphic structure on H (see below), and Ker D0 is an anti-holomorphic bundle with a flat anti-holomorphic connection r induced by D00. If we wish to work with holomorphic bundle, we can thus consider the (1) conjugate bundle H00 = Ker D0, that we equip with the holomorphic flat connection

= D00Ker D .AfiltrationofH by anti-holomorphic sub-bundles is by definition a r | 0 filtration F 00•H by C1-sub-bundles on which D0 =0. It corresponds to a filtration of H00 by holomorphic sub-bundles F •H00. Conversely, given a flat C1 bundle (H,D),wedecomposetheflatconnectioninto its (1, 0) part D0 and its (0, 1) part D00.Byconsideringtypes,onechecksthatflatness is equivalent to the three properties 2 2 (D0) =0, (D00) =0,D0D00 + D00D0 =0.

(1) The precise definition is as follows. Let OX denote the sheaf of anti-holomorphic functions on X and regard OX as an OX -module : the action of an anti- g on a holomorphic function f is by definition g f := gf.ThenanyO -module E determines an O -module E by · X 00 X 00 setting E00 := OX E00. ⌦OX 4.1. VARIATIONS OF HODGE STRUCTURE 73

2 Since, by flatness, (D00) =0,theKoszul-Malgrangetheorem[KM58]impliesthat Ker D00 is a holomorphic bundle H0, that we can equip with the restriction to r Ker D00 of the connection D0.FlatnessofD also implies that is a flat holomorphic r connection on H0. The conjugate C1 bundle H is equipped with the conjugate connection D, which is also flat. Conjugation exchanges of course the (1, 0)-part and the (0, 1)-part, that is, D0 = D00 and D00 = D0. The corresponding holomorphic sub-bundle is H00 := (H)0 = Ker D00.WecanalsoexpressitasH00 = Ker D0, and it is equipped with the flat holomorphic connection induced by D0 = D00. p p Similarly, we set F 0 H = F 00 H,etc. 4.1.2. Definition (Flat sesquilinear pairings).

(1) A sesquilinear pairing s on a C1 bundle H is a pairing on H with values in the sheaf CX1, which satisfies, for local sections u, v of H and a C1 function g the relation s(gu, v)=s(u, gv)=gs(u, v).WeregarditasaCX1-linear morphism

s : H C H C1. ⌦ X1 ! X (2) A flat sesquilinear pairing s on a flat C1 bundle (H,D) is a sesquilinear pairing which satisfies ds(u, v)=s(Du, v)+s(u, Dv); equivalently, decomposing into types,

d0s(u, v)=s(D0u, v)+s(u, D00v),

(d00s(u, v)=s(D00u, v)+s(u, D0v). 4.1.3. Lemma. Giving a flat sesquilinear pairing s on (H,D) is equivalent to giving an O O -linear morphism s : H0 H C1, that is, which satisfies X ⌦C X ⌦C 0 ! X s(gu, v)=gs(u, v), g OX ,u,v H0, (s(u, gv)=gs(u, v), 2 2 and d0s(u, v)=s( u, v), r (d00s(u, v)=s(u, v). r Proof. Immediate from the definitions.

4.1.4. Definition (Variation of C-Hodge structure, first definition) AvariationofC-Hodge structure H of weight w consists of the data of a flat C1 bundle (H,D), equipped with a filtration F 0•H by holomorphic sub-bundles satisfying Griffiths transversality (4.1.1), and with a filtration F 00•H by anti-holomorphic sub- bundles satisfying anti-Griffiths transversality, such that the restriction of these data at each point x X is a C-Hodge structure of weight w (Definition 2.5.2). 2 A morphism ' : H H is a flat morphism of C1-bundles compatible with both 1 ! 2 the holomorphic and the anti-holomorphic filtrations. 74 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

A polarization S is a morphism H H C1( w) of flat filtered bundles, where ⌦ ! X C1( w),isendowedwiththenaturalconnectiond and w-shifted trivial filtra- X tions, whose restriction to each x X is a polarization of the Hodge structure H 2 x (see Definition 2.5.10). We usually denote by S : H H C1 the underlying flat ⌦ ! morphism and by S : H H C its restriction to the local system H. ⌦ ! 4.1.5. Definition (Variation of C-Hodge structure, second definition) AvariationofC-Hodge structure H of weight w consists of the data of a flat C1 bundle (H,D), equipped with a Hodge decomposition by C1-sub-bundles p,w p H = H p L satisfying Griffiths transversality : p,q 1 p,q p 1,q+1 D0H ⌦X (H H ), (4.1.5 ) ⇢ ⌦ p,q 1 p,q p+1,q 1 ⇤ D00H ⌦ (H H ). ⇢ X ⌦ A morphism H H is a D-flat morphism (H ,D) (H ,D) which is compa- 1 ! 2 1 ! 2 tible with the Hodge decomposition. A polarization is a C1 Hermitian metric h on the C1-bundle H such that

• the Hodge decomposition is orthogonal with respect to h,

• The polarization form S,definedbythepropertythat(seeDefinition2.5.15)

• the decomposition is S-orthogonal and q p,w p p,w p • h H := ( 1) S H , | | induces a D-flat O O -linear pairing S : H H C1. X ⌦C X ⌦C ! X 4.1.6. Lemma. The two definitions are equivalent.

Proof. Given a C1 flat bundle (H,D) endowed with a C1 Hodge decomposition satis- p p0,w p0 p fying (4.1.5 ), we define F 0 H = H ,and(4.1.5 )impliesD00(F 0 H) ⇤ p0>p ⇤ ⇢ E1,0 F p 1H D F pH grp H X 0 ,sothat 00 inducesL a holomorphic structure on 0 and F p,q⌦p ' H is a C1 bundle. We argue similarly to obtain the properties of F 00•H.By construction, the restriction of the filtrations to any point of X is give rise to the p,w p Hodge decomposition Hx . Conversely, assume that we are given (H,D,F0•H,F00•H) as in Definition 4.1.4. L p,w p p w p The dimension of each fiber of H := F 0 H F 00 H is constant, since it is equal p \ to that of the bundle grF H.Wenowusethat,ifallfibersoftheintersectionoftwo sub-bundles of a vector bundle have the same dimension, then this intersection is also p,w p a sub-bundle. This implies that H is a sub-bundle of H,andthedecomposition follows from the Hodge property in each fiber. In order to obtain (4.1.5 ), we notice q ⇤ that F 00 H,beinganti-holomorphic,ispreservedbyD0,so p q 1 p 1 q D0(F 0 H F 00 H) ⌦ (F 0 H F 00 H). \ ⇢ X ⌦ \ We argue similarly with D00. 4.1. VARIATIONS OF HODGE STRUCTURE 75

4.1.7. Remark. While it is easy, by using a partition of unity, to construct a Hermitian metric compatible with the Hodge decomposition, the condition of flatness of S is a true constraint if dim X > 1.Forexample,anyflatC1-bundle (H,D) can be regarded as a variation of C-Hodge structure of type (0, 0),anditadmitsmanyHermitian metrics, but the polarization condition imposes that the Hermitian metric is flat, which only occurs when the monodromy representation of the flat bundle is (conjugate to) a unitary representation.

4.1.8. Remark (Polarized variation of Hodge structure as a flat filtered Hermitian pair) In analogy with Section 2.5.17,wecandescribeapolarizedvariationofHodge structure by using only one filtration. By a flat filtered Hermitian pair we mean the data ((H0, ),F•H0, S), where r (i) (H0, ) is a flat holomorphic vector bundle and F •H0 is a filtration by holo- r morphic sub-bundles satisfying Griffiths transversality, (ii) S : H0 H0 C1 is a -flat Hermitian pairing as defined in Lemma 4.1.3. ⌦C ! X r This can also be described as C1 data ((H,D),F•H, S) as in Definition 4.1.2,the equivalence being given by Lemma 4.1.3. AflatfilteredHermitianpair((H0, ),F•H0, S) is a polarized variation of Hodge r structure of weight w if its restriction to every x X is a polarized Hodge structure 2 in the sense of 2.5.17(1)–(3).

4.1.9. Definition (The categories VHS(X, C,w) and pVHS(X, C,w)) Definitions 4.1.4 and 4.1.5 produce the category VHS(X, C,w) of variations of C-Hodge structures of weight w on X. The category pVHS(X, C,w) of polarizable variations of C-Hodge structures of weight w is the full subcategory of VHS(X, C,w) whose objects admit a polarization.

We refer to Exercises 4.1 and 4.2 for the following result.

4.1.10. Proposition. (1) The category VHS(X, C,w) is abelian and each morphism is strictly compatible with the Hodge filtration. It is endowed with the operations tensor product, Hom, dual, conjugation and Hermitian dual. (2) The full subcategory pVHS(X, C,w) is abelian and stable by the previous ope- rations. It is stable by direct summand in VHS(X, C,w) and is semi-simple.

4.1.b. Variations of Q-Hodge structure. We can now mimic the definition of VHS(X, ,w) (H ,H,iso) Section 2.5.c.Anobjectof Q is a tuple Q , where

• HQ is a Q-local system on X,

• H is an object of VHS(X, C,w), ⇠ • iso is an isomorphism C H H, ⌦Q Q ! with the condition that at each x X,thesedatarestricttoaQ-Hodge structure. 2 Morphisms are the obvious ones which are compatible with the data. The definition 76 CHAPTER 4. VHS ON A COMPLEX MANIFOLD of pVHS(X, Q,w) is similar, by imposing that the polarization form S comes, after tensoring with C,fromabilinearform w S : H H (2⇡i) Q. Q Q ⌦Q Q ! 4.1.11. Proposition. (1) The category VHS(X, Q,w) is abelian and each morphism is strictly compatible with the Hodge filtration. It is endowed with the operations tensor product, Hom, and dual. (2) The full subcategory pVHS(X, Q,w) is abelian and stable by the previous ope- rations. It is stable by direct summand in VHS(X, Q,w) and is semi-simple.

4.1.12. Remark. One can define the category VHS(X, R,w) of variations of R-Hodge structure without referring to the Riemann-Hilbert correspondence, i.e., without using the local system H,byusinginsteadacomplexinvolution :(H,D) ⇠ (H, D). !

4.2. The Hodge theorem 4.2.a. The Hodge theorem for unitary representations. We will extend the Hodge theorem (Theorem 2.4.4 and the results indicated after its statement concer- ning the polarization) to the case of the cohomology with coefficients in a unitary representation. Let us start with a holomorphic vector bundle H0 of rank d on a complex projec- tive manifold X equipped with a flat holomorphic connection . The local system r H = H0r corresponds to a representation ⇡1(X,?) GLd(C),uptoconjugation. ! The unitarity assumption means that we can conjugate the given representation in such a way that it takes values in the unitary group. In other words, there exists a Hermitian metric h on the associated C1-bundle H = C1 O H0 such that, if we denote as above by D the connection on H defined ⌦ X by D(' v)=d' v + ' v,theconnectionD is compatible with the metric h ⌦ ⌦ ⌦r (i.e., is the Chern connection of the metric h). That D is a connection compatible with the metric implies that its formal adjoint (with respect to the metric) is obtained with a suitably defined Hodge ? operator ? by the formula D = ?D?. This leads to the decomposition of the space of C1 k-forms on X with coefficients in H (resp. (p, q)-forms) as the orthogonal sum of the of the with respect to D (resp. D0 or D00), that is, the space of harmonic sections, and its image. As the connection D is flat,thereisaC1 de Rham complex (E• H,D),and X ⌦ standard arguments give k k k H (X, H)=H (X, DR(H0, )) = H (X, (E• H,D)) . r X ⌦ p,q One can also define the groups by decomposingE• into E ’s p,q q p and by decomposing D as D0 + D00. Then H (X, H)=H (X, ⌦ H0). D00 X ⌦ 4.2. THE HODGE THEOREM 77

As the projective manifold X is Kähler,weobtaintheKähleridentitiesforthe various Laplace operators : D = 2D0 = 2D00 . Then, exactly as in Theorem 2.4.4,weget: 4.2.1. Theorem. Under these conditions, one has a canonical decomposition k p,q H (X, DR(H0, )) = H (X, H) D00 r p+q=k q,p p,q L (2) H (X, H) H (X, H_) H_ and D00 is identified with D00 , where is the dual bundle. The Hard Lefschetz theorem also holds in this context.

4.2.b. Harmonic bundles. If we do not assume anymore that H is unitary, but only assume that it underlies a polarized variation of Hodge structure of some weight w (so that the unitary case is the particular case of a variation of pure type (0, 0)), we have a flat connection D on the C1-bundle H associated to H0, with D = D0 +d00, and we also have a Hermitian metric h on H associated with S,butD is possibly not compatible with the metric. The argument using the Hodge ? operator is not valid anymore. We first consider a general situation. Let X be a complex manifold, let (H,D) be a flat C1 bundle on X,andleth be a Hermitian metric on H.WedecomposeD into its (1, 0) and (0, 1) parts : D = D0 +D00.

4.2.2. Lemma. Given (H,D,h), there exists a unique connection Dh = Dh0 + Dh00 on H 1 and a unique C1-linear morphism ✓ = ✓0 + ✓00 : H E H satisfying the following ! X ⌦ properties :

(1) Dh is compatible with h,i.e.,dh(u, v)=h(Dhu, v)+h(u, Dhv), or equivalently, decomposing into types,

d0h(u, v)=h(Dh0 u, v)+h(u, Dh00v), d00h(u, v)=h(Dh00u, v)+h(u, Dh0 v). (2) ✓ is self-adjoint with respect to h,i.e.,h(✓u, v)=h(u, ✓v), or equivalently, decomposing into types,

h(✓0u, v)=h(u, ✓00v), h(✓00u, v)=h(u, ✓0v).

(3) D = Dh + ✓, or equivalently, decomposing into types,

D0 = Dh0 + ✓0,D00 = Dh00 + ✓00. 4.2.3. Remark. In 4.2.2(1), we have extended the metric h in a natural way as a sesquilinear operator (E1 H) H E1 resp. H (E1 H) E1 . X ⌦ ⌦ ! X ⌦ X ⌦ ! X

Proof. Let Dh be a connection on H which is compatible with h.LetA be a CX1-linear morphism A : H E1 H which is skew-adjoint with respect to h,thatis,such ! X ⌦ that h(Au, v)= h(u, Av)=0for every local sections u, v of H. Then the connec- tion Dh + A is also compatible with the metric. So let us choose any h-compatible

(2)When we work with a polarized variation of Hodge structure, the polarization S identifies (H,D) q,p p,q and (H_,D_) and we recover the usual conjugation relation between H and H . 78 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

connection Dh (for example the Chern connection, also compatible with the holomor- + phic structure D00)andletussetA = D Dh.LetusdecomposeA as A + A, + with A self-adjointe and A skew-adjoint. We can thus set Dh = Dh + A and ✓ = A+.Uniquenessisseensimilarly. e e 4.2.4. Remarks.

(1) Iterating 4.2.2(2), we find that ✓00 ✓00 is h-adjoint to ✓0 ✓0 and ✓0 ✓00 + ^ ^ ^ 2 ✓00 ✓0 is skew-adjoint. By applying d0 or d00 to 4.2.2(1)and(2), we see that Dh00 is ^ 2 adjoint to D0 , D00(✓0) is adjoint to D0 (✓00), D00(✓00) is adjoint to D0 (✓0),and h h h h h Dh0 Dh00 + Dh00Dh0 is skew-adjoint with respect to h. (2) Let us set D0 = D0 ✓0. Then the Chern connection for the metric h on the h holomorphic bundle (H,D00) is equal to D0 + D00 (see Exercise 4.4). Similarly, setting D00 = D00 ✓00, theb Chern connection for the anti-holomorphic bundle (H,D0) is h D0 + D00. We will set b b c D = D00 D0. b We refer to Exercise 4.4 for the properties of these operators. b b

4.2.5. Definition (Harmonic bundle). Let (H,D,h) be a flat C1-bundle endowed with aHermitianmetrich.Wesaythat(H,D,h) is a harmonic bundle if the operator 1 c Dh00 + ✓0 = 2 (D + D ) has square 0.Wealsoset

(4.2.5 ) D = D0 + ✓00, D = D00 + ✓0, ⇤ h h c so that D = D0 + D00 = D + D and D = D D. By looking at types, the harmonicity condition is equivalent to

2 (4.2.6) D00 =0,D00(✓0)=0,✓0 ✓0 =0. h h ^ By adjunction, this implies

2 D0 =0,D0 (✓00)=0,✓00 ✓00 =0. h h ^ Moreover, the flatness of D implies then

D0 (✓0)=0,D00(✓00)=0,D0 D00 + D00D0 = (✓0 ✓00 + ✓00 ✓0). h h h h h h ^ ^ 4.2.7. Lemma. Let (H,D) be a flat bundle and let h be a Hermitian metric on H. Then

DDc + DcD =0, (D)2 =0= ) (DD + DD =0, D2 =0.

Proof. Since D2 =0,itisamatterofproving(Dc)2 =0.Fromthevanishingabove, 2 2 we find (D0) =0, (D00) =0.WealsogetD00D0 +D0D00 =0. The properties for D, D are obtained similarly. b b b b b b 4.2. THE HODGE THEOREM 79

4.2.8. Definition (Higgs bundle). Set E =KerD00 : H H. If (H,D,h) is harmonic, h ! then E is a holomorphic vector bundle equipped with a holomorphic End(E)-valued 1-form ✓ induced by ✓0, which satisfies ✓ ✓ =0. It is called the Higgs bundle associated ^ to the harmonic bundle, and ✓ is its associated holomorphic Higgs field.Letusalso notice that, by definition, Dh is the Chern connection for the Hermitian holomorphic bundle (E, h).

4.2.9. Kähler identities for harmonic bundles. We assume that X is a compact Kähler complex manifold. We can apply the Kähler identities to the holomorphic bundle (H,D00) with Hermitian metric h and Chern connection D0 + D00,aswellastothe holomorphic bundle (H,Dh00) with Hermitian metric h and Chern connection Dh0 +Dh00. Denoting by P ? the formal L2-adjoint of an operator P ,theclassicalKähleridentitiesb take the form ? ? D0 = i[⇤,D00],D00 = i[⇤, D0], = i[D00, [⇤, D0]], D00 ? ? D0 = i[⇤,D00],D00 = i[⇤,D0 ], D = i[D00, [⇤,D0 ]]. bh h h bh h00 h bh As a consequence, we find ? ? ? c Dh := Dh0 + Dh00 = i[⇤,Dh]. Note also the following identity (see Exercise 4.7):

(4.2.10) D = 2D = 2D.

4.2.c. Polarized variations of Hodge structure on a compact Kähler ma- nifold : the Hodge-Deligne theorem. Let us come back to the setting of Sec- tion 4.2.a.

4.2.11. Proposition. Let (H,D,h) be a flat bundle with metric underlying a polarized variation of C-Hodge structure on X. Then (H,D,h) is a harmonic bundle. Proof. This is the content of Exercise 4.3.

Let us emphasize that the h-compatible connections Dh0 resp. Dh00 of Lemma 4.2.2 are given by the first projection in Griffiths’ transversality relations (4.1.5 ), and ✓0 ⇤ resp. ✓00 as the second projections. Recall that (H,D) is the flat C1 bundle associated with the flat holomorphic bundle (H0, ). Then Dh00 defines a holomorphic structure on p,w p r H , with associated holomorphic bundle Ker Dh00 isomorphic to the holomorphic p p,w p p 1,w p+1 bundle gr H0.Moreover,✓0 : H H is the C1 morphism associated F ! with the O -linear morphism induced by : X r 1 p 1 p 1 (4.2.12) ✓ := gr : gr H0 ⌦ gr H0. F r F ! X ⌦ F The decomposition D = D0 + D00 is thus replaced with the decomposition D = D + D. The disadvantage is that we loose the decomposition into types (1, 0) and (0, 1),butwekeeptheflatnessproperty.Ontheotherhand,wealsokeeptheKähler identities (4.2.10). 80 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

We did not really loose the decomposition into types : the operator D sends a p,q 1 p 1,q+1 1 p,q section of H to a section of (⌦ H )+(⌦ H ). Counting the total X ⌦ X ⌦ type, we find (p, q + 1) for both terms. In other word, taking into account the Hodge type of a section, the operator D is indeed of type (0, 1).Asimilarargumentapplies to D.ByDefinition4.2.5, (E• H, D) is a complex. Let us set X ⌦ p m i,j k,w k F (EX H)= EX H . ⌦ i+j=m ⌦ i+Lk>p We note that DF p(Em H) F p(Em+1 H), X ⌦ ⇢ X ⌦ i,j k,w k i,j+1 k,w k i,j k,w k since Dh00 sends EX H to EX H and ✓0 sends EX H to i+1,j k 1,w k+1⌦ ⌦ ⌦ E H .Wethusgetafilteredcomplexbysetting X ⌦ p p 0 D p 1 F (E• H, D)= F (E H) F (E H) , X ⌦ X ⌦ ! X ⌦ ! ··· and the associated graded complexn has the following degree-m term : o m i,m i p i,w p+i (EX H ). i=0 ⌦ L p p0 On the other hand, we filter gr H0 by setting F gr H0 = gr H0,sothat, F F p0>p F according to (4.2.12), we obtain the holomorphic Dolbeault complex L ✓ 1 ✓ (4.2.13) Dol(gr H0, ✓):= 0 gr H0 ⌦ gr H0 . F { ! F ! X ⌦ F ! ···} which is filtered by setting

p p ✓ 1 p 1 F Dol(gr H0, ✓)= F gr H0 ⌦ F gr H0 , F F ! X ⌦ F ! ··· so that n o p p ✓ 1 p 1 gr Dol(gr H0, ✓)= gr H0 ⌦ gr H0 . F F F ! X ⌦ F ! ··· n o p 4.2.14. Proposition (Dolbeault resolution). For each p, the complex F (EX• H, D) p ⌦ p • • is a resolution of F (⌦X grF H0, ✓) and grF (EX H, D) is a resolution of p ⌦ ⌦ grF Dol(grF H0, ✓).

Proof. Since the filtration F • is finite, it is enough to prove the second statement. Due p to the relations (4.2.6), we can regard (up to signs) gr (E• H, D) as the simple F X ⌦ complex associated with the double complex

i,j p i,w p+i ✓0 i+1,j p i 1,w p+i+1 E H / E H ⌦ ⌦

Dh00 Dh00 ✏ ✏ i,j+1 p i,w p+i ✓0 i+1,j+1 p i 1,w p+i+1 E H / E H ⌦ ⌦ i p i The ith vertical complex a resolution of ⌦ gr H0. X ⌦ F 4.2. THE HODGE THEOREM 81

4.2.15. Corollary (Dolbeault Lemma). We have for each p, q Z : 2 q p q p H X, gr Dol(gr H0, ✓) H (X, gr (E• H, D) , F F ' F X ⌦ q p q p H X, F Dol(gr H0, ✓) H (X, F (E• H, D) . F ' X ⌦ This being understood, the arguments of Hodge theory apply to this situation as in the case considered in Section 4.2.a,togettheHodge-Delignetheorem. 4.2.16. Theorem (Hodge-Deligne theorem). Let (H, S) be a polarized variation of C-Hodge structure of weight w on a smooth complex projective variety X of pure dimension n and let L be an ample line bundle on X. Then H •(X, H), XL) is naturally equipped with a polarizable sl -Hodge structure with central weight w+n 2 (see Definition 3.2.7). In particular, each Hk(X, H) is endowed with a polarized C-Hodge structure of weight w + k.If(H, S) is a polarized variation of Q-Hodge w Hk(X, H ) structure of weight , then each Q is endowed with a polarized Q-Hodge structure of weight w + k. Sketch of proof. We refer to [Zuc79,§2]forthedetailedadaptationtothissetting of the Kähler identities and their consequences. One realizes each cohomology class in H •(X, H) by a unique D-harmonic section, by the arguments of , which extend if one takes into account the total type, as above. The polarization is obtained from S on H and Poincaré duality as we did for S in Section 2.4,andfromitwecookuptheformS on the cohomology. More precisely, the flat pairing S : H H C1 ⌦ ! X induces a pairing of locally constant sheaves

S : H H CX , ⌦ ! which induces for every k with k n apairingS of C-Hodge structures | | 6 n+k H n k H 2n H H H (X, H) H (X, H) H (X, C) ( (w + n)) = C ( (w + n)) ⌦ ! by the formula

(4.2.17) S(•, •) := Sgn(n, k) S(•, •). Z[X] Since the Lefschetz operator XL only acts on the forms and not on the sections of H,it is self-adjoint with respect to S since it is so for the modified Poincaré duality pairing Sgn(n, k) , . Then S is a sesquilinear pairing on the sl -Hodge structure H •(X, H) h• •i 2 (see Section 3.2.b). Due to the Kähler identities and the commutation of LL with D,aharmonic n ` section of E H (` 0) is primitive if and only if each of its components with X ⌦ > respect to the total bigrading is so, and since LL only acts on the differential form part of such a component, this occurs if an only if each of its component on Ep,q Ha,b X ⌦ is primitive, with p + q = n ` and a + b = w.Fixinganh-orthonormal basis (v ) i i of Ha,b, such a component can be written in a unique way as ⌘p,q v with ⌘p,q i i ⌦ i i P 82 CHAPTER 4. VHS ON A COMPLEX MANIFOLD primitive. Then the positivity property of S defined by (3.2.18)on⌘p,q v amounts `,` i ⌦ i to the positivity of

Sgn(n, `) ( 1)q+bS(v , v ) X` ⌘p,q ⌘p,q. i i · L i ^ i ZX b 2 By the positivity for S,thereexistsaC1 function gi such that ( 1) S(vi, vi)= gi . p,q | | Therefore, (2.4.16)ongi⌘i gives the desired positivity. Last, in the presence of a Q-structure, we define the SQ by replacing S S with Q in (4.2.17)inordertoobtainapairingofQ-Hodge structures

n+k H n k H 2n H H H (X, H ) H (X, H ) H (X, Q)( (w + n)) = Q ( (w + n)). Q ⌦ Q ! 4.2.18. Remarks. Let H be a polarizable variation of C-Hodge structure of weight w on a smooth complex projective variety X.

(1) (The Hodge filtration) Consider the deRham complex DR(H0, ).According r to the Griffiths transversality property, it comes equipped with a filtration, by setting (see Definition 8.4.1):

p p 1 p 1 n p n F DR(H0, )= 0 F H0 r ⌦ F H0 r r ⌦ F H0 0 . r { ! ! X ⌦ ! ··· ! X ⌦ ! } p The natural inclusion of complexes F DR(H0, ) , DR(H0, ) induces a morphism r ! r k p k k (4.2.18 ) H (X, F DR(H0, )) H (X, DR(H0, )) = H (X, H), ⇤ r ! r p k whose image is the filtration F 0 H (X, H). Working anti-holomorphically with the filtration F 00•H by anti-holomorphic sub-bundles and the anti-holomorphic connection k induced by D00 on Ker D0,oneobtainsthefiltrationF 00•H (X, H). The Hodge-Deligne theorem implies that these filtrations are (w + k)-opposed. (2) (Degeneration at E1 of the Hodge-to-de Rham spectral sequence) Moreover, we claim that, for every p, k, the morphism (4.2.18 ) is injective. In other ⇤ words, the filtered complex R X, F • DR(H0, ) is strict (see Section 5.1.b). The p r graded complex gr DR(H0, ) is the complex F r p p ✓ 1 p 1 ✓ ✓ n p n gr DR(H0, )= 0 gr H0 ⌦ gr H0 ⌦ gr H0 0 , F r { ! F ! X ⌦ F ! ··· ! X ⌦ F ! } that is, p p gr DR(H0, ) = gr Dol(gr H0, ✓). F r F F Since each term of this complex is OX -locally free of finite rank and since ✓ is OX -lin- k p ear, the hypercohomology spaces H (X, gr DR(H0, )) are finite-dimensional. The F r strictness property is then equivalent to p k k p p, k, gr H (X, H)=H (X, gr DR(H0, )), 8 F F r p k p,w+k p where gr H (X, H) H (X, H). This property is also equivalent to F ' k, dim Hk(X, H)=dimHk(X, Dol(gr H, ✓)). 8 F This statement is obtained by standard arguments of Hodge theory applied to the operators D, D, D and their Laplacians. 4.2. THE HODGE THEOREM 83

(3) For any smooth projective variety X,thespaceH0(X, H) is primitive (for any L)and,givenapolarizationS of H,apolarizationofthepureC-Hodge structure of weight w is obtained by taking of the polarization function against a volume form (defined from L, in order to match with the Hodge-Deligne theorem). If X is a compact Riemann surface, then H1(X, H) is also primitive, and there is no need to choose a polarization bundle L in order to obtain the polarized pure C-Hodge structure on H1(X, H).Thepolarization(4.2.17)onH1(X, H) is simply written as 1 S(1) = S( , ). 2⇡i • • Z[X] (4) (The fixed-part theorem) The maximal constant subsheaf of H is the constant subsheaf with stalk H0(X, H) at each point, by means of a natural injective morphism 0 0 H (X, H) CX H.BytheHodge-Delignetheorem4.2.16, H (X, H) CX is ⌦C ! ⌦C endowed with a constant variation of Hodge structure of weight w.Weclaimthat the previous morphism is compatible with the Hodge filtrations, i.e., is a morphism in VHS(X, C,w),thatis,theinjectivemorphism 0 ' : H (X, H) O H O = H0 ⌦C X ! ⌦C X is compatible with the Hodge filtration F • on both terms. p Since X is compact and F H0 is OX -coherent (being OX -locally free of finite rank), 0 p the space H (X, F H0) is finite dimensional, and we have a natural injective morphism 0 p p ' : H (X, F H0) O F H0 p ⌦C X ! p by sending a global section of F H0 to its germ at every point of X.Ontheother p hand, regarding F H0 as a complex with only one term in degree zero, we have an obvious morphism of complexes p p F DR H0 F H0, ! 0 p 0 p which induces a morphism H (X, F DR H0) H (X, F H0), from which, together ! with 'p,weobtainamorphism 0 p p H (X, F DR H0) O F H0. ⌦C X ! p For p small enough so that F H0 = H0,werecoverthemorphism' above. By the 0 p p 0 degeneration property (2), H (X, F DR H0) is identified with F H (X, H),hence the assertion. As a consequence, if a global horizontal section of (H0, ), i.e., a global section p r of H, regarded as a global section of H0,isinF H0 at one point, it is a global section p of F H0. Arguing similarly with the anti-holomorphic Hodge filtration, and then with the Hodge decomposition of the C1-bundle H,wefindthatthenaturalinjectivemor- phism 0 H (X, H) C1 H C1 = H ⌦C X ! ⌦C X 84 CHAPTER 4. VHS ON A COMPLEX MANIFOLD is compatible with the Hodge decomposition of each term. As a consequence, for any global horizontal section of (H,D), i.e., any global section of H, regarded as a global section of H, the Hodge (p, q)-components are also D-horizontal. In particular, if the global section is of type (p, q) at one point, it is of type (p, q) at every point of X. Concerning the polarization, let us notice that the restriction of the polarization S 0 to the constant sub local system H (X, H) CX is constant. The polarization on ⌦ H0(X, H) is thus equal, up to a positive multiplicative constant, to the restriction 0 0 of S to H (X, H) regarded as the stalk of H (X, H) CX at any chosen point of X. ⌦ 4.2.d. The L2 de Rham and Dolbeault complexes. The compactness assump- tion in Hodge theory is not mandatory. One can relax it, provided that the metric on the manifold remains complete (see e.g. [Dem96, §12]). We will indicate the new phenomena that occur in the setting of Section 4.2.a. 2 One has to work with C1 sections v of E• H which are globally L with respect X ⌦ to the metric h and to the complete metric on X, and whose differential Dv is L2. The analysis of the Laplace operator is now similar to that of the compact case. One uses a L2 de Rham complex and a L2 Dolbeault complex (i.e., one puts a L2 condition on sections and their derivatives), that we analyze in this section. We assume that the complex manifold X is endowed with a metric, hence a volume form vol,thatenablestodefinethespaceL2(U, vol) for each open set U X,and ⇢ we will omit vol in the notation from now on. The locally free sheaves of differential forms are then endowed with a metric. We can regard the sheaf of C1 functions on X at a subsheaf of the sheaf of locally integrable function, simply denoted by L1,loc (with respect to any metric on X). For any relatively compact open set U X, (U, L ):=L1 (U, vol).Onecheckseasily ⇢ 1,loc loc that the assignment U L1 (U, vol) is a sheaf, which does not depend on the choice 7! loc of the metric. Let (H, h) be a C1 vector bundle on X with a Hermitian metric h and let 1 L1,loc C H be the associated sheaf of L sections of H. The h-norm of any local ⌦ X1 loc section of the latter is a locally integrable function.

4.2.19. Definition. The space L2(X, H, h) is the subspace of (X, L H) consisting 1,loc ⌦ of sections whose h-norm belongs to L2(X).

The following lemmas are standard.

2 4.2.20. Lemma. A section u (X, L1,loc C H) belongs to L (X, H, h) if and only 2 ⌦ X1 if there exists a sequence u (X, H) L2(X, H, h) such that u u 0 in n 2 \ k nk2,h ! L2(X, H, h). In such a case, u u weakly, that is, for any (X, H), n ! 2 c (h(u, ) h(u ,)) vol 0. n ! ZX Let " be an h-orthonormal frame of H on X. Then a section u = f " , with f i i i i 2 L1 (X),belongstoL2(X, H, h)) if and only if each f belongs to L2(X).Orthonormal loc i P 4.2. THE HODGE THEOREM 85 frames may not be easy to find and in order to check the L2 property, other frames may be more convenient.

4.2.21. Definition (L2-adaptedness). Aframev of H on X is said to be L2-adapted if there exists a positive constant Cv such that , for any section u = i fivi in (X, L1 H), the following inequality holds loc ⌦ P f v C u C f v . k i ik2 6 vk k2 6 v k i ik2 i i X ⇣ X ⌘ In other words, an L2-adapted frame v gives rise to a decomposition L2(X, H, h) = 2 L (X, Hi, h), where Hi is the C1 subbundle generated by vi with induced metric. Let us state some properties.

4.2.22. Lemma. (1) An orthonormal frame " is L2-adapted. 2 (2) If h and h0 are comparable metrics on H, then a frame of H is L -adapted for h if and only if it is so for h0. 2 (3) If a frame v =(v1,...,vr) is L -adapted, any rescaled frame (h1v1,...,hrvr), 1 2 with hi Lloc(X) nowhere vanishing, is also L -adapted. 2 2 (4) A sufficient condition for a frame v to be L -adapted is that the functions vi h 1 1 k k are locally bounded and each entry of the matrix hv := (h(vi,vj)i,j) is locally bounded.

Proof. The first three points are clear. For the last one, let C>0 denote a bounding constant, let u = f v be a section of L1 H and set g := h(u, v ). Then i i i loc ⌦ i i gi 6 u h vi h 6 C u h,since vi h 6 C. The column vectors G and F (of the gi’s | | k k k k Pk k k k 1 and the fi’s respectively) are related by G = hv F ,sothatF = hv G. It follows 2 · 3 · 2 that, for each i, fi 6 rC u h and thus fivi h 6 rC u h. Then v is L -adapted | | 3 k k k k k k with constant Cv = rC .

Assume now that H is endowed with a flat connection D : H E1 H. ! X ⌦ 4.2.23. Definition. The space L2(X, H, h,D) is the subspace of L2(X, H, h) consisting of sections u such that

• Du,consideredintheweaksense,(i.e.,distributionalsense)isasectionofL1,loc 1 ⌦ (EX H), ⌦ 2 • as such, its h-norm belongs to L (X).

C1-approximation also holds in this case.

2 4.2.24. Lemma. A section u (X, L1,loc C H) belongs to L (X, H, h,D) if and 2 ⌦ X1 only if there exists a sequence u (X, H) L2(X, H, h) such that n 2 \ u u 0 in L2(X, H, h), • k nk2,h ! Du belongs to L2(X, E1 H, h) and is a Cauchy sequence in this space. • n X ⌦ 86 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

The L2 de Rham complex is then well-defined as the complex

D D (4.2.25) 0 L2(X, H, h,D) L2(X, E1 H, h,D) ! ! X ⌦ ! ··· D L2(X, E2n H, h,D) 0, ! X ⌦ ! k whose cohomology is denoted by HD,L2 (X, H).

4.2.26. Definition. The assignment U L2(U, H, h) defines a presheaf, which is a 7! sheaf on X,denotedbyL(2)(H, h). If U is relatively compact in X,wehave

2 (U, L(2)(H, h)) = L (U, H, h).

We define similarly the sheaf L(2)(H, h,D) and we set Li (H, h,D):=L (Ei H, h,D). (2) (2) X ⌦ We obtain therefore a complex of sheaves D D D (4.2.27) 0 L (H, h) L1 (H, h,D) L2n (H, h,D) 0. ! (2) ! (2) ! ··· ! (2) ! 4.2.28. Lemma (L2 Poincaré lemma). The complex (4.2.27) is a resolution of the locally constant sheaf H =KerD.

rk H Proof. Near a given point of X,wecanfindalocalisomorphism(H,D) (CX1) , d) rk H' and the metric h is equivalent to the standard metric on (CX1) in which the canonical basis is orthonormal. Then both assertions of the theorem need only to be proved for (C1, d, ) where 1 =1.Ontheotherhand,thePoincarémetricis X k·k k k locally equivalent to the standard Euclidean metric near this point. The proof is then obtained by a standard regularization procedure (see Corollary 12.2.5 for a statement in any dimension).

4.2.29. Definition. By considering the decomposition (4.2.5 )onedefinessimilarly 2 2 ⇤ L (X, H, h, D) and the L Dolbeault complexes (p =0,...,dX )

D (4.2.30) 0 L2(X, grp (E0 H), h, D) L2(X, grp (E1 H), h, D) ! F X ⌦ ! F X ⌦ D D L2(X, grp (En H), h, D) 0, ! ··· ! F X ⌦ ! p,m+w p m H (X, H) whose th cohomology is denoted by D,L2 .

The analogue of the C1-approximation lemma 4.2.24 also holds in this case, and we define the sheaves L (grp (Ei H), h, D) in a way similar to Definition 4.2.26. (2) F X ⌦ Then the Dolbeault complex (4.2.30)sheafifiesas D D (4.2.31) 0 L (grp (E0 H), h, D) L (grp (E1 H), h, D) ! (2) F X ⌦ ! (2) F X ⌦ ! ··· 4.2. THE HODGE THEOREM 87

4.2.32. Lemma (L2 Dolbeault lemma). The inclusion of complexes p p gr Dol(gr H0,✓) , L (gr (E• H), h, D), D) F F ! (2) F X ⌦ is a quasi-isomorphism. 2 Proof. We write D =d00 ✓0.Since✓0 is C1,itislocallybounded,sothelocalL - h condition on the D-derivative only concerns d00,andwecanconsider(uptosigns) the complex (4.2.31) as the simple complex associated with the double complex with differentials d00 and ✓0. For the complex with differential d00 we can apply the standard L2-Dolbeault lemma (recalled as Theorem 12.2.6 in Chapter 12). Then the statement is clear.

4.2.e. Polarized variations of Hodge structure on a complex manifold with a complete metric One missing point in the general context of noncompact complex manifolds is the finite dimensionality of the L2-cohomologies involved. In the compact case, it is ensured, for instance, by the finiteness of the Betti cohomology Hk(X, H).Sothe Hodge theorem is stated as 4.2.33. Theorem. Let (X,!) be a complete Kähler manifold and let (H, S) be a polarized variation of Hodge structure on X.Let(H,D) be the associated flat C1 bundle. Then, with the assumption that all the terms involved are finite dimensional, one has a canonical isomorphisms k p,q q,p p,q H 2 (X, H) H (X, H),H (X, H) H (X, H_). D,L D,L2 D,L2 D,L2 ' p+q=k+w ' L We refer to [Zuc79, §7] for a proof of this result. The finiteness assumption can be obtained by relating the L2 with topology. If we are lucky, then this will not only provide a relation with Betti cohomology, but the Betti cohomology will be finite-dimensional and this will also provide finiteness of the L2 de Rham cohomology. There is also a need for finiteness of the L2 Dolbeault cohomology. In the case that will occupy us later, where X is a punctured compact Riemann surface, this will be done by relating L2 Dolbeault cohomology with the cohomology of a on the compact Riemann surface. We will indicate in Sections 6.10 and 6.12 the way to solve these two problems in dimension one, by means of the L2 Poincaré lemma and the L2 Dolbeault lemma. What about the Lefschetz aspect of Hodge theory in this context ? The the complete Kähler metric acts as a Lefschetz operator on the L2 cohomology, and gives rise to a polarization of the Hodge structure. On the other hand, the theory of Hodge modules takes place on smooth complex projective varieties (or for some aspects compact Kähler manifolds). The non-compact manifold that occurs is the complement of divisor with normal crossings. Such a manifold can be endowed with a complete Kähler metric having a controlled behaviour at infinity, that is, in the neighbourhood 88 CHAPTER 4. VHS ON A COMPLEX MANIFOLD of the normal crossing divisor of the compactification : the metric has a Poincaré- like behaviour locally at infinity. However, in the theory of Hodge modules, we only consider the Lefschetz operator coming from an ample line bundle on the projective variety.(3)

4.3. Semi-simplicity 4.3.a. A review on completely reducible representations. We review here some classical results concerning the theory of finite-dimensional linear representa- tions. Let ⇧ be a group and let ⇢ be a linear representation of ⇧ on a finite-dimensional k-vector space V . In other words, ⇢ is a group homomorphism ⇧ GL(V ). We will ! say that V is a ⇧-module (it would be more correct to introduce the associative al- gebra k[⇧] of the group ⇧,consistingofk-linear combinations of the elements of ⇧, and to speak of a left k[⇧]-module). The subspaces of V stable by ⇢(⇧) correspond thus to the sub-⇧-modules of V . We say that a ⇧-module V is irreducible if it does not admit any nontrivial sub-⇧- module. Then, any homomorphism between two irreducible ⇧-modules is either zero, or an isomorphism (Schur’s lemma). If k is algebraically closed, any automorphism of an irreducible ⇧-module is a nonzero multiple of the identity (consider an eigenspace of the automorphism).

4.3.1. Proposition. Given a ⇧-module V , the following properties are equivalent : (1) The ⇧-module V is semi-simple, i.e., every sub-⇧-module has a supplementary sub-⇧-module. (2) The ⇧-module V is completely reducible, i.e., V has a decomposition (in general non unique) into the direct sum of irreducible sub-⇧-modules. (3) The ⇧-module V is generated by its irreducible sub-⇧-modules.

Proof. The only non-obvious point is (3) (1). Let then W be a sub-⇧-module of V . ) We will show the result by induction on codim W ,thisbeingclearforcodim W =0. If codim W > 1,thereexistsbyassumptionanontrivialirreduciblesub-⇧-module V V not contained in W .SinceV is irreducible, we have W V = 0 ,soW := 1 ⇢ 1 \ 1 { } 1 W V is a sub-⇧-module of V to which one can apply the inductive assumption. 1 If W 0 is a supplementary ⇧-module of W ,thenW 0 = W 0 V is a supplementary 1 1 1 1 ⇧-module of W .

It follows then from Schur’s lemma that a completely reducible ⇧-module has a unique decomposition as the direct sum o V = Vi = (Vi Ei), i i ⌦ L L

(3)See [KK87, §6.4] for the comparison between both Lefschetz operators. 4.3. SEMI-SIMPLICITY 89 in which the isotypic components V are sub-⇧-modules of the form V o E , where V o i i ⌦ i i is an irreducible ⇧-module, V o is not isomorphic to V o for i = j,andE is a trivial i j 6 i ⇧-module, i.e., on which ⇧ acts by the identity. One also notes that if W is a sub-⇧-module of a completely reducible ⇧-module V , then W is completely reducible and its isotypical decomposition is

W = (W Vi), i \ L in which W V = V o F for some subspace F of E .A⇧-module supplementary \ i i ⌦ i i i to W can be obtained by choosing for every i a k-vector space supplementary to Fi in Ei.

4.3.2. Remarks. The previous properties have easy consequences. (1) A k-vector space V is a semi-simple ⇧-module if and only if the associated complex space V = C V is a semi-simple ⇧-module (for the complexified represen- C ⌦ tation).

Indeed, let us first recall that the group Autk(C) acts on VC :if" =("1,...,"n) is some k-basis of V then, for a1,...,an C and Autk(C),onesets ai"i = 2 2 i (ai)"i.AsubspaceW of V is “defined over k”, i.e., of the form C W for some i C C ⌦P sub-space W of V ,ifandonlyifitisstablebyanyautomorphism Autk(C) : P 2 indeed, if d =dimC WC,onecanfind,uptorenumberingthebasis",abasise1,...,ed of WC such that e = " + a " + + a " + + a " 1 1 1,2 2 ··· 1,d d ··· 1,n n e = " + + a " + + a " 2 2 ··· 1,d d ··· 2,n n . . e = " + + a " , d d ··· d,n n with ai,j C ; one then shows by descending induction on i d, . . . , 1 that, if W 2 2{ } C is stable by Autk(C),thenai,j are invariant by any automorphism of C over k,i.e., belong to k since C is separable over k. Let us now prove the assertion. Let us first assume that V is irreducible and let us consider the subspace WC of VC generated by the sub-⇧-modules of minimal dimension (hence irreducible). Since the representation of ⇧ is defined over k,ifEC is a ⇧-module, so is (E ) for every Autk(C) ;thereforethespaceW is invariant C 2 C by Autk(C),inotherwordstakestheformC k W for some subspace W of V . It ⌦ is clear that W is a sub-⇧-module of V ,henceW = V .Accordingto4.3.1(3), VC is semi-simple.

Conversely, let us assume that VC is semi-simple. Let us choose a k-linear form ` : C k such that `(1) = 1. It defines a k-linear map L : V V which is ⇧- ! C ! invariant and which induces the identity on V .LetW be a sub-⇧-module of V .We have a ⇧-invariant projection V W ,henceacomposedprojectionp which is C ! C 90 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

⇧-invariant : L V / W / W OC C 4 ? p V from which one obtains a ⇧-module supplementary to W in V . (2) If ⇧00 ⇧ is a surjective group-homomorphism and ⇢00 is the composed re- ! presentation, then V is a semi-simple ⇧-module if and only if it is a semi-simple ⇧00-module. Indeed, the ⇧-module structure only depends on the image ⇢(⇧) ⇢ GL(V ). (3) Let ⇧0 C ⇧ be a normal subgroup, and let V be a ⇧-module. Then, if V is semi-simple as a ⇧-module, it so as a ⇧0-module. Indeed, if V 0 is an irreducible sub-⇧0-module of V ,then⇢(⇡)V 0 remains so for every ⇡ ⇧. If V is ⇧-irreducible 2 and if V 0 is a nonzero irreducible sub-⇧0-module, the sub-⇧0-module generated by the ⇢(⇡)V 0 is a ⇧-module, hence coincides with V .Asaconsequence,V is generated by its irreducible sub-⇧0-modules, hence is ⇧0-semi-simple, according to 4.3.1(3). (4) Arealrepresentation⇧ Aut(V ) is simple if and only if the associated ! R complexified representation ⇧ Aut(V ) has at most two simple components. [Hint : ! C Any simple component of the complexified representation can be summed with its conjugate to produce a sub-representation of the real representation.]

4.3.b. The semi-simplicity theorem. Let X be a smooth projective variety. Let H =(H,F0•H,F00•H,D,S) be a polarized variation of C-Hodge structure of weight w on X (see Definition 4.1.4), and let H =KerD be the associated complex local system.

4.3.3. Theorem. Under these assumptions, the complex local system H is semi-simple.

Let us already note that the result is easy for unitary local systems (underlying thus polarized variations of type (0, 0),asinSection4.2.a). The general case will use the objects introduced in Section 4.2.b, and will not be specific to polarized variations of Hodge structure. The proof of the semi-simplicity theorem will apply to these more general objects called harmonic bundle (see Section 4.2.b). Moreover, we can relax the property that the smooth variety is projective, and only assume that it is a compact Kähler manifold, since we will only use the Kähler identities.

H H H 4.3.4. Remark. If is obtained from a local system Q defined over Q,then Q is also semi-simple as such, according to Remark 4.3.2(1).

Let H be a C1-bundle with metric h. The group of C1 automorphisms g of H 1 1 acts on a given connection D by the formula D := g D g = D D(g) g , g where we have extended the action of D in a natural way on the bundle End(H). If D is flat, then so is D.Wethenset D = D + ✓.LetusalsosetD = D ✓ g g g h g h (see Lemma 4.2.2). b 4.3. SEMI-SIMPLICITY 91

1 1 1 4.3.5. Lemma. We have ✓ = ✓ D(g)g + g⇤ D(g⇤) , where g⇤ is the h-adjoint g 2 of g. b Proof. We have 1 1 1 1 D = D + ✓ D (g)g +[✓, g]g = D D (g)g + g ✓g. g h h h h 1 1 It follows that ✓ is the self-adjoint part of D (g)g + g ✓g,thatis,takinginto g h account that the adjoint of Dh(g) is Dh(g⇤) (by working in a local h-orthonormal basis),

1 1 1 1 1 (4.3.6) ✓ = (D (g)g + g⇤ D (g⇤)) + g ✓g + g⇤✓g⇤ . g 2 h h The lemma follows from a straightforward computation.

If we fix a metric on X, we deduce with h ametriconE1 H and then a metric X ⌦ on Hom(H, E1 H) with associated scalar product ( , ).Wethendenoteby , k·k X ⌦ · · h· ·i the integrated product using the volume form on X :

, = ( , ) vol . h· ·i · · ZX 4.3.7. Definition. The energy of g Aut(H) with respect to (H,D,h) is defined as 2 E (g):= ✓ 2 = ✓, ✓ . (H,D,h) kg k hg g i + Let ⇠ End(H) and let ⇠ = ⇠ + ⇠ be its decomposition into its self-adjoint part + 1 2 1 ⇠ = (⇠ + ⇠⇤) and its skew-adjoint part ⇠ = (⇠ ⇠⇤). 2 2 4.3.8. Proposition. For t varying in R, we have d E (et⇠) =2D ⇠+,✓ . dt (H,D,h) t=0 h h i t⇠ t⇠ 2 t⇠⇤ t⇠⇤ 2 Proof. We have D(e )e = tD⇠ mod t and e D(e )=tD⇠⇤ mod t .From Lemma 4.3.5 we deduce b b d t⇠ E (e ) = D⇠ + D⇠⇤,✓ ✓,D⇠ + D⇠⇤ dt (H,D,h) t=0 h ih i + + = D ⇠ +[✓,⇠],✓ ✓,D ⇠ +[✓,⇠] h h b ih bh i = 2Re D ⇠+,✓ = 2 D ⇠+,✓ , h h i h h i + since ✓⇠,✓ = ✓,✓⇠ , ⇠✓,✓ = ✓,⇠✓ ,andboth✓ and D ⇠ are self- h i h i h i h i h adjoint.

The property of being semi-simple or not for (H,D) is seen on the energy functional.

4.3.9. Proposition. Let (H,D) be a flat bundle. Assume that there exists a metric h such that the energy functional g E H (g) has a critical point at g =Id. Then 7! ( ,D,h) (H,D) is semi-simple. 92 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

Proof. Let us argue by contraposition and let us assume that (H,D) is not semi- simple. Let h be any metric on H. We will prove that Id is not a critical point for g E H (g). It is enough to prove that there exists ⇠ End(H) such that the 7! ( ,D,h) 2 function t⇠ f : R R,t E(H,D,h)(e ) ! 7! has no critical point at t =0.Byassumption,thereexistsasub-bundleH1 of H stable by D such that its orthogonal H2 is not stable by D.Setni =rkHi (i =1, 2). With respect to this decomposition we have D 2⌘ D = 1 , 0 D ✓ 2◆ 1 t⇠ with ⌘ : H2 E H1 nonzero. Set ⇠ = n2 IdH1 n1 IdH2 and g = e (t R). We ! X ⌦ 2 have

n2t n2t t⇠ t⇠ 02⌘ e 0 e 0 D(e )e = , 00 0 e n1t · 0 en1t ✓ ◆ ✓ ◆ ✓ ◆ 02⌘ e n2t 0 02⌘ en2t 0 = 00 0 en1t 00 0 e n1t ✓ ◆ ✓ ◆✓ ◆✓ ◆ (n +n )t 0 2(1 e 1 2 )⌘ = , 00 ✓ ◆ so (n1+n2)t t⇠ t⇠ D1 2e ⌘ D = D D(e )e = , g 0 D ✓ 2 ◆ and (n1+n2)t ✓1 e ⌘ g✓ = . e (n1+n2)t⌘ ✓ ✓ ⇤ 2 ◆ It follows that

(n1+n2)t f(t)=c0 + c1e ,c0 > 0,c1 > 0, and it is clear that f 0(0) =0. 6 Proof of the semi-simplicity theorem. In view of Propositions 4.2.11 and 4.3.9,thesemi- simplicity theorem 4.3.3 is a consequence of the following.

4.3.10. Proposition. Assume that X is compact Kähler and that (H,D,h) is a harmonic bundle. Then g E H (g) has a critical point at g =Id. 7! ( ,D,h) ? Proof. According to Proposition 4.3.8,itisenoughtoshowthatDh⇤(✓)=0, where Dh c denotes the formal adjoint of Dh.SettingDh := Dh00 Dh0 ,theKähleridentitiesfor ? c aHermitianvectorbundle(seeSection4.2.9)implythatDh is a multiple of [⇤,Dh]. Since ✓ is a matrix of 1-forms and ⇤ is an operator of type ( 1, 1),wehave⇤(✓)=0. c On the other hand, by the properties after Definition 4.2.5,wehaveDh(✓)=0. 4.3. SEMI-SIMPLICITY 93

4.3.c. Structure of polarized variations of C-Hodge structure Let X be a complex manifold. We will say that two polarized variations of C-Hodge structures are equivalent if one is obtained from the other one by a twist (k, `) (see Exercise 2.10)andbymultiplyingthepolarizationformbyapositiveconstant.

4.3.11. Lemma. There exists at most one equivalence class of polarized variations of C-Hodge structure on a simple (i.e., irreducible) C-local system H on a compact com- plex manifold X.

4.3.12. Remark. Acriterionfortheexistence of a polarized variation of C-Hodge structure on a simple C-local system H is given in [Sim92,§4]intermsofrigidity.

Proof. If we are given two polarizable variations of C-Hodge structure on an irre- ducible local system H,wededucesuchapolarizablevariationonEnd(H) (Remark 5.4.5), and the dimension-one vector space End(H):=H0(X, End(H)) is endowed with a C-Hodge structure of some type (k, `) by the Hodge-Deligne theorem 4.2.16. The identity morphism IdH End(H) defines thus a morphism of type (k, `) between 2 the two variations. Therefore, the first one is obtained from the second one by a twist (k, `). It remains to check that, on a given polarizable variation of C-Hodge structure on an irreducible local system H,thereexistsexactlyonepolarizationuptoapositive multiplicative constant. Note that such a polarization is an isomorphism H ⇠ H⇤,so ! one polarization is obtained from another one by multiplying by a nonzero constant. This constant must be positive, by the positivity property of the associated Hermitian form.

Let X be a compact complex manifold and let H =(H,F0•H,F00•H,D,S) be a polarized variation of C-Hodge structure of weight w on X. If the associated local system H is semi-simple, which is the case when X is Kähler according to Theo- n↵ rem 4.3.3,itdecomposesasH = ↵ A H↵ , where H↵ are irreducible and pair- 2 wise non isomorphic, and Hn↵ means the direct sum of n copies of H .Simi- ↵ L ↵ ↵ n↵ larly, (H,D)= ↵ A(H↵,D) ,andthepolarizationS,beingD-horizontal, de- 2 o n↵ composes with respect to ↵ A as S = S↵,n↵ .LetussetH↵ := C and let us L o 2 write H = ↵ A H↵ H↵. If we are given a basis S↵ of the dimension-one vector 2 ⌦ L o o o space Hom(H , H⇤ ),thereexistsauniquemorphismS Hom(H , H ⇤) such that L↵ ↵ ↵ 2 ↵ ↵ S =So S . ↵,n↵ ↵ ⌦ ↵ 4.3.13. Theorem. Under these conditions, the following holds : (1) For every ↵ A, there exists a unique equivalence class of polarized variation 2 of C-Hodge structure of weight w on H↵. (2) For every ↵ A, let us fix a representative H =(H ,F0•H ,F00•H ,D,S ) 2 ↵ ↵ ↵ ↵ ↵ of such an equivalence class. There exists then a polarized C-Hodge structure

o o • o • o o H↵ =(H↵,F0 H↵,F00 H↵, S↵) 94 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

o of weight 0 with dim H↵ = n↵ such that

o (4.3.13 ) H = (H↵ C H↵). ⇤ ↵ A ⌦ L2 4.3.14. Remark. Let us emphasize the following statement : given a polarized variation of Hodge structure on a compact complex manifold X such that the underlying local system is semi-simple (which is the case if X is Kähler), then each irreducible com- ponent of this local system underlies a polarized variation of Hodge structure (which is essentially unique). The proposition also explains how to reconstruct the original variation from its irreducible components.

Proof of Theorem 4.3.13. (1) The uniqueness statement is given by Lemma 4.3.11. In order to prove the existence in 4.3.13(1), it is enough to exhibit for every ↵ A asub-variationofHodge 2 structure of H of weight w with underlying local system H↵. The polarization S will then induce a polarization S↵,accordingtoExercise4.2(1). For that purpose, it is enough to exhibit H as the image of an endomorphism H H which is compatible ↵ ! with the Hodge structures : by abelianity (Proposition 4.1.10), this image is an object of VHS(X, C,w).LetusthereforeanalyzeEnd(H)=H0(X, End(H)). If we set Ho = Cn↵ ,sothatH = (Ho H ),wehaveanalgebraisomorphism ↵ ↵ ↵ ⌦C ↵ End(H) End(Ho ) (where x x =0if x Ho , x Ho and ↵ = ). We know ' ↵ ↵ ↵ L ↵ 2 ↵ 2 6 that the local system End(H) underlies a polarized variation of -Hodge structure of Q C weight 0.Therefore,End(H)=(X, End(H)) underlies a C-Hodge structure of weight o 0 by the Hodge-Deligne theorem 4.2.16. It is then enough to show that each H↵ under- o o lies a C-Hodge structure H↵ of weight 0 such that the equality End(H)= ↵ End(H↵) is compatible with the Hodge structures on both terms. Indeed, choose then any rank- o,(k, k) Q one endomorphism p↵ of some nonzero vector space H↵ .Extenditasarank-one o o,(`, `) endomorphism of H↵ of type (0, 0) by mapping every other summand H↵ to zero, o and extend it similarly as a rank-one endomorphism of H of type (0, 0).Oneob- tains thus a rank-one endomorphism in End(H)0,0.Withrespecttothisidentification, L its image is (Im p ) H H ,aswanted. ↵ ⌦C ↵ ' ↵ Let us prove the assertion, which reduces to proving the existence of a grading o of each H↵ giving rise to the Hodge grading of End(H).Bytheproductformula above, the C-algebra End(H) is semi-simple, with center Z = C IdHo .Analge- ↵ · ↵ bra automorphism ' of End(H) induces an automorphism of the ring Z, whose matrix Q in the basis above only consists of zeros and ones. By the Skolem-Noether theorem (see e.g. [Bou12,§14,No 5, Th. 4]), algebra automorphisms for which the correspon- ding matrix is the identity are interior automorphisms, that is, products of interior o automorphisms of each End(H↵). Any algebra automorphism can be composed with an automorphism with matrix having block entries Id or 0 in order that the matrix on Z is the identity. As a consequence, the identity component of the group of algebra alg o automorphisms Aut (End(H)) is identified with ↵ A(Aut(H↵)/C⇤ Id↵). 2 Q 4.4. EXERCISES 95

As in Section 2.5.8,theC-Hodge structure of weight 0 on End(H) defines a conti- 1 p p, p nuous representation ⇢ : S Aut(End(H)),suchthat⇢()= on End(H) . ! Since the grading is compatible with the algebra structure, the continuous repre- sentation ⇢ takes values in the group of algebra automorphisms Autalg(End(H)). Since ⇢(1) = Id,ittakesvaluesintheidentitycomponentofAutalg(End(H)),i.e.,in o ↵ A(Aut(H↵)/C⇤ Id↵).BytheargumentgiveninSection2.5.8,itdefinesagrading, 2 o up to a shift, on each H↵,aswanted. Q o (2)LetusnowendowH↵ with a polarized C-Hodge structure of weight 0 so that (4.3.13 )holds.Wealreadyhaveobtainedagrading,i.e.,aC-Hodge struc- ⇤ ture of weight 0. In order to obtain a polarization of this C-Hodge structure satis- fying (4.3.13 ), we note the equality Ho = Hom(H , H),andsinceHom(H , H) ⇤ ↵ ↵ ↵ underlies a polarized variation of Hodge structure of weight 0 according to 4.3.13(1), o H↵ comes equipped with a polarized Hodge structure of weight 0 by the Hodge- Deligne theorem 4.2.16.Bydefinition,thenaturalmorphismHo H H underlies ↵ ⌦ ↵ ! amorphismofpolarizedvariationsofHodgestructure.

4.4. Exercises Exercise 4.1. (1) Show that the category VHS(X, C,w) is abelian. [Hint :Usethat,accordingto Definition 4.1.5, any morphism is bigraded with respect to the Hodge decomposition, hence so are its kernel, image and cokernel.] (2) Let H be an object of VHS(X, C,w).Showthatapolarizationisanisomor- phism H ⇠ H⇤( w).[Hint :UseRemark2.5.16(2).] ! (3) Define the tensor product

VHS(X, C,w1) VHS(X, C,w2) VHS(X, C,w1 + w2) ⌦ ! and the external Hom

VHS(X, C,w1) VHS(X, C,w2) VHS(X, C,w2 w1). ⌦ ! (4) Show that these operations preserve the subcategories of polarizable objects.

Exercise 4.2 (Abelianity and semi-simplicity). Let H, S be a polarized variation of Hodge structure of weight w on X. (1) Show that any subobject of H in VHS(X, C,w) is a direct summand of the given variation, and that the polarization S induces a polarization. [Hint :UseExercise2.12.] (2) Conclude that the full subcategory pVHS(X, C,w) of polarizable variations of Hodge structure is abelian and semi-simple (i.e., any object decomposes as the direct sum of its irreducible components). [Hint :UsetheC1 interpretation of Definition 4.1.5.]

Exercise 4.3. Let (H, S) be a polarized variation of Hodge structure of weight w on X (see Definition 4.1.4). Let h be the Hermitian metric deduced from S and let 96 CHAPTER 4. VHS ON A COMPLEX MANIFOLD

p,q D =D0 +D00 be the flat C1 connection. Let H= p+q=w H be the Hodge decom- position (which is h-orthogonal by construction). Show the following properties. L (1) In the Griffiths transversality relations (4.1.5 ), the composition of D0 ⇤ (resp. D00) with the projection on the first summand defines a (1, 0) (resp. (0, 1))- connection Dh0 (resp. Dh00), and that the projection to the second summand defines a C1-linear morphism ✓0 (resp. ✓00).

(2) The connection Dh := Dh0 +Dh00 is compatible with the metric h,butispossibly not flat. (3) The morphism ✓00 is the h-adjoint of ✓0.

(4) The connection D := Dh00 + ✓0 has square zero, as well as the connection D := Dh0 + ✓00.

Exercise 4.4. Let (H,D) be a flat bundle and let h be a Hermitian metric on H.

(1) Show that there exist a unique (1, 0)-connection D0 and a unique (0, 1)-con- nection D00 such that D0 + D00 and D0 + D00 preserve the metric h. (2) Show that D0 = D0 ✓0 and D00 = D00 ✓00. b h h (3) Concludeb that D0 + Db00 is theb Chern connection of the Hermitian holomorphic bundle (H,D00, h).b b c b c 1 c (4) We set D := D00 D0.ShowthatD = D D and 2 (D + D )=Dh00 + ✓0 = D. 2 (5) Let Rh =(D0 + D00) be the curvature of the Hermitian holomorphic bundle (H, h,D00).Showthatb b b 2 R = 2(D + ✓0 ✓00 + ✓00 ✓0). h ^ ^

(6) Show that, if (H,D,h) is harmonic,thentr Rh =0.[Hint :Usethat,inany case, tr(✓0 ✓00 + ✓00 ✓0)=0.] ^ ^ Exercise 4.5.

(1) Given C1 bundles with flat connection and Hermitian metric (H1,D1, h1) and (H ,D , h ),endowtheC1 bundles H H and Hom(H , H ) with a natural flat 2 2 2 1 ⌦ 2 1 2 connection D and a natural Hermitian metric h,andidentifyconnectionandmetric on H⇤ H and Hom(H , H ).[Hint :UseExercise2.2.] 1 ⌦ 2 1 2 (2) Show that, for a section ' of Hom(H , H ),wehaveD(')=D ' ' D . 1 2 2 1 (3) Prove that if (H ,D , h ) and (H ,D , h ) are harmonic, then (H H ,D,h) 1 1 1 2 2 2 1 ⌦ 2 and (Hom(H1, H2),D,h) are also harmonic. (4) Show that ✓ on Hom(H1, H2) is given by the following formula. For an open set U and a local section ' : H1 U H2 U of Hom(H1, H2) on U, | ! | V U, u (V,H ), [✓(')](u )=✓ ('(u )) '(✓ (u )) (V,E H ). 8 ⇢ 8 1 2 1 1 2 1 1 1 2 X ⌦ 2 Exercise 4.6 (Formal adjoint). Let (H, h) be a Hermitian vector bundle on a complex manifold X endowed with a Hermitian metric on its tangent bundle, which induces a Hermitian metric on the sheaves Ek ,simplydenotedby , . Then the sheaves Ek H X h i X ⌦ 4.5. COMMENTS 97 are endowed with a natural Hermitian metric denoted by , (see Exercise 2.2). For h ih the differential operator of order one D : Ek H Ek+1 H X ⌦ ! X ⌦ induced by a connection D : H E1 H,theformaladjointD? : Ek+1 H Ek H ! X ⌦ X ⌦ ! X ⌦ is the operator defined by

Du, v vol = u, D?v vol h ih X h ih X ZX ZX for any pair of local sections with compact support of Ek H and Ek+1 H. X ⌦ X ⌦ ? (1) Show that the formal adjoint ' of a C1-linear morphism ' Hom(H , H ) X 2 1 2 is nothing but its h-adjoint '⇤. (2) Show that, if ' is self-adjoint with respect to h,thenregardingD(')=[D,'] k k+1 ? as a C1-linear morphism E H E H,wehaveD(')⇤ = [D ,']. X X ⌦ ! X ⌦ Exercise 4.7. The goal of this exercise is to prove the identity (4.2.10). Recall D =

Dh0 + ✓00 and D = Dh00 + ✓0. ? ? (1) Prove that ✓0 = i[⇤,✓00] and ✓00 = i[⇤,✓0]. (2) Deduce from the Kähler identities of Section 4.2.9 that ? ? ? D00 = i[⇤,D0], D = i[⇤, D], D = i[⇤, D]. (3) Conclude by using Lemma 4.2.7 and the computation of standard Kähler iden- b tities. (4) Show also that D? = i[⇤,Dc].

Exercise 4.8 (Formulas for a holomorphic subbundle). We keep the setting of Exer- cise 4.4.Let(H1,D1) be a flat holomorphic subbundle of (H,D),i.e.,H1 is stable by D0 and D00.Let⇡ : H H be the orthogonal projection (so that ⇡ ⇡ = ⇡). We ! 1 still denote by D the connection on End(H),sothatD⇡ ⇡D = D(⇡). (1) Show the following relations for D, D1 and ⇡ : (a) ⇡D(⇡)=D(⇡) and D(⇡)⇡ =0.[Hint :forthefirstone,usethat⇡ D ⇡ = D ⇡ ;forthesecondone,usethat,forasectionv of H , D(v) is a section of 1 E1 H ,sothat⇡(D(v)) = D(v).] X ⌦ 1 (b) D = ⇡ D ⇡ = D ⇡ = ⇡ D + D(⇡). 1 (2) Show that (D0 + D00)=⇡ (D0 + D00) ⇡.[Hint :recallthat(D0 + D00) is the 1 1 Chern connection for (H, h,D00) and use [GH78,Lem.p.73]).] c (3) Show a similarb relation for (Db00 + D0) and deduce a similar relationb for D . c c c c c (4) Conclude that D1D1 + D1D1 = ⇡(DD + D D)⇡ + D(⇡)D (⇡). b

4.5. Comments Although one can trace back the notion of variation of Hodge structure to the study of the Legendre family of elliptic curves in the nineteenth century, the modern 98 CHAPTER 4. VHS ON A COMPLEX MANIFOLD approach using the Gauss-Manin connection goes back to the fundamental work of Griffiths [Gri68, Gri70a, Gri70b]motivatedbythepropertiesoftheperioddomain (see also [Del71c], [CMSP03]), a subject that is not considered in the present text. In the work of Griffiths, the transversality property (4.1.1)hasbeenemphasized. From the point of view of D-modules, this property is now encoded in the notion of acoherentfiltration,andisattheheartofthenotionoffilteredD-module, which is part of a Hodge module as defined by Saito. The notion of a polarized variation of Hodge structure can be regarded as equivalent to the notion of a smooth polarized Hodge module. However, this equivalence is not obvious since the definition of a polarized Hodge module imposes properties on nearby cycles along any germ of holomorphic function, while the notion of variation only requires to consider coordinate functions. The C1 approach as in Definition 4.1.5 proves useful for extending the Hodge theorem on smooth complex projective varieties and constant coefficients to the case when the coefficient system is a unitary local system (see [Dem96]) and the more general case when it underlies a polarized variation of Hodge structure (Hodge-Deligne theorem 4.2.16 explained in the introduction of [Zuc79]). It is also well-adapted to the extension of this theorem to harmonic bundles, as explained by Simpson in [Sim92]. In this smooth context, the flat sesquilinear pairing s gives rise in a natural way to the (non-flat in general) Hermitian Hodge metric. The fixed-part theorem, proved in Remark 4.2.18(4), is originally due to Griffiths [Gri70a]inageometricsetting,and has been proved in a more general context by Deligne [Del71b, Cor. 4.1.2], and also by Schmid [Sch73, Th. 7.22]. We have also mentioned the case of complete Kähler manifolds, going back to Andreotti and Vesentini [AV65]andHörmander[Hör65, Hör66]. Theorem 4.2.33 is taken from [Dem96,§12B].TheyareusefulforunderstandingtheL2 approach as in Zucker’s theorem 6.9.1 of [Zuc79]. It is remarkable that the local system underlying a polarized variation of Hodge structure on a smooth complex projective variety (or a compact Kähler manifold) is semi-simple. This property, proved by Deligne in the presence of a Z-structure (see [Del71b, Th. 4.2.6]), can be regarded as a special case of a result of Corlette [Cor88]and[Sim92], since the Hodge metric is a pluri-harmonic metric on the cor- responding flat holomorphic bundle. These articles are at the source of Sections 4.2.b and 4.3.b.Exercises4.4 and 4.8 are extracted from [Sim90]and[Sim92]. Last, the structure theorem for polarized variations of Hodge structure (Theorem 4.3.13)isnothingbut[Del87,Prop.1.13].