Mixed Hodge Structures

Mixed Hodge structures

8.1 Deﬁnition of a mixed Hodge structure · Deﬁnition 1. A mixed Hodge structure V = (VZ,W·,F ) is a triple, where:

(i) VZ is a free Z-module of ﬁnite rank.

(ii) W· (the weight filtration) is a finite, increasing, saturated filtration of VZ (i.e. Wk/Wk−1 is torsion free for all k). · (iii) F (the Hodge filtration) is a finite decreasing filtration of VC = VZ ⊗Z C, such that the induced filtration on (Wk/Wk−1)C = (Wk/Wk−1)⊗ZC defines a Hodge structure of weight k on Wk/Wk−1. Here the induced filtration is

p p F (Wk/Wk−1)C = (F + Wk−1 ⊗Z C) ∩ (Wk ⊗Z C).

Mixed Hodge structures over Q or R (Q-mixed Hodge structures or R-mixed Hodge structures) are deﬁned in the obvious way. A weight k Hodge structure V is in particular a mixed Hodge structure: we say that such mixed Hodge structures are pure of weight k.

The main motivation is the following theorem:

Theorem 2 (Deligne). If X is a quasiprojective variety over C, or more generally any separated scheme of finite type over Spec C, then there is a · mixed Hodge structure on H (X; Z) mod torsion, and it is functorial for k morphisms of schemes over Spec C. Finally, W`H (X; C) = 0 for k < 0 k k and W`H (X; C) = H (X; C) for ` ≥ 2k. More generally, a topological structure definable by algebraic geometry has a mixed Hodge structure: relative cohomology, punctured neighborhoods, the link of a singularity, . . . Two extreme cases of the theorem are compact but possibly singular varieties or smooth but not necessarily compact varieties. The mixed Hodge structures look somewhat dual to each other in these two cases. In the first k k k case, the weight filtration on H (X; C) satisfies: WkH (X; C) = H (X; C), k k k and hence W`H (X; C) = H (X; C) for all ` ≥ k. Also, W`H (X; C) = 0 if ` < 2k − 2n. Of course, this is only interesting if k > n. In case X k is smooth but not necessarily compact, W`H (X; C) = 0 for all ` < k. k k k k Moreoer, W2nH (X; C) = H (X; C), and hence W`H (X; C) = H (X; C) for all ` ≥ 2n; again, this is only interesting if k > n.

1 We describe the mixed Hodge structure of a curve C in more detail: Nodal curves: Suppose that, for simplicity, C is an irreducible curve with at worst nodes as singularities. (It is not hard to modify the above discussion to the case where C is not necessarily irreducible.) Let ν : Ce → C be the −1 normalization map. For each node xi of C, write ν (xi) = {pi, qi}. Then we have an exact sequence

1 1 1 0 → W0H (C; Z) → H (C; Z) → H (C; Z) → 0, 1 ∗ 1 1 where W0H (C; Z) = Ker{ν : H (C; Z) → H (C; Z)}. In terms of hypercohomology, let C· be the complex

1 OC → ν∗Ω . Ce · 1 Then C is a subcomplex of ν∗Ω , which is quasi-isomorphic to ν∗ , and we Ce C have an exact sequence

· 1 0 → C → ν∗Ω → ν∗O /OC → 0. Ce Ce Here ν O /O is a skyscraper sheaf with stalk at each of the double ∗ Ce C C points. Taking hypercohomology gives

0 0 d 1 1 H (C; C) → H (Ce; C) → C → H (C; C) → H (Ce; C) → 0. 0 0 Since C is irreducible, and hence Ce is connected, H (C; C) → H (Ce; C) is 1 1 1 ∼ an isomorphism, and hence W0H (C; C) = Ker{H (C; C) → H (Ce; C)} = d C . Of course, a similar statement holds over Z, and it is easy to identify 1 1 W0H (C; Z) with H1(C; Z)/ν∗(H (Ce; Z)); a set of representatives for this quotient is given by paths γi on Ce with γi(0) = pi and γi(1) = qi. Open smooth curves: Suppose that C = C − S, where C is a smooth projective curve and S is a ﬁnite set. Let S = {x1, . . . , xs}. Then we have an exact sequence

1 1 s−1 0 → H (C; Z) → H (C; Z) → Z → 0, s−1 where we can identify Z with the subgroup of the free abelian group Z[S] Ps P consisting of formal sums i=1 ni · xi such that i ni = 0. In this case, we 1 1 set W1 = H (C; Z) and W2 = H (C; Z). The quotient W2/W1 is dual to the subspace of H1(C; Z) generated by small loops in C around the xi (which are 1 homologous to 0 in C). As we shall show, H (C; C) is the hypercohomology of the complex Ω· (log S) on C deﬁned by C 1 OC → ΩC (x1 + ··· + xs).

2 The inclusion Ω· → Ω· (log S) gives an exact sequence C C

s · · M 0 → ΩC → ΩC (log S) → Cxi [−1] → 0, i=1 where the map Ω1 (log S) → Ls [1] is given by taking the residue C √ i=1 Cxi −1 at xi (up to a factor of (2π −1) if we want to make sure the integral 1 1 Ls structure is correct). Here, the image of H (C; C) in H (C; i=1 Cxi [−1]) = 0 Ls P H (C; i=1 Cxi ) consists of s-tuples (t1, . . . , ts) such that i ti = 0, by the residue theorem: the exact sequence

s 0 1 0 1 M 1 1 H (C;ΩC ) → H (C;ΩC (x1 + ··· + xs)) → Cxi → H (C;ΩC ) → 0 i=1 implies that the image of H0(C;Ω1 (x + ··· + x )) in [S] has codimension C 1 s C one and is otherwise constrained by the residue theorem.

8.2 The category of mixed Hodge structures

Definition 3. Let V1 and V2 be two mixed Hodge structures. A morphism of mixed Hodge structures is a homomorphism ϕ:(V1)Z → (V2)Z such that ϕ preserves the weight filtrations (i.e. ϕ(WkV1) ⊆ WkV2 for all k), and the extension of ϕ to a linear map (V1)C → (V2)C preserves the Hodge filtrations. k In particular, the induced maps on grW are morphisms of Hodge structures for all k. Sub-mixed Hodge structures are then defined in the obvious way: V 0 ⊆ V is a sub-mixed Hodge structure of V if it carries a mixed Hodge structure and the inclusion is a morphism of mixed Hodge structures. As we will see, this implies that the mixed Hodge structure on V 0 is the one 0 0 induced from the mixed Hodge structure on V , i.e. WkV = WkV ∩ V and p 0 p 0 F V = F V ∩ V . For example, WkV is a sub-mixed Hodge structure of V for all k. One could also define morphisms of degree r, generalizing the above (morphisms of degree 0), by requiring that ϕ(WkV1) ⊆ Wk+2rV2 for all k and p p+r k ϕ(F V1) ⊆ F V2 for all p, so that the induced maps on grW are morphisms of Hodge structures of type (r, r). However, as in the case of morphisms of Hodge structures, we can always reduce to morphisms of weight 0 via Tate twists, as we describe below.

Convention: we will write V for VZ, VQ, VR, or VC, relying on the context to make precise which we mean, unless we have to avoid confusion. We

3 can also speak of Q-morphisms or R-morphisms of mixed Hodge structures. p Finally, we shall often write Wk for either WkV or WkVC, and similarly F for F pV , when V is clear from the context. Given the definition of morphisms, mixed Hodge structures form a category, and we can carry out the usual linear algebra operations. For example, the direct sum V1 ⊕ V2 is defined in the obvious way: Wk(V1 ⊕ V2) = WkV1 ⊕ WkV2, and similarly for the Hodge filtration. Likewise, tensor product: X Wk(V1 ⊗ V2) = WaV1 ⊗ WbV2; a+b≤k p X a b F (V1 ⊗ V2) = F V1 ⊗ F V2. a+b≥p

Example 4. If Z(k) is the Tate Hodge structure of weight −2, then V (k) = V ⊗ Z(k) is the Tate twist of V . For Hom, we deﬁne:

Ws Hom(V1,V2) = {ϕ ∈ Hom(V1,V2): ϕ(WkV1) ⊆ Wk+sV2}; r p p+r F Hom(V1,V2) = {ϕ ∈ Hom(V1,V2): ϕ(F V1) ⊆ F V2}.

0 In particular, W0 Hom(V1,V2)Z ∩ F Hom(V1,V2) is the group of morphisms of mixed Hodge structures from V1 to V2.

Theorem 5. If ϕ: V1 → V2 is a morphism, or R-morphism, of mixed Hodge structures, then ϕ is strict with respect to both the weight and Hodge ﬁltra- tions, i.e.

Im ϕ ∩ WkV2 = ϕ(WkV1); p p Im ϕ ∩ F V2 = ϕ(F V1).

To prove the theorem, we use the existence of a Deligne decomposition:

Theorem 6. Let V be an R-mixed Hodge structure. Then there exist vector p,q subspaces I ⊆ VC such that: L p,q L p,q (i) WkVC = p+q≤k I , and in particular VC = p,q I . p L r,q (ii) F V = r≥p I . Hence

p,q q,p I = I mod Wp+q−1.

4 (iii) The construction is functorial with respect to morphisms of mixed Hodge structures. Proof. Deﬁne   p,q p q X q−i+1 I = (F ∩ Wp+q) ∩ F ∩ Wp+q + F ∩ Wp+q−i . i≥2

p,q p,q p By deﬁnition, I ⊆ Wp+q, I ⊆ F , and

p,q p q I = F ∩ F ∩ Wp+q mod Wp+q−2.

The construction is also clearly functorial with respect to morphisms of mixed Hodge structures. p,q p,q Claim 7. The natural map I → (Wp+q/Wp+q−1) is an isomorphism. First note that Claim 7 implies the theorem: in fact, by the deﬁnition of a Hodge structure,

M p,q ∼ I = WkVC/Wk−1VC; p+q=k M r,s ∼ p I = F (WkVC/Wk−1VC). r+s=k,r≥p

L p,q L r,q Then an easy induction shows that p+q≤k I = WkVC and r≥p I = F pV . p,q To prove Claim 7, clearly the image of the map I → Wp+q/Wp+q−1 is p q p,q contained in F (Wp+q/Wp+q−1) ∩ F (Wp+q/Wp+q−1) = (Wp+q/Wp+q−1) , and we must show this map is an isomorphism. p,q p Injectivity: Given α ∈ I , write α = β1 + γ1 ∈ F ∩ Wp+q, where β1 ∈ q P q−i+1 F ∩ Wp+q and γ1 ∈ i≥2 F ∩ Wp+q−i ⊆ Wp+q−2. If the image of α q in Wp+q/Wp+q−1 is 0, then α, β1 ∈ Wp+q−1, and hence β1 ∈ F ∩ Wp+q−1. Under the induced map Wp+q−1 → Wp+q−1/Wp+q−2, β1 and α have the same image in Wp+q−1/Wp+q−2, which thus lies in

p q p,q F (Wp+q−1/Wp+q−2) ∩ F (Wp+q−1/Wp+q−2) = (Wp+q−1/Wp+q−2) = 0.

P q−i+1 0 Thus α, β1 ∈ Wp+q−2. Write γ1 ∈ i≥2 F ∩ Wp+q−i as γ + γ2, where 0 q−1 P q−i+1 γ ∈ F ∩ Wp+q−2 and γ2 ∈ i≥3 F ∩ Wp+q−i ⊆ Wp+q−3. Then

0 q q−1 q−1 β2 = β1 + γ ∈ (F ∩ Wp+q−2) + (F ∩ Wp+q−2) ⊆ F ∩ Wp+q−2.

5 p q−1 Thus α = β2 + γ2, where α ∈ F ∩ Wp+q−2, β2 ∈ F ∩ Wp+q−2, and γ2 ∈ Wp+q−3. Hence α and β2 have the same image in Wp+q−2/Wp+q−3, p,q−1 and in fact their common image lies in (Wp+q−2/Wp+q−3) = 0. Thus β2 ∈ Wp+q−3, and hence α = β2 + γ2 ∈ Wp+q−3. Iterating this argument, we see that α = βr + γr, with

q−r+1 βr ∈ F ∩ Wp+q−r; X q−i+1 γr ∈ F ∩ Wp+q−i ⊆ Wp+q−r−1. i≥r+1

As above, we conclude that α, βr have the same image in Wp+q−r/Wp+q−r−1, p,q−r+1 which must lie in (Wp+q−r) = 0. Thus, α, βr ∈ Wp+q−r−1. By induction, α ∈ Wp+q−r for all r ≥ 0. Choosing r 0, we see that α = 0. p,q Surjectivity: suppose that x ∈ (Wp+q/Wp+q−1) . Then there exist α1 ∈ p q F ∩ Wp+q, β1 ∈ F ∩ Wp+q, both of which map to x under the quotient map Wp+q → Wp+q/Wp+q−1. In particular, there exists a w1 ∈ Wp+q−1 such that α1 = β1 + w1. Since Wp+q−1/Wp+q−2 has a Hodge structure of weight · p q ∼ p + q − 1 induced by F , F (Wp+q−1/Wp+q−2) ⊕ F (Wp+q−1/Wp+q−2) = 0 p 0 q Wp+q−1/Wp+q−2. Hence there exists α ∈ F ∩ Wp+q−1, β ∈ F ∩ Wp+q−1, 0 0 such that w1 = α + β + w2 with w2 ∈ Wp+q−2. Then

0 0 α2 = α1 − α = β1 + β + w2 = β2 + w2,

p q say, where α2 ∈ F ∩ Wp+q, β2 ∈ F ∩ Wp+q, and w2 ∈ Wp+q−2. Likewise, p q−1 ∼ using F (Wp+q−2/Wp+q−3) ⊕ F (Wp+q−2/Wp+q−3) = Wp+q−2/Wp+q−3, 00 p q−1 there exist α ∈ F ∩ Wp+q−2, γ2 ∈ F ∩ Wp+q−2, and w3 ∈ Wp+q−3 such 00 that w2 = α + γ2 + w3. Then

00 α3 = α2 − α = β2 + γ2 + w3.

Repeating this construction, we arrive at

r−1 X αr = β2 + γi + wr, i=2

p q−i+1 with αr ∈ F ∩ Wp+q, γi ∈ F ∩ Wp+q−i and wr ∈ Wp+q−r. For p r 0, Wp+q−r = 0, and thus we have found an αr ∈ (F ∩ Wp+q) ∩ q P q−i+1 p,q F ∩ Wp+q + i≥2 F ∩ Wp+q−i whose image in (Wp+q/Wp+q−1) is x as claimed.

6 p,q Proof of Theorem 5. Given ϕ: V1 → V2, let I1 be the summands in the p,q Deligne decomposition for V1 and deﬁne I2 similarly. By functoriality, p,q p,q ϕ(I1 ) ⊆ I2 . We show that Im ϕ ∩ WkV2 = ϕ(WkV1). If ϕ(v) ∈ WkV2, L p,q P p,q p,q p,q then ϕ(v) ∈ p+q=k I2 . Write v = p,q v with v ∈ I1 , and let 0 P p,q 0 v = p+q=k v . Then ϕ(v) = ϕ(v ) ∈ ϕ(WkV1). Hence ϕ is strict with · respect to W·. The argument for F is similar.

Corollary 8. Let ϕ: V1 → V2 be a morphism of mixed Hodge structures. Then there are natural induced mixed Hodge structures on Ker ϕ, Im ϕ, and Coker ϕ (mod torsion).

Remark 9. Thus the category of, say Q-mixed Hodge structures is an abelian category. However it is deﬁnitely not semisimple, as we shall see below–in fact, this is an essential feature. Furthermore, there doesn’t seem to be a natural deﬁnition of polarizations for mixed Hodge structures.

Deﬁnition 10. An extension of mixed Hodge structures is an exact sequence π 0 → V1 → E −→ V2 → 0, where V1, V2, E are all mixed Hodge structures and all maps are morphisms of mixed Hodge structures. Two such extensions E,E0 are isomorphic extensions if there exists an isomorphism of mixed Hodge structures from E 0 to E inducing the identity on V1 and V2. Note that every mixed Hodge structure is an iterated extension of pure Hodge structures, via the exact sequence

0 → Wk−1V → WkV → WkV/Wk−1V → 0.

1 We can deﬁne the set of isomorphism classes extensions Ext (V2,V1) of two mixed Hodge structures.

Proposition 11. If V1 and V2 are two mixed Hodge structures, then the set 1 Ext (V2,V1) of all extensions of V2 by V1 is isomorphic to

0 W0 Hom(V2,V1)C/(F Hom(V2,V1) ∩ W0 Hom(V2,V1)C) + W0 Hom(V2,V1)Z.

Proof. Let E be an extension with π : E → V2 the given map. Let r : E → V1 be an integral retraction, i.e. a homomorphism r : EZ → (V1)Z such that r|(V1)Z = Id, such that r preserves the weight filtration. Note that two 0 different choices r, r differ by an element a: EZ → (V1)Z which is 0 on (V1)Z and hence (using strictness of morphisms) defines an elementa ¯ ∈

7 W0 Hom(V2,V1)Z. We continue to write r for the extension to a linear map EC → (V1)C. Let s:(V2)C → EC be a section of π (i.e. s ◦ π = Id) preserving the weight and Hodge filtrations; this is possible using the existence of the Deligne decomposition. Two different choices s, s0 differ by an element b:(V2)C → EC whose image lies in V1, and which then (again using strict- 0 ness of morphisms) defines an element of F Hom(V2,V1)∩W0 Hom(V2,V1)C. Define e = r ◦ s ∈ Hom(V2,V1). Since both r and s preserve the weight filtrations, e ∈ W0 Hom(V2,V1)C. Replacing r by r + a and s by s + b as above replaces e by e + a ◦ s + r ◦ b + a ◦ b.

Since a is 0 on V1 and Im b ⊆ V1, a ◦ b = 0. Also, we can identify a ◦ s with 0 a¯ ∈ W0 Hom(V2,V1)Z and r ◦ b = b ∈ F Hom(V2,V1) ∩ W0 Hom(V2,V1)C. Thus the image of e in 0 W0 Hom(V2,V1)C/(F Hom(V2,V1) ∩ W0 Hom(V2,V1)C) + W0 Hom(V2,V1)Z is well-defined. Note that the map ϕ = (r, π): E → V1⊕V2 is an isomorphism of Z-modules and ϕ(WkE) = WkV1 ⊕ WkV2. Via the isomorphism ϕ, p p p ϕ(F E) = {(v1, v2) ∈ V1 ⊕ V2 : v2 ∈ F V2, v1 − e(v2) ∈ F V1}. Conversely, given a class in 0 W0 Hom(V2,V1)C/(F Hom(V2,V1) ∩ W0 Hom(V2,V1)C) + W0 Hom(V2,V1)Z, lift it to an element e of W0 Hom(V2,V1)C. Then define a mixed Hodge structure on V1 ⊕V2 as follows: the integral structure is given by (V1 ⊕V2)Z = (V1)Z ⊕ (V2)Z, the weight filtration is given by Wk(V1 ⊕ V2) = WkV1 ⊕ WkV2, and the Hodge filtration is given by p p p F (V1 ⊕ V2) = {(v1, v2) ∈ V1 ⊕ V2 : v2 ∈ F V2, v1 − e(v2) ∈ F V1}.

Then one checks that this deﬁnes a mixed Hodge structure on V1 ⊕ V2, independent of the choice of lift, and that the two constructions are inverse to each other.

Corollary 12. In the above situation, suppose that all of the weights oc- W curring in V1, i.e. the k such that grp V1 6= 0, are strictly smaller than the nonzero weights occurring in V2. Equivalently, suppose that there exists an integer k such that WkV1 = V1 and WkV2 = 0. Then 1 ∼ 0 Ext (V2,V1) = Hom(V2,V1)C/F Hom(V2,V1) + Hom(V2,V1)Z.

Moreover, Hom(V2,V1)Z is a discrete subgroup of the complex vector space 0 Hom(V2,V1)C/F Hom(V2,V1).

8 Proof. First, W0 Hom(V2,V1) = Hom(V1,V2) as follows from the assumption on the weights. Thus the first statement is a consequence of Proposition 11. To see the second statement, suppose more generally that V is a mixed Hodge structure such that all of the weights occurring in V are negative, 0 0 and define the intermediate Jacobian J V = VC/F VC ⊕ VZ. We claim that 0 the projection of VZ is a discrete subgroup of the vector space VC/F VC. 0 0 0 0 By assumption, F VC ∩ F VC = 0, since a nonzero element of F VC ∩ F VC 0,0 would define a nonzero element of (Wk/Wk−1) for some k < 0. Thus, 0 the map VR → VC/F VC is an injection of real vector spaces. Since VZ is a 0 discrete subgroup of VR, its image in VC/F VC is also discrete. k,k−1 Suppose for example that V1 is pure of odd weight 2k−1, with only h k−1,k k,k k and h nonzero, and V2 is pure of weight 2k with V2 = V2 = F V2. After a Tate twist, we may as well assume that V1 is pure of weight 1 and 1,1 1 effective and V2 is pure of weight 2 with V2 = V2 = F V2. Then

0 1 −1 F Hom(V2,V1) = Hom(V2,F V1) ⊆ F Hom(V2,V1) = Hom(V2,V1).

0 1 Thus Hom(V2,V1)/F Hom(V2,V1) = Hom(V2,V1/F V1). It is then easy to 0 1 1 1 check that J Hom(V2,V1) = Hom((V2)Z,J V1), where J V1 = V1/F V1 + 0 1 r (V1)Z is a complex torus. Thus J Hom(V2,V1) = (J V1) is also a complex torus, where r is the rank of (V2)Z. k,k Dually, suppose that V1 is pure of even weight 2k with V1 = V1 , and k,k+1 k+1,k that V2 is pure of odd weight 2k + 1, with only h and h nonzero. After a Tate twist, we can assume that k = 0. Then

0 1 0 1 F Hom(V2,V1) = Hom(V2/F V2,F V1) = Hom(V2/F V2,V1) −1 ⊆ F Hom(V2,V1) = Hom(V2,V1).

0 ∼ 1 1 ∨ Then Hom(V2,V1)/F Hom(V2,V1) = Hom(F V2,V1) = (F V2) ⊗ V1, so that

0 1 ∨ 1 ∨ ∨ J Hom(V2,V1) = (F V2) ⊗ V1/ Hom(V2,V1)Z = ((F V2) /(V2)Z) ⊗Z (V1)Z. Example 13. (i) Suppose that C is an irreducible nodal curve with normalization ν : Ce → C, and that C is obtained from Ce by identifying pi, qi ∈ Ce 1 1 to a node xi ∈ C. If s is a section of H (C) → H (Ce) preserving the Hodge filtration, then s(ω) = ω under the identifications F 1H1(Ce) = H0(Ce;Ω1 ) = Ce 1 1 1 1 1 F H (C). With W0 = Ker{H (C) → H (Ce)}, a retraction of W0 → H (C) defined over the integers is the same thing as a section σ :(W )∨ → H (C; ). 0 Z 1 Z ∗ 1 ∨ Explicitly, r = σ : H (C; Z) = H1(C; Z) → (W0)Z. The section σ amounts

9 to choosing a set of paths γi in Ce with γi(0) = pi and γi(1) = qi. Then, 0 1 ∨ using the fact that the Jacobian JCe is H (Ce;Ω ) /H1(Ce; ), Ce Z 0 1 ∼ r r ◦ s ∈ J Hom(H (Ce),W0) = JCe ⊗Z (V1)Z = (JCe) is the vector deﬁned by

Z q1 Z qr ω 7→ ω, . . . , ω ∈ (JCe)r. p1 pr

(ii) Let C = C − S, where C is a smooth projective curve and S = {x1, . . . , xs}. Then W2/W1 is pure of weight two, all of type (1, 1), and P is identified with (t1, . . . , ts): i ti = 0. Fix i 6= j. Then there exists a section ϕ of Ω1 (x + ··· + x ) which is holomorphic except at x and x , C 1 s i j with 1 √ Resx ϕ = 1; 2π −1 j 1 √ Resx ϕ = −1, 2π −1 i and ϕ is determined up to adding a holomorphic form on C. Choose an identification of C with a polygon P with 4g sides identified, corresponding to a standard symplectic basis α1, . . . , αg, β1, . . . , βg for H1(C; Z), with xi, xj in the interior of P . Here the 4g sides of P , in order, are −1 −1 αi, βi, αi , βi , i = 1, . . . , g. The choice of loops then corresponds to a section σ : H1(C; Z) → H1(C; Z), and equivalently to a retraction r = ∗ 1 1 σ : H (C; Z) → H (C; Z). We need to compute r◦s on the class corresponding to xj −xi, or equivalently r(ϕ). Here, following the discussion on the last page, we want to compute r(ϕ) as a function on F 1H1(C; ) = H0(C;Ω1 )∨, Z C i.e. given ω ∈ H0(C;Ω1 )∨, we want to compute the element r(ϕ) ∧ ω. C Z xj Claim 14. −r(ϕ) ∧ ω = ω. Hence, up to sin, the extension class is xi s−1 identified with the Abel-Jacobi map α: Z → JC, which takes a divisor of degree zero supported on S to its image under the Abel-Jacobi map. ∗ ∗ ∗ ∗ 1 ∗ Proof. Let α1, . . . , αg, β1 , . . . , βg be the dual basis for H (C; Z). Thus α1(αj) = ∗ ∗ Pg ∗ δij and α1(βj) = 0 for all j, and similarly for βi . Then r(ϕ) = i=1 siαi + P2g ∗ Pg ∗ P2g ∗ i=g+1 siβi , and similarly ω = i=1 tiαi + i=g+1 tiβi . Here (R α ϕ, for i ≤ g; si = i R ϕ, for i > g, βi

10 Pg and similarly for ti and ω. Then r(ϕ) ∧ ω = i=1(sitg+i − sg+iti), since ∗ ∗ ∗ ∗ 1 α1, . . . , αg, β1 , . . . , βg is a standard symplectic basis for H (C; Z). So it is enough to show that

g X Z xj − (sitg+i − sg+iti) = ω. i=1 xi To prove this, note that the holomorphic 1-form ω has an antiderivative F in the simply connected polygon P . Applying the residue theorem to the meromorphic 1-form F ϕ gives Z √ √ F ϕ = 2π −1 Resxj (F ϕ) + 2π −1 Resxi (F ϕ) ∂P Z xj = F (xj) − F (xi) = ω. xi On the other hand, calculating the integral of F ϕ along ∂P means that we −1 −1 have to evaluate the integral along the sides αi, βi, αi , βi , i = 1, . . . , g. −1 At a point x ∈ αi, under the identiﬁcation of αi with α , ϕ is unchanged R i but F changes by the period tg+i = ω. Thus βi Z Z Z Z (F ϕ) + (F ϕ) = (F ϕ − (F + tg+i)ϕ) = −tg+i ϕ = −sitg+i. −1 αi αi αi αi

−1 Likewise, at a point of βi, using the identiﬁcation of αi with α , F changes R i by the period −1 ω = −ti. Thus αi Z Z Z Z (F ϕ) + (F ϕ) = (F ϕ − (F − ti)ϕ) = ti ϕ = sg+iti. −1 βi βi βi βi Combining, we see that

g X Z Z xj −r(ϕ) ∧ ω = (sitg+i − sg+iti) = F ϕ = ω, i=1 ∂P xi completing the proof of the claim.

(iii) Let S be a smooth algebraic surface with h2,0(S) = 0, so that H2(S) = H1,1(S). Let C be a smooth curve on S, and set S = S − C. As we shall see, the mixed Hodge structure on H2(S) is given as an extension

2 2 1 0 → H (S)/C[C] → H (S) → H (C)(−1) → 0.

11 Here, the Hodge ﬁltration on H2(S) is given by F 2H2(S) = H0(S;Ω2 (log C)), S which in our case is just the space of meromorphic 2-forms on S with at worst a pole along C:Ω2 (log C) = Ω2 (C). Moreover, there is an exact sequence S S (Poincar´eresidue)

2 2 1 0 → ΩS → ΩS(C) → ΩC → 0. In particular, since H0(S;Ω2 ) = 0 by assumption, every holomorphic 1-form S ω on C lifts uniquely to a meromorphic√ 2-form ϕ on S with at worst a simple pole along C, such that ResC ϕ = 2π −1ω. This describes the section s(ω) of the Hodge ﬁltration that we need. To determine the extension class, 2 which is an element of JC ⊗ (H (S;)/Z[C]), we need an integral retraction 2 2 H (S; Z) → H (S;)/Z[C]. Dually, we need a section of the homomorphism H2(S; Z) → H2(S; Z)0, where

H2(S; Z)0 = {α ∈ H2(S; Z): α • [C] = 0}. Here • is the intersection product in homology, dual to cup product in cohomology. Explicitly, we can ﬁnd such a section σ as follows: by the assumption that h2,0(S) = 0, all cohomology in S is algebraic. Hence, given a class α ∈ H2(S; Z) with α • [C] = 0, we can assume that α = [D2] − [D1], where the Di are smooth curves in S meeting C transversally at distinct points (i.e. D1 ∩ D2 ∩ C = ∅). In particular, we can write D1 ∩ C = {p1, . . . , pr}, D2 ∩C = {q1, . . . , qr}, where the two sets of points are disjoint. For every i, choose a path γi ⊆ C with γi(0) = pi and γi(1) = qi. Let τ(γi) be a small tube over γi in S, oriented by the orientation in γi plus the counterclockwise orientation in the normal S1. Then we can make an oriented 2-cycle σ(α), deﬁning an element of H2(S; Z), by cutting out a small disk around pi in D1, cutting out a small disk around qi in D2, and gluing in τ(γi) with the given orientation to make a 2-cycle in S. To compute Z s(ω), use the fact that, as s(ω) is holomorphic, its restriction to any σ(α) open subset of Di is 0. Then Z X Z X Z qi s(ω) = s(ω) = ω. σ(α) i τ(γi) i pi Equivalently, the extension class is given by the natural homomorphism

{L ∈ Pic S : deg(L|C) = 0} → JC deﬁned by L 7→ L|C ∈ Pico C = JC.

12 8.3 Logarithmic poles and filtration by order of pole Let X be a complex manifold, X = X − D, where X is also a complex SN manifold and D = i=1 Di a divisor in X with simple normal crossings. In other words, the Di are smooth divisors in X, and locally around each point x ∈ X, there exist local analytic coordinates z1, . . . , zn at x such that D is defined by z1 ··· zd = 0 for some d, 0 ≤ d ≤ n. Let j : X → X be the inclusion. For each subset I ⊆ {1,...,N} with I = {i1, . . . , i`}, i1 < ··· < i`, we define DI as follows:

DI = Di1 ∩ · · · ∩ Di` .

Note that DI is smooth (possibly empty) of dimension n − `. For example, [`] ` [1] `N D{i} = Di. Let D = #(I)=` DI . For example, D = i=1 Di is the normalization of D. Formally set D[0] = X. There are two sheaves of diﬀerential forms with poles on X that we shall study.

Deﬁnition 15. With notation as above, deﬁne [ Ω· (∗D) = lim Ω· (kD) = Ω· (kD) ⊆ j Ω· . X −→ X X ∗ X k k≥0

Thus Ω· (∗D) is the sheaf of meromorphic diﬀerentials on X with poles X along D. It is clearly a complex, since if ϕ is a holomorphic form in a neighborhood of x ∈ X, and f is a local equation for D, then

−k −k+1 −k · d(f ϕ) = −kf ϕ + f ϕ ∈ ΩX ((k + 1)D). In fact, this would be true for an arbitrary divisor D, not necessarily with simple normal crossings. More generally, for V any locally free sheaf on X, we deﬁne [ V(∗D) = lim V(kD) = V(kD) = jmV ⊆ j j∗V. −→ ∗ ∗ k k≥0

We shall mostly work with a smaller version of the complex Ω· (∗D): X Deﬁnition 16. Deﬁne the log complex Ω· (log D) ⊆ Ω· (∗D) as follows: X X locally in a neighborhood U of x ∈ X, if f is a local equation for D in U, then

· · ΩX (log D)(U) = {ϕ ∈ ΩX (∗D)(U): fϕ and fdϕ are holomorphic in U}.

13 Lemma 17. Let U be an open set in X where there exist local analytic coordinates z1, . . . , zn such that D ∩ U is deﬁned by f = z1 · zd = 0 Then a basis for the holomorphic sections of Ωk (log D)(U) as a module over O (U) X X is: dzi1 dzia ∧ · · · ∧ ∧ dzj1 ∧ · · · ∧ dzjb , zi1 zia where 1 ≤ i1 < ··· < ia ≤ d and d + 1 ≤ j1 < ··· < jb, with a + b = k. n In particular, Ωk (log D) is locally free of rank , and X k

k k ^ 1 ΩX (log D) = ΩX (log D).

Finally, Ω· (log D) is a diﬀerential graded algebra under wedge product and X d.

Proof. Note that dzi/zi is closed and (dzi/zi)∧(dzi/zi) = 0. If ω is a sum of

dzi1 dzia terms of the form ∧· · ·∧ ∧dzj1 ∧· · ·∧dzjb with 1 ≤ i1 < ··· < ia ≤ d, zi1 zia then so is dω, and the set of such expressions is closed under wedge product. In particular, it forms a diﬀerential graded algebra. Also, fω and fdω are holomorphic, so that such expressions lie in Ωk (log D)(U). X Conversely, suppose that ϕ ∈ Ωk (log D)(U). Then ϕ = (1/f)ω, where X P I J ω = I,J ψI,J dz ∧ dz , where I ⊆ {1, . . . , d}, J ⊆ {d + 1, . . . , n}, #(I) + #(J) = k, and the ψI,J are holomorphic. We need to show that i ≤ d, i∈ / I =⇒ zi|ψI,J . Note that 1 fdϕ = − df ∧ ω + dω, f df so that ∧ ω is holomorphic. Since df/f = d log f is the logarithmic f derivative of f, d df X dz1 dzd = + ··· + . f z z i=1 1 d Then, for every I0 ⊆ {1, . . . , d} with #(I0) = #(I) + 1, the coeﬃcient of dzI0 ∧ dzJ is X 1 FI0,J = ± ψI0−{i},J . zi i∈I0

14 Thus, since fFI0,J = (z1 ··· zd)FI0,J is holomorphic and X ±(z1 ··· zˆi ··· zd)ψI0−{i},J = (z1 ··· zd)FI0,J , i∈I0 zi divides (z1 ··· zˆi ··· zd)ψI0−{i},J . Since z1, . . . , zd are pairwise relatively prime, zi divides ψI0−{i},J . Hence, if i∈ / I, then zi|ψI,J as desired. n The remaining statements, that Ωk (log D) is locally free of rank X k and that Ωk (log D) = Vk Ω1 (log D) follow easily from the explicit descrip- X X tion.

We note that the log complex is functorial under morphisms, in the appropriate sense, and behaves well with respect to products. Lemma 18. Suppose that X = X − D, Y = Y − E, where X and Y are complex manifolds and D,E are divisors with simple normal crossings in X, Y respectively. (i) If f : X → Y is a morphism with f −1(E) ⊆ D, then f ∗ induces a morphism of differential graded algebras Ω· (log E) → f Ω· (log D). Y ∗ X (ii) The divisor on X × Y given by (D × Y ) ∪ (X × E) has simple normal crossings, and the log complex on (X × Y ) with respect to (D × Y ) ∪ (X × E) is the exterior product π∗Ω· (log D) ⊗ π∗Ω· (log E). 1 X 2 Y Proof. (i) Let x ∈ X and let y = f(x). Locally E is defined in small open sets V containing y by w1 ··· we = 0 and D is defined in small open subsets −1 −1 of f (V ) containing x by z1 ··· zd = 0. The condition f (E) ⊆ D then ∗ n1 nd implies that f wi = hiz1 ··· zd , where the nj are nonnegative integers and hi is a nowhere vanishing holomorphic function. Then

dwi dhi X dzj f ∗ = + n ∈ Ω· (log D)(f −1(V )). w h j z X i i j j This easily implies (i). (ii) If (x, y) ∈ X × Y , then there exist neighborhoods U, V of x and y such that D is locally deﬁned by z1 ··· zd = 0 and E by w1 ··· we = 0. Thus (D × Y ) ∪ (X × E) is locally deﬁned in U × V by z1 ··· zdw1 ··· we = 0. Again, it is easy to check that this implies all of the statements of (ii).

· ∼ · · Lemma 19. Restriction induces an isomorphism H (X; C) = H (X; j∗ΩX ). i · ∼ i Moreover, the cohomology sheaves H j∗ΩX = R j∗C.

15 · Proof. It suffices to prove that ΩX is j∗-acyclic, for then · · ∼ · · ∼ · H (X; j∗ΩX ) = H (X;ΩX ) = H (X; C), i ∼ i · · with the morphism induced by restriction, and R j∗C = H j∗ΩX since ΩX is a j∗-acyclic resolution of C. q p Thus, we must show that R j∗ΩX = 0 for q > 0. Let U be a Stein open subset of X and set U = U ∩ X = j∗U = U − D. Then U is also Stein since it is the complement of a hypersurface in U. (This is formally the same argument as the proof that the complement of a hypersurface V (f) in an affine variety is affine, using the definition of a Stein manifold N Z as a closed analytic submanifold of C for some N: if Z is a closed N submanifold of C and f is a holomorphic function on Z, then f extends N to a holomorphic function on C , and Z − V (f) is isomorphic to the closed N+1 holomorphic submanifold of C defined by (Z × {0}) ∩ V (fzN+1 − 1).) q q Thus, R j∗F (U) = H (U; F |U) = 0 for every coherent sheaf F , and so, q since the Stein neighborhoods U are cofinal, R j∗F = 0 for every coherent p sheaf F , in particular for ΩX . Proposition 20. The inclusions Ω· (log D) → Ω· (∗D) → j Ω· are quasi- X X ∗ X isomorphisms.

Proof. The statement is clearly true away from D. First consider the case n = 1. We may as well assume that D 6= ∅, so the question locally becomes: For X = ∆∗ ⊆ X = ∆, with D = {0}, show that the inclusions 1 dz · · · C[0] ⊕ C √ [1] → Γ(∆; Ω∆(0)) → Γ(∆; Ω∆(∗0)) → Γ(∆; j∗Ω∆∗ ) 2π −1 z are all quasi-isomorphisms, where the left-hand complex has trivial diﬀer- ential, and the inclusions are the obvious ones. By the previous lemma,

· · ∼ · ∗ H (Γ(∆; j∗Ω∆∗ ), d) = H (∆ ; C). Thus 1 dz · C[0] ⊕ C √ [1] → Γ(∆; j∗Ω∆) 2π −1 z is a quasi-isomorphism. So it is enough to prove that the inclusions C[0] ⊕ h i h i √1 dz [1] → Γ(∆; Ω· (0)) and [0]⊕ √1 dz [1] → Γ(∆; Ω· (∗0)) C 2π −1 z ∆ C C 2π −1 z ∆ · are quasi-isomorphisms. For the complex Γ(∆; Ω∆(0)), clearly Ker d = 1 dz C and Coker d is spanned by √ . A similar statement holds for 2π −1 z

16 · Γ(∆; Ω∆(∗0)), since a meromorphic 1-form g(z)dz is equal to dF , where F is a meromorphic function, ⇐⇒ the coefficient of 1/z in the Laurent expansion of g(z) is zero. For dimension n ≥ 1, locally around a point of D we can assume that ∗ a n−a n U = (∆ ) × ∆ ⊆ U = ∆ , where D is defined by z1 ··· za = 0. Consider the complex · · ^ 1 dz1 1 dza C = C √ ,..., √ , 2π −1 z1 2π −1 za 1 dz with trivial differential, where the terms √ 1 have degree 1. We will 2π −1 z1 show that the inclusions

· n · n · n · C → Γ(∆ ;Ω∆n (log D)) → Γ(∆ ;Ω∆n (∗D)) → Γ(∆ ; j∗Ω(∆∗)a×∆n−a ) are all quasi-isomorphisms. As in the case n = 1,

r n · ∼ r ∗ a n−a H (Γ(∆ ; j∗Ω(∆∗)a×∆n−a ), d) = H ((∆ ) × ∆ ; C) r ∼ ^ 1 dz1 1 dza = C √ ,..., √ . 2π −1 z1 2π −1 za So we must check that the remaining inclusions are quasi-isomorphisms. · n · We shall just consider the inclusion C → Γ(∆ ;Ω∆n (∗D)), as the other case is similar. The proof is by induction on n, and we have dealt with the case n = 1. By arguing as in the proof of the holomorphic Poincar´e lemma, it is easy to reduce to the case a = n. Also, it is easy to check that · · n · C → H (Γ(∆ ;Ω∆n (∗D)), d) is injective, and we shall just check that it is n p surjective. Suppose that η ∈ Γ(∆ ;Ω∆n (∗D)) and that dη = 0. Assume inductively that

X dzI η = dz ∧ ω + ω + a + dρ, k 1 2 I zI I where ω1 and ω2 don’t involve dzk, . . . , dzn and aI ∈ C. We shall show that, after modifying η by an exact meromorphic form, the same statement holds P J with k replaced by k − 1. Write ω1 = J fJ dz , where #(J) = p − 1 and J ⊆ {1, . . . , k − 1}. For r > k the only terms in dη involving dzr ∧ dzk come ∂fJ from d(dzk ∧ ω1) = dzk ∧ dω1. Since dη = 0, = 0 for r > k, i.e. as a ∂zr meromorphic function, fJ = fJ (z1, . . . , zk). Likewise, the coeﬃcients of ω2

17 are meromorphic functions in z1, . . . , zk. Then there exists a meromorphic function gJ = gJ (z1, . . . , zk) such that

∂gJ hJ = fJ + , ∂zk zk where hJ is a meromorphic function in z1, . . . , zk−1. Replacing η by η − P J d( J gJ dz ), we can assume that, up to an exact meromorphic form, η is a sum of a meromorphic form not involving dzk, . . . , dzn, terms of the form I I P dzk J aI dz /z , and the form J hJ ∧ dz , where J ⊆ {1, . . . , k − 1} and hJ zk is a meromorphic function in z1, . . . , zk−1. In other words, up to an exact I I form and a sum of terms of the form aI dz /z ,

dzk η = ϕ1 + ∧ ϕ2, zk where ϕ1 is a meromorphic form in z1, . . . , zk not involving dzk and ϕ2 is a meromorphic form in z1, . . . , zk−1. Comparing the terms involving dzk in

dzk 0 = dη = dϕ1 + ∧ dϕ2, zk

dzk it follows that all of the coeﬃcients of ∧ dϕ2 are of the form zk G ∂F I = I , zk ∂zk where GI is a meromorphic function in z1, . . . , zk−1 and FI is a meromorphic function in z1, . . . , zk. This is only possible if GI = FI = 0 for all I, hence dϕ2 = 0. By the inductive hypothesis, ϕ2 = dψ plus a sum of terms of the K K dzk dzk form aK dz /z . Hence, ∧ ϕ2 = d( ∧ ψ) plus a sum of terms of the zk zk K K form aK dz /z as well. Thus, modulo exact forms and a sum of terms of K K the form aK dz /z , η is a meromorphic form not involving dzk, . . . , dzn. This completes the inductive step.

Remark 21. (i) We can write the isomorphism r ^ 1 dz1 1 dza ∼ r ∗ a n−a C √ ,..., √ = H ((∆ ) × ∆ ; C) 2π −1 z1 2π −1 za h i more invariantly as follows: a basis for Vr √1 dz1 ,..., √1 dza is C 2π −1 z1 2π −1 za given by dzI /zI , where the I run over subsets of {1, . . . , a} with r elements.

18 These correspond to the r-fold intersections of the components of ∆n − ∗ a n−a r ∼ (∆ ) × ∆ . Thus R j∗C = CD[r] . · (ii) Of course, we could have√ deﬁned the complex C in the above proof without the factors of (2π −1)−1. However, if we keep these factors, we get an isomorphism over Z: r ^ 1 dz1 1 dza ∼ r ∗ a n−a Z √ ,..., √ = H ((∆ ) × ∆ ; Z), 2π −1 z1 2π −1 za which is important later in keeping track of the integral structures.

The quasi-isomorphism Ω· (∗D) → j Ω· has the following consequence: X ∗ X Theorem 22 (Grothendieck’s algebraic de Rham theorem). If X is a smooth affine algebraic variety, then H·(X; ) = H·(Γ (X;Ω· ), d) is the coho- C Zar X/C mology of the complex of algebraic forms on X. Equivalently, if A(X) is the affine coordinate ring of X, Ω1 is the module of Kählerdifferentials A(X)/C on X, and Ωk = Vk Ω1 , then H·(X; ) = H·(Ωk , d). A(X)/C A(X)/C C A(X)/C Proof. By Hironaka’s resolution of singularities, we can choose a smooth compactification X of X such that X = X − D, where D is a divisor in X with simple normal crossings. In particular, there exists an ample divisor supported on D, since X − D is affine. Taking hypercohomology, ·(X;Ω· (∗D)) ∼ ·(X; j Ω· ). Hence H·(X; ) ∼ ·(X;Ω· (∗D)). Using H X = H ∗ X C = H X the first hypercohomology spectral sequence for Ω· (∗D) then gives a spec- X tral sequence converging to H·(X; ) with Ep,q = Hq(X;Ω· (∗D)). Since C 1 X cohomology commutes with direct limits,

Hq(X;Ω· (∗D)) = lim Hq(X;Ω· (nD)). X −→ X n Since there exists an ample divisor whose support is D, Serre’s theorem on the vanishing of cohomology implies that, for all p, all q ≥ 1, and all n 0, Hq(X;Ω· (nD)) = 0. Thus, the spectral sequence reduces at E to X 1

Ep,0 = H0(X;Ω· (∗D)) = Γ (X;Ω· ), 1 X Zar X/C · with the differential d. Thus, E2 = E∞ is the associated graded to H (X; C) with the trivial filtration. Hence H·(X; ) = H·(Γ (X;Ω· ), d). C Zar X/C Next, we define the pole order filtration on Ω· (∗D); it is a decreasing X S · filtration. First, with D = i Di, define the filtration P on OX (∗D) as

19 follows: p X X P OX (∗D) = OX ( ((ni + 1)Di)). n ≥0 i P i i ni≤−p p 0 Thus P OX (∗D) = 0 for p > 0, P OX (∗D) = OX (D), and in general p p P OX (∗D) ⊆ OX ((−p + 1)D) for p ≤ 0 and P OX (∗D) = OX ((−p + 1)D) p for p ≤ 0 if D is smooth. Clearly, if f ∈ P OX (∗D), then, for all i,

∂f p−1 ∈ P OX (∗D). ∂zi Deﬁne p k p−k k P ΩX (∗D) = P OX (∗D) ⊗ ΩX . It is easy to check that d(P pΩk (∗D)) ⊆ P pΩk+1(∗D), so that P pΩ· (∗D) X X X is a subcomplex of Ω· (∗D). Note that P pO (∗D) and hence grp Ω· (∗D) X X P X are coherent OX -modules. Also, d induces a map

p · p · grP ΩX (∗D) → grP ΩX (∗D) which is O -linear, since, for f ∈ O and ω ∈ P p−kO (∗D) ⊗ Ωk , then X X X X d(fω) = df ∧ ω + fdω, and df ∧ ω ∈ P p−kO (∗D) ⊗ Ωk+1 ⊆ P p+1Ω· (∗D). X X X Lemma 23. The filtration P ·Ω· (∗D) ∩ Ω· (log D) is the naive filtration X X σ≥pΩ· (log D). In other words, X ( 0, if k < p; P pΩk (∗D) ∩ Ωk (log D) = X X Ωk (log D), if k ≥ p. X Proof. It is an easy consequence of the definition that

p k p−k k P ΩX (∗D) = P OX (∗D) ⊗ ΩX = 0, k < p; k k 0 k k P ΩX (∗D) = P OX (∗D) ⊗ ΩX = ΩX (D).

Thus, for k ≥ p,Ωk (log D) ⊆ P pΩk (∗D). The lemma is then clear. X X Proposition 24. The morphism of ﬁltered complexes (Ω· (log D), σ≥·) → X (Ω· (∗D),P ·) is a ﬁltered quasi-isomorphism, i.e. it induces a quasi-isomor- X phism on the associated graded complexes of sheaves.

20 Proof. P pΩ· (∗D) is the complex X

p 1 p−1 n p−n P OX (∗D) → ΩX ⊗ P OX (∗D) → · · · → ΩX ⊗ P OX (∗D). Thus, grp Ωk (∗D) = Ωk ⊗ P p−kO (∗D)/Ωk ⊗ P p−k+1O (∗D). Note that P X X X X X this is zero for p > k and equal to Ωp ⊗ P 0O (∗D) for p = k. We must X X show: for p ≥ 0, the kernel of d from

Ωp ⊗ P 0O (∗D) → Ωp+1 ⊗ P −1O (∗D)/Ωp+1 ⊗ P −1O (∗D) X X X X X X is Ωp (log D), and for all p, setting a = p − k, if a ≤ 1, then X Ωk−1 ⊗ P a+1O (∗D) → Ωk ⊗ P aO (∗D)/Ωk+1 ⊗ P a−1O (∗D) X X X X X X → Ωk+1 ⊗ P a−1O (∗D)/Ωk+1 ⊗ P aO (∗D) X X X X is exact. 0 For the ﬁrst statement, since P OX (∗D) = OX (D),

Ker{d: grp Ωp (∗D) → grp Ωp+1(∗D)} P X P X consists of the meromorphic p-forms ω ∈ Ωp (D) such that dω ∈ Ωp+1(D), X X i.e. the ω such that fω and fdω are holomorphic, where f is a local deﬁning equation for D. By deﬁnition, this is Ωp (log D). X We shall just prove the second statement under the assumption that D a is smooth, so that P OX (∗D) = OX ((−a + 1)D). (The proof in general is along similar but slightly more involved lines.) Then the statement becomes: for b ≥ 2, if ω ∈ Ωk ⊗ O (bD) is such that dω ∈ Ωk+1 ⊗ O (bD), then X X X X locally h i ω ∈ Ωk ⊗ O ((b − 1)D) + d Ωk−1 ⊗ O ((b − 1)D) . X X X X

We can assume that locally X is the polydisk ∆n and that D is deﬁned by z = 0. Writing ω = z−b P a dzI , up to a term in Ωk ⊗ O ((b − 1)D) we 1 1 I I X X can assume that the aI do not involve z1, i.e. that they are given by a power series in the coordinates z2, . . . , zn. Write

−b ω = z1 (dz1 ∧ α1 + α2), where α1 and α2 do not involve dz1, hence are forms in z2, . . . , zn, dz2, . . . , dzn alone. If 1 ψ = z1−bα , 1 − b 1

21 1 then dψ = z−b(dz ∧ α ) + z1−bdα , where the second term lies in 1 1 1 1 − b 1 k k Ω ⊗ OX ((b − 1)D). Thus, after modifying ω by elements of Ω ⊗ OX ((b − X h i X 1)D)+d Ωk−1 ⊗ O ((b − 1)D) , we can assume that ω is of the form z−bα , X X 1 2 where α2 only involves z2, . . . , zn, dz2, . . . , dzn. Then dω = −bz−b−1dz ∧ α + z−bdα ∈ Ωk+1 ⊗ O (bD). 1 1 2 1 2 X X

The only terms with a dz1 come from the ﬁrst summand, but these must be 0 becuase the order of pole of dω is b. Hence α = 0. Summarizing, h i 2 ω ∈ Ωk ⊗ O ((b − 1)D) + d Ωk−1 ⊗ O ((b − 1)D) . X X X X

8.4 The weight filtration Next we define the weight filtration on Ωk (log D) and the Poincaréresidue X map. Definition 25. Let W Ω· (log D) be the increasing filtration defined by · X ( Ωr (log D) ∧ Ωk−r, if r ≤ k; k X X WrΩ (log D) = X Ωk (log D), if r > k. X Thus locally a section of W Ωk (log D) is of the form r X X dzI ∧ ψ + holomorphic , zI I I⊆{1,...,d} #(I)=r

dzI where, if I = {i , . . . , i } with i < ··· < i , we use the shorthand for 1 r 1 r zI dzi1 dzir ∧ · · · ∧ , and ψI is a holomorphic (k − r)-form such that zi1 zir

X J ψI = gI,J dz . J⊆{1,...,d}−I #(J)=k−r

Clearly, W Ω· (log D) is a subcomplex of Ω· (log D) , and r X X · · · WrΩX (log D) ∧ WsΩX (log D) ⊆ Wr+sΩX (log D). Deﬁne the Poincar´eresidue map as follows: Recall that, for a subset

I ⊆ {1,...,N} with I = {i1, . . . , ir}, i1 < ··· < ir, DI = Di1 ∩ · · · ∩ Dir and

22 [r] ` [0] D = #(I)=r DI , with D = X by convention. We have the inclusions D[r] ⊆ X, but we shall suppress them in what follows. Given a local section X dzI ω of W Ωk (log D), say ω = ∧ ψ + holomorphic , deﬁne r X zI I I⊆{1,...,d} #(I)=r

PR (ω)|D = ψ |D ∈ Ωk−r. r I I I DI

Note that this definition depends on r; in fact, PR (W Ωk (log D)) = 0. It r r−1 X is easy to see that this definition is independent of the choice of coordinates, 0 since, for 1 ≤ i ≤ d, if we replace zi by zi = hizi, where hi is a holomorphic, 0 dzi dzi dzi dhi nowhere vanishing function, then is replaced by 0 = + . Thus zi zi zi hi dz0I dzI = + element of W Ωk (log D) . z0I zI r−1 X dzI dz0I Thus, PR ( ∧ ψ ) = PR ( ∧ ψ ). In particular, it is globally well- r zI J r z0I J defined.

Putting together the various PRr deﬁnes a morphism of complexes (also denoted by PRr)

W · · PRr : grr ΩX (log D) → ΩD[r] [−r]. Lemma 26. PR : grW Ω· (log D) → Ω· [−r] is an isomorphism. r r X D[r] Proof. First, we show that PR is injective. Suppose that ω ∈ W Ωk (log D) r r X and that PRr(ω) = 0. Then, using the above notation, for all I = {i1, . . . , ir}, i1 < ··· < ir, ψI |DI = 0. Locally, the ideal of DI is (zi1 , . . . , zir ). Since the coeﬃcients gJ of ψI lie in the ideal (zi1 , . . . , zir ), it follows that ψI ∈ W Ωk (log D). Thus, PR is injective. r−1 X r k−r To see that it is surjective (as a morphism of sheaves), let ηI ∈ ΩDI , X J say ηI = hI,J dz , where the hI,J are holomorphic functions on J⊆{1,...,d}−I #(J)=k−r DI . Then, at least locally, we can extend the hI,J to holomorphic functions on X, so that ηI extends to a holomorphic (k − r)-form ψI,J on X. Clearly, dzI ω = ∧ ψ ∈ W Ωk (log D) and PR (ω)|D = η , while, for I0 6= I, zI I r X r I I 0 #(I ) = r, PRr(ω)|DI0 = 0. Hence PRr is surjective.

23 By definition, W Ωk (log D) = Ωk (log D) if r ≥ k. On the other hand, r X X for the canonical filtration τ , τ Ωk (log D) = {0} if r < k. Hence there is ≤r X always a morphism τ Ωk (log D) → W Ωk (log D). ≤r X r X Proposition 27. The morphism of filtered complexes (Ω· (log D), τ ) → X ≤r (Ω· (log D),W ) is a filtered quasi-isomorphism. X · Proof. We must show that grW Ω· (log D) → grτ Ω· (log D) is a quasi- r X r X isomorphism. Since Ω· (log D) is quasi-isomorphic to j Ω· , the cohomology X ∗ X of grτ Ω· (log D) is Hij Ω· ∼ Rrj [−r]. On the other hand, r X ∗ X = ∗C W · ∼ · grr ΩX (log D) = ΩD[r] [−r], and thus its cohomology sheaf is CD[r] [−r] (as usual, we omit the inclusion r ∼ morphisms). We have seen in Remark 21(i) that R j∗C[−r] = C [r] [−r] as h D i well, and locally both sides are isomorphic to Vr √1 dz1 ,..., √1 dza C 2π −1 z1 2π −1 za via the appropriate maps.

Next we look at the filtration induced on grW Ω· (log D) by the naive r X filtration σ≥p of Ω· (log D). Clearly: X Lemma 28. The naive filtration σ≥p of Ω· (log D) induces the shifted naive X filtration σ≥p−rΩ· [−r] on grW Ω· (log D). D[r] r X This says that there is an induced Hodge structure on

k W · k · k−r [r] H (X; grr ΩX (log D)) = H (X;ΩD[r] [−r]) = H (D ; C),

k−r [r] and it is the Hodge structure H (D ; C)(−r), up to the integral√ structure, which we will also see is correct, justifying the factors (2π −1)r. In p k−r [r] other words, the filtration F induced on on H (D ; C) by the naive filtration of Ω· (log D) is the shifted Hodge filtration F p−rHk−r(D[r]; ), X C · k−r [r] k−r [r] where F H (D ; C) is the usual Hodge filtration on H (D ; C). In particular, the natural weight is k − r + 2r = k + r. Replacing W Ω· (log D) by W r, where W r = W , gives us in the usual r X −r way a decreasing filtration on Ω· (log D), and hence a spectral sequence, the X p,q weight spectral sequence W E∗ . It is a second quadrant spectral sequence whose E1 page is

p,q p+q W · p+q · W E1 = H (X; gr−p ΩX (log D)) = H (X;ΩD[−p] [p]) 2p+q · 2p+q [−p] = H (X;ΩD[−p] ) = H (D ; C).

24 k Since we will ultimately want an increasing ﬁltration on H (X; C) it is natural to replace −p by r and 2p + q by k − r, hence q = k + r, so that we −r,k+r k−r [r] get W E1 = H (D ; C). Explicitly, the E1 page looks as follows: H1(D[2]) H3(D[1]) H5(X) H0(D[2]) H2(D[1]) H4(X) H1(D[1]) H3(X) H0(D[1]) H2(X) H1(X) H0(X) p = −2 p = −1 p = 0

Suppose that X is a separated scheme of finite type over Spec C, or more generally that X is bimeromorphic to a Kählermanifold. Clearly, all of the terms in the E1 page carry Hodge structures. For all k, r, the differen- k−r [r] k−r+2 [r−1] tials d1 : H (D ; C) → H (D ; C) are alternating combinations of Gysin maps. For example, for the smooth hypersurface Di ⊆ X, we have k k+2 H (Di) → H (X), and it is a morphism of Hodge structures of type (1, 1), k k+2 i.e. a morphism of Hodge structures H (Di)(−1) → H (X). A similar −r,k+r −r+1,k+r+2 statement holds for all the differentials d1 : W E1 → W E1 . By our discussion of the canonical filtration versus the second cohomology spectral sequence, it follows that the above E1 page is the same as the E2 p,q p q page for the Leray spectral sequence for j∗, where E2 = H (X; R j∗C) = p [q] H (D ; C). That spectral sequence looks as follows: H0(D[3]) H1(D[3]) H2(D[3]) H0(D[2]) H1(D[2]) H2(D[2]) H0(D[1]) H1(D[1]) H2(D[1]) H0(X) H1(X) H2(X) p = 0 p = 1 p = 2

k [r] and, as for d1 in the weight spectral sequence, d2 goes from H (D ) to Hk+2(D[r−1]). Theorem 29 (Deligne). Let X be a separated scheme of ﬁnite type over Spec C with X = X − D, where X is a smooth proper separated scheme and D is a divisor in X with simple normal crossings.

(i) For the above spectral sequence, W E2 = W E∞, i.e. the spectral sequence degenerates at E2. For the induced (increasing) filtration W· · −r on H (X; C) defined by Wr = W , the filtration W· is defined over the integers.

25 · (ii) The spectral sequence abutting to H (X; C) deﬁned by the naive ﬁltra- tion on Ω· (log D), i.e. whose E page is Ep,q = Hq(X;Ωp (log D)), X 1 F 1 X degenerates at E1.

· · (iii) Let F be the filtration on H (X; C) defined by the naive filtration on Ω· (log D). The two filtrations F · and W [k] define a mixed Hodge X · k structure on H (X; C).

Henceforth, we will write W· for W·[k] and will show shortly that it is independent of the choice of the pair (X,D). As a consequence of the degeneration of the weight spectral sequence at E2, we have: Ker{Hk−r(D[r]) → Hk−r+2(D[r−1])} grW Hk(X) = . k+r Im{Hk−r−2(D[r+1]) → Hk−r(D[r])} This then implies in particular: Proposition 30 (Characterization of the weight k part). Let X and X be as k k k above. Then, for all k, WkH (X; Q) is the image of H (X; Q) in H (X; Q), k k k and W`H (X; Q) = 0 for ` < k. Lastly, W`H (X; Q) = H (X; Q) for ` ≥ 2k. Remark 31. (i) In fact, one can show that, if X0 is any smooth proper k separated scheme containing X as a subscheme, then WkH (X; Q) is the k 0 k image of H (X ; Q) in H (X; Q). W k k k (ii) The computation of grk+r H (X) shows that W`H (X; Q) = H (X; Q) for ` ≥ k + n as well. More generally, suppose that D[r] = ∅ for some r (this k k always holds for r ≥ n + 1). Then, for all k, W`H (X; Q) = H (X; Q) for ` ≥ k + r − 1.

8.5 Applications Proposition 32 (Functoriality 1). Let (X,D) and (Y,E) be two pairs such that X and Y are smooth proper schemes, D and E are two divisors with simple normal crossings in X, Y respectively, and X = X − D, Y = Y − E. If f : X → Y is a morphism of schemes such that f −1(E) ⊆ D, then f ∗ : H·(Y ) → H·(X) is a morphism of mixed Hodge structures. Proof. By Lemma 18, f induces a morphism of complexes f ∗ :Ω· (log E) → Y f Ω· (log D) which clearly preserves the weight ﬁltration. Thus there is an ∗ X induced morphism

∗ · · · · f : H (Y ;ΩY (log E)) → H (X;ΩX (log D))

26 which preserves the Hodge ﬁltration. Also, f ∗ acts on the weight spectral sequence, hence preserves the induced ﬁltrations W on k(X;Ω· (log D)) · H X as well as their shifts W·[k].

Proposition 33 (Independence of the compactification). For a smooth scheme X, the mixed Hodge structure on H·(X) defined above is independent of the choice of a smooth compactification X = X − D, where D has simple normal crossings. 0 00 0 00 Proof. Suppose that X = X −D0 = X −D00, where X and X are smooth proper schemes, D0 and D00 are two divisors with simple normal crossings 0 00 in X , X respectively. Then there exists a pair (X,D) which dominates 0 00 both (X ,D0) and (X ,D00) and is an isomorphism over X. For example, 0 00 one can start with a resolution of the closure in X × X of the graph of the identity X → X, and then further resolve until the inverse image of D0 has simple normal crossings. Then the two compactifications (X,D) 0 and (X ,D0) both induce mixed Hodge structures on H·(X), and by the previous proposition the identity H·(X) → H·(X) is a morphism of mixed Hodge structures, where the domain H·(X) has the mixed Hodge structure 0 coming from (X ,D0) and the range H·(X) has the mixed Hodge structure coming from (X,D). By strictness, the two mixed Hodge structures agree (i.e the identity is an isomorphism of mixed Hodge structures). Thus the mixed Hodge structure on H·(X) coming from (X,D) agrees with that 0 coming from (X ,D0), and by symmetry the same is true for the mixed Hodge structure on H·(X) coming from (X,D). Hence all three two mixed Hodge structures on H·(X) agree.

Remark 34. Serre has constructed examples of a smooth scheme X with two analytic compactifications X1 and X2 which are not birational (or equivalently, bimeromorphic). The mixed Hodge structures on H·(X) coming ∼ ∗ ∗ from X1 and X2 in this example are different. Here, X = C × C = 1 1 P × P − (f1 ∪ f2), where f1 and f2 are fibers of the two projections 1 1 1 πi : P ×P → P . Serre exhibits an analytic isomorphism from X to PE −σ, where E is an (arbitrary) elliptic curve, E is a rank two vector bundle over E which is a non-split extension:

0 → OE → E → OE → 0,

1 ∗ ∗ 1 ∗ ∗ and σ is a section of E. But H (C × C ) satisﬁes: W1H (C × C ) = 1 1 1 0, since by Proposition 30, it is the image of H (P × P ), hence 0, and

27 1 ∗ ∗ 1 ∗ ∗ 1 1 W2H (C × C ) = H (C × C ), whereas W1H (PE − σ) = H (PE − σ); it 1 1 is isomorphic to the image of H (PE), hence isomorphic to H (E) and of rank 2. To avoid this problem, we will understand a compactiﬁcation of X to be a proper scheme containing X as a subscheme, not just as an open dense subset.

Proposition 35 (Functoriality 2). If X and Y are two smooth schemes and f : X → Y is a morphism of schemes, then f ∗ : H·(Y ) → H·(X) is a morphism of mixed Hodge structures.

Proof. Choose a smooth compactification Y of Y such that Y = Y − E, where E is a divisor with simple normal crossings. We can then find a similar compactification (X,D) such that f extends to a morphism f : X → Y with f −1(E) ⊆ D: choosing any compactification of X, first resolve the closure of the graph of f in the product and then further resolve until the preimage of E becomes a divisor with simple normal crossings. The result then follows from Propositions 32 and 33.

We have the following sharpening of Proposition 30:

Proposition 36. Let X = X − D, where X is a smooth proper scheme and D is a divisor in X with simple normal crossings. Let f : Y → X be a morphism of schemes, where Y is smooth and proper. Then, for all k, k k k H (X; Q) and H (X; Q) have the same image in H (Y ; Q). Proof. The statement of the corollary is that Im f ∗ = Im(j ◦ f)∗ = Im(f ∗ ◦ ∗ ∗ k k j ), where j : X → X is the inclusion. Since j (H (X; Q)) = WkH (X; Q) ∗ k ∗ ∗ by Proposition 30, we must show that f (WkH (X; Q)) = Im f . Since f is a morphism of mixed Hodge structures, it is strict with respect to the ∗ k ∗ k weight ﬁltration. Hence f (WkH (X; Q)) = Im f ∩ WkH (Y ; Q). Since k k ∗ ∗ Y is smooth and proper, WkH (Y ; Q) = H (Y ; Q). Thus Im(f ◦ j ) = Im f ∗.

We can then strengthen the global invariant cycle theorem as follows:

Theorem 37. Let π : X → S be a smooth projective morphism of schemes, −1 with S smooth, let s ∈ S be a point, and set Xs = π (s). If X is a smooth compactiﬁcation of X such that X is the complement in X of a divisor k π (S,s) with simple normal crossings, then H (Xs; Q) 1 is equal to the image of k k H (X ; Q) in H (Xs; Q).

28 k k Proof. By Corollary 36, since Xs is proper, H (X ; Q) and H (X ; Q) have k k π (S,s) the same image in H (Xs; Q). By Deligne’s theorem, H (Xs; Q) 1 is k k equal to the image of H (X ; Q) in H (Xs; Q). Hence the same is true for k the image of H (X ; Q). Remark 38. One can weaken the hypotheses above signiﬁcantly. For example, it is enough to assume π is proper and that the scheme X is an arbitrary smooth compactiﬁcation of X . Finally, the assumption that S is smooth is not essential.

Corollary 39 (Theorem of the ﬁxed part). Let π : X → S be a smooth projective morphism of schemes, with S smooth, as above. Then, for every k π (S,s) k s ∈ S, H (Xs) 1 is a sub-Hodge structure of H (Xs), and the Hodge k π (S,s) structure on H (Xs) 1 is independent of the choice of s.

k π (S,s) Proof. By the global invariant cycle theorem, H (Xs) 1 is the image of Hk(X ) under a morphism of Hodge structures, hence is a sub-Hodge k k k structure of H (Xs). Since the kernel of H (X ) → H (Xs) is independent k of s, the sub-Hodge structure of H (Xs) is also independent of s.

k π (S,s) Corollary 40. Under the above hypotheses, given α ∈ H (Xs; C) 1 , if P p,q k α = p+q=k α is the decomposition of α ∈ H (Xs; C) into its components p,q k π (S,s) of type (p, q), then for all p, q, α ∈ H (Xs; C) 1 . Finally, also denoting k P p,q by α the corresponding ﬂat global section of R π∗C, α = p+q=k α is the k decomposition of α ∈ H (Xt; C) into its components of type (p, q) for every t ∈ S.

k π (S,s) Proof. By strictness, the Hodge structure on H (Xs; C) 1 is that in- k duced by H (Xs; C). Hence, the decomposition of α into its components of k π (S,s) type (p, q) in H (Xs; C) 1 is the same as the decomposition into (p, q)- k p,q k π (S,s) components in H (Xs; C). In particular, α ∈ H (Xs; C) 1 for all p, q. p,q k Thus the α are ﬂat global sections of R π∗C, and the ﬁnal statement is clear.

Let S be a complex manifold. We have defined polarized variations of · Hodge structure V = (VZ, F ) over S, and they form a category in a natural way. Polarized Q-variations of Hodge structure are defined similarly. Note that the category of polarized Q-variations of Hodge structure is semisimple. Definition 41. A polarized variation of Hodge structure V over S satisfies π1(S,s) the theorem of the fixed part if, for all s ∈ S, the subspace Vs is a sub-Hodge structure of V . Equivalently, for all α ∈ (V )π1(S,s), if α = s s C

29 P p,q p+q=k α is the decomposition of α ∈ Vs into its components of type (p, q), then αp,q ∈ (V )π1(S,s) for all p, q. In this case, the Hodge structure on s C π1(S,s) π1(S,s) Vs does not depend on s, under the canonical identification of Vs π1(S,s) with Vt . We say that S satisfies the theorem of the fixed part if, for every finite étalecover Sb → S and every polarized variation of Hodge structure V over Sb, V satisfies the theorem of the fixed part.

Example 42. (i) If π1(S, s) = 1, then a polarized variation of Hodge structure V over S satisfies the theorem of the fixed part ⇐⇒ it is constant. More generally, if V is a polarized variation of Hodge structure over S satis- fying the theorem of the fixed part and the monodromy image of π1(S, s) is finite, then the period map S → Γ\D is constant. Conversely, if the period map is constant, then the monodromy image is contained in the stabilizer in G(Z) of a point x ∈ D. Since the action of G(Z) is properly discontinuous, the stabilizer of a point is finite, and hence the mondromy image of π1(S, s) is finite as well. (ii) Let π : C → ∆∗ be family of elliptic curves over the punctured disk ∆∗ 1 degenerating to a nodal rational curve. If V = R π∗C is the corresponding variation of Hodge structure, then the monodromy group of V is

1 n : n ∈ . 0 1 Z

In this case, the theorem of the fixed part does not hold for V . Griffiths and Schmid have proved the following (Griffiths for S compact, Schmid for S quasiprojective): Theorem 43. If S is either compact or quasiprojective, then S satisfies the theorem of the fixed part.

Remark 44. If S is compact and Sb → S is a finite étalecover, then Sb is compact. If S is quasiprojective and Sb → S is a finite étalecover, then Sb is quasiprojective, by Grothendieck’s Riemann existence theorem. Without using the theorem of Griffiths and Schmid, Deligne’s theorem above (Theorem 39) shows that a variation of Hodge structure coming from a family of projective varieties satisfies the theorem of the fixed part. More generally, every “motivic” family (or “family of geometric origin”) will also satisfy the theorem of the fixed part. Similarly, all variations of Hodge structure obtained from such by the standard linear algebra operations satisfy

30 the theorem of the fixed part. In what follows, we shall derive some con- sequences of the theorem of the fixed part, which apply to all variations of Hodge structure over a quasiprojective base of geometric origin, by Deligne’s theorem, and in general by the Griffiths-Schmid theorem. The first corollary emphasizes again the crucial role that monodromy plays in determining a variation of Hodge structure. Corollary 45 (Rigidity). Suppose that S is quasiprojective, and that V and W are two polarized variations of Hodge structure over S. Let f ∈

Hom(VZ,WZ) be an isomorphism of local systems, and suppose that there exists an s ∈ S such that f : Vs → Ws is an isomorphism of Hodge structures. Then f is an isomorphism of variations of Hodge structure.

π (S,s) Proof. By construction, f is an element of Hom(Vs,Ws) 1 , and it is of type (0, 0). Hence f is of type (0, 0) at every point, and so is an isomorphism of variations of Hodge structure.

Theorem 46 (Semisimplicity). Suppose that S is quasiprojective, with s ∈ S, and that V is a polarized variation of Hodge structure over S. Then the π1(S, s)-action on Vs is semisimple, i.e. Vs is a direct sum of irreducible π1(S, s)-invariant subspaces. Moreover, the Zariski closure of the image of π1(S, s) in GL(Vs) is a semisimple algebraic group. Proof. Clearly, the second statement implies the ﬁrst, since a subspace of Vs is invariant under π1(S, s) ⇐⇒ it is invariant under the Zariski closure. To prove the ﬁrst statement, we introduce the generic Mumford-Tate group M(V ).

First, pull back V to the universal cover Se of S, so that VZ becomes a constant local system, and identify Vt with Vs for the ﬁxed s ∈ S. Recall that, for a given t ∈ S, M(Vt) is the set of g ∈ GL(Vt) such that g(α) = λα n,m n,m for every Hodge tensor α ∈ T (V )Q of type (k, k), where T (V ) = ⊗n ∨ ⊗m (V ) ⊗ (V ) . For α ∈ Vs, deﬁne

n,m Z(α) = {t ∈ Se : α is of type (k, k) in T (Vt)}.

Then Z(α) is a closed analytic subvariety of Se, since, assuming that the Hodge structure on T n,mV has weight 2k, α is of type (k, k) ⇐⇒ α ∈ k n,m k n,m k n,m F T (Vt) ∩ F T (Vt) ⇐⇒ α ∈ F T (Vt), since α is rational and n,m k n,m hence real, ⇐⇒ the image of α in Vs/T F T (Vt) is zero. Hence either Z(α) = Se or Z(α) is a proper analytic subvariety of Se. Deﬁne

[ n,m Te = {Z(α): α ∈ T (VQ) for some n, m, and Z(α) 6= Se}.

31 Thus Te is a countable union of proper subvarieties of Se. It is easy to see that Te is invariant under the π1(S, s) action, since, for γ ∈ π1(S, s), γ · Z(α) = Z(γ · α). Then Te ⊆ Se and T = Te /π1(S, s) ⊆ S are both second category subsets, and hence there exists a t ∈ S with t∈ / T . We shall refer to such a t as a very general t. Moreover, for every t ∈ S − T , the Mumford-Tate group M(Vt) is independent of T : M(V ) is the set of all g ∈ GL(Vt) = GL(Vs) such that g(α) = λα for all α such that Z(α) = Se. Also, since M(V ) is an algebraic group, we can just use the Hodge classes α ∈ T n,m(V ) for finitely ni,mi many n, m ≥ 0: there exists α1, . . . , αr ∈ T (V )Q such that g ∈ M(V ) ⇐⇒ g(αi) = λiαi for all i. For t ∈ S − T , we denote M(Vt) by M(V ) and call it the generic Mumford-Tate group of V . (On S, this only defines M(V ) up to an identification of Vt with Vs, and hence up to conjugation by π1(S, s), but we will see that a finite subgroup of ρ(π1(S, s)) is contained in M(Vt), so this is in fact well-defined after passing to a finite étalecover of S.) Note that, for all t ∈ S, M(Vt) ⊆ M(V ). Also, we may as well use a very general point s as a base point for π1(S, s). Next, we define the connected monodromy group H: H is the identity component of the Zariski closure ρ(π1(S, s)) of the image ρ(π1(S, s)) in GL(Vs). In particular, H ∩ ρ(π1(S, s)) is a subgroup of ρ(π1(S, s)) of finite index, since H is a finite index subgroup of ρ(π1(S, )). Theorem 47. H is a subgroup of M(V ), and in fact it is a normal subgroup.

Proof. First, we show that H is a subgroup of M(V ). As above, we choose ni,mi ﬁnitely many Hodge classes α1, . . . , αr ∈ T (V )Q such that g ∈ M(V ) ni,mi ⇐⇒ g(αi) = λiαi for all i. Consider the span of the α ∈ (T Vs)Q such n ,m that α is of Hodge type (ki, ki) in T i i Vt for all t ∈ S, i.e. Z(α) = S, where n ,m the weight of T i i V is 2ki. Since Z(γ · α) = γ · Z(α) for all γ ∈ π1(S, s), n ,m the α as above span a π1(S, s)-invariant subspace Si of T i i Vs deﬁned over Q, containing αi, and of pure type (ki, ki). In other words, Si is a n ,m k ,k polarized sub-Hodge structure Si of T i i (V ) i i . In this case, the image of the monodromy ρSi : π1(S, s) → GL(Si) is a subgroup of the orthogonal group O(di), where di = dim Si. In particular O(di) is compact, and thus

ρSi (π1(S, s))∩O(di) is a discrete subgroup of a compact group, hence is ﬁnite.

If Ki = Ker ρSi , then Ki is a ﬁnite index subgroup of π1(S, s). Doing this construction for all i produces a ﬁnite index subgroup K = K1 ∩ · · · ∩ Kr of

π1(S, s) such that, if γ ∈ K, then ρSi (γ)(αi) = αi for all i, and in particular ρ(γ) ∈ M(V ). Then the Zariski closure ρ(K) ⊆ M(V ) as well. Since ρ(K) is a subgroup of ﬁnite index in ρ(π1(S, s)), H ⊆ ρ(K) ⊆ M(V ). Note that

32 this part of the argument did not use the theorem of the fixed part. Now let us prove that H is in fact a normal subgroup of M(V ). As we saw in the section on Mumford-Tate groups, since H is an algebraic n,m subgroup of GL(Vs), there exists a tensor representation T (Vs) = T and a line L ⊆ T , defined over Q, such that H is the stabilizer of L in GL(Vs). Then π1(S, s) sends an integral generator λ of L to ±λ, hence there is an index two subgroup of π1(S, s) which restricts to the identity on L. Since H is connected, H also fixes L pointwise, i.e. L ⊆ T H , where by definition T H = {t ∈ T : h · t = t for all h ∈ H}. We claim that T H is stable under M(Vs). As we have seen, M(Vs)-invariant subspaces of T correspond to sub-Hodge structures of T . So it suffices to prove that T H is a sub-Hodge structure. The subgroup H ∩π1(S, s) is a subgroup of finite index in π1(S, s), H H∩π (S,s) and T = T 1 , since H ∩ π1(S, s) is Zariski dense in H. Let Sb → S be the finite étalecover corresponding to the subgroup H ∩ π1(S, s). Then, by the theorem of the fixed part, T H = T H∩π1(S,s) is a sub-Hodge structures of T . Since T H is stable under M(V ), we get a homomorphism ϕ: M(V ) → GL(T H ). By definition, H ⊆ Ker ϕ. Also, if g ∈ Ker ϕ, then g(L) = L, since L ⊆ T H . Then g ∈ H. So Ker ϕ = H, and in particular H is a normal subgroup of M(V ).

Finally, let us complete the proof of the semisimplicity theorem. By Theorem 47, H is reductive. To prove that it is semisimple, it suffices to prove H has no almost direct factor which is a torus, or equivalently that identity component Z(H)0 of the center Z(H) is trivial. It suffices to prove 0 that Z(H) ∩ρ(π1(S, s)) is finite. By the general theory of reductive groups, every connected reductive group is an almost direct product of its connected center (the identity component of the center) and a number of semisimple normal subgroups, and every connected normal subgroup is an almost direct product of a connected subgroup of the center together with some of the semsimple normal subgroups. Thus, since H is a normal subgroup of M(V ), Z(H)0 is a subgroup of the center ZM(V ) of M(V ). In particular, 0 0 every element of Z(H) commutes with h(S). Thus, every element of Z(H) 0 stabilizes the Hodge structure on Vs. Hence, Z(H) ∩ρ(π1(S, s)) is a discrete subgroup of the stabilizer K of the polarized Hodge structure Vs in G(R), 0 for an appropriate group G. Since K is compact, Z(H) ∩ ρ(π1(S, s)) is finite. Thus Z(H)0 is finite and hence trivial.