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MIXED HODGE STRUCTURES ON FUNDAMENTAL GROUPS

MA LUO

Contents

Introduction 11. Iterated Integrals 42. Bar Constructions 73. Chen’s π1-de Rham Theorem 174. Basics of Mixed Hodge Theory 185. Mixed Hodge Structures on Fundamental Groups 276. Deligne’s Canonical Extension 337. Limit Mixed Hodge Structures 408. Regularised Periods 44

This is the lecture notes for a graduate course given at Oxford in Michaelmasterm 2019. References can be found on the course website. The objective is to studymixed Hodge structures on fundamental groups. Most topics and the presentationare taken from a course I have learned from Richard Hain in Duke University. Sincethis is hastily typed up, any typos and mistakes are made by me. I would appreciateany corrections and suggestions. Please send them to [email protected].

Introduction

The first interesting topological invariant of a space X one learns in algebraictopology is usually the fundamental group, also known as the first homotopy group.For each point x ∈ X, define

PxX :=γ∣∣∣γ : [0, 1]→ X (piecewise) smooth, γ(0) = γ(1) = x

to be the loop space. With the appropriate1 topology, its path connected compo-nent π0(PxX,x) can be identified with the fundamental group π1(X,x). Higherhomotopy groups can be inductively defined as

πk+1(X,x) ∼= πk(PxX,x).

We want to study these homotopy groups via differential forms, but issues needto be overcome:

• π3(S2) ∼= Z, but E3(S2) = 0• π1 is non-abelian, but

∫αβω =

∫αω +

∫βω =

∫βαω. Question: how to

detect its commutators?

1compact open.

1

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mailto:[email protected]

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Kuo-Tsai Chen has discovered a generalization of the usual line integral as fol-lows. Given a path γ and differential 1-forms ω1, · · · , ωr on X. Define an iteratedline integral by

(0.1)

∫γ

ω1 · · ·ωr =

∫· · ·∫

0≤t1≤···≤tr≤1

f1(t1)f2(t2) · · · fr(tr)dt1 · · · dtr

where γ∗ωj = fj(t)dt. It is a time ordered integral.

Example 0.1. Suppose that we have lines l1, · · · , lr and a path γ on a plane. Forsimplicity, we assume that the path γ(t) transversely passes each line exactly once,at time t = aj . For each line lj , we have a generalized differential form (also knownas current or distribution) ωj , such that

εj :=

∫γ

ωj = #(lj · γ) = ±1,

which is the intersection number of lj with γ and the sign depends on orientation.More precisely, we have γ∗ωj = fj(t)dt with

fj(t) = εj δ(t− aj),

where δ(t− aj) is the Dirac delta function centered at t = aj . Now we have

∫γ

ω1 · · ·ωr =

ε1 · · · εr if a1 < a2 < · · · < ar

0 otherwise

For the path illustrated below∫γω1ω3 = 0,

∫γω3ω1ω2 = −1,

∫γω3ω4ω1ω2ω5 = 1.

γ(0) γ(1)

l3 l1 l4 l2 l5

Example 0.2. Let X = C − −1, 1. The homology group is H1(X;Z) = Z ·a1⊕Z · a2 where aj is the homology class [σj ]. The cohomology group H1(X;Z) isgenerated by the two (non-compact) cycles l1 and l2 pictured below.

MIXED HODGE STRUCTURES ON FUNDAMENTAL GROUPS 3

−1 0 1

l1 l2

σ1 σ2

x1 x2

The homology class of a cycle γ : S1 → X is

[γ] = #(l1 · γ)a1 + #(l2 · γ)a2.

From the previous example, by using iterated integrals, one knows the order anddirection in which a path γ passes through l1 and l2, thus we know its homotopyclass in π1(X, 0) ≈< x1, x2 >. The homotopy class of the path γ illustrated belowis x2

1x−12 x−1

1 x−12 .

γ

−1 0 1

l1 l2

Remark 0.3. Usual integrals (of differential forms/currents) detect non-trivial ho-mology classes of an oriented manifold X by an abelian intersection theory, whileiterated integrals detect non-trivial elements of π∗(X) by a non-abelian intersectiontheory.

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1. Iterated Integrals

1.1. Path space. For X a smooth manifold, define its path space

PX := (piecewise) smooth paths on X.

Its topology is given by the compact open topology.Given topological spacesX, Y , Z, one can defineXY := continuous maps Y → X.

The compact open topology has universal mapping property:

(Zα−→ XY continuous) ⇐⇒ (Y × Z φα−−→ X continuous)

where φα(y, z) = α(z)(y).We have an evaluation map

PXe−→ X ×X

γ 7→ (γ(0), γ(1))

Define subspaces of PX:

Px,yX := e−1(x, y); PxX := e−1(x, x).

1.2. Differential forms on PX. Question: When is Uα−→ PX(⊂ XI) smooth?

Shall we just ask φα : I × U → X to be smooth?Chen in his Bulletin paper defines “differentiable spaces”.

Definition 1.1. A differentiable space is a set M and a collection of maps

(α : U →M where U is open in Rd)

called “plots”, such that

(1) every constant map U →M is a plot

(2) if Uα−→ M is a plot and V ⊆ Re is open and θ : V → U is smooth, then

α θ is a plot(3) “local property”: φ|open covering is a plot =⇒ φ is a plot

Example 1.2. PX is a differentiable space. Plots:

α : U → PX is a plot

⇐⇒ ∃ a partition 0 = t0 < · · · < tn = 1 s.t. φα : I × U → X restricted to each

[tj−1, tj ]× U is smooth.

Definition 1.3. M , N are differentiable spaces. We say F : M → N is smooth iffor each plot α : U →M , F α is a plot.

Remark 1.4. (1) every plot is smooth(2) every manifold is a differentiable space

Definition 1.5. A k-form on a differentiable space M is a family (ωα) for every plot

Uα−→M , where ωα ∈ Ek(U) such that if θ : V → U is smooth, then θ∗ωα = ωαθ.

Denote the k-forms on M by Ek(M). For ω ∈ Ek(M), η ∈ El(M), definedω ∈ Ek+1(M) by dω = (dωα), and ω ∧ η ∈ Ek+l(M) by ω ∧ η = (ωα ∧ ηα).Therefore, E•(M) is a differential graded algebra (d.g.a.).


M N

U

F

α

Fα If F : M → N is smooth, we have pull-back F ∗ : E•(N) →

E•(M), ω 7→ F ∗ω, where (F ∗ω)α = ωFα ∈ E•(U).

1.3. Complex of iterated integrals/Chen complex Ch•(PX). Now we go

back to M = PX. Have

PX γ

X ×X (γ(0), γ(1))

e where e = (p0, p1). One can

check this is smooth. (In fact, pt is smooth ∀t ∈ [0, 1]). So we have

p∗t : E•(X)→ E•(PX)

and

p∗0 ⊗ p∗1 : E•(X)⊗ E•(X)→ E•(PX)

ω′ ⊗ ω′′ 7→ p∗0ω′ ∧ p∗1ω′′

Definition 1.6. For ωj ∈ Enj (X), define∫(ω1| · · · |ωr) = π∗ϕ

∗(1× ω1 × · · · × ωr × 1)

where

(1) ∆r = time ordered simplex = (t1, · · · , tr) : 0 ≤ t1 ≤ · · · ≤ tr ≤ 1(2) ϕ : ∆r × PX → X × Xr × X is the sampling map, ((t1, · · · , tr), γ) 7→

(γ(0), γ(t1), · · · , γ(tr), γ(1)). (This map is smooth)(3) π∗ denotes integration over the fiber of the projection π : ∆r×PX → PX.

Remark 1.7. If any ωj ∈ E0(X), then∫

(ω1| · · · |ωr) = 0. Reason: for pt : [0, 1] ×PX → X, (t, γ) 7→ γ(t), and ω ∈ E•(X), we can write p∗tω = dt ∧ ϕ(t) + ψ(t). Ifω ∈ E0(X) then ϕ = 0. But

ϕ∗(1× ω1 × · · · × ωr × 1) = ±dt1 ∧ · · · ∧ dtr ∧ ϕ1(t1) ∧ · · · ∧ ϕr(tr) + (lower degree terms)

=∫

∆r ϕ1 ∧ · · · ∧ ϕr

π∗

We will always assume nj = deg(ωj) ≥ 1.

Definition 1.8 (Variant). Let ω′, ω′′ ∈ E•(X) with degrees n′, n′′. Then have

(∗) π∗ϕ∗(ω′ × ω1 × · · · × ωr × ω′′) = p∗0ω

′ ∧∫

(ω1| · · · |ωr) ∧ p∗1ω′′ ∈ En(PX)

where n = (n1 + · · ·+ nr − r) + n′ + n′′.

Definition 1.9.

Ch•(PX) = iterated integrals = linear span of (∗)’s in E•(PX)

Remark 1.10. One can restrict this to subspaces Px,yX, PxX of PX and obtainCh•(Px,yX), Ch•(PxX).

Need

(1) Show that Ch•(PX) is a d.g.a.

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(2) Have formulas for d, ∧.

Proposition 1.11 (Formulas).

(1)

d

∫(ω1| · · · |ωr) =

r∑j=1

±∫

(ω1| · · · |dωj | · · · |ωr)

+

r∑j=2

±∫

(ω1| · · · |ωj−1 ∧ ωj | · · · |ωr)

±∫

(ω1| · · · |ωr−1) ∧ p∗1ωr ± p∗0ω1 ∧∫

(ω2| · · · |ωr)

(2)∫

(ω1| · · · |ωr) ∧∫

(ωr+1| · · · |ωr+s) =∑

σ∈Sh(r,s)

±∫

(ωσ(1)| · · · |ωσ(r+s)).

Proof. For (2): (follows by standard triangulation of ∆r × ∆s) Definition for ashuffle of type (r, s): a permutation σ such that σ−1(1) < σ−1(2) < · · · < σ−1(r)and σ−1(r + 1) < σ−1(r + 2) < · · · < σ−1(r + s). Let

∆r = (t1, · · · , tr) : 0 ≤ t1 ≤ · · · ≤ tr ≤ 1,

∆s = (tr+1, · · · , tr+s) : 0 ≤ tr+1 ≤ · · · ≤ tr+s ≤ 1.For each point (t1, · · · , tr; tr+1, · · · , tr+s) ∈ ∆r × ∆s, there is a shuffle σ of type(r, s) such that

0 ≤ tσ(1) ≤ tσ(2) ≤ · · · ≤ tσ(r+s) ≤ 1.

This σ is unique if tj ’s are distinct. So

∆r ×∆s =⋃

σ∈Sh(r,s)

(tσ−1(1), · · · , tσ−1(r+s)) : 0 ≤ t1 ≤ · · · ≤ tr+s ≤ 1

=⋃

σ∈Sh(r,s)

∆r+sσ .

For any plot α : U → PX, we have

α∗(∫

(ω1| · · · |ωr) ∧∫

(ωr+1| · · · |ωr+s))

=

∫0≤t1≤···≤tr≤1

ϕ1(t1) ∧ · · · ∧ ϕr(tr) ∧∫

0≤tr+1≤···≤tr+s≤1

ϕr+1(tr+1) ∧ · · · ∧ ϕr+s(tr+s)

=

∫∆r×∆s

ϕ1(t1) ∧ · · · ∧ ϕr+s(tr+s)

=∑

σ∈Sh(r,s)

∫∆r+sσ

ϕ1(t1) ∧ · · · ∧ ϕr+s(tr+s)

=∑

σ∈Sh(r,s)

∫0≤t1≤···≤tr+s≤1

ϕ1(tσ−1(1)) ∧ · · · ∧ ϕr+s(tσ−1(r+s))

=∑

σ∈Sh(r,s)

±∫

∆r+s

ϕσ(1)(t1) ∧ · · · ∧ ϕσ(r+s)(tr+s)

= α∗( ∑σ∈Sh(r,s)

±∫

(ωσ(1)| · · · |ωσ(r+s)))


For (1): needs basic formula (essetially uses Stoke’s formula) [Add proof/leaveas exercise]

π∗d± dπ∗ = (∂π)∗

where we have

(j-th face)× PX ' ∆r−1 × PX (∂∆r)× PX ∆r × PX

PX PX PX

πj ∂π π

and define (∂π)∗ :=∑rj=0(−1)j(πj)∗.

Remark 1.12. Formula (1) implies that Ch•(PX) is a sub-complex of E•(PX).This formula becomes simpler for Ch•(Px,yX) and Ch•(PxX) when we restrict tosubspaces of PX.

Natural Question: What is H•(Ch•(PX))?The geometric definition of iterated integrals does not help answer this question.

We will provide an answer in Section 3, after an algebraic description of iteratedintegrals up next.

2. Bar Constructions

2.1. Simplicial and cosimplicial objects. Denote the category of finite ordinalsby ∆. The objects are the finite ordinals [n] := 0, 1, · · · , n with the natural order0 < 1 < · · · < n, and the morphisms are order preserving functions f : [m] → [n].Note that f is not necessarily 1-1.

Remark 2.1. One can think of [n] as the set of vertices of ∆n.

We have for each 0 ≤ j ≤ n a face map

dj : [n− 1] −→ [n]

0 1 · · · j − 1 j j + 1 · · · n− 1

0 1 · · · j − 1 j j + 1 · · · n− 1 n

that is the unique order preserving injection that omits the value j, and a degen-eracy map

sj : [n+ 1]→ [n]

0 1 · · · j − 1 j j + 1 · · · n n+ 1

0 1 · · · j − 1 j j + 1 · · · n

that maps both j and j + 1 to j.

Remark 2.2. Every map f can be expressed as composites of dj ’s and sj ’s.

Remark 2.3. Each f : [m] → [n] induces a simplicial map |f | : ∆m → ∆n suchthat f(ej) = ef(j) and then extends linearly via barycentric coordinates (see thefollowing Side).

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Side.

(1) Standard n-simplex has barycentric coordinates

∆n = (s0, · · · , sn) : sj ≥ 0,

n∑j=0

sj = 1.

Vertices are ej = (0, · · · , 1, · · · , 0) where 1 is at the j-th position. Faces of∆n are given by

j-th face: sj = 0, ej /∈ j-th face.

(2) Time ordered n-simplex

∆n = (t1, · · · , tn) : 0 ≤ t1 ≤ · · · ≤ tn ≤ 1.

There is a 1-1 correspondence

(s0, · · · , sn)↔ (t1, · · · , tn)

given by

sj = tj+1 − tj , j = 0, · · · , nwhere we set t0 = 0, tn+1 = 1. Conversely,

tj = s0 + · · ·+ sj−1, j = 1, · · · , n.

Faces are

j-th face: sj = 0, i.e. tj = tj+1 .

Definition 2.4. A simplicial object in a category C is a contravariant functor

F : ∆→ C.

Using categorical notation, it is in C∆op

.

Diagrams: ∆0 ∆1 ∆2 · · ·d0

d1

d0

d2

d0

d3

· · ·F2 F1 F0

d0

d2

d0

d1

Example 2.5 (Simplicial set).

K• : ∆→ Sets

[n] 7→ Kn = “set of n-simplices”

Each simplicial set K• has a geometric realization

|K•| =( ∐n≥0

Kn ×∆n)/ ∼

where ∼ is a natural equivalence relation generated by identifications for each mor-phism f : [m]→ [n] of ∆.

e.g. standard n-simplex ∆n• ,

∆nm = Hom∆([m], [n]).

In particular, we define I• := ∆1•.

Claim:

|∆n• | = ∆n.


Example 2.6. Let X be a topological space, define a simplicial set Simp•X with

SimpnX = σ : ∆n → X = singular n-simplices of X.

Fact:

|Simp•X| → X

is a weakly homotopy equivalence, i.e. induces isomorphisms on all πj ’s and Hj ’s.

Example 2.7 (Simplicial space/variety/scheme). We usually use a simplicial spaceX• to model a nodal curve X, i.e. we have a simplicial covering map

X X•

ε

The geometric realization |X•| of X• is

=

which is homotopy equivalent to X.

A functor C → D induces C∆op → D∆op

on the corresponding simplicial objects.

Example 2.8. We have

Sets→ R−Mod

Σ = σ 7→⊕σ∈Σ

R = free R-module on Σ

SimpnX 7→ Sn(X;R) = singular n-chains.

This induces

Top Sets∆op

(R−Mod)∆op

(chain complexes of R-modules)

X Simp•X S•(X) (S•(X), ∂)

(M•, dj) (M•, d =∑nj=0(−1)jdj)

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Definition 2.9. A cosimplicial object in C is a covariant functor G : ∆ → C.

Diagram: G0 G1 G2 · · ·d0

d1

d0

d2

d0

d3

Example 2.10.

Sets→ R−Mod

S 7→ RS = functions S → R = (rσ : σ ∈ S)

SimpnX 7→ Sn(X;R) = singular cochains

The differential on the cosimplicial R-module Sn(X) is given by

δ =∑j

(−1)jdj

where dj are coface maps.

2.2. Cosimplicial model of PX. For X a topological space, define

P •X := XI• = Hom(I•, X),

where I• is the simplicial model of the unit interval (see Example 2.5). Note thatthere is a natural map PX → ||P •X|| which is not a homotopy equivalence exceptwhen X is simply connected.

We first look closely into I•. By definition,

I• := Hom∆([•], [1])

andIn = Hom∆([n], [1]).

In fact, In consists of (n+ 2) order preserving maps of the form

0 · · · j − 1︸︷︷︸ j · · · n︸︷︷︸0 1

For each j = 0, · · · , n+ 1, the map above corresponds to the j-th bipartition

0, · · · , j − 1 | j, · · · , nof [n] (although for j = 0, n + 1, it is not actually a bipartition). So we will alsorefer to elements/maps in In as bipartitions. These bipartitions pull back alongorder preserving maps. For example, for the face map

dj : [n− 1] −→ [n]

0 1 · · · j − 1 j j + 1 · · · n− 1

0 1 · · · j − 1 j j + 1 · · · n− 1 n

Pullback along dj induces a map

In = Hom∆([n], [1])→ In−1 = Hom∆([n− 1], [1])

which, expressed in terms of bipartitions, is the degeneracy map

sj : [n+ 1]→ [n]


0 1 · · · j − 1 j j + 1 · · · n n+ 1

0 1 · · · j − 1 j j + 1 · · · n

because both the j-th and the (j + 1)-th bipartitions of [n]

0, · · · , j − 1 | j, j + 1, · · · , n and 0, · · · , j − 1, j | j + 1, · · · , n

pull back to the j-th bipartition of [n− 1]

0, · · · , j − 1 | j, · · · , n− 1.

From the above discussion, we have

PnX = XIn = Hom(In, X) ∼= Xn+2

and the map dj : [n − 1] → [n] induces In → In−1, which in turn induces a cofacemap

dj : XIn−1 → XIn

(x0, · · · , xn) 7→ (x0, · · · , xj , xj , · · · , xn)

This is the coface map that we use to define the cosimplicial model P •X.

Example 2.11 (Another cosimplicial space). ∆• is given by

∆0 ∆1 ∆2 · · ·d0

d1

d0

d2

d0

d3

Note that the coface maps is given by

dj : ∆n−1 → ∆n

(t1, · · · , tn−1) 7→ (t1, · · · , tj , tj , · · · , tn)

which are “dual” to the face maps dj in ∆.Take

∆• × PX → P •X

with

∆n × PX → PnX = Xn+2

((t1, · · · , tn), γ) 7→ (γ(0), γ(t1), · · · , γ(tn), γ(1))

in degree n. This is compatible with coface maps, so we have a smooth map ofcosimplicial spaces.

Applying the de Rham complex, we have

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P •X simplicial d.g.a

......

deg n Xn+2 E•(X)⊗(n+2)

......

deg 2 X4 E•(X)⊗4

deg 1 X3 E•(X)⊗3

deg 0 X2 E•(X)⊗2

E•

······

······

d3d0d0 d3

d2d0d0 d2

d1d0d0 d1

In particular, the coface map dj : XIn−1 → XIn induces the face map

dj : E•(X)⊗(n+2) → E•(X)⊗(n+1)

ω0 ⊗ · · · ⊗ ωn+1 7→ ω0 ⊗ · · · ⊗ ωj ∧ ωj+1 ⊗ · · · ⊗ ωn+1

because the diagonal map diag : X → X ×X induces

E•(X)⊗ E•(X) E•(X)

E•(X ×X)

−∧−

∼ diag∗

From this, we have

simplicial d.g.a double complex

(E•(X)⊗(•+2), d) (E•(X)⊗(•+2), d, δ)

where d is the de Rham differential as usual, and

δ =∑j

(−1)jdj

is given by taking alternating sum of the face maps (cf. Example 2.8). One cancheck that d and δ commute.

Example 2.12. In this double complex, we have

d(ω0 ⊗ · · · ⊗ ωn+1) =

n+1∑j=0

± ω0 ⊗ · · · ⊗ ωj−1 ⊗ dωj ⊗ ωj+1 ⊗ · · · ⊗ ωn+1


and

δ(ω0 ⊗ · · · ⊗ ωn+1) =

n∑j=0

(−1)jω0 ⊗ · · · ⊗ ωj ∧ ωj+1 ⊗ · · ·ωn+1.

These are basically differentials in the bar construction B(E•(X), E•(X), E•(X)),which we will discuss next (cf. the differential d

∫(ω1| · · · |ωn) in Proposition 1.11

(1)).

2.3. Bar constructions. Both the bar construction and the reduced bar construc-tion can be combined easily with Hodge theory. We discuss the bar constructionfirst.

Given

(1) A• a d.g.a. (with differential d of degree +1). e.g. C•(X) = singularcochains, E•(X) = de Rham complex.

(2) M•, N• are cochain complexes(3) M• = right A•-module, N• = left A•-module, such that the structure maps

M• ⊗A• →M• and A• ⊗N• → N•

are chain maps

The bar construction B(M•, A•, N•) (will be simply denoted by B(M,A,N)) is adouble complex, with

B−s,t(M,A,N) = [M ⊗A⊗s ⊗N ]t

consisting of elements m[a1| · · · |as]n such that deg(m) +∑j deg(aj) + deg(n) = t.

Note that the total degree of m[a1| · · · |as]n ∈ B−s,t(M,A,N) in B(M,A,N) is(−s) + t = t− s.

Define an endomorphism J of each graded vector space by J : v 7→ (−1)deg vv.Define the bar differential by dB = d+ δ (see Example 2.12) so that

dB [a1| · · · |as] =∑j

±[a1| · · · |daj | · · · |as] +∑j

±[a1| · · · |aj ∧ aj+1| · · · |as].

The total differential for the double complex B(M,A,N) is

D = d⊗ + dC =: D(0,1) +D(1,0)

= dM ⊗ 1T ⊗ 1N + JM ⊗ dB ⊗ 1N + JM ⊗ JT ⊗ dN + dC

where the combinatorial differential dC is defined by

dC(m[a1| · · · |as]n) = (−1)sJm[a1| · · · |as−1]as · n+m · (−1)s deg(a1)a1[a2| · · · |as]n.There is a standard filtration, called bar filtration, on B(M,A,N) with

Bs(M,A,N) =⊕r≤s

B−r,t(M,A,N).

Remark 2.13. Note that

B(M,A,N) =⋃s≥0

Bs(M,A,N) = lim−→s

Bs(M,A,N),

so we have

H•(B(M,A,N)) = lim−→s

H•(Bs(M,A,N)).

This leads us to Eilenberg–Moore spectral sequence (EMss). It has pages

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• E−s,t0 = [M ⊗A⊗s ⊗N ]t with differential d0 = D(0,1)

• E−s,t1 = [H•(M)⊗H•(A)⊗s ⊗H•(N)]t with differential d1 = D(1,0).

Remark 2.14. E1 is B(H•(M), H•(A), H•(N)) with total differential D = D(1,0).

With the natural map

B(R, E•(X),R)→ Ch•(Px,yX)

[ω1| · · · |ωs] 7→∫

(ω1| · · · |ωs)

in mind, terms such as [f1| · · · |fn] where fj ’s are functions should be redundant,cf. Remark 1.7. Note that these terms have negative total degrees. The followingtheorem confirms our expectation.

Definition 2.15. Let A• be a d.g.a. over k. It is connected if

Aj =

0, j < 0

k, j = 0.

It is homologically connected if H•(A•) is connected, i.e.

Hj(A•) =

0, j < 0

k, j = 0.

For example, A• = E•(X) for X a connected manifold.

Theorem 2.16. If

(1) A• is homologically connected(2) Hj(M•) and Hj(N•) vanish for j < 0

then Hj(B(M,A,N)) = 0 when j < 0.

To get rid of these negative degree terms, we introduce the reduced bar con-struction B(M,A,N), it will have a nicer EMss. Suppose A• is connected andhomologically connected, then A0 = k. Write

A• = k ⊕ (IA•, d)

deg≥1

.

Set

B−s

(M,A,N) = M ⊗ (IA)⊗s ⊗N ⊆ B−s(M,A,N) = Bs(M,A,N)

then B(M,A,N) ⊆ B(M,A,N) is a subcomplex. The corresponding EMss has

E−s1 = H•(M)⊗ IH•(A)⊗s

deg≥s

⊗H•(N)

Theorem 2.17. If A• is homologically connected, then

B(H•(M), H•(A), H•(N))→ B(H•(M), H•(A), H•(N))

is a quasi-isomorphism.

Suppose that X is a manifold. Evaluating at x, y ∈ X induces augementationsex, ey : E•(X) → R. Take A• ⊆ E•(M) to be a sub dga that both augmentationsrestrict to non-trivial homomorphisms A• → R. Take M• = N• = R as A•-modules with these homomorphisms. We can form the reduced bar constructionB(R, A•,R).


Define Ch•(Px,y(X);A•) to be the subcomplex of Ch•(Px,yX) spanned by iter-ated integrals

∫ω1 · · ·ωr with ωj ∈ A•.

Theorem 2.18. Suppose that X is connected. If H0(A•) ∼= R and the natural mapH•(A•)→ H•(X) is injective, then the natural map

B(R, A•,R)→ Ch•(Px,yX;A•)

[ω1| · · · |ωr] 7→∫

(ω1| · · · |ωr)

is a well defined isomorphism of dgas.

Proof. Check the formulas for the differentials.

Corollary 2.19. If X is connected, A• ⊆ E•(X) is a sub dga for which the inclu-sion A• −→ E•(X) induces isomorphism on cohomology, then the inclusion

Ch•(Px,yX;A•) −→ Ch•(Px,yX)

induces an isomorphism on cohomology.

Proof. By the assumption, the E1 page on both sides should be the same, thestatement follows.

Remark 2.20. This is useful as we can take A• to be

(1) minimal model (smallest sub dga that computes cohomology)—this simpli-fies computation.

(2) the logarithmic de Rham complex—this leads to Hodge theory.

Suppose X is connected, it is instructive to see the E1 page of the EMss corre-sponding to the reduced bar construction B(R, E•(X),R). Note that IH•(E•(X))

is the reduced cohomology H•(X) of X. In the picture below, we will simply denote

H•(X) by H•.

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Denote by (Er, dr) the r-th page (with differential) of the EMss.

Remark 2.21. A few remarks are in order.

(1) The total differential D = D(0,1) +D(1,0), so that d0 = D(0,1), d1 = D(1,0).(2) The green line has total degree 0. Here lies all iterated line integrals.

Elements in H0(B(R, E•(X),R)) are represented by elements of the form

0

ω1 ⊗ ω2

ω12 0tot. deg 0 line

that are closed, i.e. applying D gives 0. For the element in the abovediagram, this means dω1 = dω2 = 0 and ω1 ∧ ω2 + dω12 = 0, this gives riseto a homotopy invariant iterated line integral

∫(ω1ω2 + ω12).

(3) Massey products are indicated by the red arrows, where the map (H1)⊗(r+1) →H2 is given by the differential dr of the r-th page. When r = 1, it is simplythe cup product; when r = 2, it is the first non-trivial Massey product.


3. Chen’s π1-de Rham Theorem

In this section, we first state de Rham theorem for the loop space of a simplyconnected space. This allows us to compute rational homotopy groups using it-erated integrals. In the non simply connected case, we state Chen’s π1-de Rhamtheorem.

Theorem 3.1 (Chen). If X is path connected and simply connected, then integra-tion induces an isomorphism

H•(Ch•(PxX))→ H•(PxX;R).

In fact, it is an isomorphism of Hopf algebras.

Example 3.2. Let X = Sn, n ≥ 2. Take minimal model

Aj =

R, j = 0

R · ω, j = n

with∫Snω = 1. As A• −→ E•(Sn) is a quasi-isomorphism, by Corollary 2.19, we

have isomorphism

H•(Ch•(PxSn);A•)

∼=−→ H•(Ch•(PxSn;R)).

Since

Ch•(PxSn;A•) =

⊕k≥0

∫(ω| · · · |ω︸︷︷︸

k

)

has total differential D ≡ 0 (as dω = 0, ω ∧ ω = 0), we have

H•(PxSn;R) =

R, j = (n− 1)k

0, else

Suppose that M is connected.

Definition 3.3. An element of H•(M) is decomposable if it is in the image of thecup product mapping

H>0(M)⊗H>0(M)→ H•(M).

The set of indecomposable elements is defined by

QH•(M) := H>0(M)/H>0(M)⊗2.

When X is simply connected, PxX is a connected H-space. Chen’s de Rhamtheorem and the Cartan–Serre theorem imply that

Theorem 3.4. If X is simply connected, then integration induces an isomorphism

QHj(PxX;R)≈−→ Hom(πj(PxX,x),R) ∼= Hom(πj+1(X,x),R).

Example 3.5. X = Sn, n ≥ 2. Denote θk :=∫

(ω| · · · |ω︸︷︷︸k

). By the previous

example,

H•(PxSn;R) =

⊕k≥0

R · θk.

If n is odd, we have θ21 =

∫(ω) ∧

∫(ω) = 2

∫(ω|ω) = 2θ2, θk1 = k!θk, so the

indecomposables consist of R-linear span of θ1.

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If n is even, we have θ21 = 0, θ1 ∧ θ2m = θ2m+1, θ2 ∧ θ2m = (m+ 1)θ2m+2, so the

indecomposables consist of R-linear span of θ1, θ2.By the above theorem, we have

πj(Sn)⊗ R =

R, j = n, n odd

R, j = n or 2n− 1, n even

0, else

In the non simply connected case, suppose that X is connected, x ∈ X, we have

H0(Ch•(PxX)) H0(PxX;R)

HomZ(Zπ1(X,x),R)

HomctsR (Rπ1(X,x),R)

lim−→r

H0(Ch•r(PxX)) lim−→r

HomR(Rπ1(X,x)/Ir+1,R)

(∗)

≈

≈(by definition)

Remark 3.6. (1) Here Ch•r(PxX) ∼= Br(R, E•(X),R) denotes iterated line in-tegrals of length ≤ r.

(2) For (∗): H0(M) are locally constant functions on M ; when M is thepath/loop space of X, locally constant functions on M are equivalent tohomotopy functionals on paths/loops in X, i.e. they depend only on the ho-motopy class relative to end points. Therefore, these homotopy functionalsdescends to functions on π1(X,x).

(3) The augmentation ideal I is the kernel of the augmentation of the groupalgebra

Rπ1(X,x)→ R

sending [γ] 7→ 1. Powers of I define a topology on the group algebra,making sense the notation Homcts.

Theorem 3.7 (Chen’s π1-de Rham Theorem). Integration induces an isomorphism

H0(Ch•(PxX))≈−→ Homcts

R (Rπ1(X,x),R)

Proof. See Hain [Bowdoin, §4].

Remark 3.8. The right hand side in the above theorem is, by definition, the coor-dinate ring O(πun

1 (X,x)/R) of the unipotent completion of π1(X,x) over R. Thecompletion can be defined over Q by replacing R with Q.

4. Basics of Mixed Hodge Theory

The standard reference is Deligne [Hodge II]. There are other useful sources:Voisin (Textbook), Peters–Steenbrink (Survey/Monograph), Cattani–El Zein et al(Summer school collections) ...


4.1. Pure Hodge structures. Suppose that A is a subring of R, for example, Z,Q, K ⊆ R a number field, or OK its ring of integers.

Definition 4.1. An A-Hodge structure of weight m ∈ Z consists of a finitelygenerated A-module VA and a bigrading on its complexification

VC = VA ⊗ C =⊕

p+q=m

V p,q

satisfying V p,q = V q,p, where (·) denotes the complex conjugation on C. We sayvectors v ∈ V p,q are of type (p, q).

Example 4.2 (Prototype). Let X be a compact Kahler manifold (e.g. smoothprojective over C), set A = Z (or Q, R), then

VZ = Hm(X;Z)

is a Z-Hodge structure of weight m, with

VC = Hm(X;C) =⊕

p+q=m

Hp,q(X)

where

Hp,q(X) =closed forms of type (p, q)

exact forms of type (p, q)⊆ Hm(X).

Remark 4.3. If VA is a Hodge structure of odd weight, then

dimC VC = rankA VA ≡ 0 mod 2.

Corollary 4.4. dimH2k+1(X;C) ≡ 0 mod 2 for X smooth projective.

Remark 4.5. No known topological proof thatH1(X;Z) has even rank forX smoothprojective.

Example 4.6 (Non-example). There is a diffeomorphism

S3 × R ≈ C2\(0, 0)

(ξ, λ) 7→ eλξ

The Z-action on S3 × Rn : (ξ, λ) 7→ (ξ, λ+ n)

corresponds to the Z-action on C2\(0, 0)n : (z, w) 7→ en(z, w) = (enz, enw),

which is free and properly discontinuous. We thus get a quotient

X := Z\(C2\(0, 0)) ≈ S3 × S1.

This is a compact complex manifold (in fact a Hopf manifold). Its Betti numbersbj are

deg j bj0 11 12 03 14 1

By the previous corollary, X is not Kahler, and thus not projective.

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Suppose that we have a family

Xt ⊂ X

t ∈ T

where T is a contractible smooth complex manifold (e.g. T = disk), and Xt issmooth projective for all t ∈ T . Then we have

X ≈ X0 × T ' X0

T T

and natural isomorphismsHm(Xt) ∼= Hm(X0).

Fix X := X0 and its Hm(X;C) as a reference vector space, then

Hp,q(Xt) ⊆ Hm(X;C)

is a subspace, varying with t, of dimension

hp,qt := dimHp,q(Xt) = const =: hp,q.

This gives rise to a map to the Grassmannian

ϕp,q : T → Grhp,q (Hm(X;C)).

Question: Is this map holomorphic?Note that we have ϕp,q = ϕq,p. All ϕp,q being holomorphic would imply that all

ϕp,q are constant. But this is NOT THE CASE!Take T = h the upper half plane and a family

Eτ := C/(Z⊕ Zτ) ⊂ E

τ ∈ h

We have H1,0(Eτ ) ⊆ H1(X;C) = Ca⊕ Cb where a, b are duals of a, b.

0 a 1

b

τ

The canonical 1-form ωτ ∈ H1(Eτ ) can be written as

ωτ = dz = a+ τ b,

which varies holomorphically in the family.In general, define the Hodge filtration

F pVC =⊕s≥p

V s,m−s.

These vary holomorphically in families!


Definition 4.7 (Hodge structure defined by F •). An A-Hodge structure of weightm consists of a finitely generated A-module VA, whose complexification VC = VA⊗Chas a decreasing filtration (called the Hodge filtration)

· · · ⊇ F p ⊇ F p+1 ⊇ · · ·satisfying

F p⊕

Fm−p+1 ≈ VCfor all p.

Remark 4.8. This definition is equivalent to the previous one. One just needs tonote that

V p,q = F p ∩ F q.

Definition 4.9 (1-dimensional Hodge structures/Tate twists). Define A(p) theHodge structure of weight −2p, whose underlying A-module is VA. Its complexifi-cation VC = V −p,−p. It is commonly written as

A(p) = (2πi)p ·A ⊆ C,

for example, Z(p) = (2πi)p Z ⊆ C.

Remark 4.10. In the context of periods, for A(p), we pick a Betti basis eB for VAand a de Rham basis edR for VC, then we have

VA ⊗ C ≈−→ VC

eB 7→ (2πi)pedR.

Example 4.11 (Basic example: H1(Gm) ∼= Z(−1)). We view this in the contextof periods.

Betti: We have σ ∈ HB1 (Gm), and its dual eB = σ ∈ H1

B(Gm).

σ

DR: We have edR = dzz ∈ H

1dR(Gm/Q). And we have

H1B(Gm)⊗ C H1

dR(Gm)⊗ C

(γ 7→∫γω) ω ∀ω ∈ H1

dR, γ ∈ HB1

(σ 7→∫σdzz = 2πi) = 2πiσ dz

z

eB = σ = (σ 7→ 1) (2πi)−1 dzz = (2πi)−1edR

≈comp

Definition 4.12. A polarization on a Q-Hodge structure of weight m is a non-degenerate (−1)m-symmetric bilinear form

S : VQ ⊗ VQ → Q

satisfying the Riemann–Hodge bilinear relations:

(1) S(V p,q, V r,s) = 0 unless p = r, q = s

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(2) ip−qS(v, v) > 0 for all 0 6= v ∈ V p,q

Example 4.13. Let C be a compact Riemann surface, then V = H1(C) is a Hodgestructure of weight 1. Define

S(u, v) =

∫C

u ∧ v

for u, v ∈ V . It is a polarization on V :

i

∫C

ω ∧ ω > 0

for any 0 6= ω ∈ H0(Ω1C).

Theorem 4.14 (Hodge). If X is smooth projective, then Hm(X) is a polarizableHodge structure.

The category of Q-polarized Hodge structures (Q-PHS) is semisimple.

Proposition 4.15. If (V, S) is a Q-PHS of weight m and A ⊆ V a sub Hodgestructure, then

(1) S|A is non-degenerate, so that VQ = AQ ⊕A⊥Q(2) V = A⊕A⊥ as Q-PHS

There is a natural correspondence between PHS of weight one and the firstcohomology of abelian varieties.

Proposition 4.16. Suppose X, Y are abelian varieties and H1(X;Q) ∼= H1(Y ;Q)as Q-Hodge structures, then X is isogenous to Y .

Theorem 4.17. If X, Y are abelian varieties, and

H1(Y ;Q) ⊆ H1(X;Q)

then X is isogenous to Y ×Z where Z is an abelian variety associated to H1(Y ;Q)⊥

in H1(X;Q).

Remark 4.18. Every abelian variety is isogenous to a product of simple abelianvarieties.

4.2. Mixed Hodge structures. Based on knowledge on l-adic cohomology andguided by the theory of motives, Deligne in [Hodge I] indicates that analogously anatural mixed Hodge structure can be put on Betti cohomology. Even limit mixedHodge structure is discussed at the end (loc. cit.).

Definition 4.19. An (Z-)mixed Hodge structure (MHS) consists of

(1) a finitely generated Z-module VZ(2) an increasing filtration (weight filtration) W• on VQ = VZ ⊗Q

· · · ⊆WnVQ ⊆Wn+1VQ ⊆ · · ·(3) a decreasing filtration (Hodge filtration) F • on VC = VZ ⊗ C

· · · ⊇ F pVC ⊇ F p+1VC ⊇ · · ·such that GrWn V = WnV/Wn−1V with induced F •

F p GrWn VC := (F p ∩Wn +Wn−1)/Wn−1 = (F p ∩Wn)/(F p ∩Wn−1)

is a (Q-)Hodge structure of weight n.


Remark 4.20. One can define A-mixed Hodge structures similarly. Unless specified,we will mostly work with Z-mixed Hodge structures in the rest of this section, e.g.we will simply denote Hm(X;Z) by Hm(X).

Theorem 4.21 (Deligne). The category A-MHS of A-mixed Hodge structures isan abelian tensor category. If A is a field (e.g. Q, R), then the category A-MHS is

tannakian. The functors GrW• , Gr•F , Gr•F GrW• are exact.

Remark 4.22. The most essential piece of the theorem is to show that the cat-egory A-MHS is abelian. This is done by constructing a natural splitting calledDeligne splitting. See Deligne [Hodge II, Lemme (1.2.8)] or Griffiths–Schmid [Sur-vey, Lemma (1.12)].

Theorem 4.23 (Deligne). If X is a complex algebraic variety, then H•(X) has anatural Z-mixed Hodge structure.

Remark 4.24. Naturality is on algebraic morphisms, NOT on continuous maps.

Remark 4.25. If X is smooth, then weights on Hm(X) are at least m; if X isprojective, then weights on Hm(X) are at most m.

The following diagram indicates possible types (p, q) for Hm(X), with d =dimCX.

p

q

(0, 0) p+ q = m ≤ d

(m,m)

(m− d,m− d) p+ q = m ≥ d

(d, d)

projective

smooth

projective

smooth

From this we get a table of weights on Hm(X), with d = dimCX

general smooth projectivem ≤ d [0, 2m] [m, 2m] [0,m]m ≥ d [2m− 2d, 2d] [m, 2d] [2m− 2d,m]

It is instructive to first learn a couple of ad hoc elementary examples picked fromDurfee.

24 MA LUO

Example 4.26 (smooth curve). Let X = X −D be a smooth curve, where X isa smooth projective curve of genus g, and D = P,Q is a divisor with P 6= Q,P,Q ∈ X. We now describe the natural mixed Hodge structure on H1(X).

We have Gysin sequence

0→ H1(X)→ H1(X)Res−−→ H0(D)(−1)

deg−−→ H2(X)

where H0(D) = H0(P ) ⊕H0(Q), Res denotes the residue maps at P and Q, degdenotes the degree map on divisors. There is a copy of Z generated by (1,−1) insideH0(D) whose degree is zero. One pulls back this from the sequence and obtains ashort exact sequence

(4.1) 0→ H1(X)α−→ H1(X)

Res−−→ Z(−1)→ 0

There exists ωP,Q ∈ H1(X) that maps to the generator (1,−1), i.e.

ResP (ωP,Q) = 1, ResQ(ωP,Q) = −1.

To describe the mixed Hodge structure on H1 = H1(X) is equivalent to describing

the two filtrations. For the weight filtration, applying GrW• to (4.1), one easily gets

W0H1Q = 0, W1H

1Q = Imα, W2H

1Q = H1

Q.

For the Hodge filtration, applying Gr•F to (4.1) we have F 0H1C = H1

C, and F 1 =F 1H1

C ⊆ H1(X;C) needs to satisfy

F 1 ∩H1(X;C) = H1,0(X) and ResF 1 = C.

Define

F 1 = F 1H1(X;C) := H0(Ω1X

)⊕ C · ωP,Q.

This is well defined, and satisfies the above condition. In general, we would define

F 1 := H0(Ω1X

(P +Q)) = H0(Ω1X

(logD)).

Example 4.27 (nodal curve). Let C be a nodal curve, obtained by identifyingtwo distinct points P , Q on a smooth projective curve X. As in Example 2.7, wereplace C by a simplicial variety X• whose geometric realization |X•| is homotopicto C. The degree 0 component X0 of X• is X; the degree 1 component X1 is apoint.


C = X =

X• =

|X•| = (X0

∐X1 × [0, 1])/ ∼ =

We have a “cofibration sequence”

X −→ |X•|(' C)→ Γ

where

Γ =

is the dual graph of C/nerve of its (simplicial) covering. This induces a short exactsequence

0→ H1(Γ)α−→ H1(C)→ H1(X)→ 0

where H1(Γ) ∼= Z(0). To describe the mixed Hodge structure on H1 = H1(C), wedescribe the two filtrations. For the weight filtration, we have

W−1H1Q = 0, W0H

1Q = Imα, W1H

1Q = H1

Q.

For the Hodge filtration, we have

F 0H1C = H1

C, F 1H1C = classes ω ∈ H0(Ω1

X) in H1C.

Before moving on to the construction of mixed Hodge structures, we end withan example of MHS on a fundamental group.

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Example 4.28 (MHS on πun1 (C\S, x)). Suppose that S = a1, · · · , an is a finite

set of points in the complex plane C, and x ∈ C\S. Let

A = C〈〈X1, · · · , Xn〉〉

be power series in noncommuting indeterminates Xj ’s. Set ωj = dzz−aj and

Ω =

n∑j=1

ωjXj ∈ H0(Ω1(C\S))⊗A.

We have

T := 1 +

∫(Ω) +

∫(Ω|Ω) + · · ·

which is an A-valued iterated integral on C\S (as all∫

(ωj1 | · · · |ωjr ) are closed).This induces an isomorphism

Θ : Cπ1(C\S, x)∧≈−→ C〈〈X1, · · · , Xn〉〉.

of completed Hopf algebras with augmentation ideals I and IA = (X1, · · · , Xn).We may view this as a comparison isomorphism from Betti to de Rham for thefundamental group (cf. Theorem 3.7).

Define W• and F • on A by giving Xj type (−1,−1), so that XJ = Xj1 · · ·Xjr

is of type (−r,−r) = (−|J |,−|J |). Define

WmA = span of XJ : −2|J | ≤ m, i.e. |J | ≥ −m2

and

F pA = span of XJ : −|J | ≥ p, i.e. |J | ≤ −p

These filtrations carry over to the left side of Θ. On the weighted graded pieces,we have

GrW−2mQπ∧1 ∼= Im/Im+1 Θm−−→ GrW−2mA∼= ImA /I

m+1A

∼= V ⊗m where V = ⊕nj=1C ·Xj

(γ1 − 1) · · · (γm − 1) 7→ (T (γ1)− 1) · · · (T (γm)− 1) ≡∫γ1

Ω · · ·∫γm

Ω mod Im+1A

(σj1 − 1) · · · (σjm − 1) 7→ (2πi)mXj1 · · ·Xjm

as

Θ1(σj) = T (σj)− 1 ≡∫σj

ωjXj = 2πiXj mod I2A.

a1σ1

a2σ2

anσn

x

Therefore, we have

GrW−2m∼=⊕|J|=m

Z(m).


5. Mixed Hodge Structures on Fundamental Groups

In this section, we will put mix Hodge structures onH•(X) andH•(Ch•(Px,yX)).The basic setup is

MHC + DR Thm → MHS

More specifically, Deligne [Hodge II]

• defines a Mixed Hodge Complex (MHC)• H•(MHC) is a MHS• constructs MHC that computes H•(X)

thus putting a MHS on H•(X).Hain [Big Red]

• defines multiplicative MHC &• modules over multiplicative MHC• B(M,A,N) is MHC if M,A,N are

thus putting a MHS on H•(Ch•(Px,yX)).

5.1. Strictness.

Definition 5.1. A filtered morphism ϕ : (V1, F•)→ (V2, F•) is strict with respectto filtrations F• if

Imϕ ∩ FmV2 = Im(FmV1)

for all m, i.e. if v ∈ FmV2 and v ∈ Imϕ, then ∃u ∈ FmV1, such that ϕ(u) = v.

Proposition 5.2. If (A•, F •) is a filtered complex, then

d is strict with respect to F • ⇐⇒ FEr degenerates at E1

Proof. We have

Es0 = GrsF A•, Es,t1 = Hs+t(GrsF A

•)

and

Es,t1 Es+1,t1

Hs+t(GrsF ) Hs+t+1(GrsF )

Given α ∈ Hs+t(GrsF ), it is represented by a ∈ F s such that da ∈ F s+1. Thend1α = [da] ∈ Hs+t+1(Grs+1

F ) and

d1α = 0 ⇐⇒ ∃b ∈ F s+1, such that db ≡ da mod F s+2.

One concludes that

d1 = 0 ⇐⇒ ∀s & a ∈ F s, s.t. da ∈ F s+1,∃b ∈ F s+1, s.t. db ≡ da mod F s+2.

(i.e. we have d(a− b) ∈ F s+2, with a− b ∈ F s. This leads us to the next page E2.)Suppose d1 = 0 so

Es,t2 = Es,t1 = Hs+t(GrsF )

then α ∈ Es,t2 is represented by a ∈ F s such that da ∈ F s+2 (for example, (a − b)we just found above). So

d2α = 0 ⇐⇒ ∃c ∈ F s+2, s.t. dc ≡ da mod F s+3.

28 MA LUO

Then (given d1 = 0),

d2 = 0 ⇐⇒ ∀s & a ∈ F s, s.t. da ∈ F s+2,∃c ∈ F s+2, s.t. dc ≡ da mod F s+3.

etc.It is then easy to see that strictness is equivalent to

∀a ∈ F s, s.t. da ∈ F s+t,∃ b+ c+ d+ · · ·︸︷︷︸t−terms

s.t. da ≡ d(b+c+d+ · · · ) mod F s+t+1

which, in turn, is equivalent to

d1 = d2 = · · · = 0,

i.e. FEr degenerates at E1.

5.2. Hodge to DR spectral sequence. Let X be a complex manifold. Its deRham complex

EmC (X) =⊕

p+q=m

Ep,q(X)

with differential d = ∂ + ∂. Define

F pEm(X) =⊕s≥p

Es,m−s(X).

This filtration gives rise to Hodge to DR spectral sequence with

Ep,q1 = Hp+q(Ep,0∂−→ Ep,1

∂−→ · · · ∂−→ Ep,n → 0) = Hp,q

∂(X)

where n = dimCX. There is a fine/acyclic resolution of the sheaf ΩpX of differentials:

0→ ΩpX → Ep,0X∂−→ Ep,1X

∂−→ · · · ∂−→ Ep,nX → 0

By sheaf theory, we have

Hq(X,ΩpX) ∼= Hp,q

∂(X).

So we have

Ep,q1 = Hq(X,ΩpX) =⇒ Hp+q(X;C).

Note that for X compact, using Hermitian metric on X we have

Hp,q

∂(X) ∼= Hp,q

∂(X)

where Hp,q∂

(X) denotes harmonic forms for the Laplacian ∆∂ ; and using Riemann-

ian metric on X

Hp+q(X;C) ∼= Hp+qd (X)

where Hp+qd (X) denotes harmonic forms for the Laplacian ∆d.If X is compact Kahler, we have Kahler identities that give

∆d = 2∆∂ .

This implies that

Hmd (X) =⊕

p+q=m

Hp,q∂

(X).

By dimension counting, Hodge to DR spectral sequence degenerates at E1 for Xcompact Kahler. By Proposition 5.2, the differential d on E•(X) is strict withrespect to F •.

We provide an easy application for this strictness.


Example 5.3 (Hodge filtration F • on H0(B2(X))). Let X be compact Kahler.Fix a base point x ∈ X. Then

B2(X) := B2(C, E•(X),C) = ∑

ajk(ωj |ωk) + (ξ)

and H0(B2(X)) consists of∑ajk(ωj |ωk) + (ξ) with∑ajkωj ∧ ωk + dξ = 0

i.e.∑ajk[ωj ]⊗ [ωk] ∈ K where

0→ K → H1(X)⊗H1(X)∪−→ H2(X)

cf. Remark 2.21 (2). We have a short exact sequence

0→ H1(X)→ H0(B2(X))→ K → 0

ϕ 7→∫

(ϕ)∑ajk(ωj |ωk) + (ξ) 7→

∑ajk[ωj ]⊗ [ωk]

For holomorphic 1-forms ω1, ω2 ∈ F 1, we have ω1 ⊗ ω2 ∈ F 2. As ω1 ∧ ω2 = 0 wehave ∫

(ω1|ω2) ∈ F 2H0(B2(X)).

For holomorphic 1-forms ωj , ωk ∈ F 1, suppose [ωj ]∪ [ωk] = 0, then as ωj∧ωk ∈ F 1,by strictness, we can find ξ ∈ F 1 such that

ωj ∧ ωk + dξ = 0.

This gives rise to ∫(ωj |ωk) + (ξ) ∈ F 1H0(B2(X)).

5.3. Mixed Hodge Complex (MHC). Let k ⊆ R be a subfield.

Definition 5.4. A k-MHC is a pair of complexes K•k and K•C with filtrations

K = ((Kk,W•), (KC,W•, F•))

and

(1) a fixed W•-filtered quasi-isomorphism

(K•k ⊗ C,W•)← (K•1 ,W•)→ (K•2 ,W•)→ · · · ← (K•C,W•)

(2) (WE0(KC), d0) is strictly compatible with F •

(3) WEl,m1 (K) has HS of weight m with respect to induced F •

Remark 5.5. (1) implies that we have isomorphisms on WE1, and

(H•(Kk),W•)⊗ C ∼= (H•(KC),W•).

This makes sense of k-structure (weight filtration) of the HS in (3), while the Hodgefiltration F • comes from (2).

Theorem 5.6 (Deligne). If K is a k-MHC, then H•(K) is a k-MHS with

(1) induced F •

(2) WmHj(Kk) = ImHj(Wm−j(Kk))→ Hj(Kk)

(3) WE2 = WE∞, i.e. weight spectral sequence degenerates at E2.

30 MA LUO

Example 5.7. For X smooth projective over C, to construct MHS on H•(X), wedefine the complex part of a MHC

K•C = E•C(X)

with

W−1 = 0, W0 = E•C(X).

Then we have

WjHj(KC) = ImHj(W0KC) = Hj(KC)

and Wj−1Hj(KC) = 0. This is consistent with our expectations (Example 4.2).

5.4. Logarithmic de Rham complex. For X a smooth variety over C, but notnecessarily projective, we construct logarithmic de Rham complex.

By resolution of singularities (Hironaka), we can write

X = X −Dwhere D is a divisor with normal crossings, i.e. at each P ∈ X, we have localholomorphic coordinates (z1, · · · , zn) on a neighborhood U of P , and k ≤ n, suchthat U ∩D is given by

z1 · · · zk = 0.

We say D has simple normal crossings if each of its components is smooth. Onecan always blow up a divisor with normal crossings to get a divisor with simplenormal crossings.

Now we construct C∞-logarithmic de Rham complex E•(X logD) for D withnormal crossings. It has two steps:

(1) Construct sheaf of logarithmic differentials Ω•X

(logD). Define

Ω•X

(logD) := ∧•Ω1X

(logD).

For sheaf Ω1X

(logD), locally on U as above,

Ω1X

(logD)(U)

is a free O(U)-module generated by Ω1(U) anddzjzj

for j = 1, · · · , k. Define

weight filtration W• so that

WrΩmX

(logD)(U)

have elements of the form

(5.1)∑|J|=ms≤r

fJ(z)dzj1zj1∧ · · · ∧ dzjs

zjs∧ dzjs+1

∧ · · · ∧ dzjm

where fJ(z) is holomorphic. Suppose that D =⋃Nj=1Dj is a union of its

components, and for a subset J ⊆ 1, · · · , N, DJ =⋂j∈J Dj . We have

residue maps

(5.2) Res : GrWr ΩmX

(logD)(U)≈−→

⊕codim r components DJ

Ωm−rX

(DJ)

with element (5.1) mapping to∑|J|=r

fJ(z)dzjr+1∧ · · · ∧ dzjm .


(2) Construct C∞-sheaf of logarithmic differential forms

Ep,qX

(logD) := E0,q

X⊗ Ωp

X(logD).

This is a double complex, we denote the associated total complex by E•X

(logD).

A (p, q)-form, locally on U , can be written as

∑ϕJ,K

dzj1zj1∧ · · · ∧ dzjr

zjr∧ dzjr+1

∧ · · · ∧ dzjp ∧ dzk1 ∧ · · · dzkq .

Definition 5.8. Define the logarithmic de Rham complex

E•(X logD) := Γ(X, E•X

(logD)).

For weight filtration, Wr is generated locally by terms with at most r logarithmic

differential formsdzjzj

. For Hodge filtration, count the number of dz’s.

Remark 5.9. If one uses the direct image j∗Ω•X for the inclusion j : X → X,

then its weight filtration can be defined by counting the order of poles. It is W•-filtered quasi-isomorphic to the logarithmic de Rham complex, see Deligne [HodgeII, (3.1.10)–(3.1.11)]. This explains in Example 5.7, the weight filtration is concen-trated in weight 0 as there are no poles.

Theorem 5.10 (Deligne). E•(X logD) is a MHC.

Remark 5.11. Actually, we have only constructed the complex part KC of K. Thesimilar construction for the k-part Kk is omitted.

Set

D[m] :=∐|J|=m

DJ .

For example, D[1] =∐j Dj , D

[2] =∐j<kDj ∩ Dk, etc. Suppose D has simple

normal crossings, then all D[j] are smooth. By the residue maps (5.2), we have

WE−l,m1 = Hm−2l(D[l])(−l)

HS of weight m

for the weight spectral sequence for E•(X logD). The differential d1 are given byGysin maps2. And this spectral sequence degenerates at E2.

2This is the degree map that showed up in the Gysin sequence in Example 4.26. In general, if Y

is a divisor of X with dimX = d, the inclusion Y −→ X induces H2d−j(Y )→ H2d−j(X); Poincare

duality then gives Hj−2(Y )(d−1)→ Hj(X)(d), and thus the Gysin map Hj−2(Y )(−1)→ Hj(X).

32 MA LUO

Corollary 5.12. For X smooth, H•(X) = H•(E•(X logD)) has a MHS.

Remark 5.13. One can check that the MHS on H1(X) is the same as the one wesaw in Example 4.26.

Example 5.14. Let X = X −D be a smooth variety over C as before. Fix a basepoint x ∈ X, we have augmentation

ex : E•(X logD)→ C

by evaluating at x. This leads to

B(E•(X logD)) := B(C, E•(X logD),C),

the complex part of a MHC. The Hodge filtration is obvious, counting numbers ofdz’s. The weight filtration is the convolution of W• on E•(X logD) with the lengthfiltration on the (reduced) bar construction. For example,

(dz

z

∣∣dww

∣∣dξ) ∈ F 3

and it is in W2+3, where 2 = 1 + 1 + 0 comes from W• on E•(X logD) countingthe number of logarithmic forms; 3 comes from the length.

Theorem 5.15 (Hain). This is a MHC.

Remark 5.16. Again the k-part of this MHC is omitted.

Corollary 5.17. For X smooth, H•(Ch•(PxX)) = H•(B(E•(X logD))) has aMHS.

One can look back at Example 4.28 to check the understanding of the weightand Hodge filtrations.


6. Deligne’s Canonical Extension

6.1. Variations of Hodge structures.

Definition 6.1. A variation of Hodge structures (VHS), V over a smooth (complex)variety T , of weight m consists of

(1) a Q-local system VQ over T , with

VC := VQ ⊗ C.

The corresponding holomorphic vector bundle

V := V⊗OTover T has a flat connection

∇ : V → V ⊗ Ω1T .

(2) Need holomorphic sub-bundles (Hodge sub-bundles) Fp of V

· · · ⊇ Fp ⊇ Fp+1 ⊇ · · ·

such that ∪pFp = V, ∩pFp = 0.(3) For each t ∈ T , the fiber

Vt := (VQ,t, (VC,t,F•t ))

is a Hodge structure of weight m.(4) Griffiths transversality:

∇Fp ⊆ Fp−1 ⊗ Ω1T .

(Think of this as ∇ : Fp(V)→ Fp(V ⊗ Ω1T ))

Example 6.2. Let f : X → T be a family of smooth projective varieties, i.e. Xt

are smooth projective for any t ∈ T . Take

VQ = Rmf∗Q.

This is a VHS of weight m with fiber Vt and

VQ,t = Hm(Xt;Q).

Definition 6.3. A polarized variation of Hodge structures (PVHS) of weight mover T is a VHS

V = (VQ,V,Fp)with a (−1)m-symmetric flat bilinear form

Q : V⊗ V→ Q

such that Qt : Vt ⊗ Vt → Q polarizes Vt.

Remark 6.4. All VHS coming from algebraic geometry are polarizable.

Remark 6.5. The category of PVHS over T is semi-simple (globally). However, itis locally (quasi-)unipotent, cf. Thm. 7.1 (1).

Families usually come with singular fibers. It is very interesting to find out theinformation near such fibers. We will focus on the 1-dimensional families locally,use the following example as a working example, eventually compute limit mixedHodge structures in the next section.

34 MA LUO

Example 6.6 (Canonical Example). We will consider a VHS over T = D∗ apunctured (q-)disk. This comes from considering families of elliptic curves over theupper half plane h, which is the universal covering space of D∗.

Hh → HD∗

h → D∗

τ 7→ q = e2πiτ

(1) Q-local system Hh: For any τ ∈ h, define Λτ := Z⊕ Zτ , Eτ := C/Λτ . Wehave

H1(Eτ ;Z) ∼= Λτ ∼= Za⊕ Zb = HZ.

After being tensored with Q, these give rise to a Q-local system Hh over h,whose corresponding vector bundle Hh is trivialized by the flat sections aand b.

0 a 1

b

τ

a+ b

τ + 1

(2) Monodromy action on the fiber: We consider the monodromy determinedby a counter-clockwise loop around 0 in D∗, which corresponds to τ 7→ τ+1in h. Therefore, the monodromy is given by(

ba

)7→(a+ ba

)=

(1 10 1

)(ba

).

Note that a is invariant under the monodromy.(3) Poincare duality identifies H1 with H1: Using the intersection form (a ·b) =

1 = −(b · a), we define the Poincare duality

P : H1 → H1

u 7→ (u · −)

a 7→ (a · −) = b

b 7→ (b · −) = −a

(4) HD∗ , HD∗ and its sections: Define a section of Hh by

w := 2πi dz = 2πi(a+ τ b).

By Poincare duality, we can write it as

w = 2πi(−b+ τa).

One can check that it is invariant under the monodromy action as

(−(a+ b) + (τ + 1)a) = −b+ τa.


Therefore the vector bundle HD∗ , corresponding to HD∗ , is trivialized by aand w:

HD∗ = OD∗a⊕OD∗w.

The Hodge subbundle

F 1HD∗ = OD∗w

is generated by w.(5) Connection ∇ on HD∗ : On Hh, sections a and b are flat, i.e. ∇a = ∇b = 0.

And we compute

∇w = ∇2πi(τa− b) = 2πi dτ · a = adq

q.

Therefore, we have connection

∇ = d+ a∂

∂w

dq

q

on HD∗ .

Remark 6.7. As we will see in Example 6.12, we can extend HD∗ to D by

HD := ODa⊕ODw

with the same connection ∇. This is Deligne’s canonical extension. Note that theconnection has regular singularity at q = 0 and nilpotent residue

a∂

∂w=

(0 10 0

),

cf. Remark 6.11.

6.2. Deligne’s canonical extension. The standard reference is Deligne [ODE].Careful expositions have also been given by Conrad (notes on Riemann–Hilbert,available on course website) and Hain (last note on his course website). We willmostly follow Hain, adopting his notations.

Suppose that we have a local system of C-vector spaces

V→ D∗

over a punctured disk, and we assume 1 ∈ D∗ (by rescaling). Denote the corre-sponding flat vector bundle by V. Denote the fiber over t ∈ D∗ by Vt, and themonodromy

ht : Vt → Vt

is determined by a counter-clockwise loop around 0 in D∗. The local system isdetermined by any one of these. In particular, it is determined by

h := h1 ∈ AutV1.

By flatness, we have parallel transport

gt : V1 → Vt

between fibers, with g1 = id and

ht = gthg−1t .

http://people.maths.ox.ac.uk/luom/mhsfg.html

http://services.math.duke.edu/~hain/teaching/mth690/

36 MA LUO

1

t

gt

h = h1

ht

Choose a logarithm of h, set

N =1

2πilog h.

Deligne chooses N such that its eigenvalues λ satisfy

0 ≤ Re(λ) < 1.

In case h is unipotent, there is a canonical choice such that Re(λ) ≡ 0, with thefinite sum

N =1

2πi

∞∑n=1

(−1)n+1 (h− id)n

n∈ EndV1.

Set Nt = gtNg−1t ∈ EndVt, then e2πiNt = ht. For v ∈ V1, let v(t) be the flat

section of V over a neighborhood U of t, then v(t) = gtv. Set

ϕ(t) := gtt−Nv = t−Ntv(t) ∈ Vt.

A priori, ϕ(t) is multi-valued on D∗, but it is single valued: at t = 1, when weanalytically continue along the unit circle, we have

t−N = e−N log t 7→ e−N(log t+2πi) = e−N log t · e−2πiN = t−N · h−1

and v 7→ hv so that

t−Nv 7→ t−N · h−1hv = t−Nv.

We trivialize V over D∗ by

V1 × D∗ ≈−→ V(v, t) 7→ ϕ(t) = gtt

−Nv

We call this Deligne’s trivialization.

Proposition 6.8. The pullback of the connection on V to V1 × D∗ is

∇ = d−N dt

t

Proof. Since gt is a flat section of Aut(V), we have

∇(gtt−Nv) = gtd(t−N )v = gt(−Nt−N−1dt)v = −N(gtt

−Nv)dt

t.

Definition 6.9. Fix a choice of N , Deligne’s canonical extension to D of V overD∗ is the extension


V V1 × D∗ V1 × D

D∗ D∗ D

≈

using Deligne’s trivialization above.

Remark 6.10. Since the extension is a trivial bundle, the central fiber V0 is welldefined up to a unique isomorphism (Proposition 6.13), we can write the extensionas

V V0 × D

D∗ Dand N should be regarded as an element of End(V0).

Remark 6.11. In the case when the monodromy is unipotent, and N is nilpotent,Deligne’s canonical extension is characterized by the properties:

(1) (V,∇) has a regular singularity at 0:

∇ : V → V ⊗ Ω1D(log 0).

(2) Res0∇ is nilpotent.

Example 6.12 (Canonical Example, cont’d). In our Example 6.6 (2), in terms offlat sections a and b, the monodromy

h =

(1 10 1

)is unipotent. And we choose canonically

N =1

2πi

∞∑n=1

(−1)n+1 (h− id)n

n=

1

2πi(h− id)

where id =

(1 00 1

). Note that as (h − id)2 = 0, N2 = 0. In the local coordinate

t = q,

q−N = e−N log q =

∞∑n=0

(−N log q)n

n!= id +(−N log q) = id−N · 2πiτ

since q = e2πiτ and τ = log q2πi . We have

q−Na = a, and q−Nb = b− τa = − w

2πi.

Up to a constant multiple, the sections a and w that we considered before give riseto Deligne’s trivialization, and we obtain Deligne’s canonical extension by simplyextending them across 0, cf. Remark 6.7.

Proposition 6.13. Deligne’s canonical extension does not depend on the choice ofholomorphic coordinate t in D.

Proof. Suppose that s is another holomorphic coordinate in D centered at 0. Write

s(t) = tf(t)

with f(t) holomorphic and f(0) 6= 0. Without loss of generality, assume s = 1 is inD∗ (otherwise rescale or swap s↔ t).

38 MA LUO

s = t = 0t = 1

s = 1

g

Denote by g the parallel transport

g : Vt=1 → Vs=1.

Seth′ = ghg−1 ∈ AutVs=1,

N ′ = gNg−1 ∈ EndVs=1,

g′s = gt(s)g−1 : Vs=1 → Vs.

We have the trivialization

Vs=1 × D∗ V

D∗

≈

given by

(w, s) 7−−−→ g′ss−N ′w.

DefineΦ : Vt=1 × Dt

t-disk

→ Vs=1 × Dss-disk

by(v, t) 7→ (gf(t)Nv, s(t)).

This is holomorphic at t = 0 and f(0) 6= 0, so Φ is an isomorphism of holomorphicvector bundles.

Restricting to punctured disks, we have the commutative diagram

Vt=1 × D∗t Vs=1 × D∗s

V

Φ|D∗

≈ ≈

with

(v, t) (w, s) = (gf(t)Nv, s(t))

gtt−Nv = g′ss

−N ′w

since

g′ss(t)−N ′(gf(t)Nv) = gtg

−1s(t)−N′gf(t)Nv

= gts(t)−Nf(t)Nv

= gtt−Nv.


For any choice of coordinate t, centered at 0 ∈ D,

limP→0

t(P )−NP v(P )

exists in V0, where P ∈ D∗, P 7→ v(P ) is a flat section of V, and NP the monodromyat P . This induces parallel transport

VP≈−→ V0

depending on t, i.e. the map

H0(D∗, π∗V)→ V0

v 7→ limP→0

t(P )−NP v(P )

is an isomorphism for each choice of t, where π : D∗ → D∗ is the universal coveringmap. In fact, this gives a Q-structure on V0:

H0(D∗, π∗VQ)⊗ C→ V0

v ⊗ 1 7→ limP→0

t(P )−NP v(P )

Proposition 6.14. This Q-structure depends only on the tangent vector ∂/∂t ∈T0D. Denote it by V∂/∂t. If ∂/∂s = λ∂/∂t, then the Q-structure V∂/∂s on V0 is

V∂/∂s = λNV∂/∂t

where N = Res0∇.

Proof. Suppose that s is another holomorphic coordinate in D centered at 0. Write

s(t) = tf(t)

with f(t) holomorphic and f(0) 6= 0. Taking differential at 0, we have

ds = f(0) dt.

Taking dual, we get

∂/∂s = f(0)−1∂/∂t and λ = f(0)−1.

Let v ∈ H0(D∗, π∗VQ), then

limP→0

s(P )−NP v(P ) = limP→0

f(t(P ))−NP t(P )−NP v(P )

= f(0)−N limP→0

t(P )−NP v(P ) need limP→0

NP = N

= λN limP→0

t(P )−NP v(P )

i.e.

V∂/∂s = λNV∂/∂t.

40 MA LUO

7. Limit Mixed Hodge Structures

The standard reference is Schmid [VHS]. Recall the definition for PVHS fromthe beginning of last section. We summarize results in [VHS] as follows.

Theorem 7.1 (Schmid). If V→ D∗ is a PVHS, then

(1) the monodromy h is quasi-unipotent (Remark 6.5), i.e. ∃n,m, s.t. (hn −id)m = 0. This is proved by Landman in geometric situations, and Borelabstractly. Without loss of generality, we will assume that h is unipotent.One can base change to a finite cover of D∗ and pull back the local system:

p∗V V

D∗ D∗p

where p is the power map t 7→ tn.(2) (Nilpotent Orbit Theorem) If monodromy is unipotent, the Hodge sub-bundles

extend to sub-bundles of the canonical extension of V to D.(3) (SL2 Orbit Theorem) For each non-zero tangent vector ~v ∈ T0D, e.g. ~v =

∂/∂t,

(V∂/∂t, V0, F•,M•)

is a MHS. Here the Q-structure V∂/∂t on V0 is given by Proposition 6.14,the Hodge filtration F • is given by (2), and the weight filtration is themonodromy weight filtration M• that we will define next.

7.1. Monodromy weight filtration. Suppose V is a finite dimensional vectorspace over a field of characteristic zero, and N : V → V is a nilpotent endomor-phism.

Proposition 7.2. There is a unique filtration W• = W•(N) on V such that

(1) NWk ⊆Wk−2

(2) The induced map

Nk : GrWk V≈−→ GrW−k V

on the associated graded is an isomorphism.

Proof. Since N is nilpotent, it has Jordan canonical form with Jordan blocks

0 1 0 · · · 0

0 0 1. . .

...

0 0 0. . . 0

......

. . .. . . 1

0 0 · · · 0 0

It suffices to prove existence and uniqueness for a single Jordan block, such asabove. Suppose it is a m×m-matrix. We can find basis ej, j = 1, · · · ,m, so that

Nej = ej+1, j < m

Nem = 0

The following diagrams indicate the weight filtration: For m = 2n,


−(2n− 1)

e2n

−1

en+1

1

en

(2n− 3)

e2

(2n− 1)

e1

N

N

The uniqueness is given by

W2n−2l = W2n−1−2l = N l(V ), 0 ≤ l ≤ 2n− 1,

W2n−1 = W2n = W2n+1 = · · · = V and W−2n = W−2n−1 = · · · = 0.

For m = 2n+ 1,

−2n

e2n+1

−2

en+2

0

en+1

2

en

2n

e1

NN

The uniqueness is given by

W2n+1−2l = W2n−2l = N l(V ), 0 ≤ l ≤ 2n,

W2n = W2n+1 = W2n+2 = · · · = V and W−2n−1 = W−2n−2 = · · · = 0.

Remark 7.3. The weight filtration W• = W•(N) is centered at 0.

Recall that if the monodromy h is unipotent, we have a nilpotent endomorphismN on the central fiber, see Remark 6.10.

Definition 7.4. For a PVHS (over D∗) of weight m, i.e. each fiber has HS ofweight m, define the monodromy weight filtration

M• := W•−m

on the central fiber where W• = W•(N) is the filtration defined in the last propo-sition for the nilpotent endomorphism N .

Remark 7.5. The monodromy weight filtration M• = W•−m is centered at m.

Remark 7.6. In general, as can be seen from the examples in the next subsection,the monodromy weight filtration of the limit MHS does not depend on the choiceof tangent vector.

Example 7.7. We have a nilpotent endomorphism

N = a∂

∂w

on the central fiberH0 = Ca⊕ Cw

of the canonical extension of HD∗ . So

Nw = a, Na = 0,

this gives the following diagram

−1

a1

w

N

for the weight filtration W• = W•(N) given by

W−2 = 0, W−1 = W0 = Ca, W1 = H0.

Since HD∗ is a PVHS of weight 1, we have the following diagram

42 MA LUO

0

a2

w

N

for the monodromy weight filtration M• = W•−1 given by

M−1 = 0, M0 = M1 = Ca, M2 = H0.

So we have

GrM2 = Cw, GrM0 = Ca.

7.2. Examples of limit mixed Hodge structures. Now we are ready to com-pute limit mixed Hodge structures for the PVHS HD∗ .

Example 7.8 (Limit MHS at ∂/∂q). By the description before Proposition 6.14,the Q-structure (even Z-structure)

H∂/∂q = Za⊕ Zb

where

a = q−Na = a

b = q−Nb = − w

2πi

is computed in Example 6.12. We view H∂/∂q as a Betti Q-structure HB∂/∂q, and

here N is the Betti version

NB =1

2πia∂

∂b.

It is different from the de Rham version

NdR = a∂

∂w

appeared in Example 7.7. In fact, by Proposition 6.8, the residue of the connectionis −NB , so we have

NdR = −NB .

With the central fiber

H0 = Ca⊕ Cw, F 1H0 = Cw

and the weight filtration M• we just computed in Example 7.7,

GrM2 = Cw, GrM0 = Ca,

it is easy to see that (by Remark 4.10)

H∂/∂q = Z(0)⊕ Z(−1)

where the weight 0 part Z(0) is generated by a = a, and weight 2 part Z(−1) byb 7→ −(2πi)−1w.

Remark 7.9. To be very precise, NB and NdR give rise to monodromy weightfiltrations MB

• on HB∂/∂q and MdR

• on H0, respectively. These monodromy weight

filtrations are compatible under the comparison isomorphism.


Example 7.10 (Limit MHS at λ∂/∂q). We use local coordinate t = q/λ, sincethen we have

∂/∂t = λ∂/∂q.

As the previous example, we compute the Q-structure using N = NB = 12πia

∂∂b .

We have

t−N = (q/λ)−N = e−N log(q/λ) = id−N log(q/λ) = id +log λ

2πia∂

∂b− log q

2πia∂

∂b

and the Q-structure Hλ∂/∂q = Za⊕ Zb with

a = t−Na = a

b = t−Nb = b+ log λa

2πi− τa = − w

2πi+ log λ

a

2πi

Then (a b

)Betti basis

=(a w

)DR basis

(1 log λ

2πi0 − 1

2πi

).

We have the central fiber

H0 = Ca⊕ Cw, F 1H0 = Cw

and the weight filtration M• with

GrM2 = Cw, GrM0 = Ca.

The weight 0 part Z(0) is again generated by a = a, and the quotient of Hλ∂/∂q by

Z(0) is isomorphic to Z(−1) since modulo a multiple of a we have b 7→ −(2πi)−1w.Unlike H∂/∂q, this MHS does not split in general. It is an extension

0→ Z(0)→ Hλ∂/∂q → Z(−1)→ 0

of Z(−1) by Z(0), and corresponds to

λ ∈ C∗ ∼= Ext1(Z(−1),Z(0)).

Remark 7.11. To understand this limit MHS geometrically, read Hain’s KZB notes,see the picture in Section 16 (ignoring the puncture, with the small loop γ, at theidentity of the elliptic curve), and Appendix B for the construction of the fibersover tangent vectors.

7.3. Limits of VMHS. Unlike PVHS, for a variation of mixed Hodge structures(VMHS), the existence of limit is not automatic. One needs local admissiblility.Suppose locally we have a local system of MHS

(V,W•)

D∗

and the corresponding vector bundle

(V, F •)

D∗

https://arxiv.org/pdf/1309.0580.pdf

44 MA LUO

extends across 0, and the central fiber

(V0,W•)

comes with a nilpotent operator N compatible with W•. Note that

H0(D∗, π∗V)≈−→ V0

and W• is from VMHS. A particular ingredient for local admissibility is the relativeweight filtration of N on (V0,W•).

Definition 7.12. M• is a relative weight filtration of N on (V,W•) if the induced

filtration on GrWm V is W•(N) reindexed to center at m and N(Mr) ⊆Mr−2.

Remark 7.13. For generic nilpotentN on (V,W•), there is no suchM•. For example,see Hain–Matsumoto [UMEM, Appendix A, Example A.5.]

Definition 7.14. An admissible VMHS over a smooth algebraic curve T = T −Σis a graded polarizable VMHS which is locally admissible at each s ∈ Σ.

Example 7.15 (Admissible VMHS).

(1) (Steenbrink–Zucker) For locally topologically trivial familiy of smooth al-gebraic varieties f : X → T , take

V = Rmf∗Q.(2) (Gullen–Navarro–Puerta) For locally topologically trivial familiy of vari-

eties (not necessarily smooth; could be simplicial, singular, ...) f : X → T ,take

V = Rmf∗Q.(3) (Hain) For locally topologically trivial familiy of smooth varieties

X

T

fσ

with σ a smooth section. Take local system V whose fiber at t ∈ T is

Vt := O(πun1 (Xt, σ(t))).

8. Regularised Periods

8.1. Asymptotics of periods. Suppose that we have dual local systems

V V

D∗ D∗

e.g. Hj(Xt), Hj(Xt) = Rjf∗Q for a family f : X → T . Assume the mon-odromy is unipotent, and we have Deligne’s canonical extensions

V V

D Dendowed with connection ∇, ∇, respectively. We have

Res0∇ = −N, Res0 ∇ = N .



Proposition 8.1. Suppose that γ(t) is a flat section of V, and ω(t) is a holomorphicsection of V, then ∫

γ(t)

ω(t) := 〈γ(t), ω(t)〉

is a polynomiald∑j=0

aj(t)(log t)j

in log t, where aj(t) ∈ O(D). Furthermore,

limt→0〈t−Ntγ(t), ω(t)〉 = a0(0).

Remark 8.2.

(1) The first statement follows from another characterization of Deligne’s canon-ical extension (unipotent case): flat sections of V, V have logarithmicgrowth at 0. See ending remark in Conrad’s notes, or higher dimensionalgeneralization in Deligne [ODE, Ch. II, Thm 5.2].

(2) The second statement is how we will compute regularised periods.

Proof. Supposeγ1(t), · · · , γm(t)

is a basis of H0(D∗, π∗V), and

ω1(t), · · · , ωm(t)is a framing of V over D. Let

ϕj(t) = t−Ntγj(t), j = 1, · · · ,mbe Deligne’s framing3 of V over D, and

ϕ1, · · · , ϕmthe dual framing of V, i.e.

〈ϕj , ϕk〉 = δjk.

Identifying the central fiber V0 withm⊕j=1

Cϕj ,

thenN = −Res0∇

acts on V0, cf. Proposition 6.8. We write

(Nϕ1, · · · , Nϕm) = (ϕ1, · · · , ϕm)A

where A ∈Mm(C) is nilpotent, then

Nk · (ϕ1, · · · , ϕm) = (ϕ1, · · · , ϕm)Ak

andtN · (ϕ1, · · · , ϕm) = (ϕ1, · · · , ϕm)tA.

We have(ω1(t), · · · , ωm(t)) = (ϕ1, · · · , ϕm)Φ(t)

3This is the framing we used in Deligne’s trivialization.

46 MA LUO

where Φ ∈ GLm(O(D)). So

〈

γ1(t)...

γm(t)

, (ω1(t), · · · , ωm(t))〉≈〈

tNϕ1

...tNϕm

, (ϕ1, · · · , ϕm)Φ(t)〉

= tAT

〈

ϕ1

...ϕm

, (ϕ1, · · · , ϕm)〉Φ(t)

= tAT

Φ(t)

= eAT log tΦ(t) ∈Mm(O(D)[log t])

since A is nilpotent, eAT log t is a polynomial in log t. If

γ(t) = c1(t)γ1(t) + · · ·+ cm(t)γm(t),

ω(t) = f1(t)ω1(t) + · · ·+ fm(t)ωm(t),

then

〈γ(t), ω(t)〉≈(c1(t), · · · , cm(t))eAT log tΦ(t)

f1(t)...

fm(t)

.

By the Remark below, we can write

〈γ(t), ω(t)〉 =

d∑j=0

aj(t)(log t)j .

Then taking the limit as t→ 0, we have

limt→0〈t−Ntγ(t), ω(t)〉= lim

t→0〈t−Nγ(t), ω(t)〉

= limt→0

(c1(t), · · · , cm(t))Φ(t)

f1(t)...

fm(t)

= (c1(0), · · · , cm(0))Φ(0)

f1(0)...

fm(0)

= a0(0).

Remark 8.3. Here we are using the fine results of nilpotent orbit theorem, indicatedby ≈,= above. Namely, we have used (the nilpotent orbits)

tNϕj , j = 1, · · · ,mto approximate flat sections γj(t) of V. Their distance |tNϕj − γj(t)| is asymptoti-cally bounded by

C|t|α| log t|β

as t→ 0, cf. Schmid [VHS, (4.9)]. This also implies that the limit

limt→0

t−Ntγ(t) = limt→0

t−Nγ(t)


is well defined, which was used implicitly in the last section when computing limitMHS.

8.2. Regularising iterated integrals. Suppose X is a smooth projective curve,D is an effective divisor on X. Let ~v ∈ TPX be a non-zero tangent vector atP ∈ Supp(D). For Q ∈ X − D, and ω1, · · · , ωr ∈ H0(X,Ω1

X(logD)), we willregularise ∫ Q

~v

(ω1| · · · |ωr).

SetA = C〈X1, · · · , Xn〉/Ir+1.

EmbedA −→ End(A)

by left multiplication. The A-valued 1-form

Ω = ω1X1 + · · ·+ ωnXn ∈ H0(Ω1X(logD))⊗A

satisfies dΩ = Ω ∧ Ω = 0, thus defines a flat connection

∇ = d+ Ω

on the trivial bundle

A×X

X

with regular singularities along D. Note that N = −ResP Ω with

ResP Ω :=

n∑j=1

ResP ωjXj

is nilpotent in A.Define

T (z) := 〈1 +

∫(Ω) +

∫(Ω|Ω) + · · ·+

∫(Ω| · · · |Ω)︸︷︷︸

r

, γz,Q〉

where γz,Q is a path from z to Q. Then, by formula of d for iterated integrals(Proposition 1.11 (1)), we have

dT = −ΩT,

i.e. T is a flat section for ∇ = d+ Ω.Choose a holomorphic coordinate t at P such that

~v = ∂/∂t.

Set

ΩP :=

n∑j=1

ωjXj(= ResP Ωdt

t)

where ωj = ResP ωjdtt . We will view ΩP as an A-valued 1-form on the tangent

space TPX, cf. Remark 8.4. Since∫ t

1

(dt

t| · · · |dt

t)︸︷︷︸

m

=1

m!(log t)m,

48 MA LUO

for a path γ1,t from 1 to t, we have

〈1 +

∫(ΩP ) +

∫(ΩP |ΩP ) + · · · , γ1,t〉 =

∞∑m=0

∫ t

1

(ΩP | · · · |ΩP )︸︷︷︸m

=

∞∑m=0

(ResP Ω log t)m

m!= eResP Ω log t = tResP Ω = t−N .

The regularised iterated integrals are coefficients of

limt→0

t−NT (t).

For example, ∫ Q

~v

(ω1| · · · |ωr) = coefficient of X1 · · ·Xr.

This implies the formula∫ Q

~v

(ω1| · · · |ωr) = limt→0

r∑j=0

∫ t

1

(ω1| · · · |ωj)︸︷︷︸coeff. of X1···Xj

in t−N

∫ Q

t

(ωj+1| · · · |ωr)︸︷︷︸coeff. of Xj+1···Xr

in T (t)

.

Remark 8.4. This formula looks much like the composition of paths formula forusual iterated integrals:∫

γ1γ2

(ω1| · · · |ωr) =

r∑j=0

∫γ1

(ω1| · · · |ωj)∫γ2

(ωj+1| · · · |ωr).

Therefore, we can think of the path from ~v to Q as a composition of paths: a pathfrom ~v to P in the tangent space TPX, followed by a path from P to Q in X, cf.Brown [MMV, Definition 4.4 & Figure 1], which was originated from Deligne [Legroupe fondamental de la droite projective moins trois points, 15.9].

Example 8.5 (Regularised periods).

(1) Let X = P1, D = 0,∞, Q ∈ C∗, ~v = ∂/∂z at 0 ∈ P1. Then∫ Q

~v

dz

z= limt→0

(∫ t

1

dz

z+

∫ Q

t

dz

z

)= limt→0

∫ Q

1

dz

z= logQ.

(2) Let X,D,Q be the same as (1), ~v = λ∂/∂z. Take t = z/λ, thendt

t=dz

z,

and ∫ Q

~v

dz

z= limt→0

(∫ t

t=1←z=λ

dz

z+

∫ Q

t

dz

z

)= logQ− log λ.

(3) (Drinfel’d Associator Φ(X0, X1))This is the regularisation of

〈T, dch〉

where dch is the path

0 1



in P1 − 0, 1,∞ from

~v0 := ∂/∂x ∈ T0P1

to

~v1 := −∂/∂x ∈ T1P1

and

T = 1 +

∫(Ω) +

∫(Ω|Ω) + · · ·

where

Ω =dz

zX0 −

dz

1− zX1

is the 1-form that defines the KZ connection

∇KZ = d+ Ω

on

C〈〈X0, X1〉〉 × P1

P1

with regular singularities at 0, 1,∞.Since

Res0dz

z= 1, Res0(− dz

z − 1) = 0,

Res1dz

z= 0, Res1(− dz

z − 1) = 1,

we have 1-forms

Ω0 := Res0 Ωdx

x=dx

xX0, Ω1 := Res1 Ω

dx

x=dx

xX1

on the tangent spaces T0P1 and T1P1 respectively. Since∫ t

1

(Ω0) = (

∫ t

1

dx

x)X0 = (log t)X0,

∫ 1

s

(Ω1) = (

∫ 1

s

dx

x)X1 = (− log s)X1,

we have

Φ(X0, X1) = limt→0s→0

e(log t)X0〈T, [t, 1− s]〉e(− log s)X1

= limt→0s→0

tX0〈T, [t, 1− s]〉s−X1 .

Here the limit is taken with s, t > 0, and [t, 1− s] is the path that traversesthe interval. If we write

Φ(X0, X1) =∑

w wordin X0,X1

c(w)w,

then we have• c(X0) = c(X1) = 0 =⇒ c(Xp

0 ) = c(Xp1 ) = 0, p > 0.

• c(X1Xk1−10 · · ·X1X

kr−10 ) = (−1)rζ(k1, · · · , kr)

• c(Xp0X1w) = (−1)pc(X1(Xp

0 xw))• c(wX0X

p1 ) = (−1)pc((wxXp

1 )X0)

50 MA LUO

where

ζ(k1, · · · , kr) :=∑

0<n1<···<nr

1

nk11 · · ·nkrr

for integers kj ≥ 1, kr ≥ 2, and x is the shuffle product for words. Thefirst two items can be computed directly. The last two can be checked byinduction on p and using the fact that

c(vxw) = c(v)c(w).

This fact follows from a regularised version of Proposition 1.11 (2).

E-mail address: [email protected]

Mathematical Institute, University of Oxford, Andrew Wiles Building, WoodstockRoad, Radcliffe Observatory Quarter, Oxford, OX2 6GG, United Kingdom

mailto:[email protected]

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