# Degenerating complex variations of Hodge structure in dimension one

Claude Sabbah
Centre de Mathématiques Laurent Schwartz, École Polytechnique
<EMAIL_ADDRESS>Christian Schnell
Department of Mathematics, Stony Brook University
<EMAIL_ADDRESS>
##  1 Introduction

###  1.1 Summary of the paper

####  1 .

In this paper, we analyze the behavior of polarized complex variations of
Hodge structure on the punctured unit disk. For _integral_ variations of Hodge
structure, this analysis was carried out by Wilfried Schmid [Schmid] in his
famous article _Variation of Hodge structure: the singularities of the period
mapping_ , one of the central works in modern Hodge theory. The restriction to
integral variations of Hodge structure is natural from the point of view of
geometry; it also implies that the eigenvalues of the monodromy transformation
are roots of unity, and this fact makes things technically easier.
Nevertheless, there are two reasons why we want to remove this restriction
now. First, we need the results for complex variations of Hodge structure to
set up the theory of complex mixed Hodge modules, the topic of our “Mixed
Hodge Module Project”. Second, it turns out that working with complex
variations clarifies many aspects of the theory that were somewhat obscured by
the presence of an integral structure, especially the central role played by
the Hodge metric.

####  2 .

Analytically, the behavior of a polarized variation of Hodge structure on the
punctured disk turns out to be surprisingly simple. Before getting into any
details, let us therefore quickly highlight the two results that are,
conceptually, the most important. A complex variation of Hodge structure is a
smooth vector bundle $E$ with a flat connection $d$, and a decomposition into
smooth subbundles $E^{p,q}$ that interacts with the connection in a certain
way. The polarization defines a smooth metric on the bundle $E$, called the
Hodge metric. In order to compare different fibers of $E$, we use the vector
space $V$ of multi-valued flat sections as a reference frame; together with
the monodromy transformation $T$, it contains the same information as the flat
bundle $(E,d)$. The _first result_ is that, in this reference frame, the Hodge
metric degenerates in a very simple manner: near the origin, the Hodge metric
of any nonzero multi-valued flat section grows or decays like a power of
$\lvert\log\lvert t\rvert\rvert$, where $t$ is the coordinate on the disk; the
exponents are determined by the monodromy transformation $T$. This means that,
up to a constant, the asymptotic behavior of the Hodge metric only depends on
the underlying flat bundle.

####  3 .

The fact that the Hodge metric grows and decays at different rates of course
prevents the Hodge structures from having any sort of limit as $t\to 0$. We
then take the natural step of “rescaling” the Hodge metric, in order to even
out these different rates. The rescaling can be done by moving the Hodge
structures by elements of the symmetry group of the period domain; it involves
a choice of splitting for the filtration by order of growth of the Hodge
metric. The _second result_ is that the rescaled Hodge structures converge to
a limit, which is again a polarized Hodge structure of the same type. In
particular, the asymptotic behavior of the Hodge metric also controls the
asymptotic behavior of the Hodge structures in the variation. Without
rescaling, one gets a “limiting” mixed Hodge structure on the vector space
$V$, whose weight filtration is, up to a shift, the filtration by order of
growth of the Hodge metric. Each subquotient of the weight filtration is again
a polarized Hodge structure; the general principle at work, coming from
analysis, is that one can only expect to get a meaningful limit when
everything has the same order of growth.

###  1.2 What is new?

####  4 .

For the benefit of those readers who are already familiar with Schmid’s
results, we briefly summarize the new features of our approach. We mentioned
already that we treat arbitrary polarized complex variations of Hodge
structure on the punctured disk, without assuming that the eigenvalues of the
monodromy transformation are roots of unity. In this generality, we give new –
and, we think, more conceptual – proofs for all the major results in Schmid’s
paper, such as the estimates for the rate of growth of the Hodge norm; the
existence of a limiting mixed Hodge structure; the nilpotent orbit theorem;
and a simplified (but still sufficiently powerful) version of the
$\operatorname{SL}(2)$-orbit theorem. The results as such are of course just
special cases of Mochizuki’s monumental work on tame harmonic bundles
[MochizukiI], but we think that it is worthwhile to have a self-contained and
simple treatment.111In several places, for example [Zucker, p. 453], it is
claimed that Schmid’s results carry over to polarized variations of Hodge
structure whose monodromy is not quasi-unipotent; but despite considerable
effort, we have not been able to adapt Schmid’s proof of the nilpotent orbit
theorem to this setting.

####  5 .

Our starting point is a direct proof for the Hodge norm estimates, based on
calculations with harmonic bundles and on a comparison with certain model
variations of Hodge structure. Technically, the crucial point is Simpson’s
“basic estimate” for the Hodge norm of the Higgs field, which is proved using
Ahlfors’ lemma. In Schmid’s paper, the Hodge norm estimates are deduced from
the orbit theorems, and therefore appear towards the end; here, they stand at
the beginning. We feel that this gives a better conceptual explanation for the
appearance of the monodromy weight filtration. Next, we give a different proof
for the nilpotent orbit theorem, based on curvature properties of the Hodge
metric and $L^{2}$-extension theorems; analytically, the key point is that one
can make the curvature of the Hodge metric either positive or negative by
multiplying by a suitable power of $\lvert\log\lvert t\rvert\rvert$; this
general idea is due to Cornalba-Griffiths [CG] and Simpson [SimpsonVHS]. The
parameter dependence of various constants, important for extending the theory
to more than one dimension, is handled by using the maximum principle. We then
use the Hodge norm estimates and the nilpotent orbit theorem to prove that the
rescaled period mapping has a well-defined limit inside the period domain.

####  6 .

From the convergence of the rescaled period mapping, we deduce the existence
of a limiting mixed Hodge structure; the argument is a pleasant mix of linear
algebra and representation theory, and unlike in Schmid’s paper, does not rely
on the $\operatorname{SL}(2)$-orbit theorem. (This is useful because there is
a simplified proof for the $\operatorname{SL}(2)$-orbit theorem by Cattani-
Kaplan-Schmid [CKS, §6], whose input is the existence of the limiting mixed
Hodge structure.) We also show that, after a choice of splitting, the space of
multi-valued flat sections becomes what we call a “polarized
$\mathfrak{sl}_{2}$-Hodge structure”; this is the same kind of structure that
one has on the cohomology of a compact Kähler manifold. Finally, we prove a
cheap version of the $\operatorname{SL}(2)$-orbit theorem that applies to an
arbitrary splitting of the monodromy weight filtration (instead of the
$\operatorname{SL}(2)$-splitting). The degree of approximation is slightly
worse than in Schmid’s version, but in return, the proof is much easier. As an
application, we describe the asymptotic behavior of the Hodge metric in the
frame of multi-valued flat sections, something that is only implicit in
Schmid’s paper.

###  1.3 Acknowledgements

####  7 .

This paper has its immediate origins in a graduate course about variations of
Hodge structure that Ch.S. taught at Stony Brook University in Fall 2019. We
take this opportunity to thank all the students in the course for their
interest, and for their patience while the instructor worked through the
details of Schmid’s paper. Ch.S. thanks Wilfried Schmid for answering some
questions about the nilpotent orbit theorem; Takuro Mochizuki for a useful
discussion about the definition of the limiting Hodge filtration; Dingxing
Zhang for organizing a small reading seminar about Schmid’s paper back in
2014; Ruijie Yang for helping with the calculations for harmonic bundles; and
Bruno Klingler for several conversations about variations of Hodge structure
and harmonic bundles. He also thanks Ya Deng for a very useful email exchange
in 2020 about the nilpotent orbit theorem, and especially for sharing an early
version of a preprint [Deng] that suggested the use of Hörmander’s
$L^{2}$-estimates in this context.

####  8 .

During the preparation of this paper, Ch.S. was partially supported by NSF
grant DMS-1551677 and by a Simons Fellowship. He thanks the National Science
Foundation and the Simons Foundation for their financial support. Large parts
of this paper were written during a sabbatical stay at the Max-Planck-
Institute for Mathematics and at the Kavli Institute for the Physics and
Mathematics of the Universe, and Ch.S. thanks both of these institutions for
providing him with excellent working conditions.

##  2 Overview of the results

###  2.1 Complex variations of Hodge structure

####  9 .

Since we are going to be working with complex Hodge structures, let us briefly
recall the definition. For the purposes of this text, a Hodge structure of
weight $n$ on a complex vector space $V$ is a decomposition

$V=\bigoplus_{p+q=n}V^{p,q},$

and a polarization is a hermitian pairing $Q\colon
V\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C}$ such that the decomposition is orthogonal with respect to
$Q$, and such that $(-1)^{q}Q$ is positive definite on the subspace $V^{p,q}$.
The Hodge norm of a vector $v\in V$ is

$\lVert v\rVert^{2}=\sum_{p+q=n}(-1)^{q}\,Q(v^{p,q},v^{p,q}).$

Unlike in the case of real Hodge structures, it takes _two_ filtrations to
describe an arbitrary complex Hodge structure: the usual Hodge filtration

$F^{p}=F^{p}V=\bigoplus_{i\geq p}V^{i,n-i}$

and the conjugate Hodge filtration

$\overline{F}\vphantom{F}^{q}=\overline{F}\vphantom{F}^{q}V=\bigoplus_{j\geq
q}V^{n-j,j},$

because $V^{p,q}=F^{p}\cap\overline{F}\vphantom{F}^{q}$. When a polarization
is given, the Hodge structure is still determined by the Hodge filtration
$F=F^{\bullet}V$ alone: the reason is that

$\overline{F}\vphantom{F}^{q}=\bigl{\\{}\thinspace v\in
V\thinspace\thinspace\big{|}\thinspace\thinspace\text{$Q(v,x)=0$ for all $x\in
F^{n-q+1}$}\thinspace\bigr{\\}}.$

We are going to use the notation $\lVert v\rVert_{F}^{2}$ whenever we want to
emphasize the dependence of the Hodge norm on the Hodge filtration.

######  Example 1.

A Hodge structure is called real if
$V=V_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}$ for an $\mathbb{R}$-vector
space $V_{\mathbb{R}}$, and if $\overline{F}\vphantom{F}=\sigma(F)$, where
$\sigma\in\operatorname{End}_{\mathbb{R}}(V)$ is the conjugation operator
$\sigma(v\otimes z)=v\otimes\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$.
In the real case, one has the familiar Hodge symmetry
$V^{q,p}=\sigma\bigl{(}V^{p,q}\bigr{)}$ for every $p+q=n$.

####  10 .

The paper contains a certain amount of analysis, and for that reason, it will
be convenient to describe variations of Hodge structure in the language of
smooth vector bundles. Let $E$ be a smooth vector bundle on a complex manifold
$X$. We denote by $A^{k}(X,E)$ the space of smooth $k$-forms with coefficients
in $E$, and by $A^{i,j}(X,E)$ the space of smooth $(i,j)$-forms with
coefficients in $E$. With this notation, a variation of Hodge structure of
weight $n$ on $E$ is decomposition

$E=\bigoplus_{p+q=n}E^{p,q}$

into smooth subbundles, together with a flat connection $d\colon A^{0}(X,E)\to
A^{1}(X,E)$ that maps each $A^{0}(X,E^{p,q})$ into the direct sum of the
subspaces

$A^{1,0}(X,E^{p,q})\oplus A^{1,0}(X,E^{p-1,q+1})\oplus
A^{0,1}(X,E^{p,q})\oplus A^{0,1}(X,E^{p+1,q-1}).$

In the analysis that follows, the most important component of the connection
turns out to be the Higgs field, which is the linear operator

$\theta\colon A^{0}(X,E^{p,q})\to A^{1,0}(X,E^{p-1,q+1}).$

A polarization is a hermitian pairing

$Q\colon A^{0}(X,E)\otimes_{A^{0}(X)}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to A^{0}(X)$

that is flat with respect to $d$, such that the expression

$h(v,w)=\sum_{p+q=n}(-1)^{q}\,Q(v^{p,q},w^{p,q})\quad\text{for $v,w\in
A^{0}(X,E)$}$

defines a smooth hermitian metric on the bundle $E$, and the different
subbundles $E^{p,q}$ are orthogonal to each other with respect to $h$. This
metric is called the Hodge metric on the bundle $E$; for the sake of clarity,
we sometimes denote it by the symbol $h_{E}$.

####  11 .

The relation with the more familiar holomorphic description comes from
decomposing $d=d^{\prime}+d^{\prime\prime}$ into its $(1,0)$-component
$d^{\prime}\colon A^{0}(X,E)\to A^{1,0}(X,E)$ and its $(0,1)$-component
$d^{\prime\prime}\colon A^{0}(X,E)\to A^{0,1}(X,E)$. Then $d^{\prime\prime}$
gives $E$ the structure of a holomorphic vector bundle, which we denote by the
symbol $\mathscr{E}$; and $d^{\prime}$ defines an integrable holomorphic
connection
$\nabla\colon\mathscr{E}\to\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}\mathscr{E}$
on this bundle. The condition on $d$ is saying that the Hodge bundles

$F^{p}E=\bigoplus_{i\geq p}E^{i,n-i}$

come from holomorphic subbundles $F^{p}\mathscr{E}$, and that the connection
satisfies Griffiths’ transversality relation
$\nabla(F^{p}\mathscr{E})\subseteq\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}F^{p-1}\mathscr{E}$.
From this point of view, the Higgs field is simply the holomorphic operator

$\theta\colon
F^{p}\mathscr{E}/F^{p+1}\mathscr{E}\to\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}F^{p-1}\mathscr{E}/F^{p}\mathscr{E}$

induced by the action of the connection $\nabla$.

###  2.2 Multi-valued flat sections and monodromy

####  12 .

Our main concern in this paper is the asymptotic behavior of a polarized
variation of Hodge structure on the punctured disk. Let

$\Delta^{\ast}=\bigl{\\{}\thinspace
t\in\mathbb{C}\thinspace\thinspace\big{|}\thinspace\thinspace 0<\lvert
t\rvert<1\thinspace\bigr{\\}}$

be the punctured unit disk, and let $E$ be a polarized variation of Hodge
structure on $\Delta^{\ast}$. In order to compare the Hodge structures on
different fibers of the bundle $E$, we need a common reference frame; the most
natural choice is the space of all multi-valued flat sections of the flat
bundle $(E,d)$.

####  13 .

Since the fundamental group of $\Delta^{\ast}$ is cyclic, a flat bundle
$(E,d)$ on the punctured disk is uniquely determined by the pair $(V,T)$,
where $V$ is the complex vector space of all multi-valued flat sections, and
$T\in\operatorname{GL}(V)$ is the monodromy transformation. To define $V$ and
$T$ more precisely, recall that the universal covering space of
$\Delta^{\ast}$ is naturally the _left_ half-plane

$\mathbb{H}=\bigl{\\{}\thinspace
z\in\mathbb{C}\thinspace\thinspace\big{|}\thinspace\thinspace\operatorname{Re}z<0\thinspace\bigr{\\}},$

which maps to $\Delta^{\ast}$ via the exponential function
$\exp\colon\mathbb{H}\to\Delta^{\ast}$. Let $V$ be the complex vector space of
all multi-valued flat sections of $E$; concretely, $V$ is defined to be the
space of $d$-flat sections of the pullback bundle $\exp^{\ast}\\!E$. The
polarization on $E$ induces a nondegenerate hermitian form

$Q\colon V\otimes_{\mathbb{C}}\bar{V}\to\mathbb{C},$

and translation by $2\pi i$ induces the monodromy operator

$T=T_{s}\cdot T_{u}=T_{s}\cdot e^{2\pi i\,N}\in\operatorname{GL}(V).$

Here $T_{s}$ is diagonalizable and $N$ is nilpotent, and one has

$Q(T_{s}v,T_{s}w)=Q(v,w)\quad\text{and}\quad Q(Nv,w)=Q(v,Nw).$

Note that our definition of the nilpotent operator $N$ differs from Schmid’s
definition [Schmid, (4.6)] by a factor of $2\pi i$, because $\log T_{u}=2\pi
iN$. The advantage is that this makes $N$ independent of the choice of
$i=\sqrt{-1}$.

####  14 .

The fact that the flat bundle $(E,d)$ is part of a polarized variation of
Hodge structure puts the following restriction on the monodromy transformation
(see 3).

######  Proposition 1 (Monodromy theorem).

If $\lambda\in\mathbb{C}$ is an eigenvalue of $T$, then
$\lvert\lambda\rvert=1$.

For _integral_ variations of Hodge structure, it follows (because of
Kronecker’s theorem) that all eigenvalues of $T$ are roots of unity. This fact
is traditionally called the “monodromy theorem”; Schmid [Schmid, Lem. 4.5]
attributes it to Borel.

###  2.3 The Hodge norm estimates

####  15 .

Conceptually, the most important piece of the theory are the Hodge norm
estimates, which describe the rate of growth of the pointwise Hodge norm for
multi-valued flat sections. This result is the starting point of our treatment
of the theory; in Schmid’s work, it comes towards the end [Schmid, Thm. 6.6].
For a multi-valued flat section $v\in V$, the Hodge metric gives us a smooth
function

$h(v,v)=\sum_{p+q=n}(-1)^{q}\,Q(v^{p,q},v^{p,q})\colon\mathbb{H}\to[0,\infty).$

The asymptotic behavior of this function as
$\lvert\operatorname{Re}z\rvert\to\infty$ turns out to be surprisingly simple
– in fact, it is completely controlled by the following elementary
construction. The nilpotent operator $N\in\operatorname{End}(V)$ determines an
increasing filtration $W_{\bullet}=W_{\bullet}V$, called the monodromy weight
filtration. This filtration is uniquely characterized by the following two
properties:

  1. (a)

For every $k\in\mathbb{Z}$, one has $NW_{k}\subseteq W_{k-2}$.

  2. (b)

For every $k\in\mathbb{N}$, the induced operator
$N^{k}\colon\operatorname{gr}_{k}^{W}\to\operatorname{gr}_{-k}^{W}$ is an
isomorphism.

An explicit formula for the monodromy weight filtration is

$W_{k}=\sum_{j\in\mathbb{N}}N^{j}\bigl{(}\ker N^{k+2j+1}\bigr{)}\quad\text{for
$k\in\mathbb{Z}$.}$

The content of the Hodge norm estimates is that the monodromy weight
filtration precisely captures the asymptotic behavior of the Hodge norm.

######  Theorem 1 (Hodge norm estimates).

For $v\in V$ any multi-valued flat section,

$v\in W_{k}\setminus W_{k-1}\quad\Longleftrightarrow\quad
h(v,v)\sim\lvert\operatorname{Re}z\rvert^{k},$

as long as $\operatorname{Im}z$ stays in a fixed interval.

On the punctured disk, the statement is that $h(v,v)\sim\lvert\log{\lvert
t\rvert}\rvert^{k}$ on each angular sector.

####  16 .

The main ingredient of the proof is the following basic estimate [Simpson, §2]
for the pointwise Hodge norm of the Higgs field $\theta$, viewed as a smooth
section of the induced variation of Hodge structure on the bundle
$\operatorname{End}(E)$. It is analogous to the distance-decreasing property
of period mappings used in Schmid’s paper [Schmid, Cor. 3.17].

######  Theorem 2 (Basic estimate).

One has the inequality

$h_{\operatorname{End}(E)}\bigl{(}\theta_{\partial/\partial
t},\theta_{\partial/\partial_{t}}\bigr{)}\leq\frac{C_{0}^{2}}{\lvert
t\rvert^{2}(\log\lvert t\rvert)^{2}},$

where $C_{0}=\frac{1}{2}\sqrt{\binom{r+1}{3}}$ and $r=\operatorname{rk}E$.

We use some elementary calculations with harmonic bundles, already done by
Simpson [Simpson, §2], together with the fact that $\theta$ is nilpotent, to
show that the Laplacian of the smooth function
$f=h_{\operatorname{End}(E)}\bigl{(}\theta_{\partial/\partial
t},\theta_{\partial/\partial_{t}}\bigr{)}$ satisfies the differential
inequality

$\Delta\log f\geq 8f/C_{0}^{2}.$

The basic estimate then follows by applying Ahlfors’ lemma [Ahlfors]. The
value of the constant $C_{0}$ is optimal; more important than the exact value,
however, is the fact that $C_{0}$ depends on nothing but the rank of the
bundle $E$.

####  17 .

To prove the Hodge norm estimates, we first establish a special case.

######  Lemma 1.

If $v\in V$ is a multi-valued flat section with $Tv=\lambda v$ for some
$\lambda\in\mathbb{C}$, then the function $\varphi_{v}=\log h(v,v)$ remains
bounded from above as $\lvert\operatorname{Re}z\rvert\to\infty$.

This is indeed a special case of the Hodge norm estimates, because $Tv=\lambda
v$ implies $Nv=0$, and therefore $v\in W_{0}$. Some further calculations with
harmonic bundles, together with the basic estimate, show that:

  1. 1.

The function $\varphi_{v}$ is subharmonic on $\mathbb{H}$.

  2. 2.

The first derivatives of $\varphi_{v}$ are bounded by a constant times
$\lvert\operatorname{Re}z\rvert^{-1}$.

  3. 3.

One has $\varphi_{v}(z+2\pi i)=\varphi_{v}(z)$ for every $z\in\mathbb{H}$.

By a simple calculus argument, these three properties are enough to conclude
that $\varphi_{v}$ is bounded from above as
$\lvert\operatorname{Re}z\rvert\to\infty$.

####  18 .

This special case is already enough to prove that the asymptotic behavior of
the Hodge norm only depends on the underlying flat bundle $(E,d)$.

######  Theorem 3 (Comparison theorem).

Let $E_{1}$ and $E_{2}$ be two polarized variations of Hodge structure on the
punctured disk. If $(E_{1},d_{1})\cong(E_{2},d_{2})$ as flat bundles, then the
Hodge metrics $h_{1}$ and $h_{2}$ are mutually bounded, up to a constant, as
$t\to 0$.

This follows by considering the induced variation of Hodge structure on the
flat bundle $\operatorname{Hom}(E_{1},E_{2})$. To prove the Hodge norm
estimates in general, we then proceed as follows. Using representation theory,
we construct a “model” variation of Hodge structure on the given flat bundle
$(E,d)$, whose Hodge norm has the correct asymptotic behavior; the comparison
theorem guarantees that, up to a constant, the Hodge norm on $E$ has the same
behavior. The construction of the models is done by decomposing $V$ into
irreducible representations of the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{C})$, and then explicitly constructing a model
variation on each irreducible representation. The construction uses the theory
of $\mathfrak{sl}_{2}$-Hodge structures, which we study in some depth in
Section 5. (See also 35 in this chapter.)

######  Example 2.

In the model variation of Hodge structure coming from the standard
representation of $\mathfrak{sl}_{2}(\mathbb{C})$ on $\mathbb{C}^{2}$, the
Hodge metric is given by the formula

$\begin{pmatrix}\lvert x\rvert+y^{2}\lvert x\rvert^{-1}&-iy\lvert
x\rvert^{-1}\\\ iy\lvert x\rvert^{-1}&\lvert x\rvert^{-1}\end{pmatrix},$

where $z=x+iy$. Since $(1,0)\in W_{1}$ and $(0,1)\in W_{-1}$, this is in
complete agreement with the Hodge norm estimates.

###  2.4 Period domains and period mappings

####  19 .

Once we understand the behavior of the Hodge metric, the next question is what
happens to the Hodge structures as $t\to 0$. The answer is best stated in the
language of period domains and period mappings. The space of multivalued flat
sections gives us a trivialization of the bundle $\exp^{\ast}E$, and so we can
consider all the Hodge structures in the variation as living on the same
vector space $V$. From our polarized variation of Hodge structure $E$, we then
get a holomorphic period mapping

$\Phi\colon\mathbb{H}\to D.$

The period domain $D$ parametrizes all Hodge structures of weight $n$ on $V$,
with fixed Hodge numbers $\dim V^{p,q}$, that are polarized by the pairing
$Q$; a basic fact is that $D$ is a homogeneous space for the real Lie group

$G=\bigl{\\{}\thinspace
g\in\operatorname{GL}(V)\thinspace\thinspace\big{|}\thinspace\thinspace\text{$Q(gv,gw)=Q(v,w)$
for all $v,w\in V$}\thinspace\bigr{\\}}.$

The period domain is an open subset of the so-called compact dual $\check{D}$,
which parametrizes all decreasing filtrations $F^{\bullet}$ on $V$ such that
$\dim F^{p}=\dim V^{p,q}+\dim V^{p+1,q-1}+\dotsb$ for all $p\in\mathbb{Z}$.
The compact dual is a projective complex manifold, and a homogeneous space for
the complex Lie group $\operatorname{GL}(V)$; the complex structure on the
period domain $D$ comes from the embedding $D\subseteq\check{D}$.

####  20 .

The period mapping is holomorphic, and its differential is represented by the
Higgs field $\theta$; see 15 for the precise statement. Moreover, the
monodromy transformation $T\in G$ is defined in such a way that

$\Phi(z+2\pi i)=T\cdot\Phi(z).$

The eigenvalues of $T=T_{s}T_{u}$ lie on the unit circle, and so we can write
$T_{s}=e^{2\pi i\,S}$, where $S\in\operatorname{End}(V)$ is a semisimple
operator with real eigenvalues in a fixed half-open interval of length $1$;
then $T=e^{2\pi i(S+N)}$. One can interpret the operator $S+N$ as the residue
of the connection $\nabla$ on the canonical extension of the holomorphic
vector bundle $\mathscr{E}$; see 177.

###  2.5 Convergence of the rescaled period mapping

####  21 .

Now we come to a crucial point in the argument. The Hodge norm estimates

$v\in W_{k}\setminus W_{k-1}\quad\Longleftrightarrow\quad\lVert
v\rVert_{\Phi(z)}^{2}\sim\lvert\operatorname{Re}z\rvert^{k}$

suggest “rescaling” the period mapping, in order to even out the behavior of
the Hodge norm on different parts of the weight filtration. To do this
efficiently, we choose a splitting $H\in\operatorname{End}(V)$ for the weight
filtration, with the following three properties:

  1. 1.

$H$ is semisimple, with integer eigenvalues.

  2. 2.

$W_{k}=E_{k}(H)\oplus W_{k-1}$ and $[H,N]=-2N$.

  3. 3.

$[H,T_{s}]=0$ and $Q(Hv,w)+Q(v,Hw)=0$ for all $v,w\in V$.

It is not hard to show that such a splitting always exists (see 9); in
general, it is far from unique. For a nonzero multi-valued flat section $v\in
E_{k}(H)$, we then have

$\lVert
v\rVert_{\Phi(z)}^{2}\sim\lvert\operatorname{Re}z\rvert^{k}\qquad\text{and}\qquad
e^{-\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}v=\lvert\operatorname{Re}z\rvert^{-\frac{k}{2}}v$

Consequently, the Hodge norm of
$e^{-\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}v$ stays bounded as
$\lvert\operatorname{Re}z\rvert\to\infty$. We can achieve the same effect by
moving the Hodge structures by the operator
$e^{\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}$. Either way, the
conclusion is that the rescaled period mapping

$\hat{\Phi}_{S,H}\colon\mathbb{H}\to
D,\quad\hat{\Phi}_{S,H}(z)=e^{\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}e^{-\frac{1}{2}(z-\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}})(S+N)}\Phi(z),$ (21.1)

is invariant under the substitution $z\mapsto z+2\pi i$, and the Hodge norm of
every multi-valued flat section is uniformly bounded as
$\operatorname{Re}z\to-\infty$. The two exponential factors are elements of
the real Lie group $G$; this is the reason why the rescaled period mapping
stays in $D$.

####  22 .

The second key result is that, after rescaling, the polarized Hodge structures
now converge to a well-defined limit. As indicated by the notation, the limit
does depend on the choice of splitting: different splittings lead to different
limits.

######  Theorem 4 (Convergence of the rescaled period mapping).

For any choice of splitting $H\in\operatorname{End}(V)$ of the monodromy
weight filtration as above, the limit

$e^{-N}F_{H}=\lim_{\operatorname{Re}z\to-\infty}\hat{\Phi}_{S,H}(z)\in D$

exists in the period domain. Moreover, the filtration $F_{H}\in\check{D}$
satisfies

$T_{s}(F_{H}^{\bullet})\subseteq F_{H}^{\bullet},\quad
H(F_{H}^{\bullet})\subseteq F_{H}^{\bullet},\quad N(F_{H}^{\bullet})\subseteq
F_{H}^{\bullet-1}.$

Together with the Hodge norm estimates, this is saying that a polarized
variation of Hodge structure on the punctured disk behaves in a very simple
way. Namely, as $t\to 0$, the Hodge metric grows (and decays) at several
different rates, and this of course prevents the Hodge structures from
approaching any sort of limit. But once we rescale to even out the behavior of
the Hodge metric, _both_ the Hodge metric and the Hodge structures converge.

###  2.6 The nilpotent orbit theorem

####  23 .

The most difficult step in proving the convergence of the rescaled period
mapping is the nilpotent orbit theorem, an important result in its own right.
Recall that the period mapping satisfies

$\Phi(z+2\pi i)=T\cdot\Phi(z),$

and that we have $T=e^{2\pi i(S+N)}$, where $S\in\operatorname{End}(V)$ is
semisimple with real eigenvalues in a fixed interval of length $<1$.
Consequently, the mapping

$\mathbb{H}\to\check{D},\quad z\mapsto e^{-z(S+N)}\Phi(z),$

is invariant under $z\mapsto z+2\pi i$, and therefore descends to a
holomorphic mapping

$\Psi_{S}\colon\Delta^{\ast}\to\check{D},\quad\Psi_{S}(e^{z})=e^{-z(S+N)}\Phi(z),$

sometimes called the untwisted period mapping. The exponential factor
$e^{-z(S+N)}$ is now an element of the complex Lie group
$\operatorname{GL}(V)$, and so the untwisted period mapping takes values in
the compact dual $\check{D}$.

####  24 .

The nilpotent orbit theorem has two parts. The first half is a convergence
statement: it says that $\Psi_{S}$ extends holomorphically over the origin.

######  Theorem 5 (Nilpotent orbit theorem, convergence).

The holomorphic mapping

$\Psi_{S}\colon\Delta^{\ast}\to\check{D},\quad\Psi_{S}(e^{z})=e^{-z(S+N)}\Phi(z),$

extends holomorphically across the origin, and the limit filtration
$\Psi_{S}(0)\in\check{D}$ has the property that
$(S+N)\Psi_{S}^{\bullet}(0)\subseteq\Psi_{S}^{\bullet-1}(0)$.

The limit $\Psi_{S}(0)$ in the nilpotent orbit theorem depends on the choice
of $S$, but one can show that the limiting Hodge filtration

$F_{\mathrm{lim}}=\lim_{\operatorname{Re}z\to-\infty}e^{-zN}\Phi(z)=\lim_{\operatorname{Re}z\to-\infty}e^{-\lvert\operatorname{Re}z\rvert\,S}\Psi_{S}(0)\in\check{D}$

is independent of $S$. It satisfies

$T_{s}(F_{\mathrm{lim}}^{\bullet})\subseteq
F_{\mathrm{lim}}^{\bullet}\quad\text{and}\quad
N(F_{\mathrm{lim}}^{\bullet})\subseteq F_{\mathrm{lim}}^{\bullet-1}.$

The limiting Hodge filtration does not come from a polarized Hodge structure
on $V$ (except when $N=0$); instead, it is part of a _mixed Hodge structure_ ,
whose weight filtration is given by the monodromy weight filtration
$W_{\bullet-n}$ from above. We are going to discuss this point in more detail
in Section 2.7 below.

####  25 .

We can now explain why the rescaled period mapping
$\hat{\Phi}_{S,H}\colon\mathbb{H}\to D$ converges to a limit in $D$. The
argument needs both the Hodge norm estimates (in 1) and the nilpotent orbit
theorem (in 5). The identities $[H,N]=-2N$ and $[H,S]=0$ can be used to write

$\hat{\Phi}_{S,H}(z)=e^{-N}\cdot
e^{\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}\,e^{-\lvert\operatorname{Re}z\rvert\,S}\,\Psi_{S}(e^{z}).$

The nilpotent orbit theorem implies that $\Psi_{S}(e^{z})\to\Psi_{S}(0)$ at a
rate proportional to $\lvert e^{z}\rvert=e^{-\lvert\operatorname{Re}z\rvert}$.
By analyzing the effect of the two exponential factors, and by using the fact
that the eigenvalues of $S$ lie in an interval of length $<1$, one deduces the
existence of the limit

$F_{H}=\lim_{\operatorname{Re}z\to-\infty}e^{\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}\,e^{-\lvert\operatorname{Re}z\rvert\,S}\,\Psi_{S}(e^{z})\in\check{D}.$

The filtration $F_{H}\in\check{D}$ is derived from the limit
$\Psi_{S}(0)\in\check{D}$ in the nilpotent orbit theorem in two stages:

  1. 1.

The effect of the first limit

$F_{\mathrm{lim}}=\lim_{\operatorname{Re}z\to-\infty}e^{-\lvert\operatorname{Re}z\rvert\,S}\Psi_{S}(0)$

is to make the filtration $\Psi_{S}(0)$ compatible with the eigenspace
decomposition of $T_{s}$, by projecting to the subquotients of the filtration
by decreasing eigenvalues of $S$. (Decreasing because of the minus sign in the
exponent.)

  2. 2.

The effect of the second limit

$F_{H}=\lim_{\operatorname{Re}z\to-\infty}e^{\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}F_{\mathrm{lim}}$

is to make the filtration $F_{\mathrm{lim}}$ compatible with the eigenspace
decomposition of $H$, by projecting to the subquotients of the weight
filtration $W_{\bullet}$ (which is the filtration by increasing eigenvalues of
$H$).

It follows that $\hat{\Phi}_{S,H}(z)$ converges to $e^{-N}F_{H}\in\check{D}$
as $\operatorname{Re}z\to-\infty$. At the same time, the Hodge norm estimates
tell us that the rescaled Hodge metric is comparable to a constant metric as
$\operatorname{Re}z\to-\infty$. These two facts together imply quite easily
that $e^{-N}F_{H}\in D$.

####  26 .

The nilpotent orbit theorem also has a second half, which says that the
original period mapping is very closely approximated by the nilpotent orbit

$\Phi_{\mathrm{nil}}\colon\mathbb{H}\to\check{D},\quad\Phi_{\mathrm{nil}}(z)=e^{zN}F_{\mathrm{lim}}.$

In fact, the nilpotent orbit is itself the period mapping of a polarized
variation of Hodge structure (on a punctured disk of smaller radius).

######  Theorem 6 (Nilpotent orbit theorem, approximation).

There are constants $C>0$, $x_{0}<0$, and $m\in\mathbb{N}$, such that

$\Phi_{\mathrm{nil}}(z)\in D\quad\text{and}\quad
d_{D}\bigl{(}\Phi(z),\Phi_{\mathrm{nil}}(z)\bigr{)}\leq
C\lvert\operatorname{Re}z\rvert^{m}e^{-\delta(T)\lvert\operatorname{Re}z\rvert}$

for every $z\in\mathbb{H}$ with $\operatorname{Re}z\leq x_{0}$. The constants
$C,x_{0}$ only depend on the base point $o\in D$ and on the minimal polynomial
of $T\in\operatorname{GL}(V)$; the integer $m$ only depends on
$r=\operatorname{rk}E$.

Here $d_{D}$ is the $G$-invariant distance function on the period domain,
defined by the Hodge metric (see (74.2)). The distance between the period
mapping $\Phi(z)$ and the nilpotent orbit $\Phi_{\mathrm{nil}}(z)$ is
exponentially small; the constant $\delta(T)$ in the exponent is the minimal
distance between consecutive eigenvalues of $T$ (on the unit circle), divided
by $2\pi$. For developing the theory in more than one variable, it is very
important that the values of the constants are essentially independent of the
variation of Hodge structure.

####  27 .

When $N=0$, the nilpotent orbit theorem is asserting that $F_{\mathrm{lim}}\in
D$; in other words, when the weight filtration is trivial, $F_{\mathrm{lim}}$
is the Hodge filtration of a polarized Hodge structure of weight $n$. When the
weight filtration is not trivial, what we get instead is a limiting mixed
Hodge structure (see 8).

######  Note.

This is not true for $\Psi_{S}(0)$: even when $N=0$, the filtration
$\Psi_{S}(0)$ is not necessarily the Hodge filtration of a polarized Hodge
structure (see 6).

####  28 .

The geometric meaning of the nilpotent orbit theorem is the following. Recall
that the operator $S\in\operatorname{End}(V)$ depended on a choice of half-
open interval $I$ of length $1$. The holomorphic vector bundle $\mathscr{E}$
has a canonical extension to a holomorphic vector bundle $\mathscr{E}_{I}$ on
the disk. Up to isomorphism, it is characterized by the fact that the
holomorphic connection $\nabla$ becomes a logarithmic connection

$\nabla\colon\mathscr{E}_{I}\to\Omega_{\Delta}^{1}(\log
0)\otimes_{\mathscr{O}_{\Delta}}\mathscr{E}_{I}$

whose residue $\operatorname{Res}_{0}\nabla$ has real eigenvalues in the
interval $I$. There is a distinguished trivialization
$\mathscr{E}_{I}\cong\mathscr{O}_{\Delta}\otimes_{\mathbb{C}}V$ in which

$\nabla(v\otimes 1)=\frac{\mathit{dt}}{t}\otimes(S+N)v.$

The content of 5 is that the Hodge bundles $F^{p}\mathscr{E}$ extends to
holomorphic subbundles $F^{p}\mathscr{E}_{I}$ of the canonical extension; and
that the Higgs field

$\theta\colon
F^{p}\mathscr{E}_{I}/F^{p+1}\mathscr{E}_{I}\to\Omega_{\Delta}^{1}(\log
0)\otimes_{\mathscr{O}_{\Delta}}F^{p-1}\mathscr{E}_{I}/F^{p}\mathscr{E}_{I}$

has a logarithmic pole at $t=0$, with residue $S+N$. The content of 6 is that
if we replace the extended Hodge bundles $F^{p}\mathscr{E}_{I}$ by trivial
subbundles with fiber $\Psi_{S}^{p}(0)\subseteq V$, then we still get a
polarized variation of Hodge structure of weight $n$, at least on a punctured
disk of a fixed smaller radius.

####  29 .

Our proof of the nilpotent orbit theorem is quite different from Schmid’s
original argument [Schmid, §8]. It uses in a crucial way the curvature
properties of the Hodge metric. A short calculation (see 2) shows that the
curvature tensor of the Hodge metric $h_{E}$ on the bundle $E^{p,q}$ is given
by $\Theta=-(\theta\theta^{\ast}+\theta^{\ast}\theta)$, where

$\displaystyle\theta$ $\displaystyle\colon A^{0}(\Delta^{\ast},E^{p,q})\to
A^{1,0}(\Delta^{\ast},E^{p-1,q+1})$ $\displaystyle\theta^{\ast}$
$\displaystyle\colon A^{0}(\Delta^{\ast},E^{p,q})\to
A^{0,1}(\Delta^{\ast},E^{p+1,q-1})$

are the Higgs field and its adjoint. (There are similar formulas for the
curvature tensor on the Hodge bundles $F^{p}E$ and on the quotient bundles
$E/F^{p}E$.) Consequently, the expression

$h_{E}\bigl{(}\Theta_{\partial/\partial
t\wedge\partial/\partial\bar{t}}\,u,u\bigr{)}=h_{E}\bigl{(}\theta_{\partial/\partial
t}u,\theta_{\partial/\partial
t}u\bigr{)}-h_{E}\bigl{(}\theta^{\ast}_{\partial/\partial\bar{t}}u,\theta^{\ast}_{\partial/\partial\bar{t}}u\bigr{)}$

is in general neither positive nor negative definite. However, because of the
basic estimate (in 2), any metric of the form

$h_{E}\cdot\lvert t\rvert^{a}(-\log\lvert t\rvert)^{b}\quad\text{with
$a\in\mathbb{R}$ and $b\in\mathbb{Z}$}$

will have positive curvature for $b\gg 0$, and negative curvature for $b\ll
0$. This insight already appears in Simpson’s work [SimpsonVHS, §10], who
attributes it to the important paper by Cornalba and Griffiths [CG, p. 29]. We
use both of these properties in our proof.

####  30 .

Let us now outline the steps in our proof of the nilpotent orbit theorem. The
nilpotent orbit theorem can also be deduced from Mochizuki’s work on harmonic
bundles [Mochizuki-nilpotent]; another analytic proof using $L^{2}$-estimates
has recently been given by Deng [Deng].

  1. 1.

The bundle $\operatorname{End}(E)$ inherits a polarized variation of Hodge
structure of weight $0$. The Higgs field $\theta_{\partial/\partial t}$ is a
holomorphic section of $\operatorname{End}(E)^{-1,1}$, and the basic estimate
tells us that its pointwise Hodge norm is bounded by a constant multiple of
$\lvert t\rvert^{-2}(\log\lvert t\rvert)^{-2}$. The first step is to construct
a lifting of $t\theta_{\partial/\partial t}$ to a holomorphic section
$\vartheta$ of the Hodge bundle $F^{-1}\operatorname{End}(E)$, whose
$L^{2}$-norm is controlled by an inequality of the form

$\int_{\Delta^{\ast}}h_{\operatorname{End}(E)}(\vartheta,\vartheta)\lvert
t\rvert^{a}(-\log\lvert t\rvert)^{b}\,d\mu\leq C$

where $a>-2$ and $b\gg 0$. This uses Hörmander’s $L^{2}$-estimates, for which
we include an elementary proof (in Section 7.1); the key point is that the
metric $h_{\operatorname{End}(E)}\cdot\lvert t\rvert^{a}(-\log\lvert
t\rvert)^{b}$ has positive curvature for $b\gg 0$. Using $L^{2}$-estimates to
extend holomorphic bundles goes back to Cornalba-Griffiths [CG], and in the
setting of harmonic bundles, to Simpson [SimpsonVHS].

  2. 2.

Pulling back to $\mathbb{H}$ gives us a holomorphic mapping
$\vartheta\colon\mathbb{H}\to\operatorname{End}(V)$ that satisfies
$\vartheta(z+2\pi i)=T\vartheta(z)T^{-1}$. After untwisting, we get

$B\colon\Delta^{\ast}\to\operatorname{End}(V),\quad
B(e^{z})=e^{-z(S+N)}\vartheta(z)e^{z(S+N)}.$

Since the eigenvalues of $S$ lie in an interval of length $<1$, the
$L^{2}$-estimate from the first step can be used to show that $B$ is square-
integrable on a punctured neighborhood of the origin, and therefore extends
holomorphically to the entire disk.

  3. 3.

We can now describe the differential of the untwisted period mapping
$\Psi_{S}$ using holomorphic data. The holomorphic tangent space to the
complex manifold $\check{D}$ at the point $\Phi(z)$ is isomorphic to
$\operatorname{End}(V)/F^{0}\operatorname{End}(V)_{\Phi(z)}$, and the
differential of the period mapping $\Phi\colon\mathbb{H}\to\check{D}$ is equal
to

$\theta_{\partial/\partial z}\mod F^{0}\operatorname{End}(V)_{\Phi(z)}.$

A short calculation with derivatives shows that the differential of the
mapping $z\mapsto\Psi_{S}(e^{z})$ is therefore equal to

$\displaystyle e^{-z(S+N)}\theta_{\partial/\partial z}e^{z(S+N)}$
$\displaystyle-(S+N)$ $\displaystyle\equiv B(e^{z})$ $\displaystyle-(S+N)\mod
F^{0}\operatorname{End}(V)_{\Psi_{S}(e^{z})}.$

The point is that the operator on the right-hand side is holomorphic.

  4. 4.

Let $g\colon\mathbb{H}\to\operatorname{GL}(V)$ be the unique holomorphic
solution of the ordinary differential equation

$g^{\prime}(z)=\Bigl{(}B(e^{z})-(S+N)\Bigr{)}g(z),$

subject to the initial condition $g(-1)=\operatorname{id}$. A short
calculation shows that $g(z)^{-1}\Psi_{S}(e^{z})$ must be constant, which
means that

$\Psi_{S}(e^{z})=g(z)\cdot\Psi_{S}(e^{-1}).$

The differential equation has a _regular_ singular point at $t=0$, and by the
basic theory of Fuchsian differential equations, one has

$g(z)=M(e^{z})\cdot e^{zA},$

with $M\colon\Delta^{\ast}\to\operatorname{GL}(V)$ meromorphic and
$A\in\operatorname{End}(V)$ constant. Since the untwisted period mapping
$\Psi_{S}(e^{z})$ is single-valued, it follows that

$\Psi_{S}(t)=M(t)\cdot\Psi_{S}(e^{-1}),$

and because $\check{D}$ is a projective complex manifold, this is enough to
conclude that $\Psi_{S}$ extends holomorphically across the origin. (The close
relationship between the nilpotent orbit theorem and regularity was already
observed by Griffiths and Schmid, see [GS, §9a] and the references cited
there.)

  5. 5.

The remainder of the argument consists in deriving effective estimates for the
rate of convergence. Let $P_{\lambda}$ denote the projection to the eigenspace
$E_{\lambda}(T_{s})$. The fact that $\Psi_{S}(t)$ converges as $t\to 0$ can be
used to show that

$\displaystyle\lVert\theta_{\partial/\partial
z}-N^{-1,1}\rVert_{\Phi(z)}^{2}+\sum_{k\leq-2}\lVert
N^{k,-k}\rVert_{\Phi(z)}^{2}$ $\displaystyle\leq
C\lvert\operatorname{Re}z\rvert^{2m}e^{-2\delta(T)\lvert\operatorname{Re}z\rvert}$
$\displaystyle\sum_{k\leq-1}\lVert P_{\lambda}^{k,-k}\rVert_{\Phi(z)}^{2}$
$\displaystyle\leq
C\lvert\operatorname{Re}z\rvert^{2m}e^{-2\delta(T)\lvert\operatorname{Re}z\rvert}$

for $\lvert\operatorname{Re}z\rvert\gg 0$; the exponent $m$ only depends on
$r=\operatorname{rk}E$, but we have no control over the value of the constant
$C$. We then use the fact that the metric
$h_{\operatorname{End}(E)}\cdot(-\log\lvert t\rvert)^{b}$ has negative
curvature for $b\ll 0$, together with the maximum principle for subharmonic
functions, to deduce that the above estimates actually hold on any half-plane
of the form $\operatorname{Re}z\leq x<0$, with a constant $C$ that only
depends on $x$, $r$, and on the minimal polynomial of
$T\in\operatorname{GL}(V)$.

  6. 6.

Finally, we consider the curve

$[0,\infty)\to\check{D},\quad x\mapsto e^{xN}\Phi(z-x),$

joining the two points $\Phi(z)$ and $\Phi_{\mathrm{nil}}(z)$. Its derivative
is controlled by the inequalities in the previous step. After integration, we
obtain a bound for the distance in $\check{D}$ between these two points, and
together with the Hodge norm estimates, this gives us the desired bound for
their distance in $D$, provided that $\lvert\operatorname{Re}z\rvert$ is
sufficiently large.

####  31 .

The estimates in the fifth step are saying that, modulo
$F^{0}\operatorname{End}(V)_{\Phi(z)}$, the Higgs field
$\theta_{\partial/\partial z}$ converges to the nilpotent operator $N$ at a
rate that is exponential in $\lvert\operatorname{Re}z\rvert$. Moreover, the
eigenspace decomposition

$V=\bigoplus_{\lvert\lambda\rvert=1}E_{\lambda}(T_{s})$

becomes orthogonal in the limit, at a rate that is exponential in
$\lvert\operatorname{Re}z\rvert$. Since the limiting Hodge filtration
$F_{\mathrm{lim}}$ is compatible with this decomposition, both of these facts
are of course also contained in the statement of the nilpotent orbit theorem
itself.

###  2.7 The limiting mixed Hodge structure

####  32 .

Probably the most widely known result from Schmid’s paper [Schmid, §6] is
that, in the limit, a polarized variation of Hodge structure on the punctured
disk gives rise to a mixed Hodge structure on $V$. The weight filtration of
this “limiting” mixed Hodge structure is the shifted monodromy weight
filtration $W_{\bullet-n}$, and the Hodge filtration is the limiting Hodge
filtration $F_{\mathrm{lim}}$. Schmid obtained this result as a consequence of
his $\operatorname{SL}(2)$-orbit theorem.

####  33 .

We take a somewhat different approach here, and deduce the existence of the
limiting mixed Hodge structure directly from the convergence of the rescaled
period mapping $\hat{\Phi}_{S,H}\colon\mathbb{H}\to D$. In fact, we prove that
the vector space $V$ is an example of a polarized $\mathfrak{sl}_{2}$-Hodge
structure; this is the same kind of structure that one finds on the cohomology
of a compact Kähler manifold. The proof is elementary and only uses linear
algebra and some basic representation theory of the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{C})$, but no further analysis.

####  34 .

Let us briefly recall how $\mathfrak{sl}_{2}(\mathbb{C})$ enters into the
picture. Recall that this Lie algebra is $3$-dimensional, and is generated by
the three matrices

$\mathsf{H}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\mathsf{X}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\quad\mathsf{Y}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},$

subject to the relations $[\mathsf{H},\mathsf{X}]=2\mathsf{X}$,
$[\mathsf{H},\mathsf{Y}]=-2\mathsf{Y}$, and
$[\mathsf{X},\mathsf{Y}]=\mathsf{H}$. In any finite-dimensional
representation, the operator $\mathsf{H}$ acts semisimply with integer
eigenvalues, and the eigenspace $E_{k}(\mathsf{H})$ are called the weight
spaces. The splitting $H$ that we chose in 21 uniquely determines a
representation

$\rho\colon\mathfrak{sl}_{2}(\mathbb{C})\to\operatorname{End}(V)$

of this Lie algebra such that $\rho(\mathsf{H})=H$ and $\rho(\mathsf{Y})=-N$;
the minus sign is dictated by our sign conventions for Hodge structures. It
lifts to a representation of the Lie group
$\operatorname{SL}_{2}(\mathbb{C})$; in particular, we have the Weil element

$\mathsf{w}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{C}),$

which can also be written as
$\mathsf{w}=e^{\mathsf{X}}e^{-\mathsf{Y}}e^{\mathsf{X}}$. The Weil element is
a sort of linear algebra analogue of the Hodge $\ast$-operator on a compact
Kähler manifold: it satisfies $\mathsf{w}^{2}=(-1)^{\mathsf{H}}$, and induces
isomorphisms between the two weight spaces $E_{\pm k}(\mathsf{H})$. Using a
dagger to denote the adjoint of an operator with respect to the nondegenerate
pairing $Q$, it is easy to see that $\mathsf{H}^{{\dagger}}=-\mathsf{H}$,
$\mathsf{Y}^{{\dagger}}=\mathsf{Y}$, $\mathsf{X}^{{\dagger}}=\mathsf{X}$, and
hence also $\mathsf{w}^{{\dagger}}=\mathsf{w}$.

####  35 .

We can now define $\mathfrak{sl}_{2}$-Hodge structures and their
polarizations. The definition is modeled on the cohomology
$H^{n+\bullet}(X,\mathbb{C})$ of an $n$-dimensional compact Kähler manifold,
but perhaps surprisingly, it is equally relevant in our context. An
$\mathfrak{sl}_{2}$-Hodge structure on a finite-dimensional complex vector
space $V$ is a representation of $\mathfrak{sl}_{2}(\mathbb{C})$ on $V$ with
the following properties:

  1. (a)

Each weight space $V_{k}=E_{k}(\mathsf{H})$ has a Hodge structure of weight
$n+k$; the integer $n$ is called the weight of the $\mathfrak{sl}_{2}$-Hodge
structure.

  2. (b)

The two operators

$\mathsf{X}\colon V_{k}\to V_{k+2}(1)\quad\text{and}\quad\mathsf{Y}\colon
V_{k}\to V_{k-2}(-1)$

are morphisms of Hodge structures.

In the literature, this kind of structure is also called a “Hodge-Lefschetz
structure” [Cattani] or an “$\mathbb{R}$-split mixed Hodge structure” [CKS],
but we feel that the name “$\mathfrak{sl}_{2}$-Hodge structure” is more
descriptive.

####  36 .

The polarization on the cohomology of a compact Kähler manifold is usually
discussed only for the primitive cohomology (which is the kernel of
$\mathsf{Y}$, in our notation). The standard definition is that, for every
$k\geq 0$, the hermitian pairing

$Q\circ\bigl{(}\operatorname{id}\otimes\mathsf{X}^{k}\bigr{)}\colon
V_{-k}\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C}$

should polarize the Hodge structure of weight $n-k$ on the primitive subspace
$\ker\mathsf{Y}\colon V_{-k}\to V_{-(k+2)}$. Since $\mathsf{w}$ acts on the
primitive subspace as $\frac{1}{k!}\mathsf{X}^{k}$, one can define
polarizations more compactly with the help of the Weil element. Following
Deligne [Deligne-signs], we say that a hermitian form

$Q\colon V\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C}$

is a polarization of the $\mathfrak{sl}_{2}$-Hodge structure $V$ if it
satisfies the four identities

$\displaystyle Q\circ(\mathsf{H}\otimes\operatorname{id})$
$\displaystyle=-Q\circ(\operatorname{id}\otimes\mathsf{H}),$ $\displaystyle
Q\circ(\mathsf{X}\otimes\operatorname{id})$
$\displaystyle=Q\circ(\operatorname{id}\otimes\mathsf{X}),$ $\displaystyle
Q\circ(\mathsf{Y}\otimes\operatorname{id})$
$\displaystyle=Q\circ(\operatorname{id}\otimes\mathsf{Y}),$ $\displaystyle
Q\circ(\mathsf{w}\otimes\operatorname{id})$
$\displaystyle=Q\circ(\operatorname{id}\otimes\mathsf{w}),$

and if $Q\circ(\operatorname{id}\otimes\mathsf{w})$ polarizes the Hodge
structure of weight $n+k$ on each $V_{k}$.

####  37 .

Because the Weil element $\mathsf{w}$ acts as $\frac{1}{k!}(-\mathsf{Y})^{k}$
on the “other” primitive subspace $\ker\mathsf{X}\colon V_{k}\to V_{k+2}$ for
$k\geq 0$, an equivalent formulation is that

$Q\circ\bigl{(}\operatorname{id}\otimes(-\mathsf{Y})^{k}\bigr{)}\colon
V_{k}\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C}$

should polarize the Hodge structure of weight $n+k$ on $\ker\mathsf{X}$. Since
$\rho(-\mathsf{Y})=N$ in our case, this matches Schmid’s definition [Schmid,
Lem. 6.4].

####  38 .

We now describe how the limiting $\mathfrak{sl}_{2}$-Hodge structure on $V$
comes about. The convergence of the rescaled period mapping (in 4) gives us a
filtration $F_{H}\in\check{D}$ with the following properties:

  1. (a)

The filtration $e^{-N}F_{H}\in D$ is the Hodge filtration of a Hodge structure
of weight $n$, polarized by the pairing $Q$.

  2. (b)

One has $N(F_{H}^{\bullet})\subseteq F_{H}^{\bullet-1}$,
$H(F_{H}^{\bullet})\subseteq F_{H}^{\bullet}$, and
$T_{s}(F_{H}^{\bullet})\subseteq F_{H}^{\bullet}$.

From these properties, we deduce by a mix of representation theory and linear
algebra that $F_{H}$ is the “total” Hodge filtration of an
$\mathfrak{sl}_{2}$-Hodge structure of weight $n$ on $V$.

######  Theorem 7 (Limiting $\mathfrak{sl}_{2}$-Hodge structure).

The vector space $V$ has a polarized $\mathfrak{sl}_{2}$-Hodge structure of
weight $n$, whose Hodge filtration is $F_{H}$. Concretely, this means the
following:

  1. 1.

Each weight space $V_{k}=E_{k}(H)$ has a Hodge structure of weight $n+k$,
whose Hodge filtration is $V_{k}\cap F_{H}$.

  2. 2.

The operator $N\colon V_{k}\to V_{k-2}(-1)$ is a morphism of Hodge structures.

  3. 3.

The Hodge structure on each primitive subspace $\ker N^{k+1}\colon V_{k}\to
V_{-k-2}$ is polarized by the hermitian pairing $Q(-,N^{k}-)$.

  4. 4.

The operator $T_{s}$ is an endomorphism of the polarized
$\mathfrak{sl}_{2}$-Hodge structure.

####  39 .

Recall from 25 that the filtration $F_{H}$ is obtained from the limiting Hodge
filtration $F_{\mathrm{lim}}$ by the following procedure: $F_{\mathrm{lim}}$
induces a filtration on each subquotient $W_{k}/W_{k-1}$, and $V_{k}\cap
F_{H}$ is gotten by pulling back along the isomorphism $V_{k}\cong
W_{k}/W_{k-1}$. The fact that each weight space $V_{k}$ has a Hodge structure
of weight $n+k$ therefore means that $W_{\bullet-n}$ is the weight filtration
of a mixed Hodge structure [Schmid, Thm. 6.16].

######  Theorem 8 (Limiting mixed Hodge structure).

The vector space $V$ has a mixed Hodge structure, with weight filtration
$W_{\bullet-n}$ and Hodge filtration $F_{\mathrm{lim}}$; the associated graded
of this mixed Hodge structure is a polarized $\mathfrak{sl}_{2}$-Hodge
structure of weight $n$ with $\mathsf{Y}=-N$ and polarization $Q$. The two
operators

$N\colon V\to V(-1)\quad\text{and}\quad T_{s}\colon V\to V$

are morphisms of mixed Hodge structures.

The $\mathfrak{sl}_{2}$-Hodge structure depends on the choice of splitting
$H$; the limiting mixed Hodge structure is independent of it. Without choosing
a splitting, one gets a canonical polarized $\mathfrak{sl}_{2}$-Hodge
structure on the associated graded $\operatorname{gr}_{\bullet}^{W}$ with
respect to the monodromy weight filtration. In the terminology of [CKS, Def.
2.26], the limiting mixed Hodge structure is therefore “polarized” by the
pairing $Q$ and the nilpotent operator $N$.

###  2.8 Asymptotic behavior of the Hodge metric

####  40 .

Finally, we describe the asymptotic behavior of the Hodge metric
$h(v,v^{\prime})$ where $v,v^{\prime}\in V$ are two multi-valued flat
sections. The Hodge norm estimates cover the case $v=v^{\prime}$, but
something new happens when $v\neq v^{\prime}$, namely that the Hodge metric
$h(v,v^{\prime})$ grows more slowly than expected as
$\lvert\operatorname{Re}z\rvert\to\infty$. This result does not appear as such
in Schmid’s paper, but it is implicitly contained in the statement about
Siegel sets in [Schmid, Cor. 5.29].

######  Theorem 9 (Asymptotic behavior of the Hodge metric).

For any two multi-valued flat sections $v,v^{\prime}\in V$, there is a
constant $C(v,v^{\prime})>0$ such that

$\bigl{\lvert}\langle v,v^{\prime}\rangle_{\Phi(z)}\bigr{\rvert}\leq
C(v,v^{\prime})\cdot\min\Bigl{(}\lVert v\rVert_{\Phi(z)}^{2},\lVert
v^{\prime}\rVert_{\Phi(z)}^{2}\Bigr{)}$

for all sufficiently large values of $\lvert\operatorname{Re}z\rvert$.

An equivalent formulation is that if $v\in V_{k}$ and $v^{\prime}\in
V_{k^{\prime}}$, then

$\langle
v,v^{\prime}\rangle_{\Phi(z)}=O\bigl{(}\lvert\operatorname{Re}z\rvert^{\min(k,k^{\prime})}\bigr{)}$

which is typically much smaller than the
$\lvert\operatorname{Re}z\rvert^{\frac{1}{2}(k+k^{\prime})}$ one would expect
based on the Hodge norm estimates. We can also restate the theorem in terms of
the rescaled period mapping from (21.1): the result is that if $v\in V_{k}$
and $v^{\prime}\in V_{k^{\prime}}$ belong to different weight spaces, then

$\langle
v,v^{\prime}\rangle_{\hat{\Phi}_{S,H}(z)}=O\bigl{(}\lvert\operatorname{Re}z\rvert^{-\frac{1}{2}\lvert
k-k^{\prime}\rvert}\bigr{)}.$ (40.1)

Since different weight spaces are orthogonal to each other in the Hodge
structure $F_{\sharp}=e^{-N}F_{H}\in D$, this is not unexpected; but not only
are the weight spaces becoming orthogonal in the limit, they are doing so at a
rate that depends on the difference between their weights.

####  41 .

We end the introduction with a brief sketch of the proof. Up to an error of
size $e^{-\varepsilon\lvert\operatorname{Re}z\rvert}$, which is negligible in
this context, the nilpotent orbit theorem allows us to assume that the period
mapping has the form $\Phi(z)=e^{zN}F_{\mathrm{lim}}$. The rescaled period
mapping then looks like

$\hat{\Phi}_{H}(z)=e^{-N}\cdot
e^{\frac{1}{2}\log\lvert\operatorname{Re}z\rvert\,H}F_{\mathrm{lim}}.$

This expression only depends on $\lvert\operatorname{Re}z\rvert$; since we are
interested in its behavior as $\lvert\operatorname{Re}z\rvert\to\infty$, we
make the substitution $u=\lvert\operatorname{Re}z\rvert^{-\frac{1}{2}}$, so
that

$\hat{\Phi}_{H}\bigl{(}1/u^{2}\bigr{)}=e^{-N}\cdot e^{-(\log
u)H}F_{\mathrm{lim}}.$

Recall from 25 that we have

$\lim_{u\to 0}e^{-(\log u)H}F_{\mathrm{lim}}=F_{H},$ (41.1)

and hence that $\hat{\Phi}_{H}\bigl{(}1/u^{2}\bigr{)}$ converges to
$F_{\sharp}=e^{-N}F_{H}$ as $u\to 0$.

####  42 .

To get a handle on the rate of convergence, we prove the following cheap
version of the $\operatorname{SL}(2)$-orbit theorem. Schmid’s original result
[Schmid, Thm. 5.13] has two functions: it provides a distinguished splitting
$H$ for the weight filtration, called the “$\operatorname{SL}(2)$-splitting”
(or “canonical splitting”); and it gives a very precise description, using
power series with values in the real group $G$, for the rate of convergence of
(41.1). Here we focus on the second aspect, using an arbitrary splitting $H$
chosen as in 139.

######  Theorem 10 (Cheap $\operatorname{SL}(2)$-orbit theorem).

There is a convergent power series

$g\colon(-\varepsilon,\varepsilon)\to G,\quad
g(u)=\operatorname{id}+ug_{1}+u^{2}g_{2}+\dotsb,$

defined for some $\varepsilon>0$, such that for $0<\lvert
u\rvert<\varepsilon$, one has

$e^{-N}\cdot e^{-(\log u)H}F_{\mathrm{lim}}=g(u)^{-1}\cdot e^{-N}F_{H}.$

The coefficients $g_{n}\in\operatorname{End}(V)$ of the series are
$(S_{n}\oplus S_{n-2}\oplus\dotsb)$-isotypical with respect to the action of
$\mathfrak{sl}_{2}(\mathbb{C})$ on $\operatorname{End}(V)$.

The point is that we can then write the rescaled period mapping

$\hat{\Phi}_{H}\bigl{(}1/u^{2}\bigr{)}=g(u)^{-1}\cdot F_{\sharp}$

in terms of the real group $G$. Now suppose that $v\in V_{k}$ and
$v^{\prime}\in V_{k^{\prime}}$. Since different weight spaces are orthogonal
in the Hodge structure $F_{\sharp}$, and since 10 gives us some control on how
much the coefficients $g_{n}$ can change the weight of a given vector, it is
an easy matter to deduce that

$\displaystyle\langle v,v^{\prime}\rangle_{\hat{\Phi}_{H}(1/u^{2})}$
$\displaystyle=\bigl{\langle}g(u)\cdot v,\,g(u)\cdot
v^{\prime}\bigr{\rangle}_{F_{\sharp}}$ $\displaystyle=\sum_{m+n\geq\lvert
k-k^{\prime}\rvert}u^{m+n}\,\bigl{\langle}g_{m}v,\,g_{n}v^{\prime}\bigr{\rangle}_{F_{\sharp}}=O\bigl{(}u^{\lvert
k-k^{\prime}\rvert}\bigr{)},$

which is equivalent to (40.1).

####  43 .

The original proof of the $\operatorname{SL}(2)$-orbit theorem in [Schmid, §9]
is very difficult; and even the simplified argument in [CKS, §6] (assuming the
limiting mixed Hodge structure) is quite involved. By contrast, our proof of
the cheap $\operatorname{SL}(2)$-orbit theorem is short and elementary. The
crucial point is that

$F_{\mathrm{lim}}=h\cdot F_{H},$

where $h=\operatorname{id}+h_{1}+h_{2}+\dotsb\in\operatorname{GL}(V)$ has the
property that

$h_{n}\in E_{-n}(\operatorname{ad}H)\cap\ker(\operatorname{ad}N).$

Such an element $h$ can be constructed by comparing Deligne’s decomposition of
the limiting mixed Hodge structure (whose Hodge filtration is
$F_{\mathrm{lim}}$) and the Hodge decomposition of the
$\mathfrak{sl}_{2}$-Hodge structure (whose Hodge filtration in $F_{H})$, and
using the fact that, up to a Tate twist, $N$ is a morphism in both structures.
10 follows from this by a formal argument. The crucial fact from the
representation theory of the Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ is
that the tensor product

$S_{k}\otimes S_{\ell}\cong\bigoplus_{i=0}^{\min(k,\ell)}S_{k+\ell-2i}$

of two irreducible representation decomposes in a very simple way.

##  3 Variations of Hodge structure and harmonic bundles

###  3.1 Harmonic bundles and harmonic metrics

####  44 .

Since we are going to do a certain amount of analysis, it will be convenient
to describe variations of Hodge structure in terms of smooth vector bundles.
Let $E$ be a smooth vector bundle on a complex manifold $X$. We denote by
$A^{k}(X,E)$ the space of smooth $k$-forms with coefficients in $E$, and by
$A^{i,j}(X,E)$ the space of smooth $(i,j)$-forms with coefficients in $E$.

######  Definition 1.

Let $E$ be a smooth vector bundle on a complex manifold $X$. A variation of
Hodge structure of weight $n$ on $E$ is a decomposition

$E=\bigoplus_{p+q=n}E^{p,q}$

into smooth subbundles, together with a flat connection $d\colon A^{0}(X,E)\to
A^{1}(X,E)$ that maps each $A^{0}(X,E^{p,q})$ into the direct sum of the
subspaces

$A^{1,0}(X,E^{p,q})\oplus A^{1,0}(X,E^{p-1,q+1})\oplus
A^{0,1}(X,E^{p,q})\oplus A^{0,1}(X,E^{p+1,q-1}).$

####  45 .

The definition gives us a decomposition
$d=\partial+\theta+\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}+\theta^{\ast}$ into four operators

$\displaystyle\partial$ $\displaystyle\colon A^{0}(X,E^{p,q})\to
A^{1,0}(X,E^{p,q})$ $\displaystyle\theta$ $\displaystyle\colon
A^{0}(X,E^{p,q})\to A^{1,0}(X,E^{p-1,q+1})$
$\displaystyle\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$
$\displaystyle\colon A^{0}(X,E^{p,q})\to A^{0,1}(X,E^{p,q})$
$\displaystyle\theta^{\ast}$ $\displaystyle\colon A^{0}(X,E^{p,q})\to
A^{1,0}(X,E^{p+1,q-1}).$

It is easy to see that $\partial$ and $\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$
are again connections of type $(1,0)$ and $(0,1)$, whereas $\theta$ and
$\theta^{\ast}$ are linear over smooth functions. As usual, we can extend all
four operators to the space of all $E$-valued forms

$A^{\ast}(X,E)=\bigoplus_{i,j,p,q}A^{i,j}(X,E^{p,q})$

by enforcing the Leibniz rule. With respect to the
$\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}$-grading on this space, $\partial$ is an
operator of bidegree $(1,0)\otimes(0,0)$, because it maps $A^{i,j}(X,E^{p,q})$
into $A^{i+1,j}(X,E^{p,q})$; similarly, $\theta$ has bidegree
$(1,0)\otimes(-1,1)$, $\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$
has bidegree $(0,1)\otimes(0,0)$, and $\theta^{\ast}$ has bidegree
$(0,1)\otimes(1,-1)$. By decomposing the identity $d^{2}=0$ according to
bidegree, one can deduce the following set of identities for these four
operators.

######  Lemma 2.

The four operators $\partial$, $\theta$, $\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$,
and $\theta^{\ast}$ satisfy the following identities:

$\displaystyle\partial^{2}=\theta^{2}=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}^{2}=(\theta^{\ast})^{2}$ $\displaystyle=0$
$\displaystyle\partial\theta+\theta\partial=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\theta^{\ast}+\theta^{\ast}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$
$\displaystyle=0$ $\displaystyle\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\theta+\theta\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}=\partial\theta^{\ast}+\theta^{\ast}\partial$ $\displaystyle=0$
$\displaystyle\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}+\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\partial+\theta\theta^{\ast}+\theta^{\ast}\theta$ $\displaystyle=0$

####  46 .

In order to define polarizations, we consider a hermitian pairing

$Q\colon A^{0}(X,E)\otimes_{A^{0}(X)}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to A^{0}(X).$

It takes as input two smooth sections of the bundle $E$, and outputs a smooth
function on $X$. We require that $Q$ is hermitian symmetric and
$A^{0}(X)$-linear in its first argument, meaning that for $v,w\in A^{0}(X,E)$
and $f\in A^{0}(X)$, one has

$Q(w,v)=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\quad\text{and}\quad Q(f\cdot v,w)=f\cdot Q(v,w).$

Of course, it follows that $Q$ is also conjugate $A^{0}(X)$-linear in its
second argument. We say that $Q$ is flat with respect to the connection $d$ if

$dQ(v,w)=Q(dv,w)+Q(v,dw)\quad\text{for $v,w\in A^{0}(X,E)$.}$

######  Definition 2.

Let $E$ be a variation of Hodge structure of weight $n$ on a complex manifold
$X$. A polarization is a flat hermitian pairing

$Q\colon A^{0}(X,E)\otimes_{A^{0}(X)}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to A^{0}(X),$

such that the formula

$h(v,w)=\sum_{p+q=n}(-1)^{q}\,Q(v^{p,q},w^{p,q})\quad\text{for $v,w\in
A^{0}(X,E)$}$

defines a smooth hermitian metric on the bundle $E$, and the subbundles
$E^{p,q}$ are orthogonal to each other with respect to this metric.

The hermitian metric in the definition is called the Hodge metric; it is the
most important object in the theory. For the sake of clarity, we sometimes
denote the Hodge metric on $E$ by the symbol $h_{E}$.

####  47 .

The fact that $Q$ is flat with respect to $d$ gives us the following
additional information about the operators $\theta$ and $\theta^{\ast}$.

######  Lemma 3.

With respect to the Hodge metric on $E$, the operator $\theta^{\ast}$ is the
adjoint of the operator $\theta$, in the sense that

$h(\theta v,w)=h(v,\theta^{\ast}w)\quad\text{for $v,w\in A^{0}(X,E)$.}$

###### Proof.

Let $v\in A^{0}(X,E^{p,q})$ and $w\in A^{0}(X,E^{p-1,q+1})$ be two arbitrary
sections. Then $Q(v,w)=0$, and therefore

$0=dQ(v,w)=Q(dv,w)+Q(v,dw)=Q(\theta v,w)+Q(v,\theta^{\ast}w)$

because all other terms have the wrong type. But then

$h(\theta v,w)=(-1)^{q+1}Q(\theta
v,w)=(-1)^{q}Q(v,\theta^{\ast}w)=h(v,\theta^{\ast}w).$

Since the decomposition of $E$ is orthogonal with respect to $h$, the result
follows. ∎

####  48 .

Let us briefly recall the relation with the (more familiar) holomorphic
description of variations of Hodge structure. As usual, we decompose the flat
connection into $d=d^{\prime}+d^{\prime\prime}$, where $d^{\prime}\colon
A^{0}(X,E)\to A^{1,0}(X,E)$ is a connection of type $(1,0)$, and
$d^{\prime\prime}\colon A^{0}(X,E)\to A^{0,1}(X,E)$ is a connection of type
$(0,1)$. The identity $d^{2}=0$ then becomes
$(d^{\prime})^{2}=d^{\prime}d^{\prime\prime}+d^{\prime\prime}d^{\prime}=(d^{\prime\prime})^{2}=0$.
The operator $d^{\prime\prime}$ gives $E$ the structure of a holomorphic
vector bundle, which we denote by the symbol $\mathscr{E}$, and the operator
$d^{\prime}$ defines a flat holomorphic connection

$\nabla\colon\mathscr{E}\to\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}\mathscr{E}.$

Moreover, for each $p\in\mathbb{Z}$, the Hodge bundle

$F^{p}E=E^{p,q}\oplus E^{p+1,q-1}\oplus E^{p+2,q-2}\oplus\dotsb$

is preserved by the operator $d^{\prime\prime}$, and therefore defines a
holomorphic subbundle $F^{p}\mathscr{E}$. Since $d^{\prime}=\partial+\theta$,
we also get Griffiths’ transversality condition

$\nabla(F^{p}\mathscr{E})\subseteq\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}F^{p-1}\mathscr{E}.$

From this perspective, we can see the other operators by looking at the
quotient bundles $F^{p}\mathscr{E}/F^{p+1}\mathscr{E}$. The operator
$\bar{\partial}$ gives each smooth bundle $E^{p,q}$ the structure of a
holomorphic vector bundle $\mathscr{E}^{p,q}$, and because
$d^{\prime\prime}=\bar{\partial}+\theta^{\ast}$,

$\mathscr{E}^{p,q}\cong F^{p}\mathscr{E}/F^{p+1}\mathscr{E}$

are isomorphic as holomorphic vector bundles. Since
$\bar{\partial}\theta+\theta\bar{\partial}=0$, the operator $\theta$ defines a
morphism of holomorphic vector bundles

$\theta\colon\mathscr{E}^{p,q}\to\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}\mathscr{E}^{p-1,q+1},$

and one checks that the following diagram is commutative:

${\mathscr{E}^{p,q}}$${\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}\mathscr{E}^{p-1,q+1}}$${F^{p}\mathscr{E}/F^{p+1}\mathscr{E}}$${\Omega_{X}^{1}\otimes_{\mathscr{O}_{X}}F^{p-1}\mathscr{E}/F^{p}\mathscr{E}}$$\scriptstyle{\theta}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\nabla}$

The two remaining operators $\theta^{\ast}$ and $\partial$ in the
decomposition of $d$ have the following interpretation: $\theta^{\ast}$ is the
adjoint of the Higgs field with respect to the Hodge metric (3), and
$\partial$ is the Chern connection on the bundle $E^{p,q}$.

####  49 .

We are now going to do some computations with the Hodge metric; these become
easier if we use the language of harmonic bundles. In fact, a polarized
variation of Hodge structure is an example of a harmonic bundle, with the
harmonic metric being the Hodge metric. We briefly recall the general
definition. Let $E$ be a smooth vector bundle on a complex manifold $X$, let
$d=d^{\prime}+d^{\prime\prime}$ be a flat connection on $E$, and let $h$ be a
hermitian metric on $E$. As in the construction of the Chern connection, there
are unique operators

$\displaystyle\delta^{\prime}$ $\displaystyle\colon A^{0}(X,E)\to
A^{1,0}(X,E)$ $\displaystyle\delta^{\prime\prime}$ $\displaystyle\colon
A^{0}(X,E)\to A^{0,1}(X,E)$

with the property that $d^{\prime}+\delta^{\prime\prime}$ and
$\delta^{\prime}+d^{\prime\prime}$ are metric connections on $E$. Then

$\partial=\frac{1}{2}(d^{\prime}+\delta^{\prime})\quad\text{and}\quad\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}=\frac{1}{2}(d^{\prime\prime}+\delta^{\prime\prime})$

are again connections of type $(1,0)$ and $(0,1)$, and

$\theta=\frac{1}{2}(d^{\prime}-\delta^{\prime})\quad\text{and}\quad\theta^{\ast}=\frac{1}{2}(d^{\prime\prime}-\delta^{\prime\prime})$

are linear over smooth functions. The metric $h$ is called a harmonic metric,
and the triple $(E,d,h)$ is called a harmonic bundle, if the identity
$(\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}+\theta)^{2}=0$ holds. This is equivalent to the identities in 2. The
associated Higgs bundle is $E$, with the holomorphic structure defined by
$\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$,
and with Higgs field $\theta$. As suggested by the notation, in the case of a
polarized variation of Hodge structure, $\partial$,
$\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$,
$\theta$, and $\theta^{\ast}$ agree with the operators that we had defined
above.

######  Lemma 4.

Let $E$ be a variation of Hodge structure with polarization $Q$. Then

$d^{\prime}=\partial+\theta,\quad
d^{\prime\prime}=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}+\theta^{\ast},\quad\delta^{\prime}=\partial-\theta,\quad\delta^{\prime\prime}=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}-\theta^{\ast},$

and so $E$ is a harmonic bundle and the Hodge metric $h$ is a harmonic metric.
The associated Higgs bundle is the graded holomorphic vector bundle

$\bigoplus_{p+q=n}\mathscr{E}^{p,q}$

with Higgs field
$\theta\colon\mathscr{E}^{p,q}\to\Omega_{X}^{1}\otimes\mathscr{E}^{p-1,q+1}$.

###### Proof.

The claim is that the two operators
$(\partial+\bar{\partial})\pm(\theta-\theta^{\ast})$ are metric connections.
For any two sections $v,w\in A^{0}(X,E^{p,q})$, we have

$dh(v,w)=(-1)^{q}Q(dv,w)+(-1)^{q}Q(v,dw)=h\bigl{(}(\partial+\bar{\partial})v,w\bigr{)}+h\bigl{(}v,(\partial+\bar{\partial})w\bigr{)},$

because all other terms have the wrong type. Therefore
$\partial+\bar{\partial}$ is a metric connection on $E$. Since $\theta^{\ast}$
is the adjoint of $\theta$ with respect to $h$, it follows that
$(\partial+\bar{\partial})\pm(\theta-\theta^{\ast})$ are also metric
connections. The identities in 2 imply that $(\bar{\partial}+\theta)^{2}=0$,
and so $E$ is a harmonic bundle with harmonic metric $h$. The remaining
assertions are clear from the discussion in the previous paragraph. ∎

####  50 .

The Hodge metric $h$ gives us a hermitian metric on the holomorphic vector
bundle $\mathscr{E}$ and on the subbundles $F^{p}\mathscr{E}$. It is not hard
to derive a formula for the curvature of this metric, using the identities
among the operators $\partial$, $\bar{\partial}$, $\theta$, and
$\theta^{\ast}$; Schmid does this, in a more cumbersome way, in [Schmid, §7].
The curvature of the Hodge bundles will be important later, in Section 7, when
we prove the nilpotent orbit theorem. We need one more piece of notation to
state the result: we denote by

$\pi^{p,q}\colon E\to E^{p,q}$

the projection operator to the direct summand $E^{p,q}$; of course,
$\pi^{p,q}$ is also orthogonal projection with respect to the Hodge metric.

######  Proposition 2.

Let $E$ be a variation of Hodge structure with polarization $Q$.

  1. (a)

On the holomorphic vector bundle $\mathscr{E}$, the curvature operator of the
Hodge metric $h$ is equal to $-2(\theta\theta^{\ast}+\theta^{\ast}\theta)$.

  2. (b)

On the holomorphic vector bundle $\mathscr{E}^{p,q}$, the curvature operator
of the Hodge metric $h$ is equal to
$-(\theta\theta^{\ast}+\theta^{\ast}\theta)$.

  3. (c)

On the holomorphic quotient bundle $\mathscr{E}/F^{p}\mathscr{E}$, the
curvature operator of the induced metric is equal to
$-2(\theta\theta^{\ast}+\theta^{\ast}\theta)+\theta\theta^{\ast}\pi^{p-1,q+1}$.

###### Proof.

Since the holomorphic structure on $\mathscr{E}$ is defined by the operator
$d^{\prime\prime}$, the Chern connection of $h$ is nothing but the metric
connection
$\delta^{\prime}+d^{\prime\prime}=\partial+\bar{\partial}-\theta+\theta^{\ast}$.
A short computation shows that the curvature operator is

$(\delta^{\prime}+d^{\prime\prime})^{2}=(\partial+\bar{\partial}-\theta+\theta^{\ast})^{2}=(\partial\bar{\partial}+\bar{\partial}\partial)-(\theta\theta^{\ast}+\theta^{\ast}\theta)=-2(\theta\theta^{\ast}+\theta^{\ast}\theta),$

using the identities in 2. This proves (a). The proof of (b) is very similar.
The holomorphic structure on $\mathscr{E}^{p,q}$ is defined by the operator
$\bar{\partial}$, and so the Chern connection of $h$ is the metric connection
$\partial+\bar{\partial}$, and the curvature operator is

$(\partial+\bar{\partial})^{2}=(\partial\bar{\partial}+\bar{\partial}\partial)=-(\theta\theta^{\ast}+\theta^{\ast}\theta).$

The derivation of (c) is slightly more complicated, because the operator
$d^{\prime\prime}=\bar{\partial}+\theta^{\ast}$ does not preserve the
individual summands $E^{p,q}$. We can correct this with the help of the
projection $\pi^{p,q}\colon E\to E^{p,q}$. Note that $\pi^{p,q}$ commutes with
$\partial$ and $\bar{\partial}$, whereas

$\pi^{p,q}\theta=\theta\pi^{p+1,q-1}\quad\text{and}\quad\pi^{p,q}\theta^{\ast}=\theta^{\ast}\pi^{p-1,q+1}.$

Now for the proof. The Hodge decomposition is orthogonal with respect to the
Hodge metric, and so, as smooth vector bundles with metric,

$E/F^{p}E\cong E^{p-1,q+1}\oplus E^{p-2,q+2}\oplus E^{p-3,q+3}\oplus\dotsb$
(50.1)

Under this isomorphism, the holomorphic structure on $E/F^{p}E$ is represented
by the operator $d^{\prime\prime}-\theta^{\ast}\pi^{p-1,q+1}$, which preserves
the bundle on the right-hand side. A short computation shows that the modified
operator

$\delta^{\prime}+d^{\prime\prime}-\theta^{\ast}\pi^{p-1,q+1}=\partial+\bar{\partial}-\theta+\theta^{\ast}-\theta^{\ast}\pi^{p-1,q+1}$

is a metric connection, and therefore equal to the Chern connection. It
follows that the curvature operator of the induced metric is

$\displaystyle\bigl{(}\delta^{\prime}+d^{\prime\prime}$
$\displaystyle-\theta^{\ast}\pi^{p-1,q+1}\bigr{)}^{2}$
$\displaystyle=(\delta^{\prime}+d^{\prime\prime})^{2}-(\partial+\bar{\partial}-\theta+\theta^{\ast})\theta^{\ast}\pi^{p-1,q+1}-\theta^{\ast}\pi^{p-1,q+1}(\partial+\bar{\partial}-\theta+\theta^{\ast})$
$\displaystyle=-2(\theta\theta^{\ast}+\theta^{\ast}\theta)+\theta\theta^{\ast}\pi^{p-1,q+1},$

using the identities in 2 and the fact that
$\pi^{p-1,q+1}\theta=\theta\pi^{p,q}=0$ on the vector bundle in (50.1). ∎

###  3.2 Multivalued flat sections and the Hodge metric

####  51 .

Now we are ready to start doing some computations specifically for polarized
variations of Hodge structure over the punctured disk. As usual, let

$\Delta^{\ast}=\bigl{\\{}\thinspace
t\in\mathbb{C}\thinspace\thinspace\big{|}\thinspace\thinspace 0<\lvert
t\rvert<1\thinspace\bigr{\\}}$

be the punctured unit disk, with coordinate $t$. Let $E$ be a polarized
variation of Hodge structure of weight $n$ on $\Delta^{\ast}$. We are going to
denote by $V$ the space of all multi-valued flat sections of $(E,d)$. More
precisely, we use the letter

$\mathbb{H}=\bigl{\\{}\thinspace
z\in\mathbb{C}\thinspace\thinspace\big{|}\thinspace\thinspace\operatorname{Re}z<0\thinspace\bigr{\\}}$

for the _left_ half-plane; the exponential function
$\exp\colon\mathbb{H}\to\Delta^{\ast}$ makes it into the universal covering
space of $\Delta^{\ast}$, independently of the choice of $i=\sqrt{-1}$. To
keep the notation simple, let us denote the pullbacks of $E$, $d$, and $h$ by
the same symbols; then $V\subseteq A^{0}(\mathbb{H},E)$ is exactly the kernel
of the operator $d$.

####  52 .

If $v\in V$ is a nontrivial multi-valued flat section, the pointwise Hodge
norm $h(v,v)$ is a smooth function on $\mathbb{H}$ that is everywhere nonzero.
The first observation is that the Higgs field $\theta_{\partial/\partial z}$
controls the first derivatives of $h(v,v)$.

######  Lemma 5.

For every nonzero $v\in V$, one has $\partial h(v,v)=-2h(\theta v,v)$.

###### Proof.

This is true for arbitrary harmonic bundles. By assumption, we have
$d^{\prime}v=d^{\prime\prime}v=0$, and therefore $\partial v=-\theta v$. Since
$\delta^{\prime}+d^{\prime\prime}$ is a metric connection,

$\partial\,h(v,v)=h(\delta^{\prime}v,v)+h(v,d^{\prime\prime}v)=h(\partial
v-\theta v,v)=-2h(\theta v,v),$

as claimed. ∎

####  53 .

Another important property of the function $h(v,v)$, discovered by Griffiths
and Schmid [Schmid, Lem 7.19], is that the logarithm of $h(v,v)$ is
subharmonic.

######  Lemma 6.

If $v\in V$ is nonzero, the function $\log h(v,v)$ is subharmonic on
$\mathbb{H}$.

###### Proof.

This is again true for arbitrary harmonic bundles. From
$d^{\prime}v=d^{\prime\prime}v=0$, we get $\partial v=-\theta v$ and
$\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}v=-\theta^{\ast}v$. For the same reason as in 5, we have

$\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\,h(v,v)=-2h(v,\theta v).$

Since $\partial+\bar{\partial}$ is a metric connection, we then get

$\displaystyle\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\,h(v,v)=-2h(\partial v,\theta v)-2h(v,\bar{\partial}\theta v)$
$\displaystyle=2h(\theta v,\theta v)+2h(v,\theta\bar{\partial}v)$
$\displaystyle=2h(\theta v,\theta v)-2h(\theta^{\ast}v,\theta^{\ast}v),$

using the identity $\bar{\partial}\theta+\theta\bar{\partial}=0$ and 3. To
shorten the next couple of formulas, we introduce the two operators

$A=\theta_{\partial/\partial z}\quad\text{and}\quad
A^{\ast}=\theta^{\ast}_{\partial/\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}},$

which are both smooth sections of the bundle $\operatorname{End}(E)$. The
notation is justified because $A$ and $A^{\ast}$ are adjoints under the Hodge
metric $h$, according to 3. We then get

$\frac{\partial}{\partial
z}\frac{\partial}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\,h(v,v)=2\Bigl{(}h(Av,Av)+h(A^{\ast}v,A^{\ast}v)\Bigr{)}.$

We also know from 5 that

$\frac{\partial}{\partial z}\,h(v,v)=-2h(v,Av)=-2h(A^{\ast}v,v).$

An application of the Cauchy-Schwarz inequality now gives

$\left\lvert\frac{\partial}{\partial
z}\,h(v,v)\right\rvert^{2}\leq\frac{1}{2}\Bigl{(}4h(v,v)h(Av,Av)+4h(v,v)h(A^{\ast}v,A^{\ast}v)\Bigr{)}=h(v,v)\cdot\frac{\partial}{\partial
z}\frac{\partial}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\,h(v,v).$

Since the Laplacian of $\varphi=\log h(v,v)$ satisfies

$\frac{1}{4}\Delta\varphi=\frac{\partial}{\partial
z}\frac{\partial}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log h(v,v)=\frac{1}{h(v,v)}\cdot\frac{\partial}{\partial
z}\frac{\partial}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\,h(v,v)-\frac{1}{h(v,v)^{2}}\left\lvert\frac{\partial}{\partial
z}\,h(v,v)\right\rvert^{2}$

this inequality is what we need to conclude that $\Delta\varphi\geq 0$. ∎

####  54 .

The argument above also shows that the first derivatives of $\log h_{E}(v,v)$
are controlled by the operator $A=\theta_{\partial/\partial z}$, in the
following sense. From 5 and the triangle inequality, we get

$\displaystyle\left\lvert\frac{\partial}{\partial
z}\,h_{E}(v,v)\right\rvert=2\bigl{\lvert}h_{E}(Av,v)\bigr{\rvert}$
$\displaystyle\leq 2h_{E}(v,v)^{\frac{1}{2}}h_{E}(Av,Av)^{\frac{1}{2}}$
$\displaystyle\leq 2h_{E}(v,v)\,h_{\operatorname{End}(E)}(A,A)^{\frac{1}{2}},$

where $h_{\operatorname{End}(E)}(A,A)$ is the Hodge norm of $A$ under the
induced hermitian metric on the bundle $\operatorname{End}(E)$ (which is an
upper bound for the operator norm of $A$ with respect to the Hodge metric
$h_{E}$). After dividing by $h_{E}(v,v)$, this says that

$\left\lvert\frac{\partial}{\partial z}\log
h_{E}(v,v)\right\rvert=\left\lvert\frac{\partial}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log h_{E}(v,v)\right\rvert\leq
2h_{\operatorname{End}(E)}\bigl{(}\theta_{\partial/\partial
z},\theta_{\partial/\partial z}\bigr{)}^{\frac{1}{2}}.$ (54.1)

The Hodge norm of the Higgs field therefore gives us an upper bound for the
derivatives of the function $\log h_{E}(v,v)$.

###  3.3 A universal upper bound for the Higgs field

####  55 .

Clearly, our next task is to bound the quantity
$h_{\operatorname{End}(E)}(\theta_{\partial/\partial
z},\theta_{\partial/\partial z})$. This is analogous to the distance-
decreasing property of period mappings, and the proof naturally uses Ahlfors’
lemma, a generalization of the Schwarz-Pick lemma from complex analysis
[Ahlfors]. The result is usually stated for the open unit disk; for the sake
of completeness, we include a proof that covers both cases.

######  Lemma 7 (Ahlfors).

Let $f\colon\mathbb{H}\to(0,+\infty)$ be a positive smooth function such that

$\frac{\partial^{2}}{\partial z\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log f\geq Cf$

for a positive constant $C>0$. Then

$f(z)\leq\frac{(2C)^{-1}}{\lvert\operatorname{Re}z\rvert^{2}}\quad\text{for
all $z\in\mathbb{H}$.}$

###### Proof.

We start by proving the result on the open unit disk $\Delta$, where the
technique is easier to understand. Here the statement of Ahlfors’ lemma is the
following: for any positive smooth function $g\colon\Delta\to(0,+\infty)$, we
have the implication

$\frac{\partial^{2}}{\partial t\partial\bar{t}}\log g\geq
g\quad\Longrightarrow\quad g(t)\leq\frac{2}{(1-\lvert
t\rvert^{2})^{2}}\quad\text{for $t\in\Delta$.}$

Let $\Delta_{R}$ denote the open disk of radius $R>0$. We are going to argue
that

$g(t)\leq\frac{2R^{2}}{(R^{2}-\lvert t\rvert^{2})^{2}}\quad\text{for
$t\in\Delta_{R}$}$ (55.1)

for every $R<1$; this is enough, because we can then let $R\to 1$ to get the
result. Define an auxiliary function $u\colon\Delta_{r}\to(0,+\infty)$ by the
formula

$g=u\cdot\frac{2R^{2}}{(R^{2}-\lvert t\rvert^{2})^{2}}.$

We observe that $u$ goes to zero near the boundary of $\Delta_{R}$, because
$g$ is bounded on $\Delta_{R}$, whereas $2R^{2}/(R^{2}-\lvert
t\rvert^{2})^{2}$ goes to infinity near the boundary. Therefore $u$ must
achieve its maximum at some interior point $t_{0}\in\Delta_{R}$. In
particular, the function $\log u$ has a local maximum at $t_{0}$, and by
looking at the second derivatives, we get

$\frac{\partial^{2}}{\partial t\partial\bar{t}}\log
u=\frac{1}{4}\left(\frac{\partial^{2}}{\partial x\partial
x}+\frac{\partial^{2}}{\partial y\partial y}\right)\log u\leq 0$

for $t=t_{0}$, where $t=x+iy$. It follows that

$0\geq\frac{\partial^{2}}{\partial t\partial\bar{t}}\log
u=\frac{\partial^{2}}{\partial t\partial\bar{t}}\log
g-\frac{\partial^{2}}{\partial
t\partial\bar{t}}\left(\log\frac{2r^{2}}{(r^{2}-\lvert
t\rvert^{2})^{2}}\right)\geq g-\frac{2r^{2}}{(r^{2}-\lvert t\rvert^{2})^{2}}$

for $t=t_{0}$, which says exactly that $u(t_{0})\leq 1$. But $u$ had a maximum
at $t_{0}$, and therefore $u\leq 1$ on the entire disk $\Delta_{R}$, which is
the content of (55.1).

It remains to deduce the analogous result for a smooth function
$f\colon\mathbb{H}\to(0,+\infty)$. Let us write the hypothesis in the symbolic
form $\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\log f\geq f\mathit{dz}\wedge d\bar{z}$. After pulling back both
$2$-forms along the isomorphism

$\Delta\to\mathbb{H},\quad t\mapsto\frac{t+1}{t-1},$

this becomes the condition that

$\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\log f\left(\frac{t+1}{t-1}\right)\geq\frac{4}{\lvert
t-1\rvert^{4}}\cdot f\left(\frac{t+1}{t-1}\right)\mathit{dt}\wedge d\bar{t}.$

If we now set $g(t)=4\lvert t-1\rvert^{-4}\cdot f\bigl{(}(t+1)/(t-1)\bigr{)}$,
then we get

$\frac{\partial^{2}}{\partial t\partial\bar{t}}\log g\geq g,$

and therefore, according to the discussion above, the inequality

$g(t)\leq\frac{2}{(1-\lvert t\rvert^{2})^{2}}\quad\text{for $t\in\Delta$.}$

If we convert this back into a statement about $f$, the result follows. ∎

####  56 .

The following lemma establishes the fundamental inequality that we need in
order to apply Ahlfors’ lemma to our setting [Simpson, Thm. 1].

######  Lemma 8.

With the notation $A=\theta_{\partial/\partial z}$ and
$A^{\ast}=\theta^{\ast}_{\partial/\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}$, we have

$\frac{\partial^{2}}{\partial z\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log
h_{\operatorname{End}(E)}(A,A)\geq\frac{h_{\operatorname{End}(E)}\bigl{(}[A^{\ast},A],[A^{\ast},A]\bigr{)}}{h_{\operatorname{End}(E)}(A,A)}.$

###### Proof.

It is best to view this as a special case of a more general result. Let
$(E,d,h)$ be a harmonic bundle on $\mathbb{H}$, and let $s$ be a smooth
section of $E$ that satisfies $\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}s=0$ and $\theta s=0$; in other words, $s$ is a holomorphic section of
the Higgs bundle $E$ that lies in the kernel of the Higgs field. Then we claim
that

$\frac{\partial^{2}}{\partial z\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log h(s,s)\geq\frac{h(A^{\ast}s,A^{\ast}s)}{h(s,s)},$ (56.1)

where again
$A^{\ast}=\theta^{\ast}_{\partial/\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}$. The calculations are very similar to those in the proof of 6. The
identity
$\partial\bar{\partial}+\bar{\partial}\partial+\theta\theta^{\ast}+\theta^{\ast}\theta=0$
in 2 gives us $\bar{\partial}\partial s=-\theta\theta^{\ast}s$. Since
$\partial+\bar{\partial}$ is a metric connection, we have

$\bar{\partial}\,h(s,s)=h(\bar{\partial}s,s)+h(s,\partial s)=h(s,\partial s).$

Applying the same reasoning again, we get

$\partial\bar{\partial}\,h(s,s)=h(\partial s,\partial
s)+h(s,\bar{\partial}\partial s)=h(\partial s,\partial
s)-h(s,\theta\theta^{\ast}s)=h(\partial s,\partial
s)-h(\theta^{\ast}s,\theta^{\ast}s).$

As before, we can now use the Cauchy-Schwarz inequality to conclude that

$\frac{\partial}{\partial
z}\frac{\partial}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log h(s,s)\geq\frac{h(A^{\ast}s,A^{\ast}s)}{h(s,s)},$

which is of course just (56.1).

Now let us return to our specific situation. The vector bundle
$\operatorname{End}(E)$, with the flat connection induced by $d$ and the
metric induced by $h$, is again a harmonic bundle; moreover, it is easy to see
that $\theta_{\partial/\partial z}$ acts on $\operatorname{End}(E)$ as the
commutator with $A$, and
$\theta^{\ast}_{\partial/\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}}$
acts as the commutator with $A^{\ast}$. If we consider $A$ as a section of
$\operatorname{End}(E)$, it is holomorphic and trivially commutes with $A$,
hence belongs to the kernel of both $\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$
and $\theta$. As such, the calculation from above applies to it, and so

$\frac{\partial^{2}}{\partial z\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log
h(A,A)\geq\frac{h\bigl{(}[A^{\ast},A],[A^{\ast},A]\bigr{)}}{h(A,A)}.$

This is the desired inequality. ∎

####  57 .

At each point of $\mathbb{H}$, the fiber of $E$ is a vector space of dimension
$r=\operatorname{rk}E$ with a hermitian inner product, and
$\theta_{\partial/\partial z}$ is a nilpotent endomorphism. The following
lemma bounds the pointwise norm of the commutator.

######  Lemma 9.

Let $V$ be a complex vector space of finite dimension $r\geq 1$, with a
hermitian inner product. If $A\in\operatorname{End}(V)$ is a nilpotent
endomorphism, and if $A^{\ast}\in\operatorname{End}(V)$ denotes its adjoint
with respect to the inner product, then

$\bigl{\lVert}[A^{\ast},A]\bigr{\rVert}^{2}\leq 2\lVert
A\rVert^{4}\leq\binom{r+1}{3}\bigl{\lVert}[A^{\ast},A]\bigr{\rVert}^{2}.$

###### Proof.

Choose an orthonormal basis $e_{1},\dotsc,e_{r}\in V$ such that the matrix
representing $A$ is upper triangular; the matrix representing $A^{\ast}$ is
the conjugate transpose, hence lower triangular. Setting $a_{i,j}=\langle
Ae_{i},e_{j}\rangle$, we get

$\lVert A\rVert^{2}=\sum_{i<j}\lvert a_{i,j}\rvert^{2}.$

The $k$-th diagonal entry of the matrix representing $[A^{\ast},A]$ is easily
seen to be

$d_{k,k}=\bigl{(}\lvert a_{1,k}\rvert^{2}+\dotsb+\lvert
a_{k-1,k}\rvert^{2}\bigr{)}-\bigl{(}\lvert a_{k,k+1}\rvert^{2}+\dotsb+\lvert
a_{k,r}\rvert^{2}\bigr{)},$

and this gives us the rather simple-minded inequality

$\bigl{\lVert}[A^{\ast},A]\bigr{\rVert}\geq\sqrt{\lvert
d_{1,1}\rvert^{2}+\dotsb+\lvert d_{r,r}\rvert^{2}}.$

Let $x\in\mathbb{R}$ be arbitrary. It is easy to see that

$\sum_{k=1}^{r}(x+k)d_{k,k}=\sum_{i<j}\bigl{(}(x+j)-(x+i)\bigr{)}\lvert
a_{i,j}\rvert^{2}=\sum_{i<j}(j-i)\lvert a_{i,j}\rvert^{2},$

and from the trivial lower bound $j-i\geq 1$, we therefore get

$\sum_{i<j}\lvert a_{i,j}\rvert^{2}\leq\sum_{k=1}^{r}(x+k)\lVert
d_{k,k}\rVert\leq\left(\sum_{k=1}^{r}(x+k)^{2}\right)^{\frac{1}{2}}\left(\sum_{k=1}^{r}\lvert
d_{k,k}\rvert^{2}\right)^{\frac{1}{2}}.$

The right-hand side becomes minimal for $x=-\frac{1}{2}(r+1)$, with the result
that

$\lVert
A\rVert^{2}\leq\sqrt{\frac{r(r+1)(r-1)}{12}}\bigl{\lVert}[A^{\ast},A]\bigr{\rVert}.$

This inequality is actually sharp: equality is achieved for the $r\times
r$-matrix with

$\lvert a_{k,k+1}\rvert^{2}=k(r-k)\quad\text{for $k=1,\dotsc,r-1$,}$

and all other entries zero. Interestingly, this is exactly the case where $V$
is an irreducible representation of the Lie algebra
$\mathfrak{sl}_{2}(\mathbb{C})$, with $\rho(\mathsf{Y})=A$,
$\rho(\mathsf{X})=A^{\ast}$, and $\rho(\mathsf{H})=[A^{\ast},A]$. We leave the
verification of this claim to the reader.

The proof of the other inequality is easier. Without loss of generality, we
can assume that $\lVert A\rVert=\lVert A^{\ast}\rVert=1$. The operator
$[A^{\ast},A]$ is self-adjoint; by the spectral theorem, there is an
orthonormal basis $e_{1}^{\prime},\dotsc,e_{r}^{\prime}$ such that
$[A^{\ast},A]e_{i}^{\prime}=\lambda_{i}e_{i}^{\prime}$ with
$\lambda_{1},\dotsc,\lambda_{n}\in\mathbb{R}$. Therefore

$\bigl{\lVert}[A^{\ast},A]\bigr{\rVert}^{2}=\sum_{i=1}^{r}\lambda_{i}^{2}=\sum_{i=1}^{r}\bigl{(}\lVert
Ae_{i}^{\prime}\rVert^{2}-\lVert
A^{\ast}e_{i}^{\prime}\rVert^{2}\bigr{)}^{2}\leq\sum_{i=1}^{r}\lVert
Ae_{i}^{\prime}\rVert^{4}+\sum_{i=1}^{r}\lVert
A^{\ast}e_{i}^{\prime}\rVert^{4},$

because
$\lambda_{i}=\bigl{\langle}(A^{\ast}A-AA^{\ast})e_{i}^{\prime},e_{i}^{\prime}\bigr{\rangle}=\lVert
Ae_{i}^{\prime}\rVert^{2}-\lVert A^{\ast}e_{i}^{\prime}\rVert^{2}$. At the
same time,

$1=\lVert A\rVert^{2}=\sum_{i=1}^{r}\lVert Ae_{i}^{\prime}\rVert^{2},$

and so both sums on the right-hand side are $\leq 1$. ∎

####  58 .

We can now apply the version of Ahlfors’ lemma in 7, and deduce the following
very striking upper bound for the Hodge norm of the Higgs field.

######  Corollary 1.

Let $E$ be a polarized variation of Hodge structure of rank $r\geq 1$ on the
left half-plane $\mathbb{H}$. Then the Higgs field $\theta_{\partial/\partial
z}$ satisfies the inequality

$h_{\operatorname{End}(E)}\bigl{(}\theta_{\partial/\partial
z},\theta_{\partial/\partial
z}\bigr{)}\leq\frac{C_{0}^{2}}{(\operatorname{Re}z)^{2}},$

where $C_{0}=\frac{1}{2}\sqrt{\binom{r+1}{3}}$.

###### Proof.

Since the Higgs field is trivial when $r=1$, we can assume that $r\geq 2$. Let
us again use the abbreviations $A=\theta_{\partial/\partial z}$ and
$A^{\ast}=\theta_{\partial/\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}$. If we put the inequalities in 8 and 9 together, we get

$\frac{\partial^{2}}{\partial z\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\log
h_{\operatorname{End}(E)}(A,A)\geq\frac{12}{r(r+1)(r-1)}h_{\operatorname{End}(E)}(A,A).$

Now apply 7 to reach the desired conclusion. ∎

From an analytic point of view, this is really the central fact about
polarized variations of Hodge structure. The amazing thing is that the
constant depends on nothing but the rank of the variation of Hodge structure.
This fact turns out to be especially useful for the theory in several
variables.

####  59 .

The induced variation of Hodge structure on the bundle $\operatorname{End}(E)$
has rank $r^{2}$, but in that case, there is a bound on the Higgs field that
is much better than simply replacing $r$ by $r^{2}$ is the result above.

######  Corollary 2.

Let $E$ be a polarized variation of Hodge structure of rank $r\geq 1$ on the
left half-plane $\mathbb{H}$. Then the Higgs field $\theta_{\partial/\partial
z}$ of the induced variation of Hodge structure on $\operatorname{End}(E)$
satisfies the inequality

$h_{\operatorname{End}(\operatorname{End}(E))}\bigl{(}\theta_{\partial/\partial
z},\theta_{\partial/\partial
z}\bigr{)}\leq\frac{2rC_{0}^{2}}{(\operatorname{Re}z)^{2}},$

where again $C_{0}=\frac{1}{2}\sqrt{\binom{r+1}{3}}$.

###### Proof.

Let $A=\theta_{\partial/\partial z}$ be the Higgs field acting on $V$. The
induced Higgs field on $\operatorname{End}(V)$ is then equal to the commutator
$(\operatorname{ad}A)(X)=[A,X]$. It is therefore enough to prove that for any
nilpotent matrix $A\in\operatorname{End}(V)$, one has

$\lVert\operatorname{ad}A\rVert^{2}\leq 2r\lVert A\rVert^{2}.$

Let $e_{1},\dotsc,e_{r}\in V$ be an orthonormal basis such that the matrix for
$A$ is strictly upper triangular. Then the matrices $E_{i,j}$, whose only
nonzero entry is a $1$ in the position $(i,j)$, form an orthonormal basis for
$\operatorname{End}(V)$, and a short calculation gives

$(\operatorname{ad}A)(E_{i,j})=\sum_{k<i}a_{k,i}E_{k,j}-\sum_{j<k}a_{j,k}E_{i,k}.$

We therefore get

$\displaystyle\lVert\operatorname{ad}A\rVert^{2}$
$\displaystyle=\sum_{i,j}\lvert(\operatorname{ad}A)(E_{i,j})\rvert^{2}$
$\displaystyle=\sum_{i,j}\Bigl{(}\lvert a_{1,i}\rvert^{2}+\dotsb+\lvert
a_{i-1,i}\rvert^{2}+\lvert a_{j,j+1}\rvert^{2}+\dotsb+\lvert
a_{j,r}\rvert^{2}\Bigr{)}\leq 2r\lVert A\rVert^{2},$

as claimed. ∎

###  3.4 Proof of the monodromy theorem

####  60 .

Let us summarize the results so far. We have shown that if $v\in V$ is a
nontrivial multi-valued flat section of $E$, then the function $\varphi=\log
h(v,v)$ is smooth and subharmonic on $\mathbb{H}$, and its first derivatives
are bounded by

$\left\lvert\frac{\partial\varphi}{\partial
z}\right\rvert=\left\lvert\frac{\partial\varphi}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}\right\rvert\leq\frac{2C_{0}}{\lvert\operatorname{Re}z\rvert},$ (60.1)

where $C_{0}=\frac{1}{2}\sqrt{\binom{r+1}{3}}$ and $r=\operatorname{rk}E$.
Everything up to this point, maybe except for the optimal value of the
constant, was known to Simpson [Simpson, §2] back in the 1990s. Now we add a
small – but crucial! – new insight, namely that the uniform bound on the first
derivatives can be used directly to control the behavior of the function
$\varphi$ for $\operatorname{Re}z\ll 0$.

####  61 .

Before we can explain how this works, we first need to introduce the monodromy
operator $T\in\operatorname{Aut}(V)$. Following Schmid [Schmid, (4.4)], we
define this by the formula

$(Tv)(z)=v(z-2\pi i),$

where $i=\sqrt{-1}$. That is to say, for any $v\in V$, the function $z\mapsto
v(z-2\pi i)$ is again a flat section of the bundle $E$ on $\mathbb{H}$, and we
define $Tv\in V$ to be this section. The flat pairing on $E$ induces a
hermitian pairing

$Q\colon V\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C},$

and one checks easily that $T$ preserves $Q$, in the sense that
$Q(Tv,Tw)=Q(v,w)$ for every $v,w\in V$. We write the Jordan decomposition of
$T$ in the form

$T=T_{s}\cdot e^{2\pi iN},$

with $T_{s}\in\operatorname{Aut}(V)$ semisimple and
$N\in\operatorname{End}(V)$ nilpotent. Then

$Q(T_{s}v,T_{s}w)=Q(v,w)\quad\text{and}\quad Q(Nv,w)=Q(v,Nw),$

for every $v,w\in V$, because $Q$ is conjugate-linear in the second argument.
Note that $T$ depends on the choice of $i=\sqrt{-1}$, but $N$ is independent
of it.

####  62 .

The space $V$ of multi-valued flat sections gives us a trivialization of the
vector bundle $E$ on the left half-plane $\mathbb{H}$. We can therefore
consider the polarized variation of Hodge structure as a family of Hodge
structures on $V$; as usual, we denote by the letter $\Phi(z)$ the Hodge
filtration of the Hodge structure at the point $z\in\mathbb{H}$. The
definition of the monodromy operator has the following consequence.

######  Lemma 10.

For every $z\in\mathbb{H}$, we have the equality

$\Phi(z+2\pi i)=T\,\Phi(z),$

where both sides are filtrations on $V$.

###### Proof.

This may be a bit confusing, so let us write out the argument. The
trivialization of the pullback of the bundle $E$ along
$\exp\colon\mathbb{H}\to\Delta^{\ast}$ gives us, for every point
$z\in\mathbb{H}$, an isomorphism of complex vector spaces

$\phi_{z}\colon V\to(\exp^{\ast}E)\big{|}_{z}\cong
E_{t},\quad\phi_{z}(v)=v(z).$

Here $t=e^{z}$. Since $v(z+2\pi i)=(T^{-1}v)(z)$, the diagram

${V}$${E_{t}}$${V}$$\scriptstyle{T^{-1}}$$\scriptstyle{\phi_{z+2\pi
i}}$$\scriptstyle{\phi_{z}}$

commutes. The way the period mapping is constructed, we have

$F_{\Phi(z+2\pi i)}^{p}=\phi_{z+2\pi
i}^{-1}\bigl{(}F^{p}E_{t}\bigr{)}=(T^{-1})^{-1}\phi_{z}^{-1}\bigl{(}F^{p}E_{t}\bigr{)}=TF_{\Phi(z)}^{p},$

which is what we wanted to prove. ∎

####  63 .

Given a multi-valued flat section $v\in V$, we can write the value of the
function $h(v,v)$ at the point $z\in\mathbb{H}$ in the equivalent form

$h(v,v)(z)=\lVert v\rVert_{\Phi(z)}^{2},$

using the notation for the Hodge norm from 9. Since
$T\in\operatorname{Aut}(V,Q)$, we have

$h(v,v)(z+2\pi i)=\lVert v\rVert_{T\Phi(z)}^{2}=\lVert
T^{-1}v\rVert_{\Phi(z)}^{2}=h(T^{-1}v,T^{-1}v)(z).$ (63.1)

It is this relation that connects the asymptotic behavior of the Hodge norm to
the monodromy transformation $T$.

####  64 .

To illustrate the power of the derivative bound in (60.1), we are now going to
prove the monodromy theorem [Schmid, Lem. 4.5].

######  Proposition 3.

If $\lambda\in\mathbb{C}$ is an eigenvalue of $T$, then
$\lvert\lambda\rvert=1$.

###### Proof.

Let $v\in V$ be a nonzero eigenvector with $Tv=\lambda v$, and consider the
smooth function $\varphi=\log h(v,v)$. The relation in (63.1) implies that

$\varphi(z+2\pi i)=\log\lvert\lambda\rvert^{-2}+\varphi(z)$

for every $z\in\mathbb{H}$. On the other hand, we clearly have

$\varphi(z+2\pi
i)-\varphi(z)=\int_{0}^{2\pi}\left(i\frac{\partial\varphi}{\partial
z}(z+iy)-i\frac{\partial\varphi}{\partial\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}}(z+iy)\right)\mathit{dy},$

and on account of (60.1), this leads to the inequality

$\bigl{\lvert}\varphi(z+2\pi i)-\varphi(z)\bigr{\rvert}\leq\frac{8\pi
C_{0}}{\lvert\operatorname{Re}z\rvert},$

where $C_{0}=\frac{1}{2}\sqrt{\binom{r+1}{3}}$. Letting
$\lvert\operatorname{Re}z\rvert\to\infty$, we conclude that
$\log\lvert\lambda\rvert=0$, whence $\lvert\lambda\rvert=1$. ∎

###  3.5 Effective bounds for the monodromy transformation

####  65 .

We can use the derivative bound in (60.1) to control the Hodge norms of the
nilpotent operator $N$ and of the projections $P_{\lambda}\colon V\to
E_{\lambda}(T_{s})$ to the eigenspaces of $T_{s}$. The fact that $N$ and
$\theta_{\partial/\partial z}$ are both of order
$\lvert\operatorname{Re}z\rvert^{-1}$ is no accident: during the proof of the
nilpotent orbit theorem, we will see that the difference between these two
operators, modulo $F^{0}\operatorname{End}(V)_{\Phi(z)}$, is of order
$e^{-\varepsilon\lvert\operatorname{Re}z\rvert}$. (The precise statement is in
16.)

######  Proposition 4.

There is a constant $C>0$, whose value only depends on $\dim V$ and on the
minimal polynomial of the monodromy transformation $T\in\operatorname{GL}(V)$,
such that:

  1. (a)

For every $v\in V$ and every $z\in\mathbb{H}$ with $\operatorname{Re}z\leq-1$,
one has

$\lVert Nv\rVert_{\Phi(z)}\leq\frac{C}{\lvert\operatorname{Re}z\rvert}\lVert
v\rVert_{\Phi(z)}.$

  2. (b)

For every $v\in V$ and every $z\in\mathbb{H}$ with $\operatorname{Re}z\leq-1$,
one has

$\lVert P_{\lambda}v\rVert_{\Phi(z)}\leq C\lVert v\rVert_{\Phi(z)}.$

####  66 .

Given the bound on the derivative of the function $\log\lVert
v\rVert_{\Phi(z)}^{2}$, the proof is just linear algebra. By the same argument
as in the proof of 3, we have

$\Bigl{\lvert}\log\lVert Tv\rVert_{\Phi(z)}^{2}-\log\lVert
v\rVert_{\Phi(z)}^{2}\Bigr{\rvert}=\Bigl{\lvert}\log\lVert
v\rVert_{\Phi(z-2\pi i)}^{2}-\log\lVert
v\rVert_{\Phi(z)}^{2}\Bigr{\rvert}\leq\frac{8\pi
C_{0}}{\lvert\operatorname{Re}z\rvert};$

after exponentiating, this becomes

$e^{-8\pi C_{0}/\lvert\operatorname{Re}z\rvert}\leq\frac{\lVert
Tv\rVert_{\Phi(z)}^{2}}{\lVert v\rVert_{\Phi(z)}^{2}}\leq e^{8\pi
C_{0}/\lvert\operatorname{Re}z\rvert}.$

If we restrict to $\operatorname{Re}z\leq-1$ for simplicity, it follows that

$\Bigl{\lvert}\lVert Tv\rVert_{\Phi(z)}^{2}-\lVert
v\rVert_{\Phi(z)}^{2}\Bigr{\rvert}\leq\frac{C^{\prime}}{\lvert\operatorname{Re}z\rvert}\lVert
v\rVert_{\Phi(z)}^{2},$ (66.1)

where $C^{\prime}=(e^{8\pi C_{0}}-1)/(8\pi C_{0})$.

####  67 .

The following elementary lemma, applied to the inner products
$\langle-,-\rangle_{\Phi(z)}$, turns the inequality in (66.1) into a bound for
the operator norm of the nilpotent operator $N$.

######  Lemma 11.

Let $V$ be a finite-dimensional complex vector space with an inner product
$\langle-,-\rangle$. Let $T\in\operatorname{GL}(V)$ be an endomorphism whose
eigenvalues have absolute value $1$. Suppose that $T$ is close to unitary, in
the sense that there is a constant $\varepsilon>0$ such that

$\Bigl{\lvert}\lVert Tv\rVert^{2}-\lVert
v\rVert^{2}\Bigr{\rvert}\leq\varepsilon\lVert v\rVert^{2}\quad\text{for all
$v\in V$.}$

Then there are two constants $C>0$ and $m\in\mathbb{N}$, whose values only
depends on the integer $\dim V$ and on the minimal polynomial of the operator
$T$, such that

$\lVert Nv\rVert\leq C\varepsilon(1+\varepsilon^{m})\cdot\lVert
v\rVert\quad\text{for all $v\in V$,}$

where $N=(2\pi i)^{-1}\log T_{u}$ is the logarithm of the unipotent part of
$T$.

###### Proof.

Set $r=\dim V$, and choose an orthonormal basis $e_{1},\dotsc,e_{r}\in V$ such
that the matrix representing $T$ is upper triangular. Write
$T=U(\operatorname{id}+B)$, where $U$ is a diagonal matrix and $B$ is strictly
upper triangular. The entries of $U$ are the eigenvalues of $T$, hence of
absolute value $1$, and so $U$ is unitary. Pick any two integers $1\leq
i<j\leq r$, and let $b_{i,j}=\langle Be_{j},e_{i}\rangle$ denote the
corresponding entry of the matrix representing $B$. By applying the hypothesis
to vectors of the form $v=e_{i}+ze_{j}$ with $\lvert z\rvert=1$, we get

$2\operatorname{Re}(zb_{i,j})=2\operatorname{Re}\bigl{\langle}B(e_{i}+ze_{j}),e_{i}+ze_{j}\bigr{\rangle}\leq\varepsilon\lVert
e_{i}+ze_{j}\rVert^{2}=2\varepsilon,$

and by choosing $z$ appropriately, this gives $\lvert
b_{i,j}\rvert\leq\varepsilon$. Consequently, the $L^{2}$-norm of the operator
$B\in\operatorname{End}(V)$ is bounded by $\lVert
B\rVert^{2}\leq\binom{r}{2}\varepsilon^{2}$, and therefore

$\lVert Bv\rVert\leq r\varepsilon\lVert v\rVert\quad\text{for all $v\in V$.}$

Now it is a basic fact from linear algebra that the nilpotent operator
$N=(2\pi i)^{-1}\log T_{u}$ can be written as a polynomial in $T$, of the form

$N=P(T)\cdot\prod_{\lambda}(T-\lambda\operatorname{id}),$

where the product runs over the distinct eigenvalues of $T$; the polynomial
$P(t)\in\mathbb{C}[t]$ is uniquely determined by the minimal polynomial of
$T$. Consequently,

$N=P(T)\cdot\prod_{\lambda}\bigl{(}(U-\lambda\operatorname{id})+UB\bigr{)},$

and because the operator norm of $P(T)$ is easily bounded using the inequality
$\lVert Tv\rVert^{2}\leq(1+\varepsilon)\lVert v\rVert^{2}$, it suffices to
estimate the operator norm of the product. Given our bound $r\varepsilon$ for
the operator norm of $B$, this comes down to

$\prod_{\lambda}(U-\lambda\operatorname{id})=0,$

which holds because $U$ is a diagonal matrix. ∎

####  68 .

The same argument also proves that the projection operators

$P_{\lambda}\colon V\to E_{\lambda}(T_{s})$

are uniformly bounded, meaning that there is a constant $C>0$, whose value
again only depends on the integer $\dim V$ and on the minimal polynomial of
$T\in\operatorname{GL}(V)$, such that

$\lVert P_{\lambda}v\rVert_{\Phi(z)}\leq C\lVert
v\rVert_{\Phi(z)}\quad\text{for $v\in V$ and $\operatorname{Re}z\leq-1$.}$
(68.1)

The reason is that $P_{\lambda}$ can again be expressed as a certain
polynomial in $T$, whose operator norm can then be bounded using (66.1). This
finishes the proof of 4.

##  4 Period domains and period mappings

####  69 .

In this chapter, we briefly review the basic properties of period domains and
period mappings, in the setting of _complex_ Hodge structures. The theory is
actually simpler than in the case of rational Hodge structures [Schmid, §3],
because the Lie groups that are involved are just real forms of the general
linear group $\operatorname{GL}(V)$.

###  4.1 The period domain and its compact dual

####  70 .

Fix a finite-dimensional complex vector space $V$ and a hermitian pairing
$Q\colon V\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C}$. Let $D$ be the period domain parametrizing Hodge
structures of the type we are interested in: each point $o\in D$ corresponds
to a Hodge structure

$V=\bigoplus_{p+q=n}V_{o}^{p,q}$

on the vector space $V$, with fixed Hodge numbers $\dim V_{o}^{p,q}$, and
polarized by the hermitian pairing $Q$. We assume that there is at least one
such Hodge structure; in particular, $Q$ is nondegenerate and of the correct
signature. In this setting, the real Lie group

$G=\bigl{\\{}\thinspace
g\in\operatorname{GL}(V)\thinspace\thinspace\big{|}\thinspace\thinspace\text{$Q(gv,gw)=Q(v,w)$
for all $v,w\in V$}\thinspace\bigr{\\}}$

acts transitively on $D$, according to the rule $V_{g\cdot
o}^{p,q}=g(V_{o}^{p,q})$. (This is easily proved by chosing an orthonormal
basis adapted to the Hodge decomposition, and then mapping one such
orthonormal basis to another.) The inner products at the two points $o$ and
$g\cdot o$ are related by the formula

$\langle gv,gw\rangle_{g\cdot o}=\langle v,w\rangle_{o},$

which follows from the fact that $g\in G$.

####  71 .

The Lie algebra of $G$ is easily seen to be

$\mathfrak{g}=\bigl{\\{}\thinspace
A\in\operatorname{End}(V)\thinspace\thinspace\big{|}\thinspace\thinspace\text{$Q(Av,w)+Q(v,Aw)=0$
for all $v,w\in V$}\thinspace\bigr{\\}}.$

Note that $\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}=\operatorname{End}(V)$,
which means that $G$ is a non-compact real form of the complex Lie group
$\operatorname{GL}(V)$. This also gives us a real structure on
$\operatorname{End}(V)$; the “complex conjugate” of an endomorphism
$A\in\operatorname{End}(V)$ is $-A^{\dagger}$, where the adjoint $A^{\dagger}$
with respect to the hermitian pairing $Q$ is defined by the formula

$Q(Av,w)=Q(v,A^{\dagger}w)\quad\text{for all $v,w\in V$.}$

This makes sense because $Q$, being a polarization, is nondegenerate.

####  72 .

The stabilizer of a point $o\in D$ is the compact subgroup

$H_{o}=\bigl{\\{}\thinspace g\in
G\thinspace\thinspace\big{|}\thinspace\thinspace\text{$g(V_{o}^{p,q})=V_{o}^{p,q}$
for all $p,q$}\thinspace\bigr{\\}}\subseteq G,$

and this gives us an isomorphism $D\cong G/H_{o}$. We can use it to describe
the real tangent space $T_{o}D$ in terms of Hodge structures. The Hodge
structure on $V$ induces a Hodge structure of weight $0$ on
$\operatorname{End}(V)$, by setting

$\operatorname{End}(V)_{o}^{j,-j}=\bigl{\\{}\thinspace
A\in\operatorname{End}(V)\thinspace\thinspace\big{|}\thinspace\thinspace\text{$A(V_{o}^{p,q})\subseteq
V_{o}^{p+j,q-j}$ for all $p,q$}\thinspace\bigr{\\}}.$

It is an easy exercise to check that the decomposition

$\operatorname{End}(V)=\bigoplus_{j\in\mathbb{Z}}\operatorname{End}(V)_{o}^{j,-j}$
(72.1)

is an $\mathbb{R}$-Hodge structure (with respect to the real structure
$\mathfrak{g}$), and that the induced polarization on $\operatorname{End}(V)$
is equal to the trace pairing

$(A,B)\mapsto\operatorname{tr}(AB^{\dagger}).$

Since the conjugate of $B$ is $-B^{\dagger}$, we can view the polarization on
$\operatorname{End}(V)$ as coming from the $\mathbb{R}$-valued bilinear
pairing $(A,B)\mapsto-\operatorname{tr}(AB)$ on $\mathfrak{g}$.

####  73 .

Here is the proof that the trace pairing is indeed a polarization for the
induced Hodge structure on $\operatorname{End}(V)$. The same argument is
needed in Section 5, and so we give some details.

######  Lemma 12.

The pairing

$\operatorname{End}(V)\otimes_{\mathbb{C}}\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}\to\mathbb{C},\quad(A,B)\mapsto\operatorname{tr}(AB^{\dagger}),$

polarizes the Hodge structure on $\operatorname{End}(V)$.

###### Proof.

It is easy to see that
$\operatorname{tr}A^{\dagger}=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$
for every $A\in\operatorname{End}(V)$. Therefore

$\operatorname{tr}(BA^{\dagger})=\operatorname{tr}\bigl{(}(AB^{\dagger})^{\dagger}\bigr{)}=\mathchoice{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}{\macc@depth\char