# Shafarevich mappings and period mappings

Mark Green, Phillip Griffiths and Ludmil Katzarkov  To our friend Enrico
Arbarello, whose zest for life and enjoyment of mathematics continue to
inspire us  Department of Mathematics, University of California at Los
Angeles, Los Angeles, CA 90095 E-mail address<EMAIL_ADDRESS>Institute for
Advanced Study, Einstein Drive, Princeton, NJ 08540 E-mail address<EMAIL_ADDRESS>University of Miami, Coral Gables, FL and Institute of Mathematics and
Informatics, Bulgarian Academy of Sciences NRU HSE, Moscow E-mail address:
<EMAIL_ADDRESS>
###### Abstract.

We shall show that a smooth, quasi-projective variety $X$ has a
holomorphically convex universal covering $\widetilde{X}$ when (i)
$\pi_{1}(X)$ is residually nilpotent and (ii) there is an admissable variation
of mixed Hodge structure over $X$ whose monodromy representation has a finite
kernel, and where in each case a corresponding period mapping is assumed to be
proper.

## Outline

  1. I.

_Introduction and statements of results_

  2. II.

_The nilpotent case_

  3. III.

_The variation of Hodge structure and mixed Hodge structure cases_

  4. IV.

_Further directions_

##  I. Introduction and statements of results

In [Sh72] Shafarevich posed a beautiful question, a variant of which has
become known as Shafarevich’s conjecture (cf. [Ko93] and [Ko95]):

######  I.1:

_Let $X$ be a smooth projective variety. Is the universal covering
$\widetilde{X}$ of $X$ holomorphically convex?_

This means that there is a Stein analytic variety $\widetilde{S}$ and a proper
holomorphic mapping $\widetilde{X}\to\widetilde{S}$ that contracts the
connected compact analytic subvarieties of $\widetilde{X}$ to points. This
mapping is the Cartan-Remmert reduction of $\widetilde{X}$ ([Car79]).

In practice for a quasi-projective variety $X$ one seeks a _Shafarevich
mapping_ ([Ko93], [Ko95] and [Cam94])

(I.2) $\mathop{\text{\it Sh}}\nolimits:X\to S$

onto an analytic variety with the properties

  1. (i)

If $Y\subset X$ is a connected complex analytic subvariety of $X$ that is
contracted to a point by $\mathop{\text{\it Sh}}\nolimits$, then $Y$ is
compact and the image

$\pi_{1}(Y)\to\pi_{1}(X)$

is a finite subgroup (we shall say that the mapping $\pi_{1}(Y)\to\pi_{1}(X)$
is finite).

  2. (ii)

In the diagram

(I.3) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Sh}}\nolimits}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
31.53471pt\raise-31.02777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.53471pt\raise-31.02777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}\vspace*{-4pt}$

the universal cover $\widetilde{S}=\widetilde{X}\times_{{}_{\\!X}}S$ is Stein.

We note that if (i) is satisfied, then the connected fibres of
$\widetilde{\mathop{\text{\it Sh}}\nolimits}$ are the connected components of
$\pi^{-1}(Y)$ where $Y\subset X$ is a connected fibre of $\mathop{\text{\it
Sh}}\nolimits$. Moreover the components of $\pi^{-1}(Y)$ are compact.

The Shafarevich conjecture has been established when

  1. (a)

$X$ is projective and $\pi_{1}(X)$ is residually nilpotent;

  2. (b)

$X$ is projective and $\pi_{1}(X)$ has a faithful linear representation.

The initial proof of (a) is in [Ka97]. Subsequently there have been numerous
further works (cf. [Cam95] and [Cl08]). The proof of (b) is in [Ey04] and
[EKPR12]; it is the culmination of a series of works drawing on both classical
and non-abelian Hodge theory (cf. [CM-SP17] and [Si88], [Si92]). Some of the
history and references to the literature are given in [EKPR12]. Further
related references are [BdO06], [EF21], [KR98], and [Ey09].

The purpose of this partly expository paper is to extend (a) and a special
case of (b) to the case when $X$ is quasi-projective. The essence of what will
be proved here may be informally expressed as saying

proper period mappings give Shafarevich mappings.

In order to understand what is meant by proper for the period mappings to be
considered we introduce some standard terminology and notations. By a
_completion_ of $X$ we mean a smooth projective variety $X$ in which
$\overline{X}$ is a Zariski open set with complement $Z=\cup Z_{i}$ a reduced
normal crossing divisor with $Z_{i}$ irreducible.

In case (a) there are two choices for a period mapping. One is the classical
Albanese mapping

(I.4a) $\alpha:X\to{\rm Alb}(X)$ where ${\rm
Alb}(X)=H^{0}({\Omega}^{1}_{X}(\log Z))^{\ast}/H_{1}(X,\mathbb{Z})$ is a semi-
abelian variety. The other choice is to use a higher Albanese mapping ([Ha87a]
and [Cl08]) (I.4b) $\alpha_{s}:X\to{\rm Alb}^{s}(X),\quad s=1,2,3,\dots.$

This is a period mapping for a unipotent variation of mixed Hodge structure
(UVMHS) ([HZ85]). It reduces to (I.4a) when $s=1$, and when $\pi_{1}(X)$ is
residually nilpotent the mappings (I.4b) are all essentially the same for
$s\gg 0$.

The properness of the Albanese mapping (I.4a) implies that the image of the
residue mapping

$H^{0}({\Omega}^{1}_{\overline{X}}(\log
Z))\to\mathop{\oplus}\limits^{i}H^{0}({\mathscr{O}}_{Z_{i}})(-1)$

projects non-trivially to each summand $H^{0}({\mathscr{O}}_{Z_{i}})(-1)$.
These conditions are independent of the completion of $X$. Additional for each
stratum $Z_{I}:=\cap_{i\in I}Z_{i}$ with $|I|\geqq 2$ and each ray ${\lambda}$
emanating from $Z_{I}^{\ast}$ out into $X$ there should be an ${\omega}\in
H^{0}({\Omega}^{1}_{\overline{X}}(\log Z)$ having a logarithmic singularity at
${\lambda}\cap Z^{\ast}_{I}$. If the second of these conditions is not
satisfied, then only the closure of the image of the Shafarevich map will be
Stein. From the second of the three proofs of Theorem A below it will follow
that $\alpha$ is proper if, and only if, the $\alpha_{s}$ are proper. In fact
as a consequence of the proof of Theorem II.4 below we will have the

######  Theorem I.5:

The images $\alpha_{s}(X)$ are algebraic varieties and the mapping
$\alpha_{s}(X)\to\alpha(X)$ is a proper surjective morphism. If $Y\subset X$
is a compact subvariety with $\alpha\big{|}_{Y}=$ constant, then
$\alpha_{s}\big{|}_{Y}=$ constant for $s\gg 0$.

To state the result in case (a) we will use the diagram

(I.6) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Alb}(X)}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

where $\widetilde{{\rm Alb}(X)}$ is the universal cover ${\rm Alb}(X)$,
together with similar diagrams (I.6s) for the higher Albanese mappings.

######  Theorem A:

Assume that $\pi_{1}(X)$ is residually nilpotent and that $\alpha$ is proper.

  1. (i)

The mapping $\widetilde{\alpha}$ is proper;

  2. (ii)

the image $\widetilde{\alpha}(\widetilde{X})$ is Stein;

  3. (iii)

the connected components of the fibres of $\widetilde{\alpha}$ are the
connected components of $\pi^{-1}(Y)$ where $Y\subset X$ is a fibre of
$\alpha$;

  4. (iv)

the connected components of the fibres of $\alpha$ are characterized by the
condition that the map

$H^{1}(X,{\mathbb{Q}})\to H^{1}(Y,{\mathbb{Q}})$

be trivial.

The substantive parts of this theorem are (ii), (iii) and (iv). We note that
(iii) and (iv) imply

(I.7) $\pi_{1}(Y)\to\pi_{1}(X)\text{ is finite }\iff H^{1}(X,{\mathbb{Q}})\to
H^{1}(Y,{\mathbb{Q}})\text{ is trivial.}$

The implication $\Rightarrow$ is clear. The proof of the converse will use
Hodge theory. One argument will be an extension of that used in [Ka97]. A
second argument, one that is more in the conceptual framework of general
period mappings, will use that the implication $\Leftarrow$ is a consequence
of Theorem I.5. A third argument, different from the first two, will describe
a construction of the minimal model of $\widehat{\pi_{1}}(X)$ as in [Mo78] and
[Ha87a]. For use elsewhere this construction will give a bound on the
singularities along $Z$ of the differential forms in the minimal model. We
will also give an extension to the quasi-projective case of the classical
$\partial\overline{\partial}$-lemma.

For case (b) we shall first give a result where we have a standard variation
of Hodge structure (VHS) whose underlying local system $\mathbb{V}\to X$ has a
monodromy representation

$\rho:\pi_{1}(X)\to{\Gamma}\subset\mathop{\rm Aut}\nolimits(V)$

with a finite kernel ([CM-SP17]). We denote by

(I.8) $\Phi:X\to{\Gamma}\backslash D$

the corresponding period mapping (loc. cit.). The properness of $\Phi$ is
equivalent to the logarithm $N_{i}$ of monodromy around each $Z_{i}$ being
non-zero. Again this condition is independent of the smooth completion of $X$.
In this case it is well known that the image

$\Phi(X):=P\subset{\Gamma}\backslash D$

is a closed analytic subvariety of ${\Gamma}\backslash D$.111It is in fact a
quasi-projective algebraic subvariety ([BBT06]). We then have a diagram

(I.9) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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D}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

where $\widetilde{P}$ is the image in $D$ of $\widetilde{X}$.

######  Theorem B:

Assume that $\Phi$ is proper and that $\rho$ has a finite kernel.

  1. (i)

$\widetilde{\Phi}$ is a proper holomorphic mapping;

  2. (ii)

$\widetilde{P}$ is a Stein subvariety of $D$;

  3. (iii)

the connected fibres of $\widetilde{\Phi}$ are the connected components of
$\pi^{-1}(Y)$ where $Y\subset X$ is a fibre of $\Phi$;

  4. (iv)

these $Y$ are characterized by the condition that $\rho\big{|}_{\pi_{1}(Y)}$
is finite.

Given our assumptions the main substantive statement in the theorem is (ii).
As will be seen in Section III this will be a direct consequence of classical
results about the geometry of period domains.

Theorem B may be summarized by saying that $\Phi$ is a Shafarevich mapping for
$X$. What is missing is a cohomological, rather than a homotopy-theoretic,
description of the fibres of $\widetilde{\Phi}$. One would like some analogue
of (iv) in Theorem A. This will be further discussed in Section III.

Our final main result is an amalgam of Theorems A and B. To state it we assume
given an admissible variation of mixed Hodge structure (VMHS) with the local
system $V_{M}\to X$ having corresponding monodromy representation
$\rho_{M}:\pi_{1}(X)\to{\Gamma}\subset\mathop{\rm Aut}\nolimits(V_{M})$ (cf.
[SZ85] and [Us83]). We denote by

(I.10) $\Phi_{M}:X\to{\Gamma}_{M}\backslash D_{M}$

the corresponding period mapping where $D_{M}$ is now a mixed period domain
(loc. cit.).

######  Theorem C:

Assume that $\Phi_{M}$ is proper and that $\rho_{M}$ has a finite kernel. Then
$\Phi_{M}$ is a Shafarevich map for $X$.

The obvious idea behind the proof is to consider the factorization

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{M}\quad}$$\scriptstyle{\Phi}$$\textstyle{{\Gamma}_{M}\backslash
D_{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Gamma}\backslash
D}$

where $\Phi$ is the period mapping obtained by passing to the associated
graded of the mixed Hodge structures parametrized by $D_{M}$. For this period
mapping we have a diagram (I.9) where the image
$\widetilde{\Phi}(\widetilde{X})\subset D$ will be a Stein analytic variety.
Note that we are not saying that $\widetilde{\Phi}$ is proper.

Along the fibres of $\Phi$ we will have an admissible VMHS such that the
associated graded polarized Hodge structures are locally constant on the
irreducible components of a fibre $Y$.222This is the situation that one
encounters in the Satake-Baily-Borel completion of a classical period mapping
([GGLR20], [GGR22] and [Gr22]). In these references a more general version of
Theorem 1.5 appears; also given there are further details of some arguments
that are only sketched here. Then a strengthening of (I.5) will give that the
image $\Phi_{M}(Y)$ of the fibre will map finitely to the image under
$\Phi_{M}$ of the extension data of levels $\leqq 2$ (this condition will be
explained below). The proof of Theorem A can be adapted to this situation to
show $\widetilde{\Phi}(\pi^{-1}(Y))$ is Stein. We will then have an analytic
variety mapping to a Stein analytic variety with Stein fibres. In the
situation at hand the total space can then be shown to be a Stein variety (cf.
[St56]).

Following the proof in Section III of Theorems B and C we will discuss the
question of describing cohomologically the fibres of the Shafarevich mapping
in case B. This leads naturally into the description of the mixed Hodge
structure on the completion relative to $\rho$ of $\pi_{1}(X,x)$ ([Ha98] and
[EKPR12]). The cohomology groups that enter here have one interpretation as
defining the extension classes of that mixed Hodge structure, and among these
are the classes of first order deformations of the local system $\mathbb{V}\to
X$ underlying a VHS. Both of these are central ingredients in the proof of the
main theorem in [EKPR12]. In [Le19] and [Le21] these results have been
extended to the quasi-projective case. Here for expository purposes we will
discuss very briefly special cases of the results in these references.

In Section IV we will discuss some questions and conjectures related to
different perspectives on the Shafarevich conjecture.

### Acknowledgements

We are very grateful to P. Eyssidieux, T. Pantev and C. Simpson for useful
discussions.

The third author was partially supported by NSF Grant, Simons Investigator
Award HMS, Simons. Collaboration Award HMS, National Science Fund of Bulgaria,
National Scientific Program “Excellent Research and People for the Development
of European Science” (VIHREN), Project No. KP-06-DV-7, HSE University Basic
Research Program.

##  II. The nilpotent case

Referring to the diagram (I.6) and statement of Theorem A there are two things
to be proved:

(II.1) if $Y\subset X$ is a connected fibre of $\alpha$, then the map
$\pi_{1}(Y)\to\pi_{1}(X)$ is finite; (II.2) there is a plurisubharmonic (psh)
exhaustion function $\varphi:\widetilde{\alpha}(\widetilde{X})\to\mathbb{R}$.

For (II.1) we will give two different arguments. The first is basically to
extend the proof in [Ka97] to the quasi-projective case (cf. also [EKPR12] and
[Cl08]). The steps are

  1. $\bullet$

reduce to the case where $Y\subset X$ has dimension 1;

  2. $\bullet$

further reduce to the case where $Y$ has nodes.

These steps are essentially the same as in [Ka97] and that are reviewed in
Section 3 of [EKPR12].

  1. $\bullet$

analyze the case when $Y$ is a cycle; i.e., the dual graph of $Y$ is
topologically an $S^{1}$.

In this case what (II.1) means is that the inverse image
$\pi^{-1}(Y)\subset\widetilde{X}$ is not an infinite chain of irreducible
curves. Equivalently, some multiple of the circuit ${\gamma}$ maps to the
identity in $\pi_{1}(X)$. The general case of a nodal curve can be done by
extending the argument in this case.

Our assumption is that the map $H_{1}(Y,{\mathbb{Q}})\to
H_{1}(X,{\mathbb{Q}})$ is trivial. We want to use this statement about
homology to infer one about homotopy. For this we shall use that the unipotent
completions

$\widehat{\mathbb{Z}\pi_{1}}(X;x)=\lim_{s\to\infty}\pi_{1}(X,x)/J^{s+1}$

of $\pi_{1}(X,x)$ and similarly for $Y$ have mixed Hodge structures ([Mo78]
and [Ha87b]). Here $\mathbb{Z}{\pi_{1}}(X,x)$ is the group ring of
$\pi_{1}(X,x)$ and $J$ is the augmentation ideal. The weights of the
generators coming from $H_{1}(X)$ and $H_{1}(Y)$ are

  1. $\bullet$

$\widehat{\pi_{1}(X)}$ is generated in weights $-1,-2$ (cf. [Mo78]);

  2. $\bullet$

$\widehat{\pi_{1}(Y)}$ is generated in weights $-1,0$ (cf. [CG14]).

If $Y=\cup Y_{\alpha}$, the -1 part comes from the
$H_{1}(Y_{\alpha},{\mathbb{Q}})$. The $0$-part, which is only well defined
modulo the $-1$ parts, corresponds to the circuit ${\gamma}$. This may be seen
by considering the exact sequence of the pair $(Y,D)$

$H_{1}(Y)\to H_{1}(Y,D)\xrightarrow{\partial}H_{0}(D)$

where $D$ is the set of nodes. Here ${\gamma}$ should be considered as a class
in $\ker\partial$ modulo the images of the $H_{1}(Y_{\alpha})$. From this
weight considerations give that the class of ${\gamma}$ maps to zero in
$\widehat{\pi_{1}(X)}$. Since $\pi_{1}(X)$ is assumed to be residually
nilpotent we may conclude that ${\gamma}$ is of finite order in $\pi_{1}(X)$.
∎

We will now give the second argument for the proof of (II.1). This argument is
more conceptual. It takes place within the general framework of period
mappings and will be used again in the proof of Theorem C. We begin by
recalling from (I.4b) the higher Albanese mappings constructed in [Ha87a] and
[HZ87]. For each $s\geqq 1$ there is a diagram of mappings

(II.3) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.53471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 31.53471pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm
Alb}^{s}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.18752pt\raise-38.57915pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.07639pt\hbox{$\scriptstyle{\pi_{s}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
52.18752pt\raise-66.65831pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
24.50139pt\raise-24.30833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.07639pt\hbox{$\scriptstyle{\alpha_{s}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 44.78078pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.99722pt\raise-62.45692pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.93459pt\raise-66.65831pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
49.18752pt\raise-38.74165pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-77.15831pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.84721pt\raise-77.15831pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Alb}(X)}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

where ${\rm Alb}(X)={\rm Alb}^{1}(X)$ and $\alpha=\alpha_{1}$ is the usual
Albanese mapping. These maps may be viewed as period mappings associated to
unipotent variations of mixed Hodge structure whose underlying local systems
are the groups

$\mathbb{V}^{s}_{x}=\mathbb{Z}\pi_{1}(X,x)/J^{s+1}.$

A basic property of the induced mappings in (II.3) is given by Theorem I.5
above. Here we only need the following part of that result.

######  Theorem II.4:

For $Y\subset X$ a compact subvariety and $s\gg 0$

$\alpha\big{|}_{Y}=\text{constant }\iff\alpha_{s}\big{|}_{Y}=\text{constant}.$

The proof of Theorem II.4 will be a consequence of a general result about the
variation of the extension data in a VMHS that we will now explain and give a
sketch of the proof.

Let

(II.5) $\Phi:X\to{\Gamma}\backslash D$

be any admissible variation of mixed Hodge structures whose associated graded
is a constant family $\\{H^{0},\dots,H^{n}\\}$ of weight $k$ Hodge structures
$H^{k}$ on a fixed vector space $\mathop{\rm Gr}\nolimits^{W}_{k}V$. What is
then varying is the extension data for the mixed Hodge structures. This
extension data comes in various levels (cf. II.8 below), and associated to
(II.5) there are maps

(II.6a) $\displaystyle X$
$\displaystyle\xrightarrow{\Phi_{1}}\\{\text{extension data of level 1}\\},$
(II.6b) $\displaystyle\\{\text{fibre of }\Phi_{1}\\}$
$\displaystyle\xrightarrow{\Phi_{1,2}}\\{\text{extension data of level 2}\\},$
(II.6c) $\displaystyle\\{\text{fibre of }\Phi_{2}\\}$
$\displaystyle\xrightarrow{\Phi_{2,3}}\\{\text{extension data of level 3}\\}.$
$\displaystyle\qquad\qquad\vdots$

The right-hand sides of the above maps are isomorphic to

level 1 $\displaystyle\quad\mathop{\oplus}\limits^{\ell}\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(H^{\ell+1},H^{\ell}),$ level 2
$\displaystyle\quad\mathop{\oplus}\limits^{\ell}\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(H^{\ell+2},H^{\ell}),$ level 3
$\displaystyle\quad\mathop{\oplus}\limits^{\ell}\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(H^{\ell+3},H^{\ell})$
$\displaystyle\qquad\qquad\vdots$

This means that if we pick a point in a fibre, then the difference of the
extension data for a variable point and the fixed point is in the
$\oplus\mathop{\rm Ext}\nolimits^{1}_{{\mathrm{MHS}}}(\bullet,\bullet)$’s.

In more detail we are here using the general fact that if we have a MHS $M$
with graded pieces $A,B,C$ that are pure Hodge structures, then there is a
fibration

$\begin{matrix}\text{Extension data for $M$}\\\ \downarrow\\\ \mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(B,A)\times\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(C,B)\end{matrix}$

whose fibres have connected components isomorphic to $\mathop{\rm
Ext}\nolimits_{{\mathrm{MHS}}}^{1}(C,A)$. To see this we have

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Ker}\nolimits(M\to
C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and a master diagram

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Ker}\nolimits(M\to
C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M/A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0.}$

From [Car80] the extension classes are

$e_{BC}\in\frac{\mathop{\rm Hom}\nolimits_{\mathbb{C}}(C,B)}{F^{0}\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(C,B)+\mathop{\rm
Hom}\nolimits_{\mathbb{Z}}(C,B)},\quad e_{AB}\in\frac{\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(B,A)}{F^{0}\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(B,A)+\mathop{\rm Hom}\nolimits_{\mathbb{Z}}(B,A)}.$

We also have the extension classes

$e_{1}\in\frac{\mathop{\rm Hom}\nolimits_{\mathbb{C}}(C,\mathop{\rm
Ker}\nolimits(M\to C))}{F^{0}+\mathop{\rm Hom}\nolimits_{\mathbb{Z}}},\quad
e_{2}\in\frac{\mathop{\rm Hom}\nolimits_{\mathbb{C}}(M/A,A)}{F^{0}+\mathop{\rm
Hom}\nolimits_{\mathbb{Z}}}$

with

$e_{1}\to e_{BC},\quad e_{2}\to e_{AB}$

induced respectively by

$\mathop{\rm Hom}\nolimits(C,\mathop{\rm Ker}\nolimits(M\to C))\to\mathop{\rm
Hom}\nolimits(C,B),\quad\mathop{\rm Hom}\nolimits(M/A,A)\to\mathop{\rm
Hom}\nolimits(B,A).$

Knowing $e_{AB}$ and $e_{1}$ or $e_{BC}$ and $e_{2}$ determines $M$ as a MHS.
For example,

  1. $\bullet$

$e_{AB}$ determines $\mathop{\rm Ker}\nolimits(M\to C)$ and

  2. $\bullet$

$e_{1}\in\\{u\in\mathop{\rm Hom}\nolimits(C,\mathop{\rm Ker}\nolimits(M\to
C))/F^{0}+H_{\mathbb{Z}}:u\to e_{BC}$ under $\ker(M\to A)\to B\\}$.

Thus if we know $e_{AB}$ and $e_{1}$, then we know the $u$ in the second
bullet and from this we know $M$.

If $e_{BC}=0$, then the second bullet reduces to

$\frac{\mathop{\rm Hom}\nolimits_{\mathbb{C}}(C,A)}{F^{0}\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(C,A)+\mathop{\rm Hom}\nolimits_{\mathbb{Z}}(C,A)}.$

In general we obtain a fibre space.

We note that even though the action of monodromy on the associated graded of
the MHS’s is trivial, its action on extension data of various levels is
abelian and will generally not be trivial.

The key observation is the use of horizontality (transversality) to prove the

######  Proposition II.7:

For $k\geqq 2$ the differentials of the maps $\Phi_{k,k+1}$ are zero.

A proof of this result is given in [GGR22] (cf. also [Gr22]). The following is
a sketch of the argument. There is a sequence of period mappings

(II.8) $\Phi_{k}:X\to{\Gamma}_{k}\backslash D_{k}$

to the extension data of levels $\leqq k$ in (II.5). Denoting by $X_{k}\subset
X$ a typical fibre of $\Phi_{k-1}$, we have

$\Phi_{k-1,k}:X_{k}\to E_{k}$

where

$E_{k}\cong\mathop{\oplus}\limits^{\ell}\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(H^{k+\ell},H^{\ell})$

with

$\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(H^{k+\ell},H^{\ell})=\frac{\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(H^{k+\ell},H^{\ell})}{F^{0}\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(H^{k+\ell},H^{\ell})+\mathop{\rm
Hom}\nolimits_{\mathbb{Z}}(H^{k+\ell},H^{\ell})}.$

By horizontality the differential of $\Phi_{k-1,k}$ gives a map

$TX_{k}\to H_{k}\subset TE_{k}$

where the following diagram depicts the complex tangent spaces $TE_{k}$ with
the subspaces $H_{k}$ being the part over the red:

$\displaystyle k=1$
$\displaystyle\quad\underbrace{(\ell-1,-\ell)\oplus\cdots\oplus(0,-1)}_{F^{0}}\oplus\underbrace{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underbrace{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1,0)}}}\oplus\cdots\oplus(-\ell,\ell-1)}_{TE_{1}}$
$\displaystyle k=2$
$\displaystyle\quad\underbrace{(\ell-2,-\ell)\oplus\cdots\oplus(0,-2)}_{F^{0}}\oplus\underbrace{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underbrace{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1,-1)}}}\oplus(-2,0)\oplus\cdots\oplus(-\ell,\ell-2)}_{TE_{2}}$
$\displaystyle k=3$
$\displaystyle\quad\underbrace{(\ell-3,-\ell)\oplus\cdots\oplus(0,-3)}_{F^{0}}\oplus\underbrace{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underbrace{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1,-2)}}}\oplus(-2,-1)\oplus\cdots\oplus(-\ell,\ell-3)}_{TE_{3}}.$

Denoting any of the $E_{k}$ by $E$, as a mapping of real manifolds we have

$T_{\mathbb{R}}E\cong\mathbb{R}^{2m}$

and the differential of $\Phi_{k-1,k}$ maps to a subspace
$H_{\mathbb{R}}\subset\mathbb{R}^{2m}$ that is invariant under the complex
structure $J:\mathbb{R}^{2m}\to\mathbb{R}^{2m}$. On the complexification of
$\mathbb{R}^{2m}$ given by the action of $J$ the complexification of
$H_{\mathbb{R}}$ is $H\oplus\overline{H}$ where $H$ is the term over the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underbrace{\quad}}$
above. Denoting by ${\Lambda}$ the discrete subgroup of $H$ induced by the
$\mathop{\rm Hom}\nolimits_{\mathbb{Z}}(H^{k+\ell},H^{\ell})$ terms above we
see that

  1. $\bullet$

$H/{\Lambda}$ is a compact complex torus for $k=1$;

  2. $\bullet$

$H/{\Lambda}\cong({\mathbb{C}}^{\ast})^{m}$ for $k=2$;333In this case we have
a fibration over a compact, complex torus whose connected components of the
fibres are isomorphic to $({\mathbb{C}}^{\ast})^{m}$’s. This does _not_ mean
that the total space is a semi-abelian variety.

  3. $\bullet$

$H\cong{\mathbb{C}}^{n}$ for $k\geqq 3$.

The maps (II.6) correspond to maps arising from a diagram

$\textstyle{\pi_{1}(X_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda}$$\textstyle{H_{1}(X_{k},\mathbb{Z}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Using a weight argument this implies that $TX_{k}$ maps to a sub-Hodge
structure of $TE_{k}$ whose complex part lies under the red. It also means
that for a desingularization $X^{\prime}_{k}$ of $X_{k}$ and smooth completion
$\overline{X}^{\prime}_{k}$ with $\overline{X}^{\prime}_{k}\backslash
X^{\prime}_{k}=Z^{\prime}_{k}$ a normal crossing divisor

  1. $\bullet$

the pullback by $\Phi_{k-1,k}$ of the dual $H^{\ast}$ of $H$ will give 1-forms
in $H^{0}({\Omega}^{1}_{\overline{X}^{\prime}_{k}}(\log Z^{\prime}_{k}))$, and

  2. $\bullet$

the mapping $\Phi_{k-1,k}$ will be given by integrating these logarithmic
1-forms taken modulo periods from $H_{1}(X^{\prime}_{k},\mathbb{Z})$.

For $k\geqq 3$ there are no periods and so $\Phi_{k-1,k}$ is locally constant.
∎

######  Corollary II.9:

In (II.5) the induced map

$\Phi(X)\to\Phi_{2}(X)$

is a finite morphism of algebraic varieties.

###### Proof.

It follows from (II.7) that the connected components of fibres of the above
map are points. An analysis of how these maps are defined (they are iterated
integrals of logarithmic 1-forms taken modulo periods) shows that both
$\Phi(X)$ and $\Phi_{2}(X)$ may be completed to algebraic varieties and that
$\Phi(X)\to\Phi_{2}(X)$ extends to a rational map on the completions (cf.
[GGR22] and [Gr22]). ∎

###### Proof of Theorem II.4.

The first basic observation is that the tower of Albanese mappings (II.3) is
_not_ the tower obtained by considering a higher Albanese mapping as a period
mapping and then taking extension data of level $\leqq k$ for each $k$. For
example, the first level of the Albanese tower is

${\rm Alb}(X)=H^{0}(X,{\Omega}^{1}_{X})^{\ast}/H_{1}(X,\mathbb{Z})$

and in general $H_{1}(X,\mathbb{Z})$ is a mixed Hodge structure with weights
$-2,-1$ and for which there is non-trivial extension data (cf. (II.15) below).

A second point is that the period mappings (II.3) are only defined upon choice
of a base point $x_{0}\in X$. Thus the relevant Hodge structures are a
${\mathbb{Q}}$ in weight zero for the point and $H^{1}(X,{\mathbb{Q}})$ for
the Albanese variety. The mapping $\alpha$ in (II.3) needs both of these (cf.
the discussion in 5.32 ff. in [SZ85]).

We now change notation and denote by (II.8) the level of extension data
mappings associated to UVMHS given by $\alpha_{s}:X\to{\rm Alb}^{s}(X)$ for
$s\gg 0$. Then $\Phi_{2}(X)$ contains the information in ${\rm
Alb}_{X,x_{0}}:\allowbreak X\to{\rm Alb}(X)$. The $D_{k}$ in the tower of
mappings (II.8) are obtained from

  1. (i)

the associated graded of the cohomology groups in $\widehat{\pi_{1}(X,x)}$;
and

  2. (ii)

the various levels of extension data among these groups, as explained above.

Since we are dealing with a _unipotent_ VMHS, the terms in (i) are constant.
The basic observation from Corollary II.9 is that the only varying extension
data in the period mapping (II.5) in this case is that arising from ${\rm
Alb}_{X,x}$. In terms of extension data arising from the UVMHS the information
in ${\rm Alb}_{X,x}$ is contained in the mapping $\Phi_{2}$ to extension data
of level $\leqq 2$. Thus

If $Y\subset X$ is a compact subvariety such that
$\alpha\big{|}_{Y}=\text{constant}$,
then $\alpha^{s}\big{|}_{Y}=\text{constant}$.

Since $\pi_{1}(X)$ is assumed to be residually nilpotent, this statement
implies Theorem II.4 and (I.7).∎

Proof of (II.2). Choosing a base point $x_{0}\in X$ and a basis
${\omega}_{\lambda}$ for $H^{0}({\Omega}^{1}_{\overline{X}}(\log Z))$ the
Albanese mapping of $X$ is given by

(II.10)
$\alpha(x)=\left(\ldots,\int^{x}_{x_{0}}{\omega}_{\lambda},\ldots\right)\text{
modulo periods}.$

The assumed completion of $X$ means that at every point $p$ of $Z$ there is an
${\omega}_{\lambda}$ having a logarithmic singularity at $p$. In more detail,
if in coordinates $Z$ is locally given by

$z_{i}=0,\qquad i\in I$

then there is an ${\omega}\in H^{0}({\Omega}^{1}_{\overline{X}}(\log Z))$ such
that

${\omega}=\sum_{i\in I}a_{i}\frac{dz_{i}}{z_{i}}+\text{ holomorphic terms}$

where all $a_{i}\neq 0$.

The pullback to $\widetilde{X}$ of ${\rm Alb}(X)$ is a trivial bundle
$\widetilde{X}\times{\mathbb{C}}^{N}$. Choosing coordinates $t_{\lambda}$ for
${\mathbb{C}}^{N}$ the pullback to $\widetilde{X}$ of ${\omega}_{\lambda}$ is
$dt_{\lambda}$. Then by the above

$\varphi=\sum|t_{\lambda}|^{2}$

is an exhaustion function on $\widetilde{X}$; the projection to $X$ of the
sets $\varphi<C$ stay a fixed distance from $Z$. Moreover, the Levi form
$\left(\frac{i}{2}\right)\partial\overline{\partial}\varphi$ is positive in
the Zariski tangent spaces to the analytic subvariety
$\widetilde{\alpha}(\widetilde{X})\subset{\mathbb{C}}^{N}$. ∎

######  II.11:

_An interpretation of the Albanese map as an extension class._

The following is intended to provide background for the discussion in the next
section of the question raised in the introduction of detecting
cohomologically the fibres of the Shafarevich mapping in the VHS case.

We have noted that the analytic group

${\rm Alb}(X)=H^{0}({\Omega}^{1}_{\overline{X}}(\log
Z))^{\ast}/H_{1}(X,\mathbb{Z})$

is a semi-abelian variety that fits in an exact sequence

(II.12) $0\to T\to{\rm Alb}(X)\to{\rm Alb}(\overline{X})\to 0$

where $T\cong({\mathbb{C}}^{\ast})^{m}$ is an algebraic torus. Completing
${\rm Alb}(X)$ by replacing the ${\mathbb{C}}^{\ast}$’s by $\mathbb{P}^{1}$’s
the Albanese mapping on $X$ extends to a rational mapping; the graph of
$\alpha$ in $X\times{\rm Alb}(X)$ closes up in
$\overline{X}\times\overline{{\rm Alb}(X)}$.

By way of an interlude we will first describe

######  II.13:

_The extension class of the semi-abelian variety ${\rm Alb}(X)$._

The exact sequence of connected abelian Lie groups (II.12) is constructed from
the exact cohomology sequence

(II.14) $0\to H^{1}(\overline{X},{\mathbb{Q}})\to H^{1}(X,{\mathbb{Q}})\to
H^{1}(T,{\mathbb{Q}})\to 0.$

The term in the middle has a mixed Hodge structure with weight filtration
$W_{1}\subset W_{2}$ where

  1. $\bullet$

$\mathop{\rm
Gr}\nolimits^{W}_{1}(H^{1}(X,{\mathbb{Q}}))=H^{1}(\overline{X},{\mathbb{Q}})$;

  2. $\bullet$

$\mathop{\rm Gr}\nolimits^{W}_{2}H^{1}(X,{\mathbb{Q}})\cong
H^{1}(T,{\mathbb{Q}})\cong\mathop{\oplus}\limits^{m}{\mathbb{Q}}(-1)$.

The extension class of (II.14) is in

$\displaystyle\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(\mathop{\oplus}\limits^{m}{\mathbb{Q}}(-1),H^{1}(\overline{X},{\mathbb{Q}}))$
$\displaystyle\cong\mathop{\oplus}\limits^{m}\left\\{\frac{H^{1}(\overline{X},{\mathbb{C}})}{F^{1}H^{1}(\overline{X},{\mathbb{C}})+H^{1}(\overline{X},\mathbb{Z})}\right\\}$
$\displaystyle\cong\mathop{\oplus}\limits^{m}H^{1}(\overline{X},{\mathscr{O}}_{\overline{X}})/H^{1}(\overline{X},\mathbb{Z})$
$\displaystyle\cong\mathop{\oplus}\limits^{m}{\mathop{\rm
Pic}\nolimits}^{\circ}(\overline{X}).$

From this we may infer that the extension class of (II.14) gives the image in
${\mathop{\rm Pic}\nolimits}^{\circ}(\overline{X})$ of the kernel of the Gysin
mapping

$\oplus^{i}H^{0}({\mathscr{O}}_{Z_{i}})\xrightarrow{\mathrm{Gy}}H^{2}(X,\mathbb{Z}).$

Next we shall describe

######  II.15:

_The extension class of ${\rm Alb}(X,x_{0})$._

Thus far we have made no reference to the choice of a base point $x_{0}\in X$.
When we include this as in [Ha87b] we obtain a variation of mixed Hodge
structure with underlying local system that we denote by $\mathbb{V}_{x_{0}}$.
When $X=\overline{X}$ is projective then for $x\in X$ the fibre
$\mathbb{V}_{x_{0},x}\cong H^{1}(X,\\{x_{0},x\\};{\mathbb{Q}})$ is a MHS with
weight filtration $W_{0}\subset W_{1}$ where

  1. $\bullet$

$\mathop{\rm Gr}\nolimits^{W}_{0}\mathbb{V}_{x_{0},x}={\mathbb{Q}}$ (constant
local system);

  2. $\bullet$

$\mathop{\rm Gr}\nolimits^{W}_{1}\mathbb{V}_{x_{0},x}\cong
H^{1}(X,{\mathbb{Q}})$.

The extension class is in

$\displaystyle\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(H^{1}(X,{\mathbb{Q}}),{\mathbb{Q}})$
$\displaystyle\cong\frac{H^{1}(X,{\mathbb{C}})^{\ast}}{F^{1}H^{1}(X,{\mathbb{C}})^{\ast}+H^{1}(X,\mathbb{Z})^{\ast}}$
$\displaystyle\cong H^{0}({\Omega}^{1}_{X})^{\ast}/H_{1}(X,\mathbb{Z}).$

It may be identified with the linear function on $H^{0}(X,{\Omega}^{1}_{X})$
given by the usual Albanese mapping defined for ${\omega}\in
H^{0}({\Omega}^{1}_{X})$ by

(II.16) $\alpha_{x_{0}}(x)({\omega})=\int^{x}_{x_{0}}{\omega}\qquad\mathop{\rm
mod}\nolimits\text{periods}.$

As in [HZ87] this defines a canonical UVMHS associated to $(X,x_{0})$.

In the quasi-projective case we again have a VMHS with local system
$\mathbb{V}_{x_{0}}\to X$ where

(II.17) $0\to{\mathbb{Q}}\to\mathbb{V}_{x_{0}}\to H^{1}(X,{\mathbb{Q}})\to 0.$

This time $\mathbb{V}_{x_{0},x}\cong H^{1}(X,\\{x_{0},x\\};{\mathbb{Q}})$ is a
MHS with weight filtration $W_{0}\subset W_{1}\subset W_{2}$ with

  1. $\bullet$

$\mathop{\rm Gr}\nolimits^{W}_{0}\mathbb{V}_{x_{0},x}={\mathbb{Q}}$;

  2. $\bullet$

$W_{2}(\mathbb{V}_{x_{0},x})/W_{0}(\mathbb{V}_{x_{0},x})\cong
H^{1}(X,{\mathbb{Q}})$.

######  Proposition II.18:

The sequence (II.17) may be constructed by the same method as in (II.15) where
the extension class is given for ${\omega}\in
H^{0}({\Omega}^{1}_{\overline{X}}(\log Z))$ by the Albanese mapping (II.16).

What this means is that the prescription for constructing the Hodge filtration
$F^{1}_{x}$ on ${\mathbb{C}}\oplus H^{1}(X,{\mathbb{C}})$ may be done using
the same formulas as in the projective case above. It is when we put in the
variable point $x\in X$ that we get a varying $F^{p}_{x}$ in the VMHS.

It is clear that the $F^{1}_{x}$ defines a holomorphic sub-bundle of
$\mathbb{V}\otimes{\mathscr{O}}_{X}$. What must be verified is that
horizontality for $F^{1}$ and conjugate horizontality for $\overline{F}^{1}$
are satisfied. If
$e_{x}\in\frac{H_{1}(X,{\mathbb{C}})}{F^{0}H_{1}(X,{\mathbb{C}})+H_{1}(X,\mathbb{Z})}$
is the extension class, then the horizontality condition is $\nabla e\in
F^{-1}H_{1}(X,{\mathbb{C}})$ which is satisfied in this case. Conjugate
horizontality is done similarly. ∎

We have put this argument in because it will be used verbatim when we replace
${\mathbb{Q}}$ by $\mathbb{V}$ below.

Third proof of I.7 and the $\partial\overline{\partial}$-lemma.

From rational homotopy theory it is known that the unipotent completion
$\widehat{\pi_{1}}(X)\otimes{\mathbb{Q}}$ is determined by
$H_{1}(X,{\mathbb{Q}})$, $H_{2}(X,{\mathbb{Q}})$ and the map

(II.19) $H_{2}(X,{\mathbb{Q}})\to{\Lambda}^{2}H_{1}(X,{\mathbb{Q}})$

that is dual to the cup product

(II.20) ${\Lambda}^{2}H^{1}(X,{\mathbb{Q}})\to H^{2}(X,{\mathbb{Q}})$

(cf. [Mo78] and the references cited there). Denoting by
${\mathscr{L}}H_{1}(X,{\mathbb{Q}})$ the free Lie algebra generated by
$H_{1}(X,{\mathbb{Q}})$, there is a surjection

(II.21)
${\mathscr{L}}H_{1}(X,{\mathbb{Q}})\to{\mathscr{L}}(\widehat{\pi_{1}(X)}\otimes{\mathbb{Q}})$

where the right-hand side is the Lie algebra/${\mathbb{Q}}$ corresponding to
the completion of the group algebra of $\pi_{1}(X)$. We will use the
construction of the Sullivan minimal model to determine the ideal
${\mathscr{I}}$ in ${\mathscr{L}}H_{1}(X,{\mathbb{Q}})$ that gives the kernel
of the mapping (II.21). Both ${\mathscr{L}}H_{1}(X)\otimes{\mathbb{Q}}$ and
${\mathscr{I}}$ are graded by degree and we denote by $(\bullet)_{k}$ the part
up to degree $k$. The key points will be that in building ${\mathscr{I}}$
step-by-step with
${\mathscr{I}}_{2}\subset{\mathscr{L}}H_{1}(X,{\mathbb{Q}})_{2}$,
${\mathscr{I}}_{3}\subset{\mathscr{L}}H_{1}(X,{\mathbb{Q}})_{3},\dots$, at
each step we will have a morphism of mixed Hodge structures

${\mathscr{L}}H_{1}(X,{\mathbb{Q}})_{k}/{\mathscr{I}}_{k}\to
H^{2}(X,{\mathbb{Q}}),$

and then a weight argument will give that ${\mathscr{I}}$ is generated in
degrees $2,3,4$ ([Mo78]). When $X=\overline{X}$ is projective the filtrations
of ${\mathscr{L}}H_{1}(X,{\mathbb{Q}})$ by degree and weight conincide; this
is not the case when $X$ is only quasi-projective.

For
$\mathcal{G}_{\mathbb{Q}}:={\mathscr{L}}H_{1}(X,{\mathbb{Q}})/{\mathscr{I}}$
we set

$\mathcal{G}=\mathcal{G}_{\mathbb{Q}}\otimes{\mathbb{C}}.$

We next denote by $A^{\bullet}(\overline{X},\log Z)$ the $C^{\infty}$ log
complex generated by the smooth forms on $\overline{X}$ adjoined by the
$dz_{i}/z_{i}$’s along $z_{1}\cdots z_{k}=0$. From [GS75] or [Mo78] the
inclusion $A^{\bullet}(\overline{X},\log Z)\hookrightarrow A^{\ast}(X)$ is a
quasi-isomorphism so that we have

$H^{\ast}(X;{\mathbb{C}})\cong H^{\ast}(A^{\bullet}(\overline{X},\log Z)).$

Assuming that $\pi_{1}(X)$ is residually nilpotent and with terms to be
explained in the proof we have

######  Proposition II.22 ([Ha87b]):

There exists a form ${\omega}\in\mathcal{G}\otimes A^{1}(\overline{X},\log Z)$
that satisfies

  1. (i)

$d{\omega}+\frac{1}{2}[{\omega},{\omega}]=0$;

  2. (ii)

integrating ${\omega}$ defines the higher Albanese mapping;

  3. (iii)

the entries of ${\omega}$ give the minimal model of
$\widehat{\pi_{1}(X)}\otimes{\mathbb{C}}$;

  4. (iv)

${\omega}_{-k}$ has $(\log z)^{k}$ singularities along $Z$; and

  5. (v)

the ideal ${\mathscr{I}}\subset{\mathscr{L}}H_{1}(X)$ is generated in degrees
$(2,3,4)$.

######  Lemma II.23 ($\partial\overline{\partial}$ Lemma):

Let $\eta\in A^{2}(\overline{X},\log Z)$ be $\partial,\overline{\partial}$
closed and with cohomology class $[\eta]=0$ in $H^{2}(X,{\mathbb{C}})$. Then
there exists a function $f\in A^{0}(X)$ that satisfies

$\partial\overline{\partial}f=\eta.$

We may choose this function to have logarithmic singularities along $Z$. Any
such function is then unique up to $\mathop{\rm
Gr}\nolimits^{W}_{2}H^{1}(X,{\mathbb{C}})$.

The proof of the lemma will be given following that of Proposition II.22.
Before giving that argument we will make some preliminary observations. The
first is that we shall use the standard identifications

(II.24)
$\begin{cases}{\rm(a)}&H^{1}(X,{\mathbb{C}})\cong\mathbb{H}^{1}(\Omega^{\bullet}_{X}(\log
Z))\cong\overbrace{H^{0}({\Omega}^{1}_{\overline{X}}(\log Z))}^{F^{1}}\oplus
H^{1}(\overline{X},{\mathscr{O}}_{\overline{X}})\\\\[6.0pt]
{\rm(b)}&H^{2}(X,{\mathbb{C}})\cong\mathbb{H}^{2}(\Omega^{\bullet}_{\overline{X}}(\log
Z))\cong\overbrace{\overbrace{H^{0}({\Omega}^{2}_{\widetilde{X}}(\log
Z))}^{F^{2}}\oplus H^{1}({\Omega}^{1}_{\overline{X}}(\log Z))}^{F^{1}}\oplus
H^{2}({\mathscr{O}}_{\overline{X}}).\end{cases}$

In both cases

$H^{q}({\mathscr{O}}_{\overline{X}})\cong\overline{H^{0}({\Omega}^{q}_{\overline{X}})}.$

A result of Deligne is that

$H^{0}({\Omega}^{q}_{\overline{X}}(\log Z))\text{ is a direct summand of
}H^{1}(\overline{X},{\mathbb{C}});$

i.e., forms in $H^{0}({\Omega}^{q}_{\overline{X}}(\log Z))$ are closed and
never exact. A proof of this using $A^{\bullet}(\overline{X},\log Z)$ is in
[GS75].

The Lie algebra $\mathcal{G}$ is nilpotent and may be realized as a Lie
algebra of lower triangular matrices. We then have

${\omega}={\omega}_{-1}+{\omega}_{-2}+{\omega}_{-3}+\cdots$

where ${\omega}_{-k}$ are $1$-forms that on a diagonal that is $k$-steps below
the principal diagonal. The equation (i) in Proposition II.22 is then a
sequence of equations

$d{\omega}_{-k}=-\left(\frac{1}{2}\right)\sum_{i+j=k}{\omega}_{-i}\wedge{\omega}_{-j},\quad
k\geqq 0\;$

We will argue that since

  1. (i)

$\widehat{\pi_{1}}(X)\oplus{\mathbb{Q}}$ is determined by
$H_{2}(X,{\mathbb{Q}})\to\wedge^{2}H_{1}(X,{\mathbb{Q}})$;

  2. (ii)

the minimal model is isomorphic to the dual of
$\widehat{\pi_{1}}(X)\otimes{\mathbb{Q}}$; and

  3. (iii)

constructing the minimal model entails solving equations $\alpha=d\beta$

we will be able to inductively solve the equations (II.25k). Using the mixed
Hodge structures on $H^{1}(X,{\mathbb{Q}})$ and
$\ker\\{\wedge^{2}H^{1}(X,{\mathbb{Q}})\to H^{2}(X,{\mathbb{Q}})\\}$, from a
weight argument we will see that the only “new” equations (II.25k) arise for
$k=2,3,4$ ([Mo78]).

###### Proof of Proposition II.22.

We take the entries of ${\omega}_{-1}$ to be bases $\varphi_{i}\in
H^{0}({\Omega}^{1}_{\overline{X}}(\log Z))$,
$\overline{\psi}_{\alpha}\in\overline{H^{0}({\Omega}^{1}_{\overline{X}})}$.
The (II.25k) is

$d{\omega}_{-2}=-\left(\frac{1}{2}\right)\left\\{\sum
a_{ij}\varphi_{i}\wedge\varphi_{j}+\sum
b_{i\alpha}\varphi_{i}\wedge\overline{\psi}_{\alpha}+\sum
c_{\alpha\beta}\overline{\psi}_{\alpha}\wedge\overline{\psi}_{\beta}\right\\}.$

The entries of ${\omega}_{-2}$ will correspond to a basis of the kernel of
(II.20)$\otimes{\mathbb{C}}$. From (II.24b) the first and third entries in the
right-hand side of (II.252) are zero as differential forms. This leaves the
middle term and by (II.23) we may take

${\omega}_{-2}=-\overline{\partial}f_{-2}\;$

and we have defined the part
$({\mathscr{L}}H_{1}(X)_{2}/{\mathscr{I}}_{2})^{\ast}$ of the minimal model up
through degree 2.

For the next step (II.253) is

$d{\omega}_{-3}=-[{\omega}_{-1},{\omega}_{-2}].$

We think of the right-hand side as giving a map

$({\mathscr{L}}H_{1}(X))_{3}/{\mathscr{I}}_{2}\cdot H_{1}(X)\to H^{2}(X).$

Since $\pi_{1}\widehat{(X)}\otimes{\mathbb{Q}}$ is accounted for by the kernel
of (II.21), this map must be zero which means we can solve for
${\omega}_{-3}$. Using Lemma II.23 we may take

${\omega}_{-3}=\overline{\partial}f_{-3}.$

This process continues inductively giving us (i) and (iii).

The proof of (ii) follows from the construction of the higher Albanese
varieties in [Ha87a].

The proof of (iv) will follow from the proof of Lemma II.23.

Finally, for (v) the maps

${\mathscr{L}}H_{1}(X)_{k}/{\mathscr{I}}_{k}\to H^{2}(X)$

that are used to inductively to define ${\omega}_{-k}$ are morphisms of MHS’s.
In (II.25k) the weights of ${\omega}_{-k}$ are the weights on the right-hand
side. In (II.25k) the correspondence on weights between the left- and right-
hand sides is

$\displaystyle k=2\qquad$ $\displaystyle 2,3,4$
$\displaystyle\longleftrightarrow 2,3,4$ $\displaystyle k=3\qquad$
$\displaystyle 3,4,5,6$ $\displaystyle\longleftrightarrow 2,3,4$
$\displaystyle k=4\qquad$ $\displaystyle 4,5,6,7,8$
$\displaystyle\longleftrightarrow 2,3,4$ $\displaystyle k=5\qquad$
$\displaystyle 5,6,7,8,9,10$ $\displaystyle\longleftrightarrow 2,3,4.$

It follows that these maps are zero for $k\geqq 5$. Thus for $k\geqq 5$

${\mathscr{I}}_{k}={\mathscr{I}}_{5}\\{\mathop{\otimes}\limits^{k-5}H_{1}(X)\\}.\qed$

###### Proof of Lemma II.23.

We will use (II.24)(b). Writing $\eta=\eta^{2,0}+\eta^{1,1}+\eta^{0,2}$ we
have $\overline{\partial}\eta^{2,0}=0$. Thus

$\eta^{2,0}\in H^{0}\left({\Omega}^{2}_{\overline{X}}(\log
Z)\right).\vspace*{-3pt}$

Since the cohomology class $[\eta]=0$ in $H^{2}(X,{\mathbb{C}})$, it follows
from the result of Deligne that the differential form $\eta^{2,0}=0$.
Similarly, $\eta^{0,2}=0$ and

$\eta=\eta^{1,1}\in A^{1,1}(\overline{X},\log Z)\vspace*{-3pt}$

is $\partial,\overline{\partial}$ closed and with cohomology class

$[\eta]=0\enspace\hbox{in}\enspace H^{1}\left({\Omega}^{1}_{\overline{X}}(\log
Z)\right).\vspace*{-3pt}$

From the exact sequence

$\oplus H^{0}(Z_{i},{\mathbb{C}})(-1)\xrightarrow{\rm
Gy}H^{2}(\overline{X},{\mathbb{C}})\to H^{2}(X,{\mathbb{C}})$

since $[\eta]$ maps to zero in $H^{2}(X,{\mathbb{C}})$, it follows that in
$H^{2}(\overline{X},{\mathbb{C}})$

$[\eta]=\sum c_{i}\mathop{\rm Gy}\nolimits(1_{Z_{i}}),\quad
c_{i}\in{\mathbb{C}}.$

It is well known, and a proof will be recalled below, that there is $g_{i}\in
A^{0}(X)$ such that $\partial\overline{\partial}g_{i}$ gives a de Rham
representative of $\mathop{\rm Gy}\nolimits(1_{Z_{i}})$. Then

$\alpha:=\eta-\sum c_{i}\partial\overline{\partial}g_{i}\enspace\hbox{is
a}\enspace\partial,\overline{\partial}\enspace\hbox{closed (1,1) form whose
cohomology class $[\alpha]=0$ in}\enspace H^{2}(\overline{X},{\mathbb{C}}).$

A standard Kähler fact is that then $\alpha=\partial\overline{\partial}h$ for
some $h\in A^{0}(\overline{X})$. This follows from the result that the
inclusion

(II.26) $A^{\bullet}(\overline{X})\cap\ker\partial\hookrightarrow
A^{\bullet}(\overline{X})$

is a quasi-isomorphism (Section 5 in [DGMS75]).

For the statement about $\mathop{\rm Gy}\nolimits(1_{Z_{i}})$, in the line
bundle $L_{i}:=[Z_{i}]$ we choose a metric and a section $\sigma_{i}\in
H^{0}(\overline{X},L_{i})$ with divisor $(\sigma_{i})=Z_{i}$. If $\rho_{i}$ is
a $C^{\infty}$ bump function with $\rho_{i}\big{|}_{Z_{i}}=1$ and that is
compactly supported in a neighborhood of $Z_{i}$, then

$\left(\frac{\sqrt{-1}}{2\pi}\right)\partial\overline{\partial}\log\|\rho_{i}\sigma_{i}\|^{2}$

represents $\mathop{\rm Gy}\nolimits(1_{Z_{i}})$ in $H^{2}(X,{\mathbb{C}})$. ∎

The statement about the singularities of $f$ is then clear.

If we have $f$ with $\partial\overline{\partial}f=0$, then $\partial f\in
A^{1}(\overline{X},\log Z)$ gives a cohomology class in $H^{1}(X)$ such that
$\mathop{\rm Res}\nolimits[\partial
f]\in\ker\\{\mathop{\oplus}\limits^{i}H^{0}(C_{Z_{i}})(-1)\to
H^{2}(\overline{X})\\}\cong\mathop{\rm
Gr}\nolimits^{W}_{2}H^{1}(X,{\mathbb{C}})$, which implies the uniqueness
statement.

Finally to prove Theorem A as in the first argument we may reduce to showing
that for a nodal curve $Y=\cup Y_{\alpha}$

$H^{1}(X,{\mathbb{Q}})\to H^{1}(Y,{\mathbb{Q}})\enspace\hbox{is
trivial}\enspace\implies\widehat{\pi_{1}}(Y)\to\widehat{\pi_{1}}(X)\enspace\hbox{is
trivial.}\enspace$

The assumption gives that all

${\omega}_{-1}\big{|}_{Y_{\alpha}}=0.$

From (II.23k) it follows that

$d{\omega}_{-2}\big{|}_{Y_{\alpha}}=0.$

From Lemma II.23 we have ${\omega}_{-2}=df_{-2}$ where
$\overline{\partial}f_{-2}=0$. Then

$\partial\overline{\partial}f_{-2}\big{|}_{Y_{\alpha}}=0\implies
f_{-2}\big{|}_{Y_{\alpha}}=\text{constant}\implies{\omega}_{-2}\big{|}_{Y_{\alpha}}=0.$

Proceeding inductively on $k$ we obtain

${\omega}\big{|}_{Y_{\alpha}}=0,$

which implies the result that all maps
$\widehat{\pi_{1}}(Y_{2})\to\widehat{\pi_{1}}(X)$ are trivial. From [Ha87b]
the higher Albanese mappings

$X\to{\rm Alb}^{s}(X)$

contract all $Y_{\alpha}$ and hence send $Y$ to a point. ∎

######  Remark:

If $\widetilde{{\omega}}$ is the pullback of ${\omega}$ to the universal cover
$\widetilde{X}\xrightarrow{\pi}X$, then it is well known that there is a
linear complex Lie group $G$ with Lie algebra $\mathcal{G}$ and a mapping

(II.27) $\widetilde{f}:\widetilde{X}\to G$

with

$\widetilde{f}^{\ast}(g^{-1}dg)=\widetilde{{\omega}}.$

If ${\Gamma}\subset G$ is the monodromy group of the flat connection
${\omega}$, then (II.27) induces a map

$f:X\to{\Gamma}\backslash G.$

From [Ha87b] there is a subgroup $F^{0}G$ such that the induced map

$X\to{\Gamma}\backslash G/F^{0}G$

is the Albanese map for $s\gg 0$. This gives another argument that

${\omega}\big{|}_{Y_{\alpha}}=0\implies\alpha_{s}(Y)=\text{point}.$

Finally one may ask about the singularities of the locally defined matrix-
valued function $g$ along $Z$. Consider the first non-trivial case where
locally on $X$

$\displaystyle g$ $\displaystyle=\begin{pmatrix}1&0&0\\\ a_{1}&1&0\\\
b&a_{2}&1\end{pmatrix}$ $\displaystyle{\omega}$
$\displaystyle=g^{-1}dg={\omega}_{-1}\oplus{\omega}_{-2}.$

From our construction, ${\omega}_{-1}$ has $dz_{i}/z_{i}$ and
$d\bar{z}_{i}/\bar{z}_{i}$ terms so that $a_{1},a_{2}$ can have linear terms
in $\log z_{i},\overline{\log z_{j}}$. From

${\omega}_{-2}=db-a_{2}da_{1}$

and the above construction of ${\omega}_{-2}$, it follows that $b$ can have
quadratic terms in $\log z_{i},\log\bar{z}_{j}$. In general

${\omega}_{-k}$ is a polynomial of degree $k$ in the $\log z_{i}$’s and their
conjugates.

This illustrates a general result ([Gr22]) about the period matrices of the
extension data in a VMHS.

##  III. The variation of Hodge structure and mixed Hodge structure cases

###### Proof of Theorem B.

The proof is basically an observation using known results.

The assumption that the monodromy representation

$\rho:\pi_{1}(X)\to\mathop{\rm Aut}\nolimits(V)$

has a finite kernel is used in two ways. One is that the covering of $X$
corresponding to $\pi_{1}(X)/\ker\rho$ is a finite quotient of the universal
covering. The other is that for a connected compact subvariety $Y\subset X$

(III.1) $\pi_{1}(Y)\to\pi_{1}(X)\text{ is finite
}\iff\rho\big{|}_{\pi_{1}(Y)}\text{ is finite.}$

It is a classical result from Hodge theory that

$\rho\big{|}_{\pi_{1}(Y)}\text{ is finite }\iff\Phi\big{|}_{Y}\text{ is
constant.}$

Thus in (I.9) the connected fibres of
$\widetilde{\Phi}:\widetilde{X}\to\widetilde{P}$ are the connected components
of $\pi^{-1}(Y)$ where (III.1) is satisfied.

It remains to prove that in (I.9) the image
$\widetilde{\Phi}(\widetilde{P})\subset D$ is Stein. We denote by $\check{D}$
the compact dual of $D$ ([CM-SP17] and [GGK13]). Then

$\displaystyle D$ $\displaystyle=G_{\mathbb{R}}/H$ $\displaystyle\cap$
$\displaystyle\check{D}$ $\displaystyle=G_{\mathbb{C}}/B=M/H$

where $G_{\mathbb{R}}$ is a real semi-simple Lie group, $G_{\mathbb{C}}$ is
its complexification, $H$ is a compact subgroup of $G_{\mathbb{R}}$ and $M$ is
the maximal compact subgroup of $G_{\mathbb{C}}$. On $D$ there is a
$G_{\mathbb{R}}$-invariant volume form ${\Omega}_{D}$, and on $\check{D}$
there is an $M$-invariant volume form ${\Omega}_{\check{D}}$. From [GGK13,
Lecture 6] the ratio

$\varphi={\Omega}_{D}/{\Omega}_{\check{D}}$

is then a function on $D$ with the property

> $\varphi:D\to\mathbb{R}$ is an exhaustion function and the Levi form
> $(i/2)\partial\overline{\partial}\varphi$ is positive in the horizontal sub-
> bundle $I\subset TD$.

From the assumed completeness of the period mapping $\Phi$ on $X$ it follows
that the image $\widetilde{\Phi}(\widetilde{X})=\widetilde{P}\subset D$ is a
closed analytic subvariety of $D$ whose Zariski tangent spaces lie in $I$.
Consequently

$\varphi\big{|}_{\widetilde{P}}\text{ is a psh exhaustion function}$

and then by [Na62] we may conclude that $\widetilde{P}$ is a Stein variety.∎

###### Proof of Theorem C.

We have outlined the proof in Section I; here we will give some further
details. To give a rigorous argument involves technical issues in the theory
of complex analytic geometry that we shall not deal with here. Ones that are
similar arise in the projective $X=\overline{X}$ case and are treated in
Section 5 in [EKPR12].

By the argument just given the image
$\widetilde{\Phi}(\widetilde{X})=\widetilde{P}$ is a Stein variety. This uses
the assumption that $\Phi_{M}$ is proper, which implies that the monodromies
of $\Phi$ around the $Z_{i}$ are of infinite order.

Using the terminology and notations from Section II, we denote by $\Phi_{M,2}$
the mapping given by taking the associated graded together with the extension
data of levels $\leqq 2$ in the MHS’s described by (I.10). The arguments given
in the proof of (II.4) apply to the VMHS along the fibres of
$\Phi_{M}(X)\to\Phi(X)$. Thus the map

$\Phi_{M}(X)\to\Phi_{M,2}(X)$

has finite fibres. Moreover, the fibres of

$\Phi_{M,2}(X)\to\Phi(X)$

are images of the mappings of the fibres to extension data of levels $\leqq
2$. Due to the assumed completeness of $\Phi_{M}$ their universal coverings
are Stein. From this we see that

(III.2)
$\widetilde{\Phi}_{M}(\widetilde{X})\to\widetilde{\Phi}(\widetilde{X})$

is holomorphic mapping onto a Stein variety and with Stein fibres. As in
Section 5 of [EKPR12] one may use [Car79] and [Se53] to conclude that the
total space is Stein. ∎

Comments regarding possible cohomological descriptions of the fibres of
$\Phi$; deformations of the monodromy representation $\rho$.

The connected components $Y\subset X$ of the fibres of a Shafarevich map are
defined homotopy-theoretically by the finiteness of the map
$\pi_{1}(Y)\to\pi_{1}(X)$. One may ask if they can be characterized
cohomologically as is possible in the nilpotent case by (iv) in Theorem A.
This is an interesting question that we shall now discuss in the case of a
VHS.

In order to be able to cohomologically describe the fibres some cohomology
groups must be non-zero, and we note that

(III.3) (i) _Given the local system $\mathbb{V}\to X$ underlying a VHS, the
group $H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))$ is zero if, and only
if, the monodromy representation_

$\rho:\pi_{1}(X)\to\mathop{\rm Aut}\nolimits(V_{\mathbb{C}})$

_is to first order rigid._

(ii) _The map $H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))\to
H^{1}(Y,\mathop{\rm End}\nolimits(\mathbb{V}))$ is trivial if, and only if,
the corresponding first order deformation of $\rho$ induces a trivial
deformation of $\rho\big{|}_{\pi_{1}(Y)}$._

As a first step we will reformulate the cohomological criterion in the
nilpotent case. Keeping our assumption that $X=\overline{X}\backslash Z$ we
denote by
${\mathscr{O}}(\widehat{\pi_{1}(X,x))}:=\lim_{s}{\mathscr{O}}(\pi_{1}(X,x)/J^{s+1})$
the Hopf algebra of functions on the unipotent completion of $\pi_{1}(X,x)$.
This algebra has a mixed Hodge structure that is generated by
$H^{1}(X,{\mathbb{Q}})$ viewed as linear functions on $\pi_{1}(X,x)$
([Ha87b]). For $Y$ a normal crossing variety the induced mapping

$\widehat{\pi_{1}(Y,y)}\to\widehat{\pi_{1}(X,x)},\qquad f(y)=x$

of a map $f:Y\to X$ is trivial if, and only if, the pullback map

$f^{\ast}:\widehat{{\mathscr{O}}(\pi_{1}(X,x))}\to\widehat{{\mathscr{O}}(\pi_{1}(Y,y))}$

is zero. This happens when this map is trival on generators; i.e., when

$H^{1}(X,{\mathbb{Q}})\xrightarrow{f^{\ast}}H^{1}(Y,{\mathbb{Q}})$

is zero. We have used this in the proof of Theorem A.

In the VHS case we let $G\subset\mathop{\rm Aut}\nolimits(V)$ be the
${\mathbb{Q}}$-Zariski closure of the image of the monodromy representation.
Then $G$ is a semi-simple ${\mathbb{Q}}$-algebraic group. For the purposes of
this discussion we shall assume that it is simple and that the members of the
set $\\{V_{\lambda}\\}$ of irreducible $G$-modules all arise as sub-quotients
of tensor products of $V$. Then each $V_{\lambda}$ gives a local system
$\mathbb{V}_{\lambda}\to X$ that underlies a VHS.

One now uses the completion of ${\pi_{1}(X,x)}$ relative to $\rho$ as in
[Ha98]. Following the notations there, except that here we use $G$ instead of
$S$, the relative completion is a pro-algebraic group $\mathcal{G}$ that fits
in a diagram

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3.0pt}}}}}}\ignorespaces{\hbox{\kern 87.47581pt\raise-3.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
104.77203pt\raise-14.02083pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\rho}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 134.8692pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\end{split}$

where ${\mathscr{U}}$ is the unipotent radical of $\mathcal{G}$. The Hopf
algebra ${\mathscr{O}}(\mathcal{G})$ of regular functions on $\mathcal{G}$ has
a functorial mixed Hodge structure with non-negative weights and relative to
which the Hopf algebra operations are morphisms.

The regular functions ${\mathscr{O}}(G)$ may be realized as the matrix
coefficients of the above representations.666From here on in this section we
will work over ${\mathbb{C}}$ and will set $G_{\mathbb{C}}=G$,
$\mathbb{V}_{\lambda},{\mathbb{C}}=\mathbb{V}_{\lambda}$ etc. We then have

${\mathscr{O}}(G)\cong\mathop{\oplus}\limits^{\lambda}V^{\ast}_{\lambda}\otimes
V_{\lambda}$

where $G$ acts on both the left and right. Referring to (III.4) we have an
inclusion ${\mathscr{O}}(G)\hookrightarrow{\mathscr{O}}(\mathcal{G})$ and

(III.5) ${\mathscr{O}}(G)=\mathop{\rm
Gr}\nolimits^{W}_{0}({\mathscr{O}}(\mathcal{G})).$

From [Ha98] it may be inferred that

(III.6) $\mathop{\rm
Gr}\nolimits^{W}_{1}({\mathscr{O}}(\mathcal{G}))\cong\mathop{\oplus}\limits^{\lambda}H^{1}(X,\mathbb{V}^{\ast}_{\lambda})\otimes
V_{\lambda}.$

The reason that $\mathbb{V}^{\ast}_{\lambda}$ is inside the parenthesis and
$V_{\lambda}$ is outside reflects the left and right actions of $G$ on
${\mathscr{O}}(G)$ (cf. loc. cit.).

Given an inclusion $j:Y\hookrightarrow X$ from (III.4) and (III.5) the
condition that

(III.7) $\widetilde{\rho}\circ j:\pi_{1}(Y)\to\mathcal{G}$

be trivial is

$(\widetilde{\rho}\circ j)^{\ast}{\mathscr{O}}(\mathcal{G})=\text{ constant
functions.}$

From (III.6) this implies that all restriction mappings

(III.8λ) $H^{1}(X,\mathbb{V}_{\lambda})\to H^{1}(Y,\mathbb{V}_{\lambda})$

are zero.

######  Question III.9:

If (III.8λ) holds for all non-trivial $V_{\lambda}$, then does (III.7) hold?

A positive answer would necessitate that

(III.10) $H^{1}(X,\mathbb{V})\neq 0$

for some $V=V_{\lambda}$. This group has a MHS with the part not in
$H^{1}(\overline{X},j_{\ast}\mathbb{V})$ having a description involving the
part of $\mathbb{V}$ that is not invariant under monodromy around the
$Z_{i}$’s (cf. [Ha98] and [Le21]). We will not get into this here. _For the
remainder of Section III we will assume that $X=\overline{X}$ is projective._
Then $H^{1}(\overline{X},\mathbb{V})$ has a Hodge structure with

(III.11) $F^{1}H^{1}(X,\mathbb{V})=H^{0}({\Omega}^{1}_{\overline{X}}).$

The non-zero vector space (III.11) gives rise to an interesting construction
that we now shall discuss (cf. loc. cit. for a related construction of locally
constant iterated integrals).

######  III.12:

_Twisted Albanese map._

We first note that for any local system $\mathbb{V}\to X$ and ${\omega}\in
H^{0}({\Omega}^{1}_{\overline{X}}\otimes\mathbb{V})$ the exterior derivative
$d{\omega}$ is well defined. If $\mathbb{V}\to X$ underlies a VHS, then it may
be shown that $d{\omega}=0$.

Now let $x_{0}\in X$ and $\widetilde{x}_{0}\in\widetilde{X}$ be base points
with $(\widetilde{X},\widetilde{x}_{0})\xrightarrow{\pi}(X,x_{0})$. Then
canonically

$\pi^{\ast}\mathbb{V}\cong\widetilde{X}\times V.$

If ${\omega}$ is as above with $\pi^{\ast}{\omega}=\widetilde{{\omega}}$, then
$\widetilde{{\omega}}\in
H^{0}({\Omega}^{1}_{\widetilde{X}}\otimes\mathbb{V})\cong
H^{0}({\Omega}^{1}_{\widetilde{X}})\otimes V$ and since
$d\widetilde{{\omega}}=0$,

(III.13) $\int^{\widetilde{x}}_{\widetilde{x}_{0}}\widetilde{{\omega}}\in V$

is well defined. Taking into account the action of $\pi_{1}(X,x)$ as deck
transformations we may descend (III.13) to $X$ to have a well-defined mapping

(III.14) ${\rm Alb}_{X,\mathbb{V}}:X\to
H^{0}({\Omega}^{1}_{X}\mathbb{V})^{\ast}\otimes
V_{x_{0}}/\widetilde{\rho}(\pi_{1}(X,x_{0})).$

This is the _twisted Albanese mapping_ referred to above.

We note that

(III.15) (III.8λ) is equivalent to ${\rm
Alb}_{X,\mathbb{V}_{\lambda}}\big{|}_{Y}=$ constant.

Consequently an affirmative answer to (III.9) would mean that we have

$\rho\big{|}_{\pi_{1}(Y)}\text{ is finite $\iff{\rm
Alb}_{X,\mathbb{V}_{\lambda}}\big{|}_{Y}=$ constant for all $V_{\lambda}$.}$

(III.16) Extensions of mixed Hodge structures and the first step in the
construction of the mixed Hodge structure on a relative completion.

This is an extension to the relative completion of a representation of the
fundamental group of the discussion in (II.13), (II.15) and of Proposition
II.18.

######  Proposition III.17:

Let $\mathbb{V}\to X$ be a local system underlying a VHS where
$\mathbb{V}_{\lambda}$ has weight zero. Choosing a base point $x_{0}\in X$
there is for each $x\in X$ a canonical mixed Hodge structure
$\mathbb{V}_{1,x}$ with

  1. $\bullet$

$\mathop{\rm Gr}\nolimits^{W}_{0}(\mathbb{V}_{1,x})={\mathbb{C}}$;

  2. $\bullet$

$\mathbb{V}_{1,x}/W_{0}(\mathbb{V}_{1,x})\cong V^{\ast}_{x_{0}}\otimes
H^{1}(X,\mathbb{V})$

and that is constructed using ${\rm Alb}_{X,\mathbb{V}}(x)$ as extension
class. Letting $x$ vary we obtain an admissible VMHS.

###### Proof.

As was the case for Proposition II.18, the construction of a mixed Hodge
structure from a class in
$H^{0}({\Omega}^{1}_{\overline{X}}\mathbb{V})^{\ast}\otimes
V_{x_{0}}/\widetilde{\rho}(\pi_{1}(X,x_{0}))$ may be done following the
prescription in [Car80]. Using the same argument as in the
$\mathbb{V}={\mathbb{C}}$ we obtain a local system (flat vector bundle) and
horzontality (transversality) of the family of mixed Hodge structures
parametrized by $x\in X$ as consequences of

$H^{1}(X,\mathbb{V}_{\mathbb{C}})\cong
H^{0}({\Omega}^{1}_{\overline{X}}\otimes\mathbb{V})\oplus\overline{H^{0}({\Omega}^{1}_{\overline{X}}\otimes\mathbb{V})}$

where the differential forms appearing in both summands on the right-hand side
are closed.

Alternatively we may proceed as follows. In general suppose we have over $X$

$0\to A\to B\to C\to 0$

where $A,C$ are Hodge structures and $B$ is a family of mixed Hodge structures
$B_{x}$ defined by

$e_{x}\in\frac{\mathop{\rm Hom}\nolimits_{\mathbb{C}}(C,A)}{F^{0}\mathop{\rm
Hom}\nolimits_{C}(C,A)+\mathop{\rm Hom}\nolimits_{\mathbb{Z}}(C,A)}.$

The connection on $B$ is

$\nabla_{B}=\begin{pmatrix}\nabla_{A}&{\omega}\\\ 0&\nabla_{C}\end{pmatrix}$

where

${\omega}\in\mathop{\rm
Hom}\nolimits_{\mathbb{C}}(C,A)\otimes{\Omega}^{1}_{X}.$

From

$\nabla^{2}_{B}=\begin{pmatrix}\nabla_{A}&\nabla_{A}{\omega}+{\omega}\nabla_{C}\\\
0&\nabla^{2}_{C}\end{pmatrix}$

to have integrability we need $\nabla_{A}{\omega}+{\omega}\nabla_{B}=0$, which
is

$\nabla_{\mathop{\rm Hom}\nolimits(C,A)}{\omega}=0.$

For horizontality a calculation shows that we need

$\nabla_{\mathop{\rm Hom}\nolimits(C,A)}e\in\frac{F^{-1}\mathop{\rm
Hom}\nolimits(C,A)\otimes{\Omega}^{1}_{X}}{\nabla F^{0}\mathop{\rm
Hom}\nolimits(C,A)}.$

This is well defined since $\nabla F^{0}\subseteq F^{-1}$. Taking

${\omega}=\frac{\nabla_{\mathop{\rm Hom}\nolimits(C,A)}e}{\nabla_{\mathop{\rm
Hom}\nolimits(C,A)}F^{0}\mathop{\rm Hom}\nolimits(C,A)}$

we see that both (i) and (ii) are satisfied.∎

For $\mathbb{V}\to X$ as in Proposition III.17 we have the associated local
system $\mathop{\rm End}\nolimits(\mathbb{V})$. We note that here there are
two interpretations of $H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))$. One
is as was just discussed in classifying extensions

(III.17a) $0\to\mathop{\rm
End}\nolimits(\mathbb{V})\to\mathbb{M}\to\mathop{\rm
End}\nolimits(\mathbb{V})\to 0.$ The other is as first order deformations of
$\mathbb{V}\to X$. One may think of the latter as local systems over a scheme
$(X,{\mathscr{O}}_{X,1})$ where ${\mathscr{O}}_{X,1}$ is an extension of
${\mathscr{O}}_{X}$ by nilpotents of order 2. In the case at hand we may take
the universal first order deformations to be the middle term in an exact
sequence (III.17b) $0\to\mathbb{V}\otimes H^{1}(X,\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast}\to\mathbb{V}_{1}\to\mathbb{V}\to 0.$

The extension class defining this sequence is the tautological class given by
the identity in

$H^{1}(X,\mathop{\rm Hom}\nolimits(\mathbb{V},\mathbb{V}\otimes
H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))^{\ast})\cong
H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))\otimes H^{1}(X,\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast}$

and ${\mathscr{O}}_{X,1}={\mathscr{O}}_{X}[H^{1}(X,\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast}]$.

######  Proposition III.18:

There is a VMHS over the scheme $(X,{\mathscr{O}}_{X,1})$ whose restriction to
$(X,{\mathscr{O}}_{X})$ is the VMHS defined by the class

${\rm Alb}_{X,\mathop{\rm End}\nolimits(\mathbb{V})}\subset\mathop{\rm
Ext}\nolimits^{1}_{{\mathrm{MHS}}}(\mathbb{V},\mathbb{V}\otimes
H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))^{\ast}).$

###### Proof.

The proof is the same as that for Proposition III.17 above. We note that the
connection form for $\mathbb{V}_{1}$ as a local system over
${\mathscr{O}}_{X}$ is

$\nabla_{\mathbb{V}}=\begin{pmatrix}\nabla_{\mathbb{V}\otimes
H^{1}(\mathop{\rm End}\nolimits(\mathbb{V}))^{\ast}}&{\Omega}\\\
0&\nabla_{\mathbb{V}}\end{pmatrix}$

where ${\Omega}\in H^{0}(X,{\Omega}^{1}_{X}\otimes\mathop{\rm
Hom}\nolimits(\mathbb{V},\mathbb{V}\otimes H^{1}(\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast})$. Then

$\nabla^{2}_{V}=0\iff\nabla_{\mathop{\rm
Hom}\nolimits}{\Omega}+\frac{1}{2}[{\Omega},{\Omega}]=0$

where the $\mathop{\rm Hom}\nolimits$ is $\mathop{\rm
Hom}\nolimits(\mathbb{V},\mathbb{V}\otimes H^{1}(\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast})$.

As a local system over ${\mathscr{O}}_{X,1}$ where $H^{1}(X,\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast}$ is treated as deformation parameters since
$[{\Omega},{\Omega}]$ is quadratic we have $\nabla^{2}_{\mathbb{V}_{1}}=0$. ∎

For a subvariety $Y\subset X$ the mapping

$H^{1}(X,\mathop{\rm End}\nolimits(\mathbb{V}))\to H^{1}(Y,\mathop{\rm
End}\nolimits(\mathbb{V}))$

is interpreted as restricting first order deformations of
$\rho:\pi_{1}(X)\to\mathop{\rm Aut}\nolimits(V)$ to $\pi_{1}(Y)$. With this
interpretation we have completed the proof of Proposition III and its
corollary.

The above may be summarized as follows:

(III.19) _Given a local system $\mathbb{V}\to X$ supporting a VHS, denote by
${\mathscr{O}}_{X,1}$ the scheme ${\mathscr{O}}_{X}[H^{1}(X,\mathop{\rm
End}\nolimits(E))^{\ast}]$ obtained by adjoining to ${\mathscr{O}}_{X}$
nilpotents of order 2. Then_

  1. (i)

${\mathscr{O}}_{X,1}$_has a mixed Hodge structure with_

$\begin{cases}\mathop{\rm
Gr}\nolimits^{W}_{0}({\mathscr{O}}_{X,1})={\mathbb{C}},\\\ \mathop{\rm
Gr}\nolimits^{W}_{1}({\mathscr{O}}_{X,1})=H^{1}(X,\mathop{\rm
End}\nolimits(\mathbb{V}))^{\ast}\end{cases}$

_and whose extension class is_ ${\rm Alb}_{X,\mathop{\rm
End}\nolimits(\mathbb{V})}$, and

  2. (ii)

_over_ $(X,{\mathscr{O}}_{X_{1}})$ there is a VMHS as described in Proposition
III.17.

In the projective case the extension of the above to all orders of deformation
of $\rho$ (Goldman-Millson theory) is carried out in [EKPR12]. Thus there are

  1. $\bullet$

a MHS on the completed local rings $\widehat{{\mathscr{O}}}_{\mathop{\rm
Def}\nolimits(\rho)}$ of the Kuranishi space $\mathop{\rm
Def}\nolimits(\rho)$; and

  2. $\bullet$

a formal VMHS over $\mathop{\rm Def}\nolimits(\rho)$.

For further discussion of these results cf. [Pr17], [Pr19], [Le19] and [Le21].

##  IV. Future directions

In this section we will very briefly outline some future research directions.
We have made an attempt to formulate a categorical version of Shafarevich
conjecture in [HKL17], [KL14]. In those papers we outline new techniques of
perverse sheaves of categories and functors between them. The categorical
information is recorded by the skeleton and the Lagrangian sheaves over it. We
give an interpretation of the infinite chain conjecture.

Allowed$C$1-core2-core$\bullet$Not Allowed$\bullet$$\bullet$$\bullet$

######  Conjecture IV.1 (Categorical Shafarevich):

Let $X$ be a quasi-projective variety with infinite (nilpotent) fundamental
group. Assume that for any curve $C$ the functor

$F_{\text{wrapped}}(C)\to F_{\text{wrapped}}(X)$

is allowed. Then $X$ is holomorphically convex.

Here allowed maps are maps those which map the core of the category of an open
Riemann surface in the connected 1 dimensional subcore of the quasipojective
variety. Details can be found in the above references.

The above conjecture suggests

######  Conjecture IV.2:

Let $X$ be quasi-projective. Assume that every proper morphism $f$ from a
smooth curve $C$ to $X$ has a meridian of infinity of infinite order in
$\pi_{1}(X)$. Then is $X$ holomorphically convex?

Successful steps in this direction were taken by P. Eyssidieux and R. Aguilar.

Other considerations naturally lead to.

######  Question IV.3:

Assume that $X$ is Picard hyperbolic and $\pi_{1}(X)$ is nilpotent. Does this
imply that $X$ is holomorphically convex?

More plausible may be a positive answer to the following:

######  Question IV.4:

Assume that $X$ is Picard hyperbolic and $\pi_{1}(X)$ is a big fundamental
group in the sense of [Br22]. Does this imply that $X$ holomorphically convex
(Stein).

This question extends the work of loc. cit. who considered the case of a
projective $X=\overline{X}$. In fact it will be interesting to investigate how
the following three notions are related.

$X$ is hyperbolic$X$ is 1-formal$\widetilde{X}$ is holomorphically convex???

Recently several outstanding results were obtained in direction of
algebraicity of the period map in the case of variations of mixed Hodge
structures; see [BBT06]. It seems likely that one can drop the condition about
infinite monodromy around $2$. We would like to pose the following.

######  Question IV.5:

Let $X$ be a quasi-projective variety such that there is an inclusion

$\pi_{1}(X)\hookrightarrow{}\mathop{\rm GL}\nolimits(n,{\mathbb{C}}).$

Then is $\widetilde{X}$ holomorphically convex?

Several results in this direction were obtained by B. Brunebarbe and Y. Deng.
Similarly in the case when the fundamental group $\pi_{1}(X)$ is subgroup of a
nilpotent group of $\mathop{\rm Nil}\nolimits p$ we have the following:

######  Question IV.6:

Let $X$ be a quasi-projective variety such that

$\pi_{1}(X)\hookrightarrow{}\mathop{\rm Nil}\nolimits p.$

Then is $\widetilde{X}$ holomorphically convex?

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