# The nilpotent quotients of normal quasi-projective varieties with proper
quasi-Albanese map

Rodolfo Aguilar Aguilar  University of Miami, Coral Gables, USA and Institute
for Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia,
Bulgaria<EMAIL_ADDRESS>https://sites.google.com/view/rodolfo-
aguilar/ and  Frédéric Campana  Université Lorraine
Institut Elie Cartan
Nancy, France<EMAIL_ADDRESS>
###### Abstract.

We show that if $X$ is a normal complex quasi-projective variety, the quasi-
Albanese map of which is proper, then the torsionfree nilpotent quotients of
$\pi_{1}(X)$ are, up to a controlled finite index, the same ones as those of
the normalisation of its quasi-Albanese image. When $X$ is quasi-Kähler
smooth, we get the same conclusion, but only for the smooth models of the
quasi-Albanese image. In this second case, the proof is elementary, as the one
given in [Cam95] for $X$ compact.

In the normal quasi-projective case, the étale Galois cover of $X$ associated
to the nilpotent completion of $\pi_{1}(X)$ is thus holomorphically convex.
This is proved in the smooth case by $3$ other methods in [GGK22], which
motivated the present text.

When $X$ is ‘special’ in the sense of [Cam11], we deduce that the torsion free
nilpotent quotients of $\pi_{1}(X)$ are abelian. Examples show that this
property fails (as first observed in [CDY22]) when the quasi-Albanese map is
not proper. This leads to replace our previous ‘Abelianity conjecture’ in the
compact case by an ‘Nilpotency conjecture’ in the non-compact quasi-Kähler
context.

The first author is partially supported by the Bulgarian Ministry of Education
and Science, Scientific Programme "Enhancing the Research Capacity in
Mathematical Sciences (PIKOM)", No. DO1-67/05.05.2022.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Proof of Theorem 1
  3. 3 Proof of Theorem 3
  4. 4 Proof of Theorem 5
  5. 5 Special Manifolds
  6. 6 Remarks and examples in the non-proper case
  7. 7 Appendix: Weight-$1$ MHS

##  1\. Introduction

A quasi-Kähler manifold $X$ is a Zariski open subset of a compact Kähler
manifold $\overline{X}$. A ‘good compactification’ $(\overline{X},D)$ of $X$
is a smooth compact Kähler manifold $\overline{X}$ together with a simple
normal crossings divisor $D$ of $\overline{X}$ such that
$\overline{X}\setminus D=X$.

We denote with $\alpha_{X}:X\to Alb(X)$ the quasi-Albanese map of $X$,
introduced by S. Iitaka (see [Iit82]), and with $Z_{0}^{\prime\prime}\subset
Alb(X)$ its image. We denote with $Z^{\prime}_{0}\to Z_{0}^{\prime\prime}$ the
normalisation of $Z_{0}^{\prime\prime}$, and still write $\alpha_{X}:X\to
Z^{\prime}_{0}$ for the lift of $\alpha_{X}$ to $Z^{\prime}_{0}$.

Main assumption: As in [GGK22], we always assume here that $\alpha_{X}:X\to
Alb(X)$ is proper, so that $Z_{0}^{\prime\prime}\subset Alb(X)$ is closed and
quasi-Kähler.

By modifying $X$, which does not change $\pi_{1}(X)$, we get the following
commutative diagram, where $\sigma^{\prime}:=d\circ\alpha$ is the Stein
factorisation of $\alpha_{X}=\varphi^{\prime}\circ\sigma^{\prime}$, so that
$Z^{\prime}$ is normal, the fibres of $\sigma^{\prime}:X\to Z^{\prime}$ are
connected, and those of $\varphi^{\prime}:Z^{\prime}\to Z^{\prime}_{0}$ are
finite. Moreover, $d:Z\to Z^{\prime}$ and $d_{0}:Z_{0}\to Z^{\prime}_{0}$ are
desingularisations. The maps $\varphi$ and $\varphi^{\prime}$ are thus finite,
of the same degree $e$.

$\begin{matrix}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.53471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.61382pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.93472pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
7.43579pt\raise-24.79692pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\alpha_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
31.53471pt\raise-31.216pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
32.93472pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.73969pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 71.87639pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.70555pt\raise-19.64554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
39.70555pt\raise-29.45776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.87639pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
79.41722pt\raise-19.64554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{\varphi^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
79.41722pt\raise-27.9022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.53471pt\raise-39.29108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.73969pt\raise-33.21608pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{d_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
71.87639pt\raise-39.29108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.87639pt\raise-39.29108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z^{\prime}_{0}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{matrix}$

We may assume that all these maps extend holomorphically to suitable
compatible compactifications
$\overline{Z},\overline{Z^{\prime}},\overline{Z_{0}},\overline{Z^{\prime}_{0}}$
of $Z,Z^{\prime},Z_{0},Z^{\prime}_{0}$, which are good for $Z,Z_{0}$.

We use the notations, and refer to the appendix of [Cam95] (see also
[Sta65],§7). In particular if $G$ is a group, and $G_{n}$ is the (nilpotent)
quotient of $G$ by the $(n+1)-th$ term of its central series, and
$G^{\prime}_{n}:=G_{n}/Torsion$, with Kernel $K^{\prime}_{n}(G)$, then we
define $G^{nilp}:=G/\cap K^{\prime}_{n}(G)$. These constructions are
functorial.

We have the following elementary property (part of [Sta65], Theorem 7.3): if
$f:G\to H$ is a morphism of groups such that , for any $n\geq 1$, the induced
morphism
$(K^{\prime}_{n}(G)/K^{\prime}_{n+1}(G))\otimes\mathbb{Q}\to(K^{\prime}_{n}(H)/K^{\prime}_{n+1}(H))\otimes\mathbb{Q}$
is isomorphic, then $G^{nilp}\to H^{nilp}$ is injective. From this, it is easy
to deduce that if, moreover, $f:G\to H$ has an image of finite index $e$, then
$G^{nilp}\to H^{nilp}$ has moreover an image of finite index at most $e$. In
particular, if $f$ is surjective, then $G^{nilp}\to H^{nilp}$ is isomorphic.

By $\pi^{X}_{nilp}:X^{nilp}\to X$, we denote the Galois étale cover of group
$\pi_{1}(X)^{nilp}$. We use a similar notation for
$Z,Z^{\prime},Z_{0},Z^{\prime}_{0}$. If $f:V\to W$ is a holomorphic map, we
denote by $f_{nilp}:V^{nilp}\to W^{nilp}$ the induced map.

######  Remark 0.

The property of $G^{nilp}$ stated above, shows that if $f:V\to W$ is a
surjective proper map with finite fibres, of degree $e$, between connected
normal complex spaces, and if
$(K^{\prime}_{n}(G)/K^{\prime}_{n+1}(G))\otimes\mathbb{Q}\to(K^{\prime}_{n}(H)/K^{\prime}_{n+1}(H))\otimes\mathbb{Q}$
is isomorphic for all $n>0$, with $G:=\pi_{1}(V),H:=\pi_{1}(W)$, then the
induced map $G^{nilp}\to H^{nilp}$ is injective with image of finite index at
most $e$. This follows from the fact that $f_{*}:G\to H$ has image of finite
index at most $e$ (see [Cam91], Proposition 1.3). We shall repeatedly apply
this remark to $\varphi:Z\to Z_{0}$ and $\varphi^{\prime}:Z^{\prime}\to
Z^{\prime}_{0}$, and thus show the results stated for either $Z$, or
$Z^{\prime}$ only, since the proofs apply to both cases.

Our first result is:

######  Theorem 1.

If $X$ is quasi-Kähler and $\alpha_{X}$ proper, then:
$\alpha_{*}:\pi_{1}(X)^{nilp}\to\pi_{1}(Z)^{nilp}$ is isomorphic,
$(\alpha_{0})_{*}:\pi_{1}(X)^{nilp}\to\pi_{1}(Z_{0})^{nilp}$ is injective with
image of finite index at most $e$.

Hence: $X^{nilp}=X\times_{Z}Z^{nilp}$, and $Z^{nilp}$ is a component of
$Z\times_{Z_{0}}Z_{0}^{nilp}$.

######  Corollary 2.

In the situation of theorem 1, we have:

1\. $X^{nilp}$ is holomorphically convex if $Z_{0}^{nilp}$ is Stein.

2\. If $\alpha_{X}^{*}:H^{2}(Alb(X),\mathbb{Q})\to H^{2}(Z_{0},\mathbb{Q})$ is
injective, $\pi_{1}(X)^{nilp}=H_{1}(X,\mathbb{Z})/Torsion$. This occurs, in
particular, if $\alpha_{X}$ is surjective.

######  Theorem 3.

If $X$ quasi-projective,and if $\alpha_{X}$ is proper:
$(d\circ\alpha)_{*}:\pi_{1}(X)^{nilp}\to\pi_{1}(Z^{\prime})^{nilp}$ is an
isomorphism, and
$\varphi_{*}:\pi_{1}(Z^{\prime})^{nilp}\to\pi_{1}(Z_{0}^{\prime})^{nilp}$ is
injective with image of finite index at most $e$.

The Claim 2 of the following Corollary is proved by 3 other methods in
[GGK22]. It was previously proved for $Z^{\prime}$ smooth in [Cam95], remarque
1.9, and in [Kat97] for $X$ projective.

######  Corollary 4.

([GGK22]) In the situation of theorem 3, we have:

1\. The connected components of the fibres of
$\pi_{1}(X)^{nilp}\to\pi_{1}(Z^{\prime})^{nilp}$ are compact.

2\. $Z^{\prime nilp}$ is Stein, and $X^{nilp}$ is thus holomorphically convex.

The Steinness of $Z^{\prime nilp}$ is clear, since it is an étale cover of
$Z^{\prime}\times_{A}\tilde{A}$, $\tilde{A}$ being the (Stein) universal cover
of $A$.

The proof of Theorem 3 shows that its conclusion holds, more generally, for
$X$ normal, instead of smooth. See Corollary 29 for the definition and
properties of the quasi-Albanese map in the normal case.

######  Theorem 5.

If $X$ is quasi-projective, normal and if $\alpha_{X}$ is proper, the induced
map $\pi_{1}(X)^{nilp}\to\pi_{1}(Z^{\prime})^{nilp}$ is an isomorphism, and
$\pi_{1}(X)^{nilp}\to\pi_{1}(Z_{0}^{\prime})^{nilp}$ is injective with image
of finite index at most $e$.

Here, $Z_{0}^{\prime}$ is, as before, the normalisation of $\alpha_{X}(X)$,
and $Z^{\prime}$ is the Stein factorisation of $X\to Z_{0}^{\prime}$. The
Corollary 4 thus remains valid for $X$ normal.

Although the preceding theorems are essentially particular cases of Theorem 5,
it is instructive to give their proofs, which relies on simpler notions and
techniques.

While the proof of Theorem 1 is elementary, directly extending to the open
case the arguments of [Cam95], the proof of Theorem 3 relies on the mixed
Hodge structures constructed in [Del74] and [Hai97] on the cohomology and
homotopy groups of complex algebraic varieties, which permit to deal with the,
possibly arbitrary, singularities occuring in $Z_{0}^{\prime\prime}$. The
proof of Theorem 5 is essentially the same one as of Theorem 3, once the
Albanese map is defined in the normal context. The arguments should work in
the quasi-Kähler case as well, but the corresponding statements have not been
written in this context, as far as we know.

Corollary 2 applies to ‘special’ quasi-Kähler manifolds as defined in [Cam11]
when the quasi-Albanese map is proper, and shows then that
$\pi_{1}(X)^{nilp}=H_{1}(X,\mathbb{Z})/Torsion$. We show however by examples
in the last section that this fails in general if $\alpha_{X}$ is not proper,
as first observed in [CDY22]. As a consequence, the ‘Abelianity conjecture’ in
the compact case needs to be weakened to the ‘Nilpotency conjecture’ when $X$
is not compact.

Examples given in Remark 27 show that the holomorphic convexity of the
universal cover of the smooth quasi-projective varieties commonly fails the
open case.

###  1.1. Acknowledgements

We thank Robert Laterveer for the example of Hilbert surfaces given in Remark
7, Morihiko Saito for the examples in Remark 14, the authors of [CDY22] for
sending us their preprint, and R. Hain for communicating his construction of
the higher-Albanese in the normal quasi-projective case (we actually use only
the case $s=1$, also constructed in [CDY22]).

##  2\. Proof of Theorem 1

When $X$ is compact Kähler, this is proved in [Cam95], Corollaire 3.1. From
the proof given there, it is sufficient to show that
$\alpha^{*}:H^{2}(Z,\mathbb{Q})\to H^{2}(X,\mathbb{Q})$ is injective, which
follows from the more general next Lemma:

######  Lemma 6.

Let $f:X\to Z$ be a proper surjective map between quasi-Kähler (connected)
manifolds.

For every $k\geq 0$, the map $f^{*}:H^{k}(Z,\mathbb{Q})\to
H^{k}(X,\mathbb{Q})$ is injective.

###### Proof.

This is the same one as for the Lemme111The hypothesis that $X$ is compact has
been forgotten in its statement. 2.4 of [Cam95], up to the non-compact version
of Poincaré duality in cohomology. With the notations of the proof of this
Lemma 2.4, it is sufficient to show that if $u\in H^{k}(Z,\mathbb{Q})\neq 0$,
there is $v\in H_{c}^{2m-k}(Z,\mathbb{Q})$ such that $[u\wedge v]\neq 0\in
H_{c}^{2m}(Z,\mathbb{Q}),m:=dim_{\mathbb{C}}(Z)$. By [BT82],Chap 1,§5,
(especially remark 5.7), this is true if $Z$ has either a good finite cover,
or more generally, if its cohomologies with compact support are finite
dimensional. But this latter property is satisfied by quasi-Kähler manifolds,
by the exact cohomology sequence with compact supports for a pair $(A,B)$ with
$A\supset B$ compact, applied to a good compactification of $Z$:
$A=\overline{Z},B=D$, $Z=\overline{Z}\setminus D$. ∎

The consequences stated in Corollary 2 are then obvious.

We give in Corollary 20 situations in which $\alpha_{X}$ is dominant, hence
surjective if proper.

######  Remark 7.

1.The Kähler hypothesis of Lemma 6 is required already when $X$ is compact as
the examples of Hopf surfaces of algebraic dimension $1$ and of the Iwazawa
threefold show, see 2.5 and 2.6 of [Cam95]. The example 24 shows that the
conclusion of lemma 6 may fail if the properness hypothesis is dropped, even
if $Z$ is an elliptic curve and $a_{X}:X\to Z$ a $\mathbb{C}^{*}$-bundle.

2\. The conclusion of Lemma 6 may also fail if $Z$, projective, is only
assumed to be normal. There indeed exist compactified normal Hilbert modular
surfaces $Z$ with elliptic singularities cusps such that the kernel of
$H^{2}(Z)\to H^{2}(X)$ is non-zero, if $X$ is their minimal resolution (see
[BPVdV84],Chap. V, §22 ). We thank Robert Laterveer for indicating us this
class of examples. These examples show that Stallings criterion (surjectivity
of $H_{2}(G)\to H_{2}(Q)$ implies $G^{nilp}=Q^{nilp}$ for two groups with the
same $H_{1}$ with $\mathbb{Q}$ coefficients if the group morphism $G\to H$ is
onto) is a sufficient, but not necessary condition. See Theorem 8 below, which
shows that the weaker condition that $H_{2}(G)$ and $H_{2}(H)$ have the same
image in $H_{2}(G^{ab}/Torsion)$ is sufficient, in the quasi Kählerian
context, to get the same conclusion, where $G^{ab}$ denotes the abelianisation
of $G$.

##  3\. Proof of Theorem 3

We start with proving the easier compact case.

######  Theorem 8.

Let $X$ be a projective manifold, $\alpha_{X}:X\to Alb(X)$ its Albanese map,
and $\sigma:X\to Z^{\prime}$ its Stein factorisation (so that $Z^{\prime}$ is
normal). Then $\pi_{1}(X)^{nilp}=\pi_{1}(Z^{\prime})^{nilp}$.

###### Proof.

We can then apply [ADH16], Theorem 1.2 and its corollary 1.3, which says that
$\pi_{1}(Z^{\prime})^{nilp}$ is $1$-formal. Since $H^{1}(X)=H^{1}(Z^{\prime})$
(coefficients $\mathbb{Q}$), it is sufficient to show that
$Ker(a_{Z^{\prime}})^{*}:H^{2}(A)\to
H^{2}(Z^{\prime})=Ker(a_{X}^{*})=Ker(a_{Z^{\prime}}\circ\sigma)^{*}:H^{2}(A)\to
H^{2}(X)$.This is a consequence of [Del74], 8.2.7. ∎

We now consider the open case. Recall the statement:

######  Theorem 9.

Let $X$ be a quasi-projective manifold, $\alpha_{X}:X\to Alb(X)$ its (proper)
Albanese map, and $\sigma^{\prime}:X\to Z^{\prime}$ its Stein factorisation
(so that $Z^{\prime}$ is normal). Then
$\pi_{1}(X)^{nilp}=\pi_{1}(Z^{\prime})^{nilp}$.

###### Proof.

By [ADH16], Lemma 4.1, dualized, and the fonctoriality of the description of
the Malc̆ev Lie algebra given there, it is sufficient to show that
$Ker(H^{2}(A)\to H^{2}(Z^{\prime}))=Ker(H^{2}(A)\to H^{2}(Z))$, with
$A:=Alb(X)$, and $H^{2}(A)=\wedge^{2}H^{1}(A)$, since $A$ is a (commutative,
complex Lie) group. By strictness, it is sufficient to check this for the
weight graduations.

We shall show this on the graded quotients for the weights $2,3,4$ of the
weight filtrations, since this is checked for the weights $0,1$ in [ADH16].

$\bullet$ Weights 3,4. For the weights $3,4$, this follows from the more
general Proposition 11, and does not use the properness of $\alpha_{X}$. The
weight $2$, which depends on this properness, will be treated after this.

The proof for the weights $3$ and $4$ relies on the statements and proofs of
[PS, 5.37 and 5.53], using the notations there.

######  Lemma 10.

Let $g:\tilde{Y}\to Y$ be a resolution of singularities of a quasi-projective
variety $Y$. Let $D\subset Y$ be the discriminant locus and $E=g^{-1}(D)$,
then the induced maps
$\operatorname{Gr}_{i}^{W}H^{2}(D)\to\operatorname{Gr}_{i}^{W}H^{2}(E)$ are
injective for $i=3,4$.

###### Proof.

We proceed by induction on the dimension of $Y$. The cases for $Y$ of
dimension $0$ or $1$ are trivial.

The Mayer-Vietoris sequence for the discriminant square of [PS, Corollary-
Definition 5.37], and strictness of MHS morphisms give:

$\operatorname{Gr}_{i}^{W}H^{k}(Y)\to\operatorname{Gr}_{i}^{W}H^{k}(\tilde{Y})\oplus\operatorname{Gr}_{i}^{W}H^{k}(D)\to\operatorname{Gr}_{i}^{W}H^{k}(E)\to\operatorname{Gr}_{i}^{W}H^{k+1}(Y)$

First suppose that $D$ is of dimension $0$. As $g$ is a proper morphism, we
have: $\operatorname{Gr}_{i}^{W}H^{k}(D)=0$ and
$\operatorname{Gr}_{i}^{W}H^{k}(E)=0$ for $i>k$ and $k\geq 0$. The result
follows for the zero dimensional case.

Now suppose that $\dim Y=n$ and that the statement is true for quasi-
projective varieties of dimension strictly less than $n$. Consider resolutions
of singularities $\tilde{E}\to E$, $\tilde{D}\to D$ and discriminant squares
for the right-vertical maps of the two following diagrams:

$\leavevmode\hbox to57.04pt{\vbox
to46.79pt{\pgfpicture\makeatletter\hbox{\hskip
28.52147pt\lower-23.44705pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.52147pt}{-23.34721pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
8.23679pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.93124pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${G}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.23679pt\hfil&\hfil\hskip 31.0833pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.77779pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\tilde{E}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.08333pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
8.21527pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.90973pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${F}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.21527pt\hfil&\hfil\hskip 32.28467pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.97916pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${E}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.2847pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.07988pt}{2.39986pt}\pgfsys@curveto{-1.69989pt}{0.95992pt}{-0.85313pt}{0.27998pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85313pt}{-0.27998pt}{-1.69989pt}{-0.95992pt}{-2.07988pt}{-2.39986pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.8479pt}{11.30557pt}\pgfsys@lineto{12.55348pt}{11.30557pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.75346pt}{11.30557pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-20.28468pt}{4.94586pt}\pgfsys@lineto{-20.28468pt}{-12.25423pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-20.28468pt}{-12.45421pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{20.23677pt}{4.94586pt}\pgfsys@lineto{20.23677pt}{-12.25423pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{20.23677pt}{-12.45421pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.86942pt}{-20.84721pt}\pgfsys@lineto{11.35211pt}{-20.84721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.5521pt}{-20.84721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}and\leavevmode\hbox
to59.18pt{\vbox to49.52pt{\pgfpicture\makeatletter\hbox{\hskip
29.59077pt\lower-24.80814pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.59077pt}{-24.7083pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
9.00677pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.70123pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${G^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
9.00677pt\hfil&\hfil\hskip 31.0833pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.77779pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\tilde{D}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
7.08333pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
8.98526pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.67972pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${F^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.98526pt\hfil&\hfil\hskip 32.58398pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.27847pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${D}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.58401pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.37723pt}{11.50003pt}\pgfsys@lineto{13.32347pt}{11.50003pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.52345pt}{11.50003pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-20.584pt}{5.14032pt}\pgfsys@lineto{-20.584pt}{-12.05977pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-20.584pt}{-12.25975pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{21.00676pt}{5.14032pt}\pgfsys@lineto{21.00676pt}{-13.61531pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{21.00676pt}{-13.81529pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.39874pt}{-22.2083pt}\pgfsys@lineto{11.82278pt}{-22.2083pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.02277pt}{-22.2083pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

The Mayer-Vietoris sequence [PS, Corollary-Definition 5.37] reads:

$\operatorname{Gr}_{i}^{W}H^{1}(G)\to\operatorname{Gr}_{i}^{W}H^{2}(E)\to\operatorname{Gr}_{i}^{W}H^{2}(\tilde{E})\oplus\operatorname{Gr}_{i}^{W}H^{2}(F)\to\operatorname{Gr}_{i}^{W}H^{2}(G)$

By induction hypothesis, we have that
$\operatorname{Gr}_{i}^{W}H^{2}(F)\to\operatorname{Gr}_{i}^{W}H^{2}(G)$ is
injective for $i=3,4$. As $H^{1}(G)$ has only weights $0,1,2$, we conclude by
a simple diagram chasing using the definition of the arrows in this exact
sequence, that
$\operatorname{Gr}_{i}^{W}H^{2}(E)\to\operatorname{Gr}_{i}^{W}H^{2}(\tilde{E})$
is injective for $i=3,4$.

Similarly, we obtain that that
$\operatorname{Gr}_{i}^{W}H^{2}(D)\to\operatorname{Gr}_{i}^{W}H^{2}(\tilde{D})$
is injective for $i=3,4$.

Now, note that the vertical lines in ${\tilde{E}}$${E}$${\tilde{D}}$${D}$ are
proper maps. As $\tilde{E}$ and $\tilde{D}$ are smooth, we can use Lemma 2
together with the above horizontal and left-vertical injectivities to conclude
that $\operatorname{Gr}_{i}^{W}H^{2}(D)\to\operatorname{Gr}_{i}^{W}H^{2}(E)$
is injective as well, for $i=3,4$.∎

######  Proposition 11.

Let $f:\tilde{Y}\to Y$ be a resolution of singularities of a quasi-projective
variety, then the induced morphism
$\operatorname{Gr}_{i}^{W}H^{2}(Y)\to\operatorname{Gr}_{i}^{W}H^{2}(\tilde{Y})$
is injective for $i=3,4$.

###### Proof.

Consider again the Mayer-Vietoris sequence for the discriminant square
${E}$${\tilde{Y}}$${D}$${Y}$, with the notation of the previous lemma.

Note that, for $i=3,4$,
$\operatorname{Gr}_{i}^{W}H^{2}(Y)\to\operatorname{Gr}_{i}^{2}H^{2}(\tilde{Y})$
is injective, if
$\operatorname{Gr}_{i}^{W}H^{2}(D)\to\operatorname{Gr}_{i}^{W}(E)$ is
injective.

Indeed, as $H^{1}(E)$ has only weights $0,1,2$, the map
$\operatorname{Gr}_{i}^{W}H^{2}(Y)\to\operatorname{Gr}_{i}^{W}H^{2}(D)\oplus\operatorname{Gr}_{i}^{W}H^{2}(\tilde{Y})$
is injective for $i=3,4$. We conclude by the previous Lemma.∎

$\bullet$ Weight 2. We still have to deal with the weight $2$ graded part of
the cohomology of $A,Z^{\prime},Z$. Let
$\overline{A},\overline{Z^{\prime}},\overline{Z}$ be compatible
compactifications of $A,Z^{\prime},Z$ respectively, the compactifications of
$A$ and $Z$ being ‘good’. Recall that $A:=Alb(X)$.

We thus have a comutative diagram, in which the maps are the obvious ones:

$\begin{matrix}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.54083pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.87875pt\raise 5.98889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.62779pt\hbox{$\scriptstyle{j}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.54083pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-18.80554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-26.22221pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.54083pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.04083pt\raise-18.80554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.98611pt\hbox{$\scriptstyle{d^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.04083pt\raise-28.93332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.32875pt\raise-31.06664pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.18335pt\hbox{$\scriptstyle{j^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
31.54083pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-56.41663pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-65.38885pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.54083pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{Z^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.04083pt\raise-56.41663pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{a^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.04083pt\raise-66.5444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.01834pt\raise-69.79161pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
31.54083pt\raise-75.22217pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.54083pt\raise-75.22217pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{A}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{matrix}$

This diagram induces the following commutative diagram:

$\begin{matrix}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
26.75148pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Gr^{W}_{2}H^{2}(Z)}$}}}}}}}{\hbox{\kern
50.75148pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{2}(\overline{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.40977pt\raise-6.45416pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.09306pt\hbox{$\scriptstyle{j^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 25.98149pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
106.22427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.45427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
146.20259pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
121.4326pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Gr^{W}_{2}H^{2}(Z^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.89583pt\hbox{$\scriptstyle{d^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.75148pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{2}(\overline{Z^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.85977pt\raise-47.68219pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.18335pt\hbox{$\scriptstyle{j^{\prime*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
26.7515pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.35948pt\raise-20.56888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.98611pt\hbox{$\scriptstyle{d^{\prime*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.10287pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.45427pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
103.96385pt\raise-20.56888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{k}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.82843pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.45427pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
146.20259pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
122.20259pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Gr^{W}_{2}H^{2}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.97223pt\hbox{$\scriptstyle{a^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-46.63776pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.75148pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{2}(\overline{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.30276pt\raise-61.70663pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0625pt\hbox{$\scriptstyle{a^{\prime*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
66.10287pt\raise-46.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.54936pt\raise-88.17134pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.89583pt\hbox{$\scriptstyle{b^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
25.96066pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
107.42564pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.45427pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
113.82843pt\raise-44.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
146.20259pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
120.23122pt\raise-82.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{matrix}$

The kernels $K,K^{\prime},L$ are naturally defined. We have replaced
$Gr_{2}^{W}H^{2}(\bullet)$ by $H^{2}(\bullet)$ in the top and bottom lines of
the central column. The map $j^{\prime*}$ is well-defined because
$H^{2}(\overline{Z^{\prime}})$ has weights $0,1,2$ only.

Moreover, the map $b^{*}$ (and $j^{*}$ as well) is surjective, by [Del71],
Corollaire 3.2.17.

We want to show that $Ker(d^{*}a^{*})=Ker(a^{*})$, or equivalently, since
$b^{*}$ is surjective, that $Ker(d^{*}a^{*}b^{*})=Ker(a^{*}b^{*})$, which
means that $Ker(j^{*}d^{\prime*}a^{\prime*})=Ker(j^{\prime*}a^{\prime*})$, or
still equivalently, that
$(d^{\prime*}a^{\prime*})^{-1}(K)=(a^{\prime*})^{-1}(K^{\prime})$.

The Gysin maps show that $L$ (resp. $K$) is the complex subspace of
$H^{2}(\overline{A})$ (resp. $H^{2}(\overline{Z}$)) generated by the Chern
classes of the irreducible components of $D_{A}:=\overline{A}\setminus A$
(resp. of $D_{Z}:=\overline{Z}\setminus Z$).

Let similarly $D_{Z^{\prime}}:=\overline{Z^{\prime}}\setminus Z^{\prime}$. We
obviously have that $K^{\prime}$ contains $K"$, the complex vector subspace of
$H^{2}(\overline{Z^{\prime}})$ generated by the Chern classes of the
components222More precisely: of sums of such components of the form
$(a^{\prime})^{-1}(D)$, for $D$ any component of $D_{\overline{A}}$, which are
Cartier.of $D_{Z^{\prime}}=(a^{\prime})^{-1}(D_{A})$ .

Indeed, since $a_{Z}:Z\to Alb(Z)=:A$ is proper,
$D_{Z}=(a^{\prime}d^{\prime})^{-1}(D_{A})$, and
$D_{Z^{\prime}}=(a^{\prime})^{-1}(D_{A})$.

From [Del74],Proposition 8.2.7, we get:
$Ker(a^{\prime*})=Ker((a^{\prime}d^{\prime})^{*}):=J$.

Thus $(d^{\prime*}a^{\prime*})^{-1}(K)=L+J$, and contains
$(a^{\prime*})^{-1}(K^{\prime})$, which itself contains $L+J$, since
$K^{\prime}$ contains $K"$. But $L+J=((a^{\prime}d^{\prime})^{*})^{-1}(K)$
contains $(a^{\prime*})^{-1}(K^{\prime})$, which contains $L+J$ and we get the
claimed equality:
$Ker(a^{\prime}d^{\prime}j)^{*}=Ker(a^{\prime}d^{\prime})^{*}+L=L+Ker(a^{\prime*})=Ker(aj^{\prime})^{*}$∎

The preceding proof shows that the injectivity statement for $Gr^{W}_{2}H^{2}$
holds more generally in the situation of the next Lemma 12, which gives a
partial extension of Deligne’s result [Del74], Proposition 8.2.7, used in an
essential way in the proof.

######  Lemma 12.

Let $f:X\to Y,g:Y\to A$ be two regular maps between quasi-projective
varieties, such that $X,A$ is smooth, $f$ is surjective, and $g\circ f$ is
proper. Then $Ker(g\circ f)^{*}:Gr^{W}_{2}H^{2}(A)\to Gr^{W}_{2}H^{2}(X))$ and
$Ker(f)^{*}:Gr^{W}_{2}H^{2}(A)\to Gr^{W}_{2}H^{2}(Y))$ coincide.

######  Remark 13.

We do not know the answer to the following question. Let $f:X\to Y$ and
$g:Y\to A$ be regular proper maps between quasi-projective varieties, such
that $X$ and $A$ are smooth and $f$ is surjective. Does $Ker(g\circ
f)*:Gr_{i}^{W}H^{k}(A)\to Gr_{i}^{W}H^{k}(X$) and
$Ker(f)*:Gr_{i}^{W}H^{k}(A)\to Gr_{i}^{W}H^{k}(Y)$ coincide for all $i$?

######  Remark 14.

The following examples, communicated to us by M. Saito [Sai], show that
Proposition 11 does not necessarily hold for higher degrees: if
$f:\tilde{Y}\to Y$ is a resolution of singularities of a quasi-projective
variety, the induced morphisms
$\operatorname{Gr}_{i}^{W}H^{n}(Y)\to\operatorname{Gr}_{i}^{W}H^{n}(\tilde{Y})$
do not need to be injective for $i=n,\ldots,2n-2$ if $2n-2\geq n$.

  * •

Assume $X=X_{1}\times X_{2}$ with $X_{1}$ smooth and $X_{2}$ compact, e.g,
$X_{1}=\mathbb{C}^{*}$ and $X_{2}$ is the union of the zero section and a
section of a very positive line bundle on $\mathbb{P}^{2}$ whose zero locus
$C$ is smooth. Then $\tilde{X_{2}}=\mathbb{P}^{2}\cup\mathbb{P}^{2}$, but
$\operatorname{Gr}^{W}_{3}H^{3}(X)$ contains
$\operatorname{Gr}^{W}_{2}H^{1}(X_{1})\times Gr^{W}_{1}H^{2}(X_{2})$ and
$Gr^{W}_{1}H^{2}(X_{2})=H^{1}(C)(-1)$.

We can also have $X_{2}$ irreducible. Let $X=X_{1}\times X_{2}$ with
$X_{1}=\mathbb{C}^{*}$ and $X_{2}$ the compactification of $f=f_{1}+f_{2}=0$
in $\mathbb{C}^{3}$ with $f_{1}$,$f_{2}$ homogeneous polynomials of degree $d$
and $d+1$ having isolated singularities at $0$. Here $\tilde{X_{2}}$ is the
blow-up of $\mathbb{P}^{2}$ along the intersections of the two curves defined
by $f_{1}$, $f_{2}$, which are assumed to be transversal.

  * •

For the case $n$ is at most $2\dim X-2$, consider for instance
$X=\mathbb{C}^{*}\times D\times A$ where $D$ is a plane curve with lots of
nodes and $A$ is an abelian variety

##  4\. Proof of Theorem 5

Before giving the proof, we need to define the quasi-Albanese map
$\alpha_{X}:X\to Alb(X)$ for $X$ normal quasi-projective, inducing an
isomorphism at $H^{1}$ level. This is done in the Appendix on Weight-$1$ MHS
(where it is denoted $\alpha_{X}:X\to A_{X}$). The crucial property used is:
$H^{1}(X)=H^{1}(Alb(X))$ for coefficients either ${\mathbb{Q}},{\mathbb{R}}$,
or ${\mathbb{C}}$ (more precisely, this is true with
${\mathbb{Z}}$-coefficients, up to torsion).

Proof of Theorem 5: We consider a smooth model $\tilde{X}$ of $X$ and a normal
model $Z_{0}^{\prime}$ of the image $Z_{0}^{\prime\prime}=\alpha_{X}(X)$.
Consider the composed map $\tilde{X}\to Z^{\prime}\to A:=Alb(X)$. This
composed map is proper, since so are $\tilde{X}\to X$ and $X\to A$, by
assumption. Moreover, $\tilde{X}$ and $A$ are smooth quasi-projective, and
$\tilde{X}\to Z^{\prime}$ is surjective. By Lemma 12, the kernels of
$\operatorname{Gr}^{W}_{i}H^{2}(A)\to\operatorname{Gr}^{W}_{i}H^{2}(\tilde{X})$
and of
$\operatorname{Gr}^{W}_{i}H^{2}(A)\to\operatorname{Gr}^{W}_{i}H^{2}(Z^{\prime})$
coincide, and so thus do the kernels of
$\operatorname{Gr}^{W}_{i}H^{2}(A)\to\operatorname{Gr}^{W}_{i}H^{2}(X)$ and of
$\operatorname{Gr}^{W}_{i}H^{2}(A)\to\operatorname{Gr}^{W}_{i}H^{2}(Z^{\prime})$.
Moreover, the injectivity for $i=3,4$ of the maps $Gr^{W}_{i}H^{2}(X)\to
Gr^{W}_{i}H^{2}(\tilde{X})$ is garanteed by Lemma 11. We have the same
property for a desingularisation $\tilde{Z^{\prime}}\to Z^{\prime}$, from
which we deduce (as in the proof of Proposition 11), the injectivity of
$Gr^{W}_{i}H^{2}(Z^{\prime})\to Gr^{W}_{i}H^{2}(X)$, and finally the equality
of the kernels of $H^{2}(A)\to H^{2}(Z^{\prime})$ and of $H^{2}(A)\to
H^{2}(X)$, which gives the conclusion of Theorem 5, since
$dimH^{1}(X,\mathbb{C})=dim(Alb(X))$.

######  Remark 15.

The kernel of the map $\pi_{1}(\widetilde{X})\to\pi_{1}(X)$ may be large, as
shown by the cone $X$, either affine or projective, over a smooth projective
manifold $E$, since $X$ is simply connected, while
$\pi_{1}(\widetilde{X})\cong\pi_{1}(E)$. More generally:

######  Question 16.

Let $d:\widetilde{X}\to X$ be the desingularisation of a normal quasi-
projective variety, and $d_{*}:\pi_{1}(\widetilde{X})\to\pi_{1}(X)$ the
induced morphism. Is the kernel $K$ of $d_{*}$ equal to the group $K^{\prime}$
generated by the images in $\pi_{1}(X)$ of the $\pi_{1}(F)$, when $F$ runs
through the fibres of $d$? It is clear that $K^{\prime}\subset K$.

The second property stated in Proposition 28 shows that this is true for the
torsionfree Abelianisations of the $\pi_{1}$’s, and Theorem 5 that it extends
to the torsionfree nilpotent completions when $\alpha_{X}$ is proper.

##  5\. Special Manifolds

We show in this section that the Albanese map, if proper, is surjective with
connected fibres if $X$ is a ‘special’ quasi-projective manifold. For
motivation and details about the class of special manifolds, see
[Cam04],[Cam11], and [Cam17]. Recall first one of the possible definitions:

######  Definition 17.

Let $X$ be a quasi-Kähler manifold together with a good compactification
$X=\overline{X}\setminus D$. We say that $X$ is special if, for any integer
$p>0$ and any $L\subset\Omega^{p}_{\overline{X}}(Log(D))$, coherent and of
rank $1$, we have: $\kappa(\overline{X},L)<p$.

This definition is short, but does not explicitely show the geometric
significance of this notion: no surjective algebraic map $f:X\to Y$ has an
‘orbifold base’ $(Y,\Delta_{f})$ of Log-general type. Special manifold should
be considered as higher-dimensional generalisations of quasi-projective curves
with Log-canonical bundle of non-positive degree (ie: rational or elliptic
curves in the compact case).

The importance of the notion of specialness comes from the existence, for any
$X$, of its ‘core fibration’ $c:X\to C(X)$, which splits $X$ into its
antithetic parts: special (the general fibres), and Log-general type (the
‘orbifold base’ $(C(X),\Delta_{c})$.

Recall that the log-canonical bundle of $X$ is
$K_{\overline{X}}+D=det(\Omega^{1}_{\overline{X}}(Log(D)))$, and that
$\overline{\kappa}(X):=\kappa(\overline{X},K_{\overline{X}}+D)$. Also, $X$ is
compact if and only if $D=0$, in which case $\overline{\kappa}(X)=\kappa(X)$.

######  Proposition 18.

Let $X$ be a quasi-Kähler manifold $X$.

1\. If $X$ is special, and if $f:X\to Z$ is a dominant meromorphic map, then
$\overline{\kappa}(Z)<dim(Z):=p>0$.

2\. if $\overline{\kappa}(X)=0$, then $X$ is special.

###### Proof.

First claim: Since
$L:=f^{*}(K_{\overline{Z}}+D_{Z}))\subset\Omega^{p}_{\overline{X}}(Log(D))$,
$f$ induces, for any $m>0$, an injective linear map
$f^{*}:H^{0}(Z,m.(K_{\overline{Z}}+D_{Z}))\to
H^{0}(X,Sym^{m}(\Omega^{p}_{\overline{X}}(Log(D)))$, where $(\overline{X},D)$
and $(\overline{Z},D_{Z})$ are good compactifications of $X,Z$ respectively.
We thus have:
$p=\kappa(\overline{Z},K_{\overline{Z}}+D_{Z})=\kappa(\overline{X},f^{*}(K_{\overline{Z}}+D_{Z}))$,
contradicting the specialness of $X$.

The second claim is [Cam11], Théorème 7.7. ∎

######  Corollary 19.

Let $X$ be a special333The proof shows that the conclusion holds more
generally if $X$ is ‘weakly special’: no finite étale cover of $X$ maps onto a
variety of Log-general type. quasi-Kähler manifold. Assume that
$\alpha_{X}:X\to Alb(X)$ is proper. Then every torsionfree nilpotent quotient
of $\pi_{1}(X)$ is abelian, quotient of $H_{1}(X,\mathbb{Z})/Torsion$.

###### Proof.

This is an immediate consequence of Corollary 2.2 and the next Lemma 20. ∎

######  Lemma 20.

Let $\alpha_{X}:X\to Alb(X)$ be the quasi-Albanese map of the quasi-
projective444Here again, the conclusion should hold in the quasi-Kähler case.
manifold $X$. Then $\alpha_{X}$ is dominant (hence surjective if it is proper)
and has connected fibres if $X$ is special, in particular if
$\overline{\kappa}(X)=0$.

###### Proof.

It is sufficient to show that any Zariski-closed subset $V\subsetneq Alb(X)$
of $Alb(X)$ admits a dominant map onto a positive-dimensional quasi-projective
$W$ of general type, that is such that $\overline{\kappa}(W)=dim(W)$.This
follows from [Kaw81], Theorem 27. ∎

######  Remark 21.

The conclusion of Corollary 19 may fail if $\alpha_{X}$ is not proper, even if
$K_{\overline{X}}+D$ is trivial. See Example 24 below.

Recall the Abelianity conjecture.

######  Conjecture 22.

([Cam04], Conjecture 7.1)Let $X$ be a compact Kähler manifold. If $X$ is
special, then $\pi_{1}(X)$ is virtually abelian.

This conjecture is shown to be true for the linear quotients of Kähler groups
([Cam04], Theorem 7.8). It is stated also in the quasi-Kähler case in [Cam11],
Conjecture 12.10. However, non-compact special quasi-Kähler manifolds with
torsionfree nilpotent, non-abelian, fundamental groups do exist, as first
noticed in [CDY22]. See examples below. In the open case, we thus restate this
as the ‘Nilpotency conjecture’ (as done in [CDY22]):

######  Conjecture 23.

If $X$ is a special quasi-Kähler manifold, $\pi_{1}(X)$ is virtually nipotent.

If $X$ is ‘slope-rationally connected’, then $\pi_{1}(X)$ is virtually
abelian.

Recall (see [Cam17]) that $X$ is ‘slope rationally connected’ if quasi-
projective, and if
$h^{0}(\overline{X},(\otimes^{m}\Omega^{1}_{\overline{X}}Log(D))\otimes H)=0$,
for any ample line bundle $H$ on $\overline{X}$, provided $m\geq
m(\overline{X},D,H)$. When $X$ is compact (i.e: $D=0$), this means that $X$ is
rationally connected.

##  6\. Remarks and examples in the non-proper case

######  Example 24.

Let $B$ be a connected complex manifold, $L$ a line bundle on $B$,
$B^{\prime}$ its zero section, and $X:=L\setminus B^{\prime}$. If $B$ is
quasi-projective, so is $X$. Notice that $X\to B$ also appears as the
complement of the zero and infinity sections of
$\mathbb{P}(L\oplus\mathcal{O}_{B})\to B$.

The Gysin exact sequence for $L,X$ reads ([MS74], Theorem 12.2, with
$\mathbb{Z}$ coefficients), the fourth map being multiplication by $c_{1}(L)$:

$0\to H^{1}(B)\to H^{1}(X)\to H^{0}(B)\to H^{2}(B)\to H^{2}(X)\to H^{1}(B)$

Hence $H^{1}(X,\mathbb{Z})=H^{1}(B,\mathbb{Z})$ if $c_{1}(L)\neq 0$, moreover
$\pi_{1}(X)$ is then a central extension of $\pi_{1}(B)$ by $\mathbb{Z}$,
hence torsionfree nilpotent if so is $\pi_{1}(B)$. If $\pi_{1}(B)$ is
torsionfree abelian, the nilpotency class of $\pi_{1}(X)$ is $2$. This happens
if $B$ is, for example, an elliptic curve.

If the Log-canonical bundle of $B$ is moreover trivial, then so is the Log-
canonical bundle of $X$, by a simple computation.

Let us first take for $B$ an elliptic curve, and $L$ of nonzero degree $d$.
Then $alb_{X}:X\to B:=Alb(X)$ is here not proper. The Logarithmic $1$-forms on
$X$ then all come from the regular $1$-forms on $B$. Finally, $\pi_{1}(X)$ is
an extension of $\pi_{1}(A)$ by $\pi_{1}(\mathbb{C}^{*})=\mathbb{Z}$, hence
torsionfree nilpotent (by [Cam95], Appendix, Lemme A.1). It is a Heisenberg
group, which is not Kähler ([Cam95]), but ‘quasi-Kähler’.

If we chose $B=\mathbb{C}^{*}\times\mathbb{C}^{*}$, with $L$ of any bidegree
$(a,b)$ on its compactification $P:=\mathbb{P}_{1}\times\mathbb{P}_{1}$,
$\pi_{1}(X)$ is abelian of rank $3$, since the natural restriction map
$H^{2}(P,{\mathbb{Z}})\to H^{2}(B)$ vanishes. In order to construct a quasi-
projective ${\mathbb{C}}^{*}$-bundle over
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$, one needs to change the algebraic
structure, and chose as a good compactification the projective bundle
$\pi:{\mathbb{P}}(E)\to C$, where $E$ is the non-trivial extension of two
trivial line bundles over the elliptic curve $C$. Then
${\mathbb{P}}(E)\setminus B^{\prime}$ is analytically, not algebraically,
isomorphic to $B:={\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$. We obtain a
${\mathbb{C}}^{*}$-bundle with non-abelian fundamental group over $B$ by
lifting one from $C$ as in the previous example.

After some versions of the current paper, the authors learned that the above
examples were also considered in [HZ87, 5.45].

######  Remark 25.

The preceding examples when $c_{1}(L)\neq 0$ show that two other properties of
fibrations $f:X\to Y$ between compact Kähler manifolds do not extend to the
case of non-proper surjective maps between quasi-Kähler manifolds, even if $Y$
is a projective curve and the map $f$ is a $\mathbb{C}^{*}$-bundle:

1\. Even if the Log-canonical bundle is trivial (which is the case of
$X=L\setminus B^{\prime}$ in the previous example), $\pi_{1}(X)$ need not be
virtually abelian. By contrast, recall that $\pi_{1}(X)$ is virtually abelian
when $X$ is a compact Kähler manifold with $c_{1}(X)=0$.

2\. If $f:X\to Y$ is a ‘neat’ fibration555‘neatness’ is achieved by suitably
blowing up $X$ and $B$. between compact Kähler manifolds, such that
$\pi_{1}(X)$ is torsionfree nilpotent, of nilpotency class $\nu(\pi_{1}(X))$,
then [Cam98], Corollaire 5.2.1 shows that

$\nu(\pi_{1}(X))=max\\{\nu(\pi_{1}(X_{y})_{X}),\nu(\pi_{1}(B^{orb}))\\}$

if $\pi_{1}(X_{y})_{X}$ is the image of $\pi_{1}(X_{y})$ in $\pi_{1}(X)$, for
$X_{y}$ any smooth fibre of $f$. Here $B^{orb}$ is the orbifold base of $f$.
In particular: $\pi_{1}(X)$ is virtually abelian if so are $\pi_{1}(Y)$ and
$\pi_{1}(X_{y})_{X}$.

For non-Kähler nilmanifolds, $\nu(\pi_{1}(X))$ is the sum, not the $Max$ of
the other two terms.

This shows that, at least for projective manifolds, one cannot construct
torsionfree nilpotent Kähler groups of higher nilpotency class by constructing
suitable fibrations. The only examples presently known seem to be of
nilpotency class $2$.

######  Question 26.

Let $G$ be a torsionfree nilpotent Kähler group. Can its nilpotency class
$\nu(G)$ be at least 3? Can $\nu(G)$ be arbitrarily large? Same questions for
the quasi-Kähler groups.

By contrast, the implication that $\pi_{1}(X)$ is virtually nilpotent if it is
polycyclic remains true for $X$ quasi-Kähler ([AN99], corollary 4.10). It
would be interesting to know if this remains true for solvable fundamental
groups of quasi-Kähler manifolds, as Delzant theorem ([Del10]) asserts that it
is the case for Kähler groups.

######  Remark 27.

The conclusions about the nilpotent cover and the universal cover being
holomorphically convex commonly fail, as the following easy examples show. In
the non-compact case, the conjecture of Shafarevich can thus extend only under
restrictive conditions.

Let $M$ be an affine, or projective, smooth variety of dimension $n\geq 2$,
and set $X:=M\setminus N$, where $N$ is a nonempty closed analytic subset of
codimension at least $2$. Then $X$ is not holomorphically convex. If the
universal cover $\widetilde{M}$ of $M$ is Stein, the universal cover
$\widetilde{X}\subset\widetilde{M}$ of $X$ is not holomorphically convex. This
happens by taking $M=\mathbb{C}^{n},\mathbb{P}_{n}$, or an Abelian variety.

A more interesting example appears in [Win90], in which ones removes a
suitable $N$ of codimension $2$ from the $4$-dimensional affine quadric $M$,
the complement $X$ being diffeomorphic to $\mathbb{C}^{4}$.

##  7\. Appendix: Weight-$1$ MHS

The constructions made here are without doubt well-known, but we were unable
to find a reference for them in the non-compact case.

We start with two holomorphic maps between irreducible quasi-projective
varieties: $f:V\to\widetilde{X},g:\widetilde{X}\to Y$, where $\widetilde{X}$
is smooth, $Y$ normal, and $g$ is surjective and proper, with connected
fibres. For $g:\widetilde{X}\to Y$, and the proof of Theorem 5 we may use
$d:\widetilde{X}\to X$, the desingularisation of a normal quasi-projective
$X$.

The maps $f,g$ induce $f^{*}:H^{1}(\widetilde{X})\to H^{1}(V)$ and
$g^{*}:H^{1}(Y)\to H^{1}(\widetilde{X})$, $g^{*}$ is injective.

We shall define the Albanese map of $Y$ and describe some of its standard
properties.

Let $H^{1}(Y):=Image(g^{*})$. This is a sub-MHS of $H^{1}(\widetilde{X})$
defined over ${\mathbb{Z}}$. It thus defines666The construction for $M\subset
H^{1}(\widetilde{X})$ is the same as for the quasi-Albanese variety: take the
quotient $A_{M}:=(F^{1}(M))^{*}/M_{{\mathbb{Z}}}$, $F^{\bullet}$ being the
Hodge filtration. a semi-Abelian variety $A_{Y}$, quotient of
$A:=Alb(\widetilde{X})$. Denote by $q_{Y}:A\to A_{Y}$ the quotient map, and by
$\alpha:\widetilde{X}\to A:=Alb(\widetilde{X})$ the Albanese map.

######  Proposition 28.

If $V$ is either compact, or smooth, the following properties are equivalent:

1\. $(g\circ f)^{*}:H^{1}(Y)\to H^{1}(V)$ vanishes.

2\. Each Logarithmic one-form $u$ on $\widetilde{X}$ which belongs to
$g^{*}(H^{1}(Y))$ vanishes on the smooth points of $V$.

3\. $q_{Y}\circ f:V\to Alb_{Y}$ maps $V$ to a point.

###### Proof.

That $2$ implies $3$ is clear, since $A_{Y}$ is obtained by integrating these
$1$-forms along paths in $\widetilde{X}$. That $3$ implies $1$ follows from
the equality $H^{1}(A_{Y})=H^{1}(Y)$, true by the construction of $A_{Y}$. The
last implication $1\Longrightarrow 2$ is clear if $V$ is smooth, since the
logarithmic $1$-forms on $\widetilde{X}$ pulled back by $f$ on $V$ are
elements of $H^{1}(V)$, and vanish cohomologically only if they vanish
identically. If $V$ is projective (but not necessarily smooth), the assertion
again follows from [Del74], 8.2.7, which says that $Ker(f^{*})=Ker(g\circ
v)^{*}:H^{1}(\widetilde{X})\to H^{1}(\widetilde{V})$, if $v:\widetilde{V}\to
V$ is a desingularisation. One can thus assume $V$ to be smooth and apply the
preceding argument. ∎

######  Corollary 29.

If $g:\widetilde{X}\to Y$ is proper, surjective, with $Y$ normal and
$\widetilde{X}$ smooth, there is a unique map $\alpha_{Y}:Y\to Alb_{Y}$ such
that $\alpha_{Y}\circ g=q_{Y}\circ g:\widetilde{X}\to Alb_{Y}$. This map is
called the Albanese map of $Y$. It does not depend on the choice of
$g,\widetilde{X}$, and enjoys the usual universal property of Albanese maps.

###### Proof.

The irreducible components of the fibres of $g$ are compact, by assumption,
and satisfy the first property of Proposition 28. They are thus mapped to
points by $q_{Y}\circ\alpha$, as this Proposition shows, which implies the
claim. We do not use the two other statements, and do not prove them (the
proof is standard). ∎

######  Corollary 30.

For each component $G$ of any fibre of $g$, denote by
$f_{G}:G\to\widetilde{X}$ the inclusion map, and
$K_{G}:=Ker(f_{G}^{*}:H^{1}(\widetilde{X})\to H^{1}(G))$. Then
$g^{*}(H^{1}(Y))=\cap_{G}K_{G}\subset H^{1}(\widetilde{X})$.

###### Proof.

The inclusion $g^{*}(H^{1}(Y))\subset\cap_{G}K_{G}$ is clear. In the opposite
direction: $H:=\cap_{G}K_{G}$ is a sub MHS of $H^{1}(\widetilde{X})$ defined
over ${\mathbb{Z}}$, and we thus have an associated quotient semi-Abelian
variety $A_{Y}^{*}$ and a quotient map $q_{g}:A\to A_{Y}^{*}$. By
construction, all $G^{\prime}s$ are mapped to points of $A_{Y}^{*}$ by
$q_{g}\circ\alpha\circ f_{G}:G\to A_{Y}^{*}$. We thus get a unique
$\alpha_{Y}^{*}:Y\to A_{Y}^{*}$ such that $\alpha^{*}_{Y}\circ
g=q_{g}\circ\alpha:\widetilde{X}\to A_{Y}^{*}$. Since
$q_{g}^{*}(H^{1}(A_{Y}^{*}))=H$ contains
$g^{*}(H^{1}(Y))=\alpha^{*}(q_{Y}^{*}(H^{1}(A_{Y})))$, we have a natural
quotient $q^{\prime}:A_{Y}^{*}\to A_{Y}$ such that
$q^{\prime}\circ\alpha_{Y}^{*}=\alpha_{Y}:Y\to A_{Y}$. The universal property
of $\alpha_{Y}:Y\to A_{Y}^{*}$ implies that $q^{\prime}$ is isomorphic. Since
we did not show this universal property, let us prove directly that
$H=g^{*}(H^{1}(Y))$: we indeed have
$(q_{g}\circ\alpha)^{*}(H^{1}(A_{Y}^{*}))=\alpha^{*}(q_{g}^{*}(H^{1}(A_{Y}^{*})))=H$.

On the other hand, since $q_{g}\circ\alpha=q_{Y}^{*}\circ g$, $H$ is also
equal to:

$g^{*}((q_{Y}^{*})^{*}(H^{1}(A_{Y}^{*})))\subset g^{*}(H^{1}(Y))$. We thus get
$H=g^{*}(H^{1}(H))$, and $A_{Y}=A_{Y}^{*}$. ∎

######  Remark 31.

The Albanese map $q_{Y}:Y\to A_{Y}$ defined here coincides for $s=1$ with the
higher Albanese maps as defined by R. Hain. The Albanese map $q_{Y}^{*}:Y\to
A_{Y}^{*}$ coincides with the one defined in [CDY22], remark 1.2. We thus
showed that all three coincide, and in particular that
$H^{1}(A_{Y}^{*})=H^{1}(Y)$, which depends on MHS considerations (at least
using the proof given here).

## References

  * [ADH16] Donu Arapura, Alexandru Dimca, and Richard Hain, _On the fundamental groups of normal varieties_ , Commun. Contemp. Math. 18 (2016), no. 4, 1550065, 17. MR 3493218
  * [AN99] Donu Arapura and Madhav Nori, _Solvable fundamental groups of algebraic varieties and Kähler manifolds_ , Compositio Math. 116 (1999), no. 2, 173–188. MR 1686777
  * [BPVdV84] W. Barth, C. Peters, and A. Van de Ven, _Compact complex surfaces_ , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
  * [BT82] Raoul Bott and Loring W. Tu, _Differential forms in algebraic topology_ , Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982\. MR 658304
  * [Cam91] Frédéric Campana, _On twistor spaces of the class $C$_, J. Differential Geom. 33 (1991), no. 2, 541–549. MR 1094468
  * [Cam95] by same author, _Remarques sur les groupes de Kähler nilpotents_ , Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 3, 307–316. MR 1326670
  * [Cam98] by same author, _Connexité abélienne des variétés kählériennes compactes_ , Bull. Soc. Math. France 126 (1998), no. 4, 483–506. MR 1693441
  * [Cam04] by same author, _Orbifolds, special varieties and classification theory_ , Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499–630. MR 2097416
  * [Cam11] by same author, _Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes_ , J. Inst. Math. Jussieu 10 (2011), no. 4, 809–934. MR 2831280
  * [Cam17] by same author, _Orbifold slope-rational connectedness_ , 2017.
  * [CDY22] Benoit Cadorel, Ya Deng, and Katsutoshi Yamanoi, _Hyperbolicity and fundamental groups of complex quasi-projective varieties_ , 2022.
  * [Del71] Pierre Deligne, _Théorie de Hodge. II_ , Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57. MR 498551
  * [Del74] by same author, _Théorie de Hodge. III_ , Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77. MR 498552
  * [Del10] Thomas Delzant, _L’invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes_ , Math. Ann. 348 (2010), no. 1, 119–125. MR 2657436
  * [GGK22] Mark Green, Phillip Griffiths, and Ludmil Katzarkov, _Shafarevich mappings and period mappings_ , 2022.
  * [Hai97] Richard Hain, _Infinitesimal presentations of the Torelli groups_ , J. Amer. Math. Soc. 10 (1997), no. 3, 597–651. MR 1431828
  * [HZ87] Richard M. Hain and Steven Zucker, _Unipotent variations of mixed Hodge structure_ , Invent. Math. 88 (1987), no. 1, 83–124. MR 877008
  * [Iit82] Shigeru Iitaka, _Algebraic geometry_ , North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982, An introduction to birational geometry of algebraic varieties. MR 637060
  * [Kat97] L. Katzarkov, _Nilpotent groups and universal coverings of smooth projective varieties_ , J. Differential Geom. 45 (1997), no. 2, 336–348. MR 1449976
  * [Kaw81] Yujiro Kawamata, _Characterization of abelian varieties_ , Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
  * [MS74] John W. Milnor and James D. Stasheff, _Characteristic classes_ , Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
  * [Sai] Morihiko Saito, Personal communication.
  * [Sta65] John Stallings, _Homology and central series of groups_ , J. Algebra 2 (1965), 170–181. MR 175956
  * [Win90] Jörg Winkelmann, _On free holomorphic $\bf C$-actions on ${\bf C}^{n}$ and homogeneous Stein manifolds_, Math. Ann. 286 (1990), no. 1-3, 593–612. MR 1032948


# Accelerating Domain-aware Deep Learning Models with Distributed Training

Aishwarya Sarkar  Chaoqun Lu and Ali Jannesari  Iowa State University, USA
{asarkar1, clu<EMAIL_ADDRESS>
Keywords: domain-knowledge, model parallelism, distributed training,
knowledge-guided neural network

##  1 Abstract and Motivation

Recent advances in data-generating techniques led to an explosive growth of
geo-spatiotemporal data. In domains such as hydrology, ecology, and
transportation, interpreting the complex underlying patterns of spatiotemporal
interactions with the help of deep learning techniques hence becomes the need
of the hour. However, applying deep learning techniques without domain-
specific knowledge tends to provide sub-optimal prediction performance.
Secondly, training such models on large-scale data requires extensive
computational resources. To eliminate these challenges, we present a novel
distributed domain-aware spatiotemporal network that utilizes domain-specific
knowledge with improved model performance. Our network consists of a pixel-
contribution block, a distributed multiheaded multichannel convolutional (CNN)
spatial block, and a recurrent temporal block. We choose flood prediction in
hydrology as a use case to test our proposed method. From our analysis, the
network effectively predicts high peaks in discharge measurements at watershed
outlets with up to 4.1x speedup and increased prediction performance of up to
93%. Our approach achieved a 12.6x overall speedup and increased the mean
prediction performance by 16%. We perform extensive experiments on a dataset
of 23 watersheds in a northern state of the U.S. and present our findings.

Domain-aware deep learning (DL) models do not solely rely on data in hand.
Having some insight about the targetted domain is what makes them stand out.
Without explicit constraints, DL models are prone to make erroneous
predictions. Research shows that these models outperform their data-driven
counterparts in various applications such as human pose estimation [1],
studying animal models [2], breast cancer diagnosis [3], and seismic inversion
[4] among others. To make a DL model domain-aware, researchers have tried to
closely integrate them with their domain-specific counterparts [5]. Despite
these advances, the bottleneck of efficiently training, testing, and deploying
these models still remains a challenge. The recent breakthrough in high-
performance computing systems shows immense potential to make domain-aware DL
models overcome bottlenecks by making them scalable, and cost-effective [6,
7].

Global floods and extreme rainfall events have surged by more than 50% this
decade and are now occurring at a rate four times higher than in 1980.
Increasing evidence shows the huge potential of predicting and mitigating
flood occurrences in reducing yield loss, improving disaster risk management,
and benefiting the environment [8, 9, 10]. This work tackles the challenges in
DL-based pixellated spatiotemporal flood prediction models by introducing a
novel approach to somewhat eliminate its black box-like nature. It exploits
domain knowledge that comes from pixelated inputs by learning their pixel-
specific spatiotemporal contributions. We further propose a novel distributed
training approach to accelerate and optimize the training time of a pixellated
domain-aware spatiotemporal network. In summary, our contributions are:

  1. 1.

A domain-aware distributed spatiotemporal network (Dom-ST) with a novel pixel-
contribution block (Pix-Con) to compute pixel-specific weights that transforms
spatiotemporal inputs based on their local contribution to the water discharge
(Figure 1).

  2. 2.

A partitioning module in the Pix-Con block that partitions spatiotemporal
pixels dynamically and distributes them to multiple devices (Figure 1).

  3. 3.

A novel domain-guided distributed training approach to accelerate training
(Figure 2).

  4. 4.

An approach to utilize domain-significant temporal input measurements (the
target day) to increase domain awareness in Dom-ST.

  5. 5.

Competitive results in improving flood prediction by upto 93% and an overall
speedup of 12.6x with domain-guided distributed training,

Figure 1: The overall workflow of the proposed distributed spatiotemporal
network Dom-ST with (a) pixel-contribution (Pix-Con) block, (b) spatial block
with parallel convolution heads, and a recurrent (c) temporal block. The blue
boxes represent data, and the orange boxes represent various modules of Dom-
ST. The arrows show the data flow during inference.

##  2 Distribution Strategy

Since the climate and other environmental factors of a region, such as soil,
land cover, temperature, and human impact, vary from one location to another,
one global hydrological model, if trained, will either give suboptimal results
or will be computationally extensive. Model performance can be improved
further from the knowledge gained by modeling targetted regions. Besides, data
scarcity in remote areas is a challenge of its own which can be tackled to
some extent by transfer learning approaches [11]. Due to the huge amount of
complex multi-dimensional spatiotemporal data, training a hydrological model
based on deep learning approaches on a single device requires more resources
in terms of time and memory. Our distributed training approach tackles this
challenge (Figure 2). Firstly, the training data of the targetted site is
split into 23 regions or watersheds in the input pipeline (I.P.), which then
distributes the set of watersheds to multiple nodes (Figure 2.a.), where each
node is responsible for a single watershed. The I.P. also creates model
replicas and distributes them for each watershed to corresponding nodes.
Secondly, in each watershed node, we partition the model and offload it to
multiple GPUs by exploiting the parallel architecture of our spatial block
(Figure 2.b.). Each head of the CNN is separated and trained on a single
separate GPU making the heads train parallelly, thus optimizing the training
time. Once the forward passes of the spatial blocks are completed, the outputs
are transferred to the next GPU, which has the temporal block containing the
stacked LSTM layers and the following final layers of the model. The temporal
GPU also receives the second input of the model directly.

Figure 2: (a) The input pipeline distributes chunks of data (watersheds) and
creates Dom-ST replicas on multiple nodes (Left). (b) Each node runs
distributed Dom-ST using the host (CPU) and multiple devices (GPU) (Right).

We use pixellated daily precipitation from the Climate Research Unit and
distances of each pixel from the nearest water source as inputs and U.S.
Geological Survey’s daily discharge measurements [12] as labels. We selected
Nash-Sutcliffe efficiency (NSE) as our model evaluation metric [13]. Our
baseline model consists of only the spatial and temporal blocks. The spatial
block consists of singlehead one-dimensional multichannel 1D-CNN without any
Pix-Con block, which implies that the baseline is purely data-driven with no
domain knowledge. We refer to this as Singlehead. The baseline runs on a
single GPU. We explore how having an additional input of the target day’s
precipitation outperforms the baseline’s performance. Figure 3 shows that in
almost all the watersheds (91%), feeding the target day’s precipitation in the
final layers of the model (Singlehead$(+P)$) improves the performance. The
target day’s precipitation, the primary contributing factor of flash floods,
provides temporally fine-grained data to Dom-ST as a domain-specific cue to
improve the prediction. Next, we evaluate our proposed Dom-ST, which has a
Pix-Con block, a distributed multihead spatial block, and a temporal block
that receives the target day’s inputs. For comparison, we refer to it as
Distributed-Multihead(+P) in Figure 3. From our results, Distributed-
Multihead(+P) outperforms both Singlehead and Singlehead$(+P)$ in almost all
the watersheds. This supports that Dom-ST is more domain-aware than its
counterparts. To evaluate the input pipeline strategy, we compare the
computation time of Singlehead+$P$ and Distributed-Multihead(+P), each running
sequentially (S) and then parallelly (IP-D). Table 1 shows the total training
time of 23 watersheds in each case.

Figure 3: Comparison of prediction performance in terms of NSE [14] between 3
approaches - singlehead, singlehead with extra precipitation inputs
(Singlehead$(+P)$), and distributed-multihead with extra precipitation inputs
(Distributed-Multihead$(+P)$). Table 1: Evaluation of the input pipeline
strategy (IP-D).

Approach | Time (S) | Time (IP-D) | Speedup
---|---|---|---
Singlehead($+P$) | 9.96 hours |  1.18 hours | 8.5x
Distributed-Multihead($+P$) | 5.49 hours |  0.44 hours | 12.6x

##  3 Conclusion

We propose a novel distributed training approach to accelerate a domain-aware
spatiotemporal network. Through rigorous experiments and ablation studies, we
further explore how domain knowledge drives a model’s prediction and can also
be utilized in finding an approach for distributed training by exploiting
pixellated spatiotemporal data to infuse pixel-level contribution. Our
findings indicate that the use of Dom-ST leads to an average speedup of
$2.02$x across all the analyzed watersheds. The pixel-contribution block in
Dom-ST utilizes domain-knowledge and improves the model’s prediction
performance. Our experiments achieved the highest individual watershed speedup
of up to $4.11$x and an overall speedup of $12.6$x with the highest increase
in individual NSE by $93\%$.

## References

  * [1] Guanghan Ning, Zhi Zhang, and Zhiquan He.  Knowledge-guided deep fractal neural networks for human pose estimation.  IEEE Transactions on Multimedia, 20(5):1246–1259, 2018.
  * [2] Tao Zhong, Fenqiang Zhao, Yuchen Pei, Zhenyuan Ning, Lufan Liao, Zhengwang Wu, Yuyu Niu, Li Wang, Dinggang Shen, Yu Zhang, et al.  Dika-nets: Domain-invariant knowledge-guided attention networks for brain skull stripping of early developing macaques.  NeuroImage, 227:117649, 2021.
  * [3] Chen Chen, Yong Wang, Jianwei Niu, Xuefeng Liu, Qingfeng Li, and Xuantong Gong.  Domain knowledge powered deep learning for breast cancer diagnosis based on contrast-enhanced ultrasound videos.  IEEE Transactions on Medical Imaging, 40(9):2439–2451, 2021.
  * [4] Jian Sun, Kristopher A Innanen, and Chao Huang.  Physics-guided deep learning for seismic inversion with hybrid training and uncertainty analysis.  Geophysics, 86(3):R303–R317, 2021.
  * [5] Aishwarya Sarkar, Jien Zhang, Chaoqun Lu, and Ali Jannesari.  Hydrodeep–a knowledge guided deep neural network for geo-spatiotemporal data analysis.  arXiv preprint arXiv:2010.04328, 2020.
  * [6] Sixing Yu, Murali Emani, Chunhua Liao, Pei-Hung Lin, Tristan Vanderbruggen, Xipeng Shen, and Ali Jannesari.  Towards seamless management of ai models in high-performance computing.  arXiv preprint arXiv:2212.06352, 2022.
  * [7] Sixing Yu, Phuong Nguyen, Waqwoya Abebe, Wei Qian, Ali Anwar, and Ali Jannesari.  Spatl: salient parameter aggregation and transfer learning for heterogeneous federated learning.  In 2022 SC22: International Conference for High Performance Computing, Networking, Storage and Analysis (SC), pages 495–508. IEEE Computer Society, 2022.
  * [8] P. E. O’Connell, J. Ewen, G. O’Donnell, and P. Quinn.  Is there a link between agricultural land-use management and flooding?  Hydrology and Earth System Sciences, 11(1):96–107, 2007.
  * [9] Roger Moussa, Marc Voltz, and Patrick Andrieux.  Effects of the spatial organization of agricultural management on the hydrological behaviour of a farmed catchment during flood events.  Hydrological Processes, 16(2):393–412, 2002.
  * [10] Howard Wheater and Edward Evans.  Land use, water management and future flood risk.  Land Use Policy, 26:S251–S264, 2009.  Land Use Futures.
  * [11] Aishwarya Sarkar, Jien Zhang, Chaoqun Lu, and Ali Jannesari.  Transfer learning approaches for knowledge discovery in grid-based geo-spatiotemporal data.  In NeurIPS 2021 AI for Science Workshop, 2021.
  * [12] Christopher S Jones, Jacob K Nielsen, Keith E Schilling, and Larry J Weber.  Iowa stream nitrate and the gulf of mexico.  PloS one, 13(4):e0195930, 2018.
  * [13] Daniel N Moriasi, Jeffrey G Arnold, Michael W Van Liew, Ronald L Bingner, R Daren Harmel, and Tamie L Veith.  Model evaluation guidelines for systematic quantification of accuracy in watershed simulations.  Transactions of the ASABE, 50(3):885–900, 2007.
  * [14] J Eamonn Nash and Jonh V Sutcliffe.  River flow forecasting through conceptual models part i—a discussion of principles.  Journal of hydrology, 10(3):282–290, 1970.