Assessing Threats in Times of War and Peace | JPMorgan Chase & Co.
Meet Caitlin McGilley, Vice President of Global Intelligence in Global Security & Investigations at JPMorgan Chase and Lieutenant. Caitlin spent seven years as a United States Navy Intelligence Officer, where she held several roles including Director of Intelligence for a Joint Special Operations Task Force in the Middle East. She is currently a reservist and may be called upon if the country is in crisis.
What are some of the challenges that you faced making the transition from the military to corporate?
Being a Navy Officer was a large part of my identity and it took some time during my transition to the civilian world to see that it wasn't the uniform that defined me but rather my drive to serve and protect people. Once I figured that out, I realized that Corporate Security was the perfect career path.
How has your work with the military complemented the role at JPMorgan Chase and vice versa?
Most of my work in the military was time-sensitive, requiring expeditious and accurate work to help keep people safe. High-stress environments bring out the best in me so when an investigation hits my desk that requires a quick turnaround, I'm always ready to prioritize my work to make sure our employees are safe. There is no such thing as an inconvenience when people need help.
What qualities do you feel women/you bring that are an asset to the workplace?
Strong leadership, integrity, competency, sound judgment – really any quality of an excellent employee will be found in all genders.
A successful career is continuously maintaining that balance of enjoying your job, learning something new every day, avoiding boredom, and making sure you get home for dinner – even if you have to work after dessert.
What advice would you give to other women transitioning from the military to corporate?
The same thing I always told my teams: "If you do the right thing, you're never wrong." It's important that people continue to use sound judgment, act with integrity, and make the choice that does the most good. You'll succeed where ever you land if you follow those principles.
Learn more about Women on the Move.


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text:
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TAMPA, Fla. (FOX 13) - Superstar Lady Gaga is in the Tampa Bay area for a concert and is helping out a lot of families for the holidays before tonight's show.
Lady Gaga's Born This Way Foundation paid off the layaway costs in conjunction with the organization 'Pay Away the Layaway' to the surprise of dozens of local families at the Burlington at University Square Mall.
In all, 25 families had their layaway balances paid for.
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Lady Gaga created the Born This Way Foundation with her mother to empower young people to create a kinder and braver world.
Her concert is Friday night at 7:30 p.m. at Amalie Arena in Tampa.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}\label{intro}

Let $S$ be an algebraic variety, and let $f:X\to Y$ be an $S$-morphism of algebraic varieties.  Let $F$ be a constructible complex
of rational vector spaces on $X$.  For each $s \in S$, the vector spaces
$H^\bullet(X_s, F_s)$ carry the perverse Leray filtrations  $P^{f_s}$ associated with the morphism $f_s:X_s\to Y_s$.
It is natural to ask how these filtered vector spaces
$(H^\bullet(X_s, F_s),P^{f_s})$ behave as $s$ varies in $S$.
The main goal of this paper is to provide criteria (Theorem \ref{finit}) for checking local constancy of these filtrations.

\subsection{Perverse filtration for the Hitchin morphism}\label{dffd}$\;$

Our motivating example arises from perverse filtrations associated to the Hitchin morphism for the moduli space of Higgs bundles for a smooth projective variety.  We refer to \S\ref{snaht} for more details on what follows.

Let $\ms{X} \to S$ be a smooth projective morphism with connected fibers over a connected quasi projective variety, and let $G$ be a reductive group.
Associated to this family is the so-called Dolbeault moduli space $\pi_D=\pi_D(\ms{X}/S,G): M_D(\ms{X}/S,G) \to S$ and the  Hitchin projective  $S$-morphism $h=h(\ms{X}/S,G): M_D(\ms{X}/S,G)\to \m{V}(\ms{X}/S,G)$, which induces, for every point $s\in S$,  the Hitchin morphism
$h_s=h(\ms{X}_s,G): M_D(\ms{X}_s,G)\to \m{V}(\ms{X}_s,G)$.

It follows from non-abelian Hodge theory that the
the structural morphism $\pi_D$ is topologically locally trivial over $S$.
It follows that the intersection cohomology groups  $I\!H^\bullet (M_D(\ms{X}_s,G), \rat)$ on the Dolbeault moduli spaces
$M_D(\ms{X}_s,G)$,
give rise to  local systems $I\!H^\bullet_D(\ms{X}/S,G)$ on $S$.

For every $s\in S$, the Hitchin morphism $h_s$ induces the perverse Leray filtration
$P^{h_s}_{D,\star}  (\ms{X}_s,G)$ on the intersection cohomology groups $I\!H^\bullet (M_D(\ms{X}_s,G), \rat)$.

It does not seem immediately clear that
these perverse Leray filtrations should match-up and give rise to locally constant  subsheaves of the local systems
$I\!H^\bullet_D(\ms{X}/S,G)$ on $S$.
A priori, the perverse filtrations could jump at special point in $S$.  For example, a priori, one could have un-expected direct summands in $D(\m{V}(\ms{X}_s,G))$ in the decomposition theorem
for the Hitchin morphism  $h(\ms{X}_s,G)$ at  special  points $s \in S$. For a brief discussion, see the beginning of \S\ref{prep}.

In this paper, we prove the following result, which can be viewed as some evidence towards the validity of the $P=W$ Question and Conjecture, see Question
\ref{4r4r} and Remark \ref{p=wdd}: the weight filtration on the intersection cohomology of the Betti moduli spaces
of a family of projective manifolds gives rise to locally constant sheaves on the base of the family; if $P=W$, then the same would be true
for the corresponding perverse Leray filtrations associated with the family of Hitchin morphisms, and this is what the following theorem
asserts.

\begin{tm}\label{ewq0} {\rm  ({\bf The perverse Leray filtration  is locally constant})}
The perverse leray filtration gives rise to locally constant subsheaves:
\begin{equation}\label{wq}
P^h_{D,\star}  (\ms{X}/S,G) \subseteq I\!H^\bullet_D(\ms{X}/S,G).
\end{equation}
\end{tm}

Theorem \ref{ewq0} follows at once from Theorem \ref{ewq} below, to the effect that specialization is a filtered isomorphism
for the  perverse Leray filtrations.

\begin{tm}\label{ewq} Let $S$ be a nonsingular connected curve, let $s\in S$ be any point, let $\Delta$ be a suitably general disk
centered at $s$ and let $t \in \Delta \setminus \{s\}$ be a suitably general point.
The specialization morphism $I\!H^\bullet (M_{D}(\ms{X}_s,G),\rat)) \to I\!H^\bullet(M_{D}(\ms{X}_t,G),\rat))$ is
well-defined and gives a filtered isomorphism for the respective perverse Leray filtrations $P^{h_s}_D(\ms{X}_s,G)$ and $P^{h_t}_D(\ms{X}_t,G).$
In particular, the dimensions
of the graded spaces ${\rm Gr}^{P^{h_s}_D}_\star I\!H^\bullet (M_{D}(\ms{X}_s,G),\rat))$ are independent of $s \in S$.
\end{tm}

\begin{rmk}\label{kkq}
Theorems \ref{ewq0} and  \ref{ewq} hold, with the same proof, if we replace the Dolbeault moduli spaces with their twisted counterparts
(cf. Remark \ref{p=wdd}) in the cases
of relative dimension $\dim \ms{X}/S=1$ and $G=GL_n, SL_n, PGL_n$.  In these cases the Dolbeault moduli spaces are integral
orbifolds, even nonsingular in the cases $G=GL_n, SL_n$.
\end{rmk}

We may say, informally, that the perverse Leray filtration for the Hitchin morphism associated with a family of projective manifolds
is independent of the members of the family. In particular, by applying this to curves and their moduli space, we may say that
the perverse Leray filtration on the intersection cohomology of the Dolbeault moduli space associated with the curve
and the reductive group $G$  is independent of the curve.

Theorem  \ref{ewq} concerning the Hitchin morphism is proved in \S\ref{gigi} as a corollary to the more general Theorem \ref{finit},
which provides  a sufficient condition for when the specialization morphism is well-defined
and is a filtered isomorphism.

$\,$
\subsection{Motivation from Gopakumar-Vafa invariants}\label{more}$\;$

One motivation for this work is to help understand a conjecture on Gopakumar-Vafa invariants
proposed in \cite{MT}, following earlier work of Kiem-Li \cite{kiem-li}.  In this section,
we briefly sketch this connection.

Let $X$ denote a Calabi-Yau threefold and let $\beta \in H_2(X,\mathbb{Z})$ denote a curve class on $X$.
There are various approaches to defining virtual counts of curves on $X$ in class $\beta$ -- Gromov-Witten or Pandharipande-Thomas invariants, for example;
While these typically produce an infinite series of invariants, a physics proposal of Gopakumar-Vafa suggests that they should be governed
by finitely many integers determined from the cohomology of a space of one-dimensional sheaves.

In \cite{MT}, the authors suggest a rigorous approach to their proposal using the perverse filtration as follows.
Consider the moduli space $M_\beta(X)$ of stable one-dimensional sheaves on $X$ with support cycle class $\beta$ and
Euler characteristic $1$, and let
$$\pi: M_\beta(X) \rightarrow \mathrm{Chow}_\beta(X)$$
denote the Hilbert-Chow morphism to the Chow variety of $X$ which remembers the support cycle.
\footnote{One needs to pass to seminormalizations to define this.}
Under certain assumptions, $M_\beta(X)$ carries a natural perverse sheaf $\phi_M$ associated to its shifted symplectic structure,
and one can study the perverse cohomology sheaves of its pushforward.
More precisely, in \cite{MT}, the authors define Gopakumar-Vafa invariants $n_{g,\beta}$ of $X$ in class $\beta$ via the identity
\begin{align}
\sum_{i \in \mathbb{Z}}
\chi(\mathop{^{p}\mathcal{H}}\nolimits^i(R \pi_{\ast}\phi_M))y^i=
\sum_{g\geq 0}n_{g, \beta}(y^{\frac{1}{2}}+y^{-\frac{1}{2}})^{2g}.
\end{align}
Conjecturally, these GV invariants should, after a universal change of variables, agree with the Gromov-Witten invariants of $X$ in class $\beta$.  Since the precise relationship is somewhat intricate, we refer the
reader there for details.

In order for this picture to be plausible, it is necessary for GV invariants to be invariant under deformations of the Calabi-Yau threefold $X$ -- indeed such deformation invariance is built into the intersection-theoretic constructions of GW/PT invariants.  One can view the main theorem of this paper as evidence
for this invariance; for example, one consequence of our main result is the deformation invariance of $n_{g,\beta}$ for local del Pezzo surfaces.
While there is no discussion of the sheaf $\phi_M$ in this paper, one might hope to extend the technique here to this more general setting.

\subsection{Outline}\label{otline}$\;$

\S\ref{notation} sets up the notation. More precisely: \S\ref{gennot} sets up the notation in the context of the constructible derived category;
\S\ref{vncf} is concerned with the formalism of the vanishing/nearby cycles functors and of the specialization morphism.

\S\ref{prep} deals with the perverse Leray  filtration and specialization: \S\ref{cdte} contains Proposition
\ref{iotanoc}, which provides a  sufficient condition for when perverse truncation and (shifted) restriction to a Cartier divisor commute;
\S\ref{cosppf} contains our main technical result,  the aforementioned Theorem \ref{finit}

\S\ref{rrr} contains our main application to the Hitchin morphism: \S\ref{snaht} is a refresher on Dolbeault and Betti moduli spaces
and also proves some preliminary facts needed in the proof, given in   \S\ref{gigi},  of  Theorem \ref{ewq}, which in turn implies
Theorem \ref{ewq0} on the local constancy of the perverse Leray filtration for the Hitchin morphism
over a base.

\subsection{Acknowledgements}
We thank: Jochen Heinloth, Luca Migliorini,  J\"org Sch\"urmann, Geordie Williamson, Zhiwei Yun
for useful suggestions. The first-named author, who is partially supported by N.S.F. D.M.S. Grant n. 1600515, would like to thank the Freiburg Research Institute for Advanced Studies  for the perfect working conditions; the research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme
(FP7/2007-2013) under REA grant agreement n. [609305].  The second-named author is supported by NSF FRG grant DMS 1159265.

$\,$
\section{Notation}\label{notation}

\subsection{General notation}\label{gennot}$\;$

By variety, we mean a separated scheme of finite type over the field of complex numbers
$\comp$.  By point, we mean a closed point. See \ci{bams} for a quick introduction and for standard references for many of the
concepts and objects in this subsection.

Given a variety $Y$, we denote by $D(Y)$ the constructible bounded derived category of  sheaves of rational vector spaces on $Y,$
endowed with the middle perversity $\frak{t}$-structure.  We denote derived functors using the un-derived notation, e.g. if $f:X\to Y$
is a morphism of varieties, then the derived direct image (push-forward) functor  $Rf_*$ is denoted by $f_*$, etc.
Distinguished triangles  in $D(Y)$ are denoted $G' \to G \to G'' \rightsquigarrow$.
The full subcategory of perverse sheaves is denoted by $P(Y)$.
We employ  the following standard notation for the objects associated with this $\frak{t}$-structure:
the full subcategories
$ ^p\!D^{\leq j}(Y)$ and ${^p\!D}^{\geq j}(Y)$, $\forall j \in \zed$, of $D(Y)$, and
${^p\!D}^{[j,k]}(Y): =    {^p\!D}^{\geq j}(Y) \cap    {^p\!D}^{\leq k}(Y),$ $\forall j\leq k \in \zed$; the truncation functors
$\ptd{j}: D(Y) \to  {^p\!D}^{\leq j}(Y)$ and  $\ptu{j}: D(Y) \to {^p\!D}^{\geq j}(Y)$;  the perverse cohomology functors
$^p\!H^j: D(Y) \to P(Y)$. At times, we drop the space variable $Y$ from the notation.

The following operations preserve constructibility of complexes:
ordinary and extraordinary push-forward and pull-backs, hom and tensor product,
Verdier duality.    The nearby and vanishing cycle functors also preserves constructibility, with the provision
that, when dealing with these functors, one is working over a disk.

The $k$-th  (hyper)cohomology
groups of $Y$ with coefficients in $G \in D(Y)$ are denoted by $H^k(Y,G)$. The complex computing
this cohomology is denoted by $R\Gamma (Y,G)$ and it lives in the bounded derived category
$D({\rm point})$
whose objects are    complexes of vector spaces with cohomology given by finite dimensional rational vector spaces.

The filtrations we consider are finite and increasing.
For every $G \in D(Y)$, the $\frak{t}$-structure defines a natural filtered object $(R\Gamma (Y,G), P)$, and $P$ is called the perverse filtration.

If a statement is valid for every value of  an index, e.g. the degree of a cohomology group, or the step of a filtration, then we denote such an index by a bullet-point symbol, or by a star symbol: $H^\bullet (X,F)$, $P_{\bullet}$, $\ptd{\bullet}$,  $P_\star H^\bullet (X,F)$, ${\rm Gr}^{P_f}_\star H^\bullet (X,F)$.

Given $G\in D(Y)$ we have the  system of truncation morphisms:
\begin{equation}\label{2wxs}
\xymatrix{
\ldots \ar[r] & \ptd{k-1} G \ar[r] & \ptd{k} G \ar[r] & \ldots \ar[r]& G
}
\end{equation}
A morphism $G\to G'$ in $D(Y)$ gives rise to a system of morphisms:
\begin{equation}\label{2w}
\xymatrix{
\ptd{\bullet} G \ar[r] & \ptd{\bullet} G'
}
\end{equation}
which are compatible with (\ref{2wxs}) is the evident manner. We say that  the system (\ref{2w}) is a  system
of  compatible  morphisms. It gives rise to a morphisms of filtered objects:
\begin{equation}\label{lmn}
\xymatrix{
\left( R\Gamma (Y, G), P) \right) \ar[r] &  \left( R\Gamma (Y, G'), P)\right) & {\rm in}\; DF({\rm pt}),
}
\end{equation}
where $DF(-)$ denotes the filtered derived category \ci{il}.

 Recall that $P(Y)$ is Artinian, so that the Jordan-Holder theorem holds in it.
The constituents of a non-zero perverse sheaf $G \in P(Y)$ are the isomorphisms classes of the
perverse sheaves appearing  in the unique and finite
collection of non-zero simple perverse sheaves appearing as  the quotients  in a Jordan-Holder filtration of $G.$
The  constituents of  a non-zero complex $G\in D(Y)$ are defined to be the constituents of all of its non-zero perverse
cohomology sheaves.

In general, we drop decorations (indices, parentheses, space variables, etc.) if it seems harmless in the context.

In the context of a Cartesian diagram of varieties, we denote parallel arrows by the same symbol. This should not lead to confusion in expressions like the base change morphism $g^*f_*  \to f_*g^*$.

We are going to make heavy use of the nearby/vanishing cycle functors. See \S\ref{vncf} for the basic facts.

We are going to use the decomposition theorem for semisimple complexes,  i.e. isomorphic to direct sums of shifted
simple perverse sheaves, due to Mochizuki, Sabbah and others; see the references in
\ci{dess}.

\begin{tm}\label{dt}
Let $f:X\to Y$ be a proper morphism of varieties and let $F \in D(X)$ be semisimple. Then $f_* F$ is semisimple.
\end{tm}

$\,$
\subsection{The vanishing and nearby cycles formalism}\label{vncf}$\;$

Standard references  for this section are \ci[XIII, XIV]{sga72}  and \ci[Ch. 8,10]{kash}.

Let $S$ be a nonsingular and connected curve and consider a morphism of varieties:
\[v=v_Y: Y\to S.\]
Let $s \in S$ and let $i:s \to S$ be the closed embedding; this is what we call the special point. Objects restricted to $s$
maybe be denoted with a subscript $s$; e.g. $Y_s:= v^{-1}(s)$.

In what follows, in Choice \ref{sch}, we
choose a disk $\Delta \subseteq S,$ centered at $s.$ For the relevance of this choice to this paper, see Remark \ref{noh}.

Let $Y_\Delta:= v_{Y}^{-1} (\Delta)$ and let $v_{Y_\Delta}:={v_Y}_{|Y_\Delta}: Y_\Delta \to \Delta.$

We have the functors $i^*, i^!, \psi:=\psi_{Y_\Delta}$ and  $\phi:=\phi_{Y_\Delta}$:
\begin{equation}\label{fctz}
i^*,i^!, \psi, \phi:  D(Y) \to D(Y_s),
\end{equation}
where  $\psi$ is the nearby cycle functor and   $\phi$ is the vanishing cycle functor.

We have  the two, Verdier-dual, canonical distinguished triangle of functors: (we denote by the same symbol the dual arrows $\s$)
\begin{equation}\label{1p}
\xymatrix{
i^*[-1]  \ar[r]^-{\s} & \psi [-1]  \ar[r] & \phi   \ar@{~>}[r]   &, &
\phi \ar[r] & \psi[-1]  \ar[r]^-{\s} & i^![1]   \ar@{~>}[r] &.
}
\end{equation}

Recall that:
\begin{equation}\label{tsh}
[\star] \circ \ptd{\bullet}= \ptd{\bullet -\star} \circ [\star], \qquad \mbox{ditto for $\ptu{\bullet}$ and $^p\!H^\bullet$}.
\end{equation}

\begin{fact}\label{psp0}

{\rm ({\bf
$\frak{t}$-exactness for $\psi [-1]$ and $\phi$})}
The functors $\psi [-1]$  and $\phi$ are exact functors of triangulated categories,  are $\frak t$-exact, and they commute with Verdier duality. In particular, they commute with the formation of the perverse cohomology sheaves functors $^p\!H^{\bullet}$ and with
the perverse truncation functors $\ptd{\bullet}$ and $\ptu{\bullet}$. We thus  have the following canonical identifications:
\begin{equation}\label{can1}
\xymatrix{
\ptd{\bullet} \phi =  \phi \ptd{\bullet}; & \ptd{\bullet} \psi [-1] =  \psi [-1] \ptd{\bullet};&
 \mbox{ditto for $ \ptu{\bullet}$ and  $^p\!H^{\bullet}$}.
}
\end{equation}
\end{fact}

 The following is a key property of the vanishing cycle functor.

 \begin{fact}\label{psp}
{\rm ({\bf
Vanishing of $\phi$ for smooth morphism and lisse  sheaves})}
If $v_{Y_\Delta}: Y_\Delta \to \Delta$ is smooth and  $G \in D(Y)$ has locally constant cohomology sheaves    on $Y_\Delta$,
then $\phi\,G=0 \in D(Y_s)$. See \ci[XIII, 2.1.5]{sga72}. Note the special case where $Y=S$ and $v_Y$ is the identity.
\end{fact}

 \begin{fact}\label{gh1}
The composition $i^*[-1] \to \psi[-1] \to i^![1]$ yields a morphism of functors $D(Y) \to D(Y_s)$:
\begin{equation}\label{gh}
i^*[-1]\longrightarrow  i^![1]
\end{equation}

The  morphism (\ref{gh})  is an isomorphism when evaluated on a complex $G\in D(Y)$ such that
$\phi \, G=0;$ in this case, we have isomorphisms:
\begin{equation}\label{ii}
\xymatrix{
i^*[-1] G \ar[r]^-\simeq &  \psi [-1] G  \ar[r]^-\simeq  & i^![1] G
}
\end{equation}

 The morphism {\rm (\ref{gh})}
coincides with the morphism obtained via Verdier's specialization functor, (cf. \ci{jorg}, for example), so that it depends  only on the closed embedding $Y_s\to Y,$
i.e. it is independent of $v_Y$ and of the choice of the disk $\Delta.$
\end{fact}

\begin{fact}\label{psphit}
({\bf
Base change diagrams for $\psi$ and $\phi$})
Let $f:X\to Y$ be an $S$-morphism and let $v_X= v_Y \circ f:X\to S$.
The  base change morphisms associated with  $i$ and $f$ give rise to
morphisms of distinguished triangles of morphisms as follows (cf. \ci[XIII, 2.1.7]{sga72}):

\begin{enumerate}

\item\label{pp3b}
\begin{equation}\label{fdtbis}
\xymatrix{
i^* [-1] f_*   \ar[r]^{\s\, \circ f_*} \ar[d] &   \psi_Y [-1]  f_*   \ar[r] \ar[d]  &   \phi_Y f_* \ar[d]  \ar@{~>}[r] & \\
f_* i^* [-1]  \ar[r]^{f_* \circ \, \s}   &  f_* \psi_X  [-1]  \ar[r]   & f_* \phi_X   \ar@{~>}[r]  &.
}
\end{equation}
When $f$ is proper,
the  morphism (\ref{fdtbis}) is an  isomorphism.

\item\label{p!}

\begin{equation}\label{fd1}
\xymatrix{
 \phi_Y f_*    \ar[r]  \ar[d]  &   \psi_Y [-1] f_*    \ar[r]^-{\s \circ f_*}  \ar[d]  &  i^! [1] f_*   \ar[d]^-=    \ar@{~>}[r] &
\\
  f_* \phi_X \ar[r]      &  f_* \psi_Y[-1]     \ar[r]^-{f_* \circ \s}      & f_*i^![1]  \ar@{~>}[r]  &,
}
\end{equation}
When $f$ is proper,
the  morphism (\ref{fd1}) is an  isomorphism.

\item
by combining  the  (\ref{fdtbis}) with   (\ref{fd1}),  we obtain the following commutative diagram:
\begin{equation}\label{r00}
\xymatrix{
i^*[-1] f_* \ar[d]    \ar[r]   & \psi[-1] f_* \ar[r]    \ar[d]    & i^![1] f_* \ar[d]
\\
f_* i^*[-1]      \ar[r]   & f_* \psi[-1]  \ar[r]        &      f_* i^![1].
}
\end{equation}
\end{enumerate}

\end{fact}

Let $t \in S$ be another point and, by abuse of notation, denote the closed embedding $t\to S$ also by $t.$
There is the natural morphism (cf. (\ref{gh})):
\begin{equation}\label{ght}
t^*[-1]\longrightarrow  t^![1].
\end{equation}
\begin{fact}\label{gogo}
Let $G\in D(Y)$ and let $t \in S$ be a  general point. Then the natural morphism $t^* [-1] G\to t^![1]G$ is an isomorphism
in $D(Y_t).$
To see this,   recall {\rm (\ref{gh})}, and use the vanishing of   vanishing cycle functor ``translated to $t$"  \ci[Rmk. 4.2.4]{jorg}. As usual, here ``general" means that it can be chosen to be any point of a suitable Zariski-dense
open subset $S^o\subseteq S$ that depends on $G \in D(Y).$
\end{fact}

The following follows from the fact that for $t$ general, $t^*[-1], t^![1]$ commute with all the functors involved in the
constructions of perverse truncation.
\begin{fact}\label{vbfz}
{\rm ({\bf $\frak{t}$-exactness and $t^*, t^!$})}
Let $G \in D(Y)$ and let $t\in S$ be general. Then
we have:
\begin{equation}\label{vbgfz}
t^*[-1] \ptd{\bullet} G = \ptd{\bullet } t^*[-1] G, \quad t^![1] \ptd{\bullet} G = \ptd{\bullet } t^![1] G,
\quad \mbox{ ditto for  $\ptu{\bullet}$ and  $^p\!H^{\bullet}$.}
\end{equation}

\end{fact}

\begin{fact}\label{gi9}
What follows is a consequence of Deligne's generic base change theorem \ci[Thm. 9.1]{sga4.5}
  and of stratification theory; see also the discussion in \ci{desomm}.
Given a finite collection of morphisms $v_i: Y_i\to S$ and complexes $G_i \in D(Y_i),$ there is a Zariski-dense open subset
$S^o \subseteq S$ such that the direct images ${v_i}_* G_i$ are lisse, and such that their formation commutes with arbitrary base change.
By shrinking $S^o$ if necessary, we can further assume that
the $G_i$ have no constituent supported on fibers of the morphisms $v_i$ over $S^o$, and that the strata on the $Y_i$, of stratifications
with respect to which the  $G_i$ are constructible, map smoothly to $S$ over $S^o$ (cf. \ci[Rmk. 4.2.4]{jorg}).
The points $t\in S^o$ are said to be general for  the finite collection $\{G_i\}.$
\end{fact}

\begin{defi}\label{tge}
We say that $t \in S$ is general (for the collection of $G_i$'s) if $t \in S^o= S^o(G)$ (cf. Fact {\rm \ref{gi9}}).
\end{defi}

\begin{choice}\label{sch}
Let $s\in S$.  Let $S^o\subseteq S$ be the Zariski-dense open subset of points
of the connected curve $S$
which are general with respect to  some finite collection of $G_i$'s as above.
 Let $t \in S^o.$  Choose a disk $s,t \in \Delta \subseteq S$ such that $\Delta^*:= \Delta \setminus \{s\} \subseteq S^o.$   Chose a pointed universal covering
$(\w{\Delta^*}, \w{t}) \to (\Delta^*,t).$
\end{choice}

\begin{rmk}\label{idspt}
In the special case where $v_Y:Y\to S$ is the identity on $S,$ we have that $i^*,i^!, \psi ,\phi: D(S) \to D(s)$ and that $t^*, t^!: D(S)\to D(t)$.
We have canonical identifications $D(s)=D({\rm pt})= D(t),$ where  ${\rm pt}$ is just a point, so that all three categories are
naturally equivalent to the bounded derived category of finite dimensional  rational vector spaces. Similarly, in the filtered case:
$DF(s)=DF({\rm pt})= DF(t)$.
\end{rmk}

When $v_Y:Y \to S$ is the identity, we have the following:

\begin{fact}\label{jo} {\rm{\bf (Fundamental isomorphism)}}
Let things be as in Choice {\rm \ref{sch}}. There are the natural isomorphisms:
\begin{equation}\label{000}
\xymatrix{
t^*[-1] G \ar[r]^-\simeq & \psi_\Delta [-1]  G  \ar[r]^-\simeq & t^![1] G,  &{\rm in}\; D(t)=D({\rm pt}) = D(s),
}
\end{equation}
where all three terms are well-defined, up to canonical isomorphism, independently of the choices, but where the isomorphisms,
which depend on the choice of $\w{t},$ are uniquely defined modulo the monodromy action of the fundamental group
$\pi_1 ({\Delta}^*,t).$
See the fundamental identity \ci[XIV, 1.3.3.1]{sga72}. Note that {\rm (\ref{000})} is Verdier self-dual.
\end{fact}

\begin{fact}\label{spmz}
{\rm ({\bf Specialization morphism})}
Consider,  in the special case where   $f$ is $v_Y$,  the morphism of distinguished triangles {\rm (\ref{fdtbis})}.
We place ourselves in the set-up of Choice {\rm \ref{sch}}.
One would like to specialize cohomology from the special  point  $s\in \Delta,$ to the general point $t \in \Delta^*.$ This is not always possible, as we now discuss.

 Diagram {\rm (\ref{fdtbis})} yields the
functorial morphism of distinguished triangles in $D({\rm pt})$:
\begin{equation}\label{jsa}
\xymatrix{
R\Gamma (s, i^*v_*G)    \ar[d] \ar[r]^-{\s_{Y_\Delta}} &  R\Gamma (s, \psi_\Delta v_*G) =  R\Gamma (Y_t, t^*G)  \ar[d] \ar[r]  &  R\Gamma (s, \phi_\Delta [1] v_* G)
\ar[d]   \ar@{~>}[r]  &
\\
R\Gamma (Y_s, i^* G)   \ar[r]^-{\s_\Delta} \ar@/0pt/@{-->}[ur]^-{\rm sp?}
&   R\Gamma (Y_s, \psi_{Y_\Delta}  G)   \ar[r] &  R\Gamma (Y_s, \phi_{Y_\Delta} [1] G) \ar@{~>}[r]  &,
}
\end{equation}
where  the  canonical identification in the middle of the first row stems from  Fact {\rm \ref{jo}} and the fact that base change is an isomorphism for general $t$ (cf. Fact {\rm \ref{gi9}}).

Note that while this identification depends on the choice of the pre-image $\w{t} \in \w{\Delta^*}$
of $t \in \Delta^*,$ the resulting restriction morphism $\s_{Y_\Delta}: R\Gamma (Y_\Delta,G) \to R\Gamma (Y_t,G)$ depends only on $t$,
and not on the choice $\w{t}$: in fact,  monodromy acts on the target, but the domain maps  into the invariants.

Let us emphasize an important piece of diagram {\rm (\ref{jsa})}: (note that the middle term below does not change
when we shrink the disk $\Delta$ centered about $s$)

\begin{equation}\label{fk1}
\xymatrix{
R\Gamma (Y_s,i^* G) && R\Gamma (Y_\Delta,G) \ar[ll]_{{\rm restriction\, to}\, s}
\ar[rr]^-{{\rm restriction\, to}\, t} &&
 R\Gamma (Y_t, t^*G).
}
\end{equation}

In general, i.e. without any further hypothesis ensuring that the restriction to $s$ is an isomorphism,   there is no resulting natural morphism
$R\Gamma (Y_s, G) \to R\Gamma (Y_t, G)$.
When restriction to $s$ is an isomorphism,  e.g. when $v_Y$ is proper, then we call the resulting morphism
the specialization morphism:
\begin{equation}\label{spmo}
\xymatrix{
R\Gamma (Y_s,G) \ar[r]^{\rm sp}&
 R\Gamma (Y_t,G).
}
\end{equation}
Of course, even if $v_Y$ is not proper, it may happen that there is a well-defined specialization morphism
for some $G \in D(Y)$.

When the specialization morphism is well-defined, we have  the distinguished triangle:
\begin{equation}\label{jsac}
\xymatrix{
 v_* i^* G \simeq i^* v_* G    \ar[r]^-{\rm sp}  &
 \psi  v_* G \ar[r] &   \phi  v_* G \ar@{~>}[r] &.
}
\end{equation}

\end{fact}

\begin{fact}\label{splisse}
If $G\in D(S)$ has locally constant cohomology sheaves, then a specialization morphism is an isomorphism. This
follows at once from Fact \ref{psp}.
\end{fact}

\begin{rmk}\label{noh}
The Choice  {\rm \ref{sch}} is harmless for our purposes: in fact, when defined, the  specialization morphism  depends only on $s$ and on $t \in S^o\setminus \{s\}$.
Note also that the morphisms {\rm (\ref{ght})} are independent of the choice of the disk $\Delta$.
\end{rmk}

$\,$
\section{Perverse filtration and specialization}\label{prep}

If one analyzes the behavior of the perverse filtration in families via the specialization morphism, there are two issues.  First, the specialization morphism is not defined in general if $X$ is not proper over $S$, which is the case for families of Dolbeault moduli spaces. The second issue is that, even when the specialization morphism is defined,  as it is in the case of the Dolbeault moduli spaces, it gives rise to a filtered morphism for the perverse Leray filtrations.  Its failure to be a filtered isomorphism is detected by the filtered cone; however, in general, this is not the filtered cone associated with the  natural morphism of functors $i^* \to \psi$ ($i^*$ the pull-back to the special fiber, $\psi$ the nearby fiber functor). This is due to the fact that perverse truncation does not commute with restriction/pull-back: e.g. when one has a direct summand supported on the special fiber.

The goal of this section is to prove
Theorem \ref{finit} in \S\ref{cosppf}, which is a criterion to have a well-defined   specialization morphism which is a filtered isomorphism
for the corresponding perverse filtrations.  To this end,
in \S\ref{cdte}, we study a bit the relationship between perverse truncation and restriction to a Cartier divisor.

$\,$
\subsection{Cartier divisors and  partial $\frak{t}$-exactness}\label{cdte} $\;$

The purpose of this section is to prove Proposition \ref{iotanoc}, especially equation (\ref{tuchm}).
We recommend that  readers skip this section at a first reading.

The next lemma records some general   $\frak{t}$-exactness properties related to embeddings of effective Cartier divisors on
varieties.

\begin{lm}\label{iotatex}
Let $\iota: T' \to T$ be a closed embedding  of varieties such that the open embedding  $T\setminus T' \to T$ is an affine morphism
(e.g. $T'$ is an effective Weil divisor supporting an effective Cartier divisor). Then: (we omit
 the space variables)

 \begin{enumerate}
 \item The functor $\iota^*$   is right $\frak{t}$-exact and the functor $\iota^!$ is left $\frak{t}$-exact, i.e.:
 \begin{equation}\label{isrt}
 \xymatrix{
 \iota^*: {^p\!D}^{\leq \bullet}  \to {^p\!D}^{\leq \bullet} , &  \iota^!: {^p\!D}^{\geq \bullet}  \to {^p\!D}^{\geq \bullet}.
 }
 \end{equation}
 \item
The functor $\iota^*[-1] $ is left  $\frak{t}$-exact and the functor $\iota^! [1]$ is
right $\frak{t}$-exact:
\begin{equation}\label{isrt1}
\xymatrix{
\iota^*: {^p\!D}^{\geq \bullet}  \to {^p\!D}^{\geq \bullet -1}, & \iota^!: {^p\!D}^{\leq \bullet}  \to {^p\!D}^{\leq \bullet +1}.
}
\end{equation}
\end{enumerate}
\end{lm}
\begin{proof}
The inequalities (\ref{isrt}) are  \ci[4.2.4]{bbd}),  in fact, they are valid for any immersion.  What follows is specific
to the situation of the Lemma.

We prove the inequalities (\ref{isrt1}).
 Recall that, by  \ci[4.1.10.ii]{bbd}, we have:  if $G \in P(T)$,  than  $\iota^* G \in {^p\!D}^{[-1,0]}$.
The desired inequality  for $\iota^*$
follows from this fact and  a  simple
descending induction on $\bullet$, by  using $\iota^*$ of  the truncation distinguished  triangles ${^p\!H}^\bullet [-\bullet] \to \ptu{\bullet} \to \ptu{\bullet +1}
\rightsquigarrow	$, and  $\iota^*{^p\!H}^{\bullet +1} \in {^p\!D}^{[-1,0]}$.

Since, for $G \in P(T)$, we have that $\iota^! G \in {^p\!D}^{[0,1]}$, the proof for $\iota^!$ is analogous. Alternatively, it can  also be deduced from the one for $\iota^*$ by Verdier duality.
\end{proof}

\iffalse
\begin{rmk}\label{lmim}
 Lemma {\rm \ref{iotatex}} implies that, for every $G \in P(T),$  we have that $\iota^* \ptd{\bullet}G  \in {^p\!D}^{[-1,0]}(T')$, and that
$\iota^! \ptd{\bullet} \in {^p\!D}^{[0,1]} (T')$.
\end{rmk}
\fi

The following lemma is a technical preliminary to Proposition \ref{iotanoc}.

\begin{lm}\label{trmeglio} {\rm ({\bf  Canonical factorization of $\ptd{\bullet -1} i^* \to \ptd{\bullet} i^*$})}
Let $\iota: T' \to T$ be as in Lemma {\rm \ref{iotatex}}.
The natural morphism $\gamma: \ptd{\bullet -1} \iota^* \to  \ptd{\bullet} \iota^*$ admits a canonical factorization:
\begin{equation}\label{nmf}
\xymatrix{
\gamma: \ptd{\bullet -1} \iota^* \ar[r]^-\delta &  \iota^* \ptd{\bullet} \ar[r]^-\e & \ptd{\bullet} \iota^*.
}
\end{equation}
Similarly, for the dual natural morphism   $\xymatrix{\ptu{\bullet +1} \iota^! &  \ptu{\bullet} \iota^!: \gamma' \ar[l]}$:
\begin{equation}\label{nmfd}
\xymatrix{
\ptu{\bullet +1} \iota^!  &  \iota^! \ptu{\bullet}  \ar[l]_-{\delta'}& \ptu{\bullet} \iota^! : \gamma'  \ar[l]_-{\e'}
}
\end{equation}
\end{lm}
\begin{proof}
We start with the following diagram of distinguished triangles in $D(T')$: (the arrows $\gamma$ and $\delta$ are not part
of either distinguished triangle; they are there to help visualize the situation)
\begin{equation}\label{tmeq1}
\xymatrix{
&&\iota^* \ptd{\bullet} \ar[r] \ar@/^0pt/@{-->}[d]^-{\e} & \iota^* \ar[r]  \ar[d]^-=& \iota^* \ptu{\bullet +1} \ar@/^0pt/@{-->}[d] \ar@{~>}[r] &
\\
(\ptd{\bullet -1} \iota^* \ar[r]^-\gamma \ar@/^0pt/@{-->}[rru]^-{\delta} &) & \ptd{\bullet} \iota^*   \ar[r]   &  \iota^*    \ar[r]  &  \ptu{\bullet +1} \iota^*  \ar@{~>}[r] &.
}
\end{equation}

By the l.h.s. of  (\ref{isrt1}), we have that:
\begin{equation}\label{rr}
\iota^* \ptu{\bullet +1}: D(T) \to {^p\!D}^{\geq \bullet}(T').
\end{equation}

By \ci[Prop. 1.1.9, p.23]{bbd}, the diagram (\ref{tmeq1}) can then  be completed uniquely to a morphism
of distinguished triangles; this is visualized by means of the dotted arrows.

The conclusion follows from the  inequality (\ref{rr}), by taking  the long exact sequence associated with
${\rm Hom}_{D(T')} (\ptd{\bullet-1} \iota^*, -)$ applied to the distinguished triangle on the top of
(\ref{tmeq1}):  in fact, for every $G\in D(Y)$, we have
${\rm Hom}_{D(T')} (\ptd{\bullet -1} \iota^*G, -)=0$, when evaluated at something in ${^p\!D}^{\geq \bullet}(T')$.

The proof for $\iota^!$ is analogous.  Alternatively, it can  also be deduced from the one for $\iota^*$ by Verdier duality.
\end{proof}

\begin{rmk}\label{ecz} In view of  the l.h.s. inequality in (\ref{isrt1}),   the r.h.s. vertex in (\ref{tmeq1})   satisfies the inequality
$\iota^* \ptu{\bullet +1}: D \to {^p\!D}^{\geq \bullet}$.
By taking the long exact sequence of perverse cohomology of  the top distinguished triangle in (\ref{tmeq1}), the aforementioned inequality yields the
natural isomorphism of functors:
\begin{equation}\label{ecz1}
\xymatrix{
\ptd{\bullet -1} \iota^* \ptd{\bullet} \ar[r]^-\simeq & \ptd{\bullet-1} \iota^*.
}
\end{equation}
\end{rmk}

\begin{pr}\label{iotanoc} {\rm ({\bf No constituents on divisors and $\frak{t}$-exactness})}
Let $\iota: T' \to T$  be as in Lemma {\rm \ref{iotatex}}.
If $G \in D(T)$ has no constituent supported on $T'$, then the morphisms $\delta$  (cfr. {\rm (\ref{nmf})}) are isomorphisms,
and we get natural isomorphisms:
\begin{equation}\label{tuch}
\xymatrix{
\ptd{\bullet -1} \iota^* G \ar[r]^-\delta_-\simeq &\iota^* \ptd{\bullet} G , & \ptu{\bullet -1} \iota^* G  \ar[r]^-{\simeq} & \iota^* \ptu{\bullet} G
  , &
{^p\!H^{\bullet -1}} \iota^* G  \ar[r]^-\simeq &
\iota^* [-1] { ^p\!H^{\bullet}} G.
}
\end{equation}
The same holds if we replace $\iota^*$ with $\iota^!$ and $(\bullet -1)$ with $(\bullet +1)$.

Equivalently, we have:
\begin{equation}\label{tuchm}
\xymatrix{
\ptd{\bullet} \iota^* [-1]  G \ar[r]^-\delta_-\simeq &\iota^*[-1] \ptd{\bullet}  G , \qquad \ptd{\bullet} \iota^! [1]  G
 & \iota^![1] \ptd{\bullet}  G   \ar[l]_-\delta^-\simeq , &
 {^p\!H^{\bullet +1}} \iota^! G   & \iota^! [1] { ^p\!H^{\bullet}} G   \ar[l]_-\simeq.
}
\end{equation}

\end{pr}
\begin{proof}
It is enough to prove (\ref{tuch}): the statement for $\iota^!$ follows by Verdier duality; The equivalent statements are mere
reformulations by means of (\ref{can1}).

It is enough to prove the first statement on the l.h.s. of (\ref{tuch}), for the remaining ones follow formally by consideration of the truncation distinguished triangles.

We have that  $\ptd{\bullet} G \in {^p\!D}^{\leq \bullet}$, so that, by (\ref{isrt}), we have that  $\iota^*\ptd{\bullet} G \in {^p\!D}^{\leq \bullet}$, and then, clearly, we have that
 \begin{equation}\label{0129}
 \iota^*\ptd{\bullet} G= \ptd{\bullet} \iota^*\ptd{\bullet} G.
 \end{equation}

 In view of (\ref{0129}) and of (\ref{ecz1}), and by considering the truncation triangle
 $\ptd{\bullet -1} \to \ptd{\bullet} \to {^p\!H^\bullet}[-\bullet] \rightsquigarrow$ applied to $\iota^*\ptd{\bullet} G$,
 in order to prove the first equality on the lhs in (\ref{tuch}), it is necessary and sufficient  to show that
 $^p\!H^{\bullet}(\iota^* \ptd{\bullet} G) =0$.

 This can be argued as follows. By taking the long exact sequence of perverse cohomology of the  distinguished triangle
 $\iota^* \ptd{\bullet -1}  G \to \iota^* \ptd{\bullet} G \to \iota^* {^p\!H^{\bullet}} G [-\bullet]  \rightsquigarrow$, we see
 that it is necessary and sufficient to  show that $\iota^* {^p\!H^{\bullet}} G [-\bullet] \in {^p\!D}^{\leq \bullet -1}$, or, equivalently, that
 $\iota^* {^p\!H^{\bullet}} G  \in {^p\!D}^{\leq -1}.$

By \ci[4.1.10.ii]{bbd}, we have the distinguished triangle  ${^p\!H^{-1}} (\iota^* {^p\!H^{\bullet}} G) [1]
\to \iota^* {^p\!H^{\bullet}} G \to {^p\!H^{0}} (\iota^* {^p\!H^{\bullet}} G)   \rightsquigarrow$ and an epimorphism
$^p\!H^{\bullet} G \to {^p\!H^{0}} (\iota^* {^p\!H^{\bullet}} G)$. Since $G$  is assumed to not have constituents supported at
$Y_s$, we must have $^p\!H^{0} (\iota^* {^p\!H^{\bullet}} G)=0$, so that $\iota^* {^p\!H^{\bullet}} G [-\bullet] \in {^p\!D}^{\leq \bullet -1}$,
as requested. The l.h.s. equality in (\ref{tuch}) follows, and we are done.
  \iffalse
 For the equality in the middle of (\ref{tuch}), we argue as follows. Since $\iota^* [-1]$ is left $\frak{t}$-exact, we have  that $\iota^*
 {\color{red} \ptu{\bullet}} G \in {^p\!D}^{\geq \bullet -1}$ (cf. \ref{isrt1}).

 Consider the distinguished triangle
 $\iota^*  \ptd{\bullet -1} G \to \iota^*G \to \iota^* \ptu{\bullet}  G \rightsquigarrow$.  By the l.h.s. equality in (\ref{tuch}),
 the distinguished triangle directly above reads
$\ptd{i-2}  \iota^* G \to \iota^* G \to  \iota^* \ptu{\bullet} G  \rightsquigarrow$  (where the first arrow is the natural one). This forces
the second equality in (\ref{tuch}). If one prefers not to verify the compatibility in the parentheses, observe that:
since $\iota^* [-1]$ is left $\frak{t}$-exact, we have  that $\iota^* \ptu{\bullet} G \in {^p\!D}^{\geq \bullet -1}$; this forces the
distinguished triangle above to be the truncation distinguished triangle and the middle equality  in (\ref{tuch})
follows again.

The r.h.s. equality in (\ref{tuch})  for the perverse cohomology sheaves follows formally from the ones we have established for the
truncations. This concludes the proof of the statements related to $\iota^*$.

The proof for $\iota^!$ is analogous. Alternatively, it can  also be deduced from the one for $\iota^*$ by Verdier duality.
\fi
\end{proof}

We shall use the following simple lemma, where we employ the notation in \S\ref{vncf}.

\begin{lm}\label{silm}
Let $G \in D(Y)$ be such that $\phi_{Y_\Delta} G=0.$ Then no constituent of $G$ is supported on $Y_s.$
\end{lm}
\begin{proof}
By chasing the definitions, we see that we may assume that $G$ is perverse. By using the fact that $\phi$ is $\frak{t}$-exact, and by an easy induction on the length of a Jordan-H\"older filtration
of $G,$ we see that $\phi G'=0$ for every constituent $G'$ of $G.$ The conclusion follows from the fact that if $G' \in D(Y)$ is a non-zero complex supported
on $Y_s$, then $i_*\phi G'=G'.$
\end{proof}

 By combining Lemma \ref{silm} with  Proposition \ref{iotanoc}, we obtain the following

\begin{cor}\label{dodo}
Let $G \in D(Y)$ be such that $\phi_{Y_\Delta} G=0.$ Then we have a natural isomorphism:
\begin{equation}\label{zzx}
\delta :\ptd{\bullet} i^*[-1] G \stackrel{\simeq}\to
i^*[-1] \ptd{\bullet} G; \quad  \mbox{similarly, for $\ptu{\bullet}$, and for $^p\!H^{\bullet}$; ditto, for $i^![1]$.}
\end{equation}
\end{cor}

$\,$
\subsection{Compatibility of the specialization morphism with  the perverse filtration}\label{cosppf}$\;$

\begin{tm}\label{finit}
Let $f:X\to Y$ be proper morphism, let  $v_Y: Y\to S$ be a  morphisms onto a nonsingular connected curve and let $F \in D(X)$.
Choose $s,t \in  S$ and a disk $\Delta$ as in Choice \ref{sch}.  Assume one of the following conditions:
\begin{enumerate}
\item[(i)]
$\phi  F=0$ and $v_Y$ is proper, or

\item[(ii)] $\phi F=0$ and the $v_* \ptd{\bullet} f_* F$ have locally constant cohomology sheaves on $S$, or

\item[(iii)] $F$ is semisimple, $\phi F=0$ and  $v_* f_* F$ has locally constant cohomology sheaves on $S$.
\end{enumerate}

Then the specialization morphism is well-defined and it is a filtered isomorphism for the respective perverse Leray filtrations:
\begin{equation}\label{5050}
 \xymatrix{
 {\rm sp}:
 \left(R\Gamma (X_s, i^* F), P^{f_s}\right) \ar[r]^-\simeq & \left(R\Gamma (X_t, t^* F), P^{f_t}\right).
 }
 \end{equation}
 \end{tm}

\begin{proof}
By applying (\ref{r00}) to the $\ptd{\bullet} f_*$,
we obtain the following commutative diagrams:
\begin{equation}\label{arce}
\xymatrix{
i^*[-1] v_* \ptd{\bullet} f_* \ar[r]^-1  \ar[d]^-3 & \psi [-1] v_* \ptd{\bullet} f_*  \ar[d]^-4   \ar[r]^-2  & i^![1] v_* \ptd{\bullet} f_* \ar[d]^-=_-5
\\
 v_* i^*[-1] \ptd{\bullet} f_* \ar[r]^-{1'}  \ar@/0pt/@{-->}[ur]^-{\rm sp?}  &  v_*  \psi [-1]  \ptd{\bullet} f_*    \ar[r]^{2'}  &  v_*   i^![1] \ptd{\bullet} f_*
}
\end{equation}
Up to shift:  the cones of $1$ and $2$ coincide with $\phi v_* \ptd{\bullet} f_*$; the cones of $1'$ and $2'$ coincide with
$v_*\phi \ptd{\bullet} f_*$.

Let us prove (i).

In this case, we only need the commutative square on the l.h.s. of (\ref{arce}).

Since $v$ is assumed to be proper, the base change morphisms $3$ and $4$ are isomorphisms. In particular,
the specialization morphism  ${\rm sp}$ is well-defined and it gives rise to a   system of compatible morphisms.

The assumption $\phi \, F=0$, which  is common to (i) and (ii),  implies,  by the $\frak t$-exactness of $\phi$ and the properness of $f$, that
  $v_*\phi \ptd{\bullet} f_*F=  v_* \ptd{\bullet} f_* \phi F =0$, i.e. that  the cone of $1'$ is zero, so that so is the cone of $1$.

  It follows that (i) is a  system of compatible isomorphisms and, as such, it gives rise to an isomorphism
  in the filtered derived category.

  The filtered complex $R\Gamma (\ms{X}_s, i^*[-1] F), P^{f_s})$ arises in connection with the cohomology of the compatible system of complexes
$v_* \ptd{\bullet} f_* i^*[-1] F)$. Similarly, the filtered complex $R\Gamma (\ms{X}_t, t^*[-1] F), P^{f_t})$ arises in connection with the cohomology of the compatible system of complexes
$v_* \ptd{\bullet} f_* t^*[-1] F)$.

It remains to identify:
\begin{enumerate}
\item[(a)]
 $v_* i^*[-1]  \ptd{\bullet} f_* F$ with $v_*  \ptd{\bullet} f_* [-1] i^*F$, and
 \item[(b)]
 $\psi [-1] v_*   \ptd{\bullet} f_* F$ with $v_*  \ptd{\bullet} f_* t^* [-1] F$.
 \end{enumerate}

To achieve (b), we argue as follows.
The choice of $t$ general for  $F,$ made in Definition
\ref{tge}, allows us to:
replace $\psi $ with $t^*$ (cf. (\ref{000})); use the identification $t^*v_*=v_* t^*$; use the identification
$t^*[-1] \ptd{\bullet} = \ptd{\bullet} t^*[-1]$ (cf. \ref{vbgfz}));  use the identification $t^*v_*=v_* t^*$. Then (b) follows.

To achieve (a), we argue as follows. We first apply Corollary \ref{dodo} to $G=f_* F$; the condition $\phi f_* F=0$ is met
in view of the properness of $f$ and of  $\phi F=0$. We then apply proper base change $i^* f_* \stackrel{\simeq}\to f_* i^*$.
Then (a) follows, and (i) is proved.

Let us prove (ii).

As it has been seen above, the assumption $\phi F=0$ implies that the cones of  $1'$ and $2'$ are zero, so that $1'$ and $2'$
are isomorphisms.

Since we are assuming that the $v_* \ptd{\bullet} f_* F$ have locally constant cohomology sheaves on $\Delta$, we have that
the cones of $1$ and $2$ are zero as well (cf. Fact \ref{psp}), so that $1$ and $2$ are isomorphisms.

Since the morphism $5$ is an isomorphism, all the morphisms in (\ref{arce}) are isomorphisms.

We conclude as in case (i).

Case (iii) is weaker than case (ii); we can also prove it without resorting to the use of Corollary \ref{dodo}. The proof is very similar, except
that in order to achieve the critical commutation property {\rm (a)} via Corollary \ref{dodo},  we use that:
$F$ semisimple implies $f_*F$ semisimple (cf. the decomposition Theorem \ref{dt}); the assumption $\phi F=0$ implies that no simple summand
of $f_* F$ is supported on $Y_s$; the commutation property for a simple, un-shifted, simple perverse summand $P$, which  we know not to be  supported
on $Y_s$ follows from \ci[4.10.1]{bbd}, to the effect that $i^*[-1]P$ is perverse.

\end{proof}

$\,$
\section{The Hitchin morphism and specialization}\label{rrr}$\;$

\subsection{The Betti and Dolbeault moduli spaces: the $P=W$ conjecture}\label{snaht}$\;$

Let $X\to S$ be a smooth  projective morphism  over a  variety $S$ and let $G$ be a reductive group.

  The Betti moduli space $M_B(\ms{X}/S,G)$ (cf. \ci[pp.12-15]{si2})  is a complex analytic space over $S$. The fiber over a point
$s\in S$ is the moduli space (a.k.a. the character variety)  $M_B(\ms{X}_s,G)$ of representations of the fundamental group of $\ms{X}_s$ into $G.$

The Dolbeault moduli space $M_D(\ms{X}/S,G)$ (cf. \ci[\S9, esp. Prop. 9.7]{si2})
is  quasi-projective over $S$; in general, it is not proper over $S$. The fiber over a point
$s\in S$ is the moduli space  $M_D(\ms{X}_s,G)$ of principal  Higgs bundles of semiharmonic type on $\ms{X}_s$ for the group $G$.

\begin{fact}\label{naht}
{\rm (\bf Non-Abelian Hodge Theorem)} There is a natural $S$-homeomorphism of the underlying topological spaces:
(cf. \ci[Thm. 9.11, and Lm. 9.14]{si2})
\begin{equation}\label{enaht}
\xymatrix{
\Psi (\ms{X}/S,G): M_B (\ms{X}/S,G) \ar[r]^-\simeq & M_D (\ms{X}/S,G).
}
\end{equation}

\end{fact}

To fix ideas, in what follows, we tacitly assume that $S$ and the fibers of the family $\ms{X}/S$ are connected; such an assumption
is for ease of exposition only; see \ci[pp. 14-15]{si2}.

Choose any point $s_o\in S$.
The structural morphism $\pi_B (\ms{X}/S,G) : M_B (\ms{X}/S,G) \to S$ is analytically locally trivial over $S$, with transition functions with values in the
group of  $\comp$-scheme
automorphisms of the fiber $M_B(\ms{X}_{s_o},G)$; see \ci[Lm 6.2, p.13]{si2}. More precisely: let $(\w{S}, \w{s_o}) \to (S, s_o)$ be a pointed universal covering space with associated identification of the deck transformation group with the fundamental group
$\pi_1 (S, s_o)$;   the fundamental group acts on $M_{s_o}$ via $\comp$-scheme automorphisms;
$M_B(\ms{X}/S,G)$
is constructed as the quotient of $M_B(\ms{X}_{s_o},G) \times \w{S}$ under the usual action of the fundamental group $\pi_1(S, s_o)$.

\begin{fact}\label{dltr}
{\rm (\bf Local triviality of the Dolbeault moduli space over the base)}
The local triviality of the Betti moduli space over the base, coupled with the Non-Abelian Hodge Theorem
$S$-homeomorphism $\Psi$ {\rm (\ref{enaht})}, implies that the structural morphism $\pi_D (\ms{X}/S,G) : M_D(\ms{X}/S,G) \to S$ is topologically  locally trivial
over the base $S$.
\end{fact}

Recall that, for irreducible varieties,  the intersection cohomology complexes/groups
are  topological invariants, independent of the stratification (cf. \ci[\S4.1]{goma2}).
Note that \ci{goma2} deals with irreducible analytic varieties;
on the other hand,  as the forthcoming  Lemma \ref{ictop} shows, if we define the intersection complex of a variety as the direct sum
of the intersection complex of its  irreducible components, then the topological invariance of the intersection cohomology complexes/groups is still valid.

In particular, given a topologically locally trivial fibration with fibers varieties, the intersection cohomology
groups of the fibers give rise to locally constant sheaves on the base.

We thank G. Williamson for suggesting the definition of the set $X'$ in the proof of the following lemma. Our original
set $X'$ was defined using the notion of local irreducibility and lead to a more cumbersome proof.

\begin{lm}\label{ictop}
{\rm ({\bf Topological invariance of   intersection cohomology for reducible varieties})}
Let $X$ and $Y$ be complex analytic set and let $g:X\simeq Y$ be a homeomorphism of the underlying topological spaces
endowed with the classical topology. Then:

\begin{enumerate}
\item
The homeomorphism $g$ induces a natural bijection $\gamma: I \simeq B $ on the sets of irreducible components
of $X$ and $Y$ such that $g$ induces homeomorphisms $X_i \simeq Y_{\gamma (i)}$, for every $i \in I$.
\item
Define the intersection complex  $IC_X$ of a complex analytic set $X$ to be the direct product of the intersection  cohomology complexes
of the irreducible components of $X$. Then the homeomorphism $u$ induces a natural  isomorphism
$IC_X = u^* IC_Y$.
\end{enumerate}
\end{lm}
\begin{proof}
The case when $X$ and $Y$ are  irreducible is proved by M. Goresky and R. MacPherson in
\ci{goma2}. It is thus clear that it is enough to prove the first statement of the lemma, which is what we do next, leaving some
elementary details to the reader.

Let $X'$ ($Y'$, resp.)  the be open subset of those points of $X$  ($Y$, resp.) admitting a classical open neighborhood all of whose points
admit a classical open neighborhood homeomorphic to a  Euclidean space of some dimension.

Since $X'$ is defined topologically, it is clear that $g(X')=Y'$.

We have the following
inclusions of classical open subsets:
\[
X^{\rm sm} \subseteq X' \subseteq X^o \subseteq X, \quad X^{\rm sm}_i \subseteq X'_i \subseteq X^o_i \subseteq X_i,
\]
where $X^{\rm sm}$ is the set of smooth points of the complex analytic set $X$, and $X^o$ is the complement of the union of all intersections $X_i \cap X_j$, $i,j \in I$, $i\neq j$, and where $-_i$ denotes intersection with the irreducible component $X_i$.

The classical open subsets $X^{\rm sm}_i$ and $X^o_i$ are also Zariski open,  irreducible and connected.
The classical open subset $X'_i$ is connected. The $X'_i$ are the connected components of $X'$. The homeomorphism
$g$ must respect the decomposition of $X'$ and $Y'$ into their connected components, so that we obtain the desired  bijection $\gamma: I \simeq B.$
The classical closure of  $X'_i$ is $X_i$ and since $g$ is a homeomorphism, we must have that $g$ induces a homeomorphism
$X_i \cong Y_{\gamma (i)}$.
\end{proof}

Lemma  \ref{ictop} implies at once the following

\begin{cor}\label{iclt}
Let $F,S$ be a varieties, let  $X$ be an $S$-variety,  and let $\tau: X \to F\times S$ be an $S$-homeomorphism. Then, for every $s \in S$, we have canonical
isomorphisms:
\[
(\tau^* pr_F^* IC_F)_{|X_s} = IC_{X_s}
\]
\end{cor}

\begin{rmk}\label{whygood}
The definition of intersection cohomology complex for reducible varieties stemming from Lemma \ref{ictop} is reasonable in view of the fact that it satisfies virtually all the usually properties of the usual intersection cohomology complex for irreducible varieties, e.g. purity,
mixed and pure Hodge structures,
Artin vanishing, Lefschetz theorems, relative hard Lefschetz, Hodge Riemann relations, decomposition theorem.  See
\ci[\S4.6]{decpf2}, for example.
\end{rmk}

The intersection cohomology groups of the  fibers of  $\pi_B(\ms{X}/S,G)$ give rise to locally constant sheaves $I\!H^\bullet_B (\ms{X}/S,G)$
on $S$. In view of the homeomorphism $\Psi (\ms{X}/S,G)$, and of the topological invariance
of intersection cohomology,
the same applies to the Dolbeault picture, and we get locally constant sheaves $I\!H^\bullet_D (\ms{X}/S,G)$
on $S$.

\begin{fact}\label{wlss} {\rm (\bf The Betti weight filtration is locally constant)}
By the local triviality of $\pi_B (\ms{X}/S,G)$ over $S$, and since the transition automorphisms are compatible with mixed Hodge structures in  (intersection) cohomology,
we see  that  the weight filtration $W$ for the mixed Hodge structure for the intersection cohomology of the fibers
of $\pi_B(\ms{X}/S,G)$ gives rise to locally constant  subsheaves $W_{\star,B} (\ms{X}/S,G) \subseteq I\!H^\bullet_B (\ms{X}/S,G) $ on $S$.
\end{fact}

The Dolbeault moduli space is endowed with a natural ${\mathbb G}_m$-action (cf. \ci[p.62, and p.17]{si2}), given by scalar multiplication on the
Higgs field.

The reference in this paragraph is  \ci[p.20-23]{si2}, which deals with the case of $G=GL_n$,  suitably adapted to
an arbitrary reductive $G$.
The Dolbeault moduli space admits the Hitchin $S$-morphism
\begin{equation}\label{hmo}
\xymatrix{
h(\ms{X}/S,G): M_D(\ms{X}/S, G) \ar[r] &  \m{V}(\ms{X}/S,G).
}
\end{equation}
Here,  $\m{V}(\ms{X}/S,G)$ is the quasi
projective $S$-variety representing the functor $(S'\to S) \mapsto \oplus_{i=1}^{{\rm rk}G} H^0
(X'/S', {\rm Sym}^{e_i(G)} \Omega^1_{X'/S'})$, where the positive integers $e_i$ are the degrees of the generators
of conjugation-invariant polynomials on the Lie algebra of $G$. The Hitchin morphism assigns to a $G$-principal Higgs bundle the symmetric polynomials
appearing as the coefficients of the  ``characteristic polynomial" of the Higgs field, viewed as sections of the appropriate sheaf.
The Hitchin morphism is proper over $S$, hence projective over $S$  (the Dolbeault moduli space is
quasi projective over $S$).
Domain and target  of the Hitchin morphism are endowed with a natural  ${\mathbb G}_m$-action (cf. \ci[p.62]{si2}), which covers the trivial action on $S$.
The ${\mathbb G}_m$-action on the target is contracting.
The Hitchin morphism is ${\mathbb G}_m$-equivariant.

We observe that the  mixed Hodge structure  on the intersection cohomology groups of each Dolbeault moduli space is pure
(the starting point is the ${\mathbb G}_m$-equivariance and the contracting action; then one can imitate the proof of     \ci[Lemma 6.1.1 and references therein, and the proof of Thm 2.4.1]{dehali}).
We do not need this fact, except to point out that  it is in sharp contrast with the  expected (known in some cases) non purity of the corresponding intersection cohomology groups of each Betti moduli space.

\begin{??}\label{4r4r}
{\rm ({\bf  Is $P=W$ in the non-Abelian Hodge theory in arbitrary dimensions?})}
Let $X$ be a connected  smooth projective variety.
Then for the weight filtration $W_B (X,G)$, do we have  $W_{B,2\star +1} (X,G) = W_{2 \star} (X.G) \subseteq I\!H^{\bullet} (M_B(X,G))$?
Via the  Non Abelian Hodge Theorem isomorphism $\Psi_*$, do we have that
$\Psi_* W_{B,2\star} (X,G) =  P^h_{D,\star}(X,G)$, where $P^h_D(X,G)$ is the perverse Leray filtration for the Hitchin morphism (suitably normalized, so that it  ``starts at zero")?
\end{??}

\begin{rmk}\label{p=wdd} Actually, the $P=W$ conjecture, which is due to M.A. de Cataldo, T. Hausel and L. Migliorini,
is concerned with a twisted version of the Betti/Dolbeault moduli spaces for curves of genus $g \geq 2$.
The paper \ci{dhm} proves the validity of the conjecture in this  twisted case
when $X$ is a curve and $G=GL_2, SL_2$ and $PGL_2$. In this twisted case over a curve of genus $g \geq 2$, the moduli spaces for  $G=GL_n, SL_n$ are nonsingular, and for   $G=PGL_n$ they are orbifolds;  Theorem \ref{ewq} applies to this situation.
\end{rmk}

In the next section, we prove Theorem \ref{ewq}, to the effect that   the perverse filtration gives rise to locally constant subsheaves  subsheaves $P^h_{D,\star}  (\ms{X}/S,G) \subseteq I\!H^\bullet (\ms{X}/S,G)$ on $S$.

We need the following

\begin{lm}\label{vaicdm}
Let things be as in Theorem \ref{ewq}.
Let $F$ be the intersection complex of the Dolbeault moduli space  $M_D (\ms{X}/S,G)$.
Then
\begin{equation}\label{zzz}
\phi F =0,
\end{equation}
the  specialization morphism is defined and it is an isomorphism:
\begin{equation}\label{zz}
\xymatrix{
{\rm sp}: R\Gamma (\ms{X}_s, i^* F) \ar[r]^-\simeq  &  R\Gamma (\ms{X}_t, t^* F).
}
\end{equation}
\end{lm}
\begin{proof}
The Betti  moduli space is topologically locally trivial over any disk contained in $S$. By the Non Abelian Hodge Theorem,
the Dolbeault moduli space is also  topologically locally trivial over our disk  $\Delta$. Let $M_D(\ms{X}_s,G) \times \Delta \cong M_D(X_\Delta/\Delta,G)$ be any topological  trivialization. Then the intersection complex on $M_D(X_\Delta/\Delta,G)$ is the pull-back
of the intersection complex of $M_D(\ms{X}_s,G)$ via the first projection associated with the chosen trivialization. Both the conclusions of the lemma follow.

\end{proof}

$\,$
\subsection{Proof of Theorem \ref{ewq} on the Hitchin morphism and specialization}\label{gigi}$\;$

\begin{proof}
We denote  the Hitchin morphism  (\ref{hmo})  simply by  $h:M\to \m{V}$,  we denote the structural $S$-morphisms
by $\pi:M \to S$ and $\rho: \m{V} \to S$.
Let $\m{IC}_M$ be the intersection complex of the Dolbeault moduli space $M$.

Our first goal is to verify that we are now in the situation of Theorem \ref{finit}.(ii).

We set $(X,Y,S, f, v,F)$ to be $(M,\m{V}, S, h, \rho, \m{IC}_M)$.

The requirement $\phi F=0$ is met by Lemma \ref{vaicdm}.

As in the proof of Lemma \ref{vaicdm}, the intersection complex $\m{IC}_M$ is, locally over the nonsingular $S$, the pull-back from
of the intersection complex of a fiber. It follows that
the direct image sheaves
$R^\bullet \pi_* \m{IC}_M = I\!H^\bullet_D(\ms{X}/S,G)$
are locally constant on $S$, with stalks the intersection cohomology groups of the fibers of $\pi:M\to S$.

By the decomposition theorem \ci{bbd}, applied to the projective $h$ and the simple perverse sheaf   $\m{IC}_M$,
we have that the truncated $\ptd{\bullet} h_* \m{IC}_M$ are direct summands of the direct image $h_*F$.
It follows that  $\rho_* \ptd{\bullet}  h_* \m{IC}_M$ are direct summands of $\pi_* \m{IC}_M= \rho_* h_* \m{IC}_M$ in $D(S)$.

By combining the two last paragraphs, we have that the $\rho_* \ptd{\bullet}  h_* \m{IC}_M$ have locally constant cohomology sheaves.

We can thus apply Theorem \ref{finit}.(ii) (or its weaker variant (iii)), the conclusion of which is that the specialization morphism
$R\Gamma (M_s, i^* \m{IC}_M) \to R\Gamma (M_t, t^* \m{IC}_M)$ is defined and it is  a filtered isomorphism
for the respective perverse Leray filtrations.

Since, as it has been observed above,   the intersection complex
of $M$ restricts to the intersection complexes of the fibers $M_s$ and $M_t$, the
 first assertion of Theorem \ref{ewq} follows.

The second assertion, i.e. the independence of $s \in S$ on a connected $S$, follows easily: pick any two points
$s,s'$ and a suitably general point $t \in S$. Then apply what we have proved to the pairs $(s,t)$ and $(s',t)$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Q: How do I exclude specific packages from installation with apt-get? I would like to exclude specific packages from installation with apt-get, that is, install a metapackage without the list of specific packages, preferably with one invocation of apt-get.
For example, in Ubuntu 14.04 LTS command-line, I am installing MATE desktop environment. In 14.04 LTS MATE is not an official flavour, so I'm adding a PPA:
sudo apt-get install software-properties-common # need them for 'apt-add-repository'
sudo apt-add-repository ppa:ubuntu-mate-dev/ppa
sudo apt-add-repository ppa:ubuntu-mate-dev/trusty-mate
sudo apt-get update

then installing:
sudo apt-get install xorg mate-core --no-install-recommends

Even without the recommended extras, mate-core installs 3 terminal emulators: xterm, uxterm and mate-terminal , the latter lacking proper fonts while installed in this minimal configuration.
Suppose I decide that 3 terminal programs would be too many and I'd like to install xorg and mate-core without xterm and mate-terminal. I could do
sudo apt-get install xorg mate-core --no-install-recommends
sudo apt-get purge xterm mate-terminal

but is it possible to do this in one go? Is there some syntax like
sudo apt-get install xorg mate-core --without xterm mate-terminal

A: So, mate-core depends on mate-desktop-environment-core which in turn depends on mate-terminal. A depend cannot be broken easily. Nor apt-get nor aptitude supports a --without or --exclude option and "holding" the packages won't help:

$ echo "mate-terminal hold" | sudo dpkg --set-selections
$ sudo apt-get install mate-core
[...]
Some packages could not be installed...
The following packages have unmet dependencies:

Sure, one can use dpkg --force-depends to install a package, but that's maybe not what you want.
A possibility would be to fullfill the Depends: flag with a dummy package:

$ sudo apt-get install equivs
$ equivs-control mate-terminal
$ vi mate-terminal
Section: misc
Priority: optional
Standards-Version: 3.9.2
Package: mate-terminal
Version: 1.8.0
Description: Dummy package for mate-terminal
:x

$ equivs-build mate-terminal
$ sudo dpkg -i mate-terminal_1.8.0_all.deb

Now mate-core should be able to install w/o mate-terminal. Repeat the same for other packages to be excluded.
Admittedly this is quite an effort and a --without option would be nice. Maybe a wishlist bug can be opened to provide such functionality in the future, but I somehow doubt that this will be implemented.
However, a more realistic option would be to petition the PPA owner to provide another meta package for MATE with lesser Depends packages set.

A: There's a special apt syntax for scenarios like this: appending a hyphen (minus sign) to a package will remove/not install it, so the closest to your sudo apt-get install xorg mate-core --without xterm mate-terminal
is
sudo apt-get install xorg mate-core xterm- mate-terminal-
(spotted on https://askubuntu.com/a/1011439/182923)
It will not help in this case, as you will run into
The following packages have unmet dependencies:
 mate-desktop-environment-core : Depends: mate-terminal (>= 1.20) but it is not going to be installed
E: Unable to correct problems, you have held broken packages.

but it seems to be the answer to your specific question. So for reference, this cross-post.

A: If you use aptitude or a graphical package manager (Synaptic, etc.), then you can unselect which packages should be installed as long as it doesn't cause a dependency issue.
In the case of aptitude, before installing a package (in the graphical view), it will show you why a dependent package is being installed. In my case, I use KDE, and so don't have any MATE packages. If I tell it to install mate-core, I get the following:

Notice that mate-core is marked as being manually installed, and many other packages are being installed. For the mate-terminal package, in the bottom, it says, "mate-desktop-environment-core [universe] depends on mate-terminal (>= 1.0.0)". Going over to the entry for mate-desktop-environment-core,

"mate-core [universe] depends on mate-desktop-environment-core (>= 1.8.0+9)". This is why mate-terminal is being installed.
If, instead, a recommended package is being installed (rather than something that is dependent on another package), aptitude will tell you so, and you can tell it not to install that package without any broken dependencies. For example:

A: Another method
Download main package which depend on the other. Extract the package, remove the unwanted dependencies, repack it. Now open the new package with gdebi package installer. It will install new dependencies for you.

*

*List of depencies can be found at DEBIAN /control ->  depends

*For help on repacing deb see my answer on how to create a PPA

redpajama_set_name: RedPajamaStackExchange


The apostle beseeches them by various considerations, to live in unity and in the spirit of the Gospel, loving each other; and each to prefer his brother to himself, Philippians 2:1 -4. He exhorts them to be like-minded with Christ, who, though in the form of God, and equal with God, made himself of no reputation, and humbled himself to the death of the cross for the salvation of man; in consequence of which he was highly exalted, and had a name above every name; to whose authority every knee should bow, and whose glory every tongue should acknowledge, Philippians 2:5 -11. They are exhorted to work out their own salvation through his power who works in them, that they may be blameless, and that the apostle's labor may not be in vain, Philippians 2:12 -16. He expresses his readiness to offer his life for the Gospel, Philippians 2:17 , Philippians 2:18 . Intends to send Timothy to them, of whom he gives a very high character; yet hopes to see them himself shortly, Philippians 2:19 -24. In the meantime sends Epaphroditus, who had been near death, and whom he begs them to receive with especial tenderness, Philippians 2:25 -30.
Fulfil ye my joy - Ye ought to complete my joy, who have suffered so much to bring you into the possession of these blessings, by being like-minded with myself, having the same love to God, his cause, and me, as I have to him, his cause, and you.Being of one accord - Being perfectly agreed in labouring to promote the honor of your Master; and of one mind, being constantly intent upon this great subject; keeping your eye fixed upon it in all you say, do, or intend.
Let nothing be done through strife - Never be opposed to each other; never act from separate interests; ye are all brethren, and of one body; therefore let every member feel and labor for the welfare of the whole. And, in the exercise of your different functions, and in the use of your various gifts, do nothing so as to promote your own reputation, separately considered from the comfort, honor, and advantage of all.But in lowliness of mind - Have always an humbling view of yourselves, and this will lead you to prefer others to yourselves; for, as you know your own secret defects, charity will lead you to suppose that your brethren are more holy, and more devoted to God than you are; and they will think the same of you, their secret defects also being known only to themselves.
Look not every man on his own things - Do nothing through self-interest in the things of God; nor arrogate to yourselves gifts, graces, and fruits, which belong to others; ye are all called to promote God's glory and the salvation of men. Labor for this, and every one shall receive the honor that comes from God; and let each rejoice to see another, whom God may be pleased to use in a special way, acquiring much reputation by the successful application of his talents to the great work.
Let this mind be in you, which was also in Christ Jesus - Christ labored to promote no separate interest; as man he studied to promote the glory of God, and the welfare and salvation of the human race. See then that ye have the same disposition that was in Jesus: he was ever humble, loving, patient, and laborious; his meat and drink was to do the will of his Father, and to finish his work.
Who, being in the form of God - This verse has been the subject of much criticism, and some controversy. Dr. Whitby has, perhaps, on the whole, spoken best on this point; but his arguments are too diffuse to be admitted here. Dr. Macknight has abridged the words of Dr. Whitby, and properly observes that, "As the apostle is speaking of what Christ was before he took the form of a servant, the form of God, of which he divested himself when he became man, cannot be any thing which he possessed during his incarnation or in his divested state; consequently neither the opinion of Erasmus, that the form of God consisted in those sparks of divinity by which Christ, during his incarnation, manifested his Godhead, nor the opinion of the Socinians, that it consisted in the power of working miracles, is well founded; for Christ did not divest himself either of one or the other, but possessed both all the time of his public ministry. In like manner, the opinion of those who, by the form of God understand the Divine nature and the government of the world, cannot be admitted; since Christ, when he became man, could not divest himself of the nature of God; and with respect to the government of the world, we are led, by what the apostle tells, Hebrews 1:3 , to believe that he did not part with even that; but, in his divested state, still continued to uphold all things by the word of his power. By the form of God we are rather to understand that visible, glorious light in which the Deity is said to dwell, 1 Timothy 6:16 , and by which he manifested himself to the patriarchs of old, Deuteronomy 5:22 , Deuteronomy 5:24 ; which was commonly accompanied with a numerous retinue of angels, Psalms 68:17 , and which in Scripture is called The Similitude, Numbers 12:8 ; The Face, Psalms 31:16 : The Presence, Exodus 33:15 ; and The Shape of God, John 5:37 . This interpretation is supported by the term μορφη, form, here used, which signifies a person's external shape or appearance, and not his nature or essence. Thus we are told, Mark 16:12 , that Jesus appeared to his disciples in another μορφη, shape, or form. And, Matthew 17:2 , μετεμορφωθη, he was transfigured before them - his outward appearance or form was changed. Farther this interpretation agrees with the fact: the form of God, that is, his visible glory, and the attendance of angels, as above described, the Son of God enjoyed with his Father before the world was, John 17:5 ; and on that as on other accounts he is the brightness of the Father's glory, Hebrews 1:3 . Of this he divested himself when he became flesh; but, having resumed it after his ascension, he will come with it in the human nature to judge the world; so he told his disciples, Matthew 16:27 : The Son of man will come in the glory of his Father, with his angels, etc,. Lastly, this sense of μορφη Θεου, is confirmed by the meaning of μορθη δουλου, Philippians 2:7 ; which evidently denotes the appearance and behavior of a servant or bondman, and not the essence of such a person." See Whitby and Macknight.Thought it not robbery to be equal with God - If we take these words as they stand here, their meaning is, that, as he was from the beginning in the same infinite glory with the Father, to appear in time - during his humiliation, as God and equal with the Father, was no encroachment on the Divine prerogative; for, as he had an equality of nature, he had an equality of rights.But the word ἁρπαγμον, which we translate robbery, has been supposed to imply a thing eagerly to be seized, coveted, or desired; and on this interpretation the passage has been translated: Who, being in the form of God, did not think it a matter to be earnestly desired to appear equal to God; but made himself of no reputation, etc. However the word be translated, it does not affect the eternal Deity of our Lord. Though he was from eternity in the form of God - possessed of the same glory, yet he thought it right to veil this glory, and not to appear with it among the children of men; and therefore he was made in the likeness of men, and took upon him the form or appearance of a servant: and, had he retained the appearance of this ineffable glory, it would, in many respects, have prevented him from accomplishing the work which God gave him to do; and his humiliation, as necessary to the salvation of men, could not have been complete. On this account I prefer this sense of the word ἁρπαγμονbefore that given in our text, which does not agree so well with the other expressions in the context. In this sense the word is used by Heliodorus, in his Ethiopics, lib. vii. cap. 19, etc., which passage Whitby has produced, and on which he has given a considerable paraphrase. The reader who wishes to examine this subject more particularly, may have recourse to Heliodorus as above, or to the notes of Dr. Whitby on the passage.
But made himself of no reputation - Ἑαυτον εκενωσε· He emptied himself - did not appear in his glory, for he assumed the form of a servant, being made in the likeness of man. And his being made in the likeness of man, and assuming the form of a servant, was a proof that he had emptied himself - laid aside the effulgence of his glory.
And being found in fashion as a man - Και σχηματι εὑρεθεις ὡς ανθρωπος. This clause should be joined to the preceding, and thus translated: Being made in the likeness of man, and was found in fashion as a man.He humbled himself - Laid himself as low as possible:1. In emptying himself - laying aside the effulgence of his glory.2. In being incarnate - taking upon him the human form.3. In becoming a servant - assuming the lowest innocent character, that of being the servant of all.4. In condescending to die, to which he was not naturally liable, as having never sinned, and therefore had a right in his human nature to immortality, without passing under the empire of death.5. In condescending, not only to death, but to the lowest and most ignominious kind of death, the death of the cross; the punishment of the meanest of slaves and worst of felons.What must sin have been in the sight of God, when it required such abasement in Jesus Christ to make an atonement for it, and undo its influence and malignity!
Wherefore God also hath highly exalted him - If by his humiliation he has merited pardon and final salvation for the whole world, is it to be wondered that the human body, in which this fullness of the Godhead dwelt, and in which the punishment due to our sins was borne upon the tree, should be exalted above all human and all created beings? And this is the fact; for he hath given him a name, το ονομα, the name, which is above every name: το is prefixed to ονομα here by ABC, 17, Origen, Dionysius Alexandrinus, Eusebius, Cyril, and Procopius. This makes it much more emphatic. According to Ephesians 1:20 , Ephesians 1:21 , the man Christ Jesus is exalted to the right hand of God, far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come. From which it appears that no creature of God is so far exalted and so glorious as the man Christ Jesus, human nature being in him dignified infinitely beyond the angelic nature; and that this nature has an authority and pre-eminence which no being, either in heaven or earth, enjoys. In a word, as man was in the beginning at the head of all the creatures of God, Jesus Christ, by assuming human nature, suffering and dying in it, has raised it to its pristine state. And this is probably what is here meant by this high exaltation of Christ, and giving him a name which is above every name. But if we refer to any particular epithet, then the name Jesus or Savior must be that which is intended; as no being either in heaven or earth can possess this name as he who is the Redeemer of the world does, for he is the only Savior; none has or could redeem us to God but he; and throughout eternity he will ever appear as the sole Savior of the human race. Hence, before his birth, Gabriel stated that his name should be called Jesus; giving for reason, he shall Save his people from their sins. The qualifications of the Savior of the world were so extraordinary, the redeeming acts so stupendous, and the result of all so glorious both to God and man, that it is impossible to conceive a higher name or title than that of Jesus, or Savior of the world.
And that every tongue should confess - That all those before mentioned should acknowledge that Jesus Christ is Lord, or absolute governor, and thus glorify God the Father, who has exalted this human nature to this state of ineffable glory, in virtue of its passion, death, resurrection, and the atonement which it has made, by which so many attributes of the Divine nature have become illustrated, the Divine law magnified and made honorable, and an eternal glory provided for man.Others by things in heaven understand the holy angels; by things on earth, human beings generally; and by things under the earth, fallen spirits of every description. Perhaps the three expressions are designed to comprehend all beings of all kinds, all creatures; as it is usual with the Hebrews, and indeed with all ancient nations, to express, by things in heaven, things on earth, and things under the earth, all beings of all kinds; universal nature. See similar forms of speech, Exodus 20:4 ; Deuteronomy 4:17 , Deuteronomy 4:18 ; Psalms 96:11 ; and Ezekiel 38:20 . But intelligent beings seem to be those which are chiefly intended by the words of the apostle; for it appears that nothing less than absolute rule over angels, men, and devils, can be designed in these extraordinary words, and by confessing him to be Lord we may understand that worship which all intelligent creatures are called to pay to God manifested in the flesh; for all should honor the Son even as they honor the Father. And the worship thus offered is to the glory of God; so that far from being idolatrous, as some have rashly asserted, it is to the honor of the Divine Being. We may add, that the tongue which does not confess thus, is a tongue that dishonors the Almighty.
As ye have always obeyed - Continue to act on the same principles and from the same motives; having the same disposition which was in Christ; laboring so as to promote his glory.Work out your own salvation - Go on, walking by the same rule, and minding the same thing, till your salvation be completed: till, filled with love to God and man, ye walk unblamably in all his testimonies, having your fruit unto holiness, and your end everlasting life.With fear and trembling - Considering the difficulty of the work, and the danger of miscarriage. If you do not watch, pray and continually depend on God, your enemies will surprise you, and your light and life will become extinct; and then consider what an awful account you must give to Him whose Spirit ye have grieved, and of whose glory ye have come short.
For it is God which worketh in you - Every holy purpose, pious resolution, good word, and good work, must come from him; ye must be workers together with him, that ye receive not his grace in vain; because he worketh in you, therefore work with him, and work out your own salvation.To will and to do - Το θελειν και το ενεργειν. The power to will and the power to act must necessarily come from God, who is the author both of the soul and body, and of all their powers and energies, but the act of volition and the act of working come from the man. God gives power to will, man wills through that power; God gives power to act, and man acts through that power. Without the power to will, man can will nothing; without the power to work, man can do nothing. God neither wills for man, nor works in man's stead, but he furnishes him with power to do both; he is therefore accountable to God for these powers.Because God works in them the power to will and the power to do, therefore the apostle exhorts them to work out their own salvation; most manifestly showing that the use of the powers of volition and action belongs to themselves. They cannot do God's work, they cannot produce in themselves a power to will and to do; and God will not do their work, he will not work out their salvation with fear and trembling.Though men have grievously puzzled themselves with questions relative to the will and power of the human being; yet no case can be plainer than that which the apostle lays down here: the power to will and do comes from God; the use of that power belongs to man. He that has not got this power can neither will nor work; he that has this power can do both. But it does not necessarily follow that he who has these powers will use them; the possession of the powers does not necessarily imply the use of those powers, because a man might have them, and not use or abuse them; therefore the apostle exhorts: Work out your own salvation.This is a general exhortation; it may be applied to all men, for to all it is applicable, there not being a rational being on the face of the earth, who has not from God both power to will and act in the things which concern his salvation. Hence the accountableness of man.Of his good pleasure - Every good is freely given of God; no man deserves any thing from him; and as it pleaseth him, so he deals out to men those measures of mental and corporeal energy which he sees to be necessary; giving to some more, to others less, but to all what is sufficient for their salvation.
Do all things without murmurings - Γογγυσμων, και διαλογισμων· Without grumblings and altercations. Be patient in, and contented with, your work; and see that ye fall not out by the way.
That ye may be blameless - In yourselves, and harmless to others.The sons of God - Showing by your holy conduct that ye are partakers of the Divine nature.Without rebuke - Persons against whom no charge of transgression can justly be laid.A crooked and perverse - Probably referring to the Jews, who were the chief opponents and the most virulent enemies which the Christian Church had.Among whom ye shine - Be like the sun and moon; bless even the perverse and disobedient by your light and splendor. Let your light shine before men; some will walk in that light, and by its shining God will be glorified. It is evident that the apostle, by φωστῃρες εν κοσμῳ, lights in the world, refers to the sun and moon particularly, and perhaps to the heavenly bodies in general.
Holding forth the word of life - An allusion, some think, to those towers which were built at the entrance of harbours, on which fires were kept during the night to direct ships into the port. Genuine Christians, by their holy lives and conversation, are the means of directing others, not only how to escape those dangers to which they are exposed on the tempestuous ocean of human life, but also of leading them into the haven of eternal safety and rest.That I have not run in vain - This appears to be a part of the same metaphor; and alludes to the case of a weather-beaten mariner who has been long tossed on a tempestuous sea, in hazy weather and dark nights, who has been obliged to run on different tacks, and labor intensely to keep his ship from foundering, but is at last, by the assistance of the luminous fire on the top of the tower, directed safely into port. Live so to glorify God and do good to men, that it shall appear that I have not run and labored in vain for your salvation.
Yea, and if I be offered upon the sacrifice and service - The metaphor appears to be still carried on. As it was customary for the weather-beaten mariner, when he had gained his port, to offer a sacrifice, θυσια, to God, of some particular animal which he had vowed while in his state of danger, and this was considered to be a religious service, λειτουργια· the apostle, pursuing the idea, states himself to be willing to become the libation, (for so much the word σπενδομαι imports), that was to be poured upon the sacrifice. Parkhurst observes that the apostle compares the faith of the Philippians to the sacrificial victim, and his own blood shed in martyrdom to the libation, i.e. the wine poured out on occasion of the sacrifice. Raphelius observes that Arrian uses the phrase σπενδειν επι τῃ θυσιᾳ for pouring out the libation after the sacrifice. The apostle had guided them safely into port; their faith in the atoning death of Christ was their sacrifice; and he was willing that his blood in martyrdom should be poured out as a libation on that sacrificial offering.
For the same cause also do ye joy - Should I be thus offered, as I shall rejoice in it, do ye also rejoice that I am counted worthy of this high honor.
But I trust in the Lord Jesus - He is governor and disposer of all events, being above all principality and power; and I humbly confide in his power and goodness that I shall be a little longer spared to visit you again, Philippians 2:24 , and to be able to send Timothy shortly to you.When I know your state - By the correct information which I shall receive from Timothy.
For I have no man like-minded - None of all my fellow helpers in the Gospel have the same zeal and affectionate concern for your prosperity in every respect as he has. He is ισοψυχος· of the same soul; a man after my own heart.
For all seek their own - This must relate to the persons who preached Christ even of envy and strife, Philippians 1:15 ; these must be very careless whether souls were saved or not by such preaching; and even those who preached the Gospel out of good will might not be fit for such an embassy as this, which required many sacrifices, and consequently much love and zeal to be able to make them.
Ye know the proof of him, that, as a son with the father, he hath served with me - The Philippians had full proof of the affectionate attachment of Timothy to Paul, for he had labored with him there, as we learn from Acts 16:1 -3; Acts 17:14 ; and we find from what is said here that Timothy was not a servant to the apostle, but that he had served with him. They both labored together in the word and doctrine; for apostles and Christian bishops, in those times, labored as hard as their deacons. There were no sinecures; every one was a laborer, every laborer had his work, and every workman had his wages.
How it will go with me - The apostle was now in captivity; his trial appears to have been approaching, and of its issue he was doubtful; though he seems to have had a general persuasion that he should be spared, see Philippians 2:19 , Philippians 2:24 .
Epaphroditus, my brother, etc - Here is a very high character of this minister of Christ; he was,1. A brother - one of the Christian family; a thorough convert to God, without which he could not have been a preacher of the Gospel.2. He was a companion in labor; he labored, and labored in union with the apostle in this great work.3. He was a fellow soldier; the work was a work of difficulty and danger, they were obliged to maintain a continual warfare, fighting against the world, the devil, and the flesh.4. He was their apostle - a man whom God had honored with apostolical gifts, apostolical graces, and apostolical fruits; and,5. He was an affectionate friend to the apostle; knew his soul in adversity, acknowledged him in prison, and contributed to his comfort and support.
Ye had heard that he had been sick - "In this passage," says Dr. Paley, "no intimation is given that the recovery of Epaphroditus was miraculous, it is plainly spoken of as a natural event. This instance, together with that in the Second Epistle to Timothy, Trophimus have I left at Miletum sick, affords a proof that the power of performing cures, and, by parity of reason, of working other miracles, was a power which only visited the apostles occasionally, and did not at all depend upon their own will. Paul undoubtedly would have healed Epaphroditus if he could; nor would he have left Trophimus at Miletum sick, had the power of working cures awaited his disposal. Had this epistle been a forgery, forgery on this occasion would not have spared a miracle; much less would it have introduced St. Paul professing the utmost anxiety for the safety of his friend, yet acknowledging himself unable to help him, which he does almost expressly in the case of Trophimus, Him have I left sick; and virtually in the passage before us, in which he felicitates himself on the recovery of Epaphroditus in terms which almost exclude the supposition of any supernatural means being used to effect it. This is a reverse which nothing but truth would have imposed." Horae Paulinae, page 234.
Lest I should have sorrow upon sorrow - The sorrows of his death, added to the sorrow he endured on account of his sickness; or he may refer to his own state of affliction, being imprisoned and maltreated.
The more carefully - Σπουδαιοτερως· With the more haste or despatch; because, having suffered so much on account of his apprehended death, they could not be too soon comforted by seeing him alive and restored.
Receive him therefore in the Lord - For the Lord's sake receive him, and as the Lord's servant; and hold such zealous, disinterested, and holy preachers in reputation - honor those whom ye perceive God hath honored.
For the work of Christ - Preaching the Gospel, and ministering to the distressed.He was nigh unto death - Having labored far beyond his strength.Not regarding his life - Instead of παραβουλευσαμενος τῃ ψυχῃ, not regarding his life, παραβολευσαμενος, risking his life, is the reading of ABDEFG, and is received by Griesbach into the text. His frequent and intense preaching, and labouring to supply the apostle's wants, appear to have brought him nigh to the gates of death.The humiliation and exaltation of Christ are subjects which we cannot contemplate too frequently, and in which we cannot be too deeply instructed.1. God destroys opposites by opposites: through pride and self-confidence man fell, and it required the humiliation of Christ to destroy that pride and self-confidence, and to raise him from his fall. There must be an indescribable malignity in sin, when it required the deepest abasement of the highest Being to remove and destroy it. The humiliation and passion of Christ were not accidental, they were absolutely necessary; and had they not been necessary, they had not taken place. Sinner, behold what it cost the Son of God to save thee! And wilt thou, after considering this, imagine that sin is a small thing? Without the humiliation and sacrifice of Christ, even thy soul could not be saved. Slight not, therefore, the mercies of thy God, by underrating the guilt of thy transgressions and the malignity of thy sin!2. As we cannot contemplate the humiliation and death of Christ without considering it a sufficient sacrifice, oblation, and atonement for sin, and for the sin of the whole world; so we cannot contemplate his unlimited power and glory, in his state of exaltation, without being convinced that he is able to save them to the uttermost that come unto God through him. What can withstand the merit of his blood? What can resist the energy of his omnipotence? Can the power of sin? - its infection? -its malignity? No! He can as easily say to an impure heart, Be thou clean, and it shall be clean; as he could to the leper, Be thou clean, and immediately his leprosy was cleansed. Reader, have faith in Him; for all things are possible to him that believeth.3. There are many ungodly men in the world who deny the inspiration of God's Holy Spirit, and affect to ridicule those who profess to have received what they know Christ has purchased and God has promised, and which, in virtue of this, they have claimed by faith; because, say these mockers, "If you had the Spirit of God, you could work miracles: show us a miracle, and we will believe you to be inspired." Will these persons assert that St. Paul had not God's Spirit when he could neither heal himself, nor restore his friends and fellow helpers from apparent death? What then doth their arguing prove? Silly men, of shallow minds!


German prosecutors have imposed a fine of NOK 83 million on BMW to produce diesel cars with higher emissions than permitted by law.
BMW has not done this with deliberation. Rather, it is about exceeding the emission limits as a result of an error, the prosecuting authority points out.
"The Munich prosecuting authority has issued a fine of 8.5 million euros for an administrative offense that led to negligence in quality control," a statement said.
Just under 8,000 cars are covered by the error. The prosecution has reason to believe that "misleading of the part of the engine software that controlled the exhaust system" was the reason why the emissions turned out to be higher than the law allows.
– Thorough investigation has neither found evidence that cheating software was being overlaid with deliberation, nor that BMW employees with deliberations have been cheating, adds the prosecution.
– But the company had not introduced an appropriate quality control system that could have prevented the error or revealed it after it appeared, writes the prosecutor.
BMW's share price appeared to be untouched by the fine on Monday morning and rose 1.3 percent on the Frankfurt stock exchange.


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Q: How can we access multiple variables using XPath in Jemter? I am making scripts using JMeter to perform some tasks in which I want to access first request's response in the second request as a value parameter value using XPath extractor.
For Ex.
Here is my response of the first request in XML format:
<a>
  <b>
    <c>

       <my_details>
          <first_name>A</first_name>
          <middle_name>B</middle_name>
          <last_name>C</last_name>
       </my_details>

    </c>
  </b>
</a>

Now, I want to use first name and last name in the second request's value. For that, I am using XPath extractor to get values from the response as given below:

But I got the response like:
        JMeterVariables:
        **full_name=A
        full_name_1=A**
        ...
        ...
        __jm__Thread Group__idx=0
        __jmeter.USER_TOKEN__=Thread Group 1-1

So, my question is, how can I get the full name using single XPath extractor?

A: Try to extract "first_name" and last "last_name" separately using XPath extractor , Then use Beanshell post-processor to create Jmeter variable for fullname , Like
vars.put("full_name",vars.get("FIRST NAME JMETER VARIABLE")+" "+vars.get("LAST NAME JMETER VARIABLE"));

A: I was able to achieve the desired result using the concat() function (https://www.w3.org/2005/xpath-functions/).
I utilized XPath2 Extractor with Xpath query set to:

concat(//my_details/first_name, " ", //my_details/last_name)

This leads to output variable being assigned the value "A C":

Please, note that it is currently advised to use XPath2 Extractor
(https://jmeter.apache.org/usermanual/component_reference.html#XPath_Extractor):