\section{Introduction}
Due to their large masses, heavy flavor ($c$ and $b$) quarks are produced in the early stages of heavy ion collisions \cite{ref1}.
A precise measurement of heavy flavor production could be achieved by identifying the decay of charmed mesons using direct topological reconstruction and thus disentangling the $c$ and $b$ contributions. The HFT has the necessary resolution for such a measurement that requires high precision. It is the assembly of the existing Silicon Strip Detector (SSD) and 2 new detecting devices: the Intermediate Silicon Tracker (IST) and the PIXEL detector. The PIXEL is composed by 2 layers of monolithic CMOS Active Pixel sensors \cite{ref2} which measure with great accuracy the position of a particle within a few centimeters of the interaction region. These very thin layers
minimize the multiple coulomb scaterring.
The intermediate tracking system is made by the IST and the existing SSD (Table \ref{tab:fit-res}). The purpose of the IST and the SSD is to link tracks found in the STAR Time Projection Chamber (TPC) to the PIXEL detector.
\begin{table}[ht]
\caption{Characteristics of each silicon layer of the HFT\label{tab:fit-res}}
\begin{center}
\item[]\begin{tabular}{c c c c c c}
\hline
Detector&Radius&Technology&Si thickness&Hit resolution&Material Budget\\
                &             &                     &                              &R/$\phi-Z$                     & in radiation length $X_\mathrm{0}$  \\
                &(cm)&                     &      ($\mu$m)       &  ($\mu$m - $\mu$m)       &                  \\
\hline
SSD&23&double sided strips&300&30 - 857&1\%\\
IST&14&Si Strip Pad sensors&300&170 -1700&1.2\% \\
PIXEL&2.5, 8&Active Pixels& 50&8.6 - 8.6&0.37\%\\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Simulation details}
Simulations presented in this proceedings were performed using the full STAR geometry package with 10k AuAu HIJING central events at $\sqrt{s_{NN}}$ = 200 GeV embedded with $D^\mathrm{0}$ and $\Lambda_\mathrm{c}$ particles, forced to decay to their hadronic channels ($D^\mathrm{0}$$\rightarrow$$K^\mathrm{-}$$\pi^\mathrm{+}$,  $\Lambda_\mathrm{c}$$\rightarrow$$K^\mathrm{-}$$\pi^\mathrm{+}p$). Their reconstruction efficiencies are based on particle identification of daughter particles provided by the TPC and extended to higher $p_\mathrm{T}$ with the Time of Flight detector (TOF) : $K$-$\pi$ and ($K$+$\pi$)-$p$ separations were done up to $p_\mathrm{T}$ $\leq$ 1.6 GeV/c and $p_\mathrm{T}$ $\leq$ 3 GeV/c, respectively. Topological cuts have been also applied to the $D^\mathrm{0}$ candidates. The effect of {\it out of time} events is included in the PIXEL simulation at a rate corresponding to RHIC-II luminosity.

\section{Estimated $D^\mathrm{0}$ and $\Lambda_\mathrm{c}$ reconstruction performances}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.28\textwidth]{D0_v2.eps}
\includegraphics[width=0.28\textwidth]{D0_Rcp.eps}
\includegraphics[width=0.28\textwidth]{updated_Lc_over_D0.eps}
\end{center}
\caption{Projections of key measurements with HFT\label{fig:keytest}}
\end{figure}
Figures \ref{fig:keytest} show the statistical error projections for the key measurements with the HFT in 500 M minimum bias AuAu collisions. Fig. \ref{fig:keytest}(left) is the flow parameter $v_{2}$, shown for two extreme scenarios [charm quark flow equal to light quark flow (red circles) and charm quark does not flow (green triangles)]. The HFT will be able to distinguish with great accuracy these 2 cases. Figure \ref{fig:keytest}(middle) shows the suppression factor $R_\mathrm{CP}$ of $D^\mathrm{0}$: the HFT will be able to measure it directly for $p_\mathrm{T}$ $\leq$ 10 GeV/c via the hadronic channel thus avoiding the indirect method using non-photonic electrons. A measurement of $\Lambda_\mathrm{c}$ is important to perform since the $\Lambda_\mathrm{c}$/$D^\mathrm{0}$ ratio may be enhanced, indicating a similar pattern to the baryon/meson ratio involving light quarks in the intermediate $p_\mathrm{T}$ region \cite{ref3}. Two scenarios are investigated for the $\Lambda_\mathrm{c}$/$D^\mathrm{0}$ ratio \cite{ref4} : no enhancement and same enhancement as $\Lambda$/$K_\mathrm{s}^\mathrm{0}$.
We see from Fig. \ref{fig:keytest}(right) that the statistical errors are sufficiently small, making a measurement of baryon/meson ratio in charm sector with good precision in heavy ion collisions.
 \section{Summary}
The HFT, by using low mass CMOS sensors, will be able to directly reconstruct charm hadrons over a large momentum range and, thus, study flow and energy loss of heavy flavor particles. Several physics capabilities such as baryon/meson ratio in the charm sector have been studied.


\section{Introduction}

Image processing and analysis has been revolutionized by the rise of deep learning. For semantic segmentation, deep learning approaches use mainly convolutional neural networks (CNN) \cite{fukushima_neural_1979,lecun_backpropagation_1989}. In the biomedical field, U-Net \cite{ronneberger_u-net:_2015} has become the state-of-the-art method for this task, but other solutions exist, such as SegNet \cite{badrinarayanan_segnet:_2015}. These networks are fully convolutional. Their receptive fields are of limited size. Therefore, they cannot intrinsically process global information, such as topological information \cite{rosenfeld_digital_1979}. We recall that, unlike in networks containing fully connected layers, where the value of each unit depends on the entire input to the network, a unit in a convolutional networks only depends on a region of the input. This region in the input is the receptive field for that unit.

In the following, we present a practical real-world situation where the segmentation result depends on topological information. We show that a classical fully convolutional CNN does not give satisfactory results and propose a solution to this problem.

During the past years, the way of working in the histological field has changed due to the emergence of Whole Slide Imaging solutions that are now available and useful for pathologists but also for dermatologists and cosmetologists. These automated scanners improve digital histology by allowing several hundred slides a day to being acquired through imaging process, and stored, used for second opinions or for automated analysis. This well-established technology provides large sets of image data that need to be automatically processed whatever the amount of generated data. Image analysis methods based on Deep Learning have proved to be extremely useful in this field \cite{bejnordi_diagnostic_2017,wang_deep_2016}.

To circumvent this, we propose a method that allows neural networks based on convolutional layers to take into account non-local information. It is based on a geodesic operator, a popular tool from mathematical morphology. We apply it to the segmentation of whole slide images of pigmented reconstructed epidermis, as described hereafter.

\section{Material}

The images that were collected include pigmented reconstructed epidermis samples used to evaluate and identify the de-pigmenting or pro-pigmenting efficiency of cosmetic ingredients. They have been colored using a Fontana Masson staining, a silver stain that is used to highlight melanin and to also reveal skin layer morphology. Their sizes are diverse and can reach up to 20 million pixels. The ground truth (GT) has been obtained by an automatic method developed by the ADCIS company, whose results were manually edited and modified by L’Oréal experts, when needed. On those images, the goal of the segmentation is to identify two regions corresponding to two specific skin layers: the \textit{stratum corneum} (SC) and the living epidermis.

The boundary between the SC and the background is usually rather difficult to determine, mainly because it can be composed of different layers separated by gaps due to the desquamation process that happens in the SC. Such layers are only considered as part of the SC if they constitute an unbroken boundary between the background and the sample. This feature is highly non local, and as such a convolutional neural network cannot enforce it.

The resulting database contains 120 color images, coming from 46 different slides. The original images are of variable size, and can contain more than 20 million pixels. The test database has been built with approximately 20\% of the images. The remaining images were used for learning and validation. We took care to transfer all the images from a given slide into the same database.

\begin{figure}[t!]
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{15ERP11_17_18X3Y1_sub03_modified_interface.png}}
\end{minipage}
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{16ERP04_09_10_X2Y1_sub04_modified_interface.png}}

\end{minipage}
\caption{Examples of original images with overlayed contours of the reference segmentation (best viewed in colour). Red/top contour: frontier between background and SC. Note that its position can be completely shifted if the detached layer is unbroken (top) or broken (bottom). Green/middle contour: frontier between SC and living epidermis. Cyan/bottom contour: frontier between living epidermis and collagen scaffold (here considered as background).}
\label{fig:examples}
\end{figure}

Given the layered structure of the images, segmenting them into four regions is equivalent to finding three frontiers (see Fig.~\ref{fig:examples}). The top one, between background and SC, has a variable appearance. In some images it corresponds to a contrasted contour; in other to a soft irregular contour. More importantly, its exact position depends on non-local information. Indeed, when a layer of the SC is separated from the rest of the tissue, it will belong to the SC region only if it is connected to the rest of the SC, or if it constitutes an unbroken frontier, going from the left side of the image to the right. Therefore it is not possible to make that decision based uniquely on local information. This kind of situation is illustrated by Fig. \ref{fig:examples}.

The second frontier separates SC from living epidermis. On our images, the distinction between those regions raises from different textures. The third frontier corresponds to the limit between living epidermis and collagen scaffold. Note that the fourth region to be segmented is made up of two compartments: the collagen scaffold that supports the reconstructed skin, and background. The collagen scaffold contains some large “holes” that can locally look as the “holes” within the SC.

The ground truth was generated using an automatic method developed by the ADCIS company, whose results were manually edited and modified by L’Oréal experts, when needed. Given that the top and bottom regions of the ground truth contained both a large white region, which could not be locally differentiated, we decided to only consider three labels: label 1 corresponds to the background (both at the top and the bottom of the images) and collagen scaffold; label 2 to the SC; label 3 to the living epidermis.

\section{Methods}

It was decided to use convolutional neural networks to tackle this problem. During the learning phase, we worked with crops of a given size. After running some tests, the final size of the crops was $512 \times 512$.

The ground truth segmentations, as they contained three labels, were classically represented as an image with three binary channels. Given that background white regions covered the majority of the images, for training we only used the crops that contained at least label 2 or label 3. This procedure is illustrated in Fig.~\ref{fig:comparison}. The resulting set of crops contains 1458 elements. 80\% are randomly picked for learning; the other 20\% are used for validation.

Concerning the loss functions, based on our experience with segmentation using convolutional neural networks, we chose the following loss function between two same length vectors $X$ and $Y$, containing values included between $0$ and $1$:

\[
J_2 = 1 - \frac{XY +  \epsilon}{ X^2 + Y^2 -XY + \epsilon}
\]

where $\epsilon$ is a small constant, used for numerical reasons. This loss is based on the Jaccard index, also called ``intersection over union'' similarity measure, often used to evaluate the quality of a segmentation.

\subsection{Neural network architecture and first results}

After testing several network architectures, the one that gave the best results was U-Net \cite{ronneberger_u-net:_2015} (using a sigmoid in the last activation layer, and using zero-padding in the convolutional layers). Full details are given in section~\ref{ssec:hyper}. The validation loss of the resulting model was $0.085$.

The networks output contains 3 channels. Each one can be interpreted as the probability of a given pixel of belonging to each region. In order to obtain a segmentation, we naturally gave to each pixel the label corresponding to the channel with the highest probability.

\begin{figure}[t!]
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{1.png}}
\end{minipage}
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{2-1.png}}

\end{minipage}
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{3-1.png}}

\end{minipage}
\caption{Top: original image, showing the selected crops. Middle: results without using global information. Bottom: result using global information, thanks to the presented method. These segmentation results have not been postprocessed. They are overlayed on the original data using the following colour code: SC (magenta), living epidermis (orange) and other regions (cyan).}
\label{fig:comparison}
\end{figure}

A qualitative analysis of the first results showed that the resulting frontiers between SC and living epidermis, on the one hand, and between living epidermis and collagen scaffold, on the other hand, were very satisfactory. However, the frontier between background and SC, in some cases, contained errors. In Fig.~\ref{fig:comparison} (middle) we see that this frontier is incorrectly detected and that the gap between the detached SC layer, on the right, is incorrectly considered as belonging to the SC. These errors span from the fact, as we previously said, that the definition of this frontier is based on non local information.

\subsection{Taking into account non local information in a convolutional neural network}

A new method is proposed here to cope with non local information within convolutional neural networks. It is based on a geodesic reconstruction of the input image from the top and bottom of the image, channel-wise.

We recall that the geodesic reconstruction \cite{vincent_morphological_1993} of a grey level image $I$ from an image $J_0$ (often called marker) is obtained by iterating the following operation until idempotence:
\[
J_{n+1} = \delta(J_n) \bigwedge I
\]

where $\delta$ is a morphological dilation, here with the cross structuring element, corresponding to the 4-connectivity. Our marker image $J_0$ is equal to $I$ on the first and last rows of the image domain, and to zero elsewhere. The process will ``fill'' the holes within the tissue, and preserve the intensity of the pixels that belong to the connected components of the background that touch the top and bottom of the image. This geodesic operation thus allows to bring topological information, which is essentially global, to a local scope.

In order to recover some of the bright details of the tissue sample, the result of this reconstruction is combined with the initial image by computing their mean. If we call $J$ the result of the above reconstruction, the output image is simply:

\[
F = ( J + I ) / 2 \text{.}
\]

\begin{figure}[t!]
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{full.png}}
\end{minipage}
\begin{minipage}[b]{1.0\linewidth}
  \centering
\centerline{\includegraphics[width=10.1cm]{full_filled.png}}

\end{minipage}
\caption{Top: original image. Bottom: image after pre-processing based on the geodesic reconstruction. Differences are mainly visible on the holes within the tissue sample.}
\label{fig:example_rec}
\end{figure}

This operator is illustrated in Fig.~\ref{fig:example_rec}. All images follow the same pre-processing (before computing the crops). Learning is done as before, with the same parameters.

The new CNN suppresses the segmentation errors due to the lack of global information on the background / SC boundary. Fig.~\ref{fig:comparison} clearly shows this improvement.

The validation loss of the model is $0.028$, to be compared with the previous value of $0.085$.

\subsection{Hyper-parameters optimization and data augmentation}
\label{ssec:hyper}

We tuned the hyper-parameters of our system through manual grid search using the validation dataset. The parameters of the final model are: Optimizer: adadelta \cite{zeiler_adadelta:_2012}, with default parameters (learning rate: 1; rho: 0.95; epsilon: 10-8; decay: 0); epochs: 200; patience: 20, batch size: 4. The initial number of convolutional filters of the U-Net network is $16$ (instead of $64$ in the original paper), resulting in a network with  1,962,659 parameters.

We also tested several standard data augmentation methods, but they did not bring any improvement. We believe that this result means that our database constitutes a representative sampling of our image domain.

\subsection{Post-processing}

The current results are already satisfactory. There are however a few defects in the resulting segmentation (as can be seen in Fig.~\ref{fig:comparison}), most of which can be corrected with the following post-processing method:

\begin{enumerate}
\item For the SC and the living epidermis, keep only the largest connected component; for the background, keep the connected components that touch the top and the bottom of the image.
\item Pixels without label are given the label of the closest labelled pixel.
  \end{enumerate}

\section{Results}

\begin{figure}[t!]
\begin{minipage}[b]{0.48\linewidth}
  \centering
\centerline{\includegraphics[width=5cm]{crop1-1.png}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
  \centering
\centerline{\includegraphics[width=5cm]{crop2-1.png}}

\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
  \centering
\centerline{\includegraphics[width=5cm]{crop3.png}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
  \centering
\centerline{\includegraphics[width=5cm]{crop4.png}}

\end{minipage}
\caption{Zoom-in on some test images to illustrate the results, as well as its robustness to acquisition artifacts. The contours of the segmentation computed with the final model are overlayed on the original images.}
\label{fig:good}
\end{figure}

It is interesting to note that once a convolutional neural network has been trained (with crops of constant size, as previously stated) it can be applied to images of almost arbitrary sizes.  There are only two limitations: the system memory has to be large enough and neural network architectures that use downsampling layers impose that the dimensions be multiple of some $2n$ (where $n$ is the number of such layers, supposing that the sampling steps are equal to $2$). This approach is interesting not only for practical reasons (no need to compute any more crops and stitch them back together at prediction time) but also significantly alleviates border effects .

There are $23$ images in the test database. Globally, the results were considered as very good by the final users. They are illustrated in Fig.~\ref{fig:good}. Only two errors were visible at first sight among the $23$ images. They are shown in Fig.~\ref{fig:errors}. Other errors are less visible. They correspond most of the time to a slight displacement of the obtained contour \footnote{One image in Fig.~\ref{fig:good} contains such an error; let the reader try to find it!}.

Tab.~\ref{tab_res} gives some quantitative results on the test database. Accuracy values (the proportion of pixels that are correctly classified) show that incorrectly classified pixels are three times less numerous with our proposed method than with the standard approach. The Jaccard index\footnote{The Jaccard index of two sets is the ratio between the size of their intersection and the size of their union} of the living epidermis region shows almost no improvement: this is natural, as this region can be correctly segmented based solely on local information. On the contrary, the Jaccard index of the \textit{stratum corneum} shows a significant improvement, as the definition of this region heavily relies on non-local information. Finally the mean distance between predicted contours and ground truth (GT) contours confirms this improvement.

\begin{table}[htp]
\centering
\caption{Quantitative results on test database.}
\label{my-label}
\begin{tabular}{|l|c|c|c|c|}
\hline
                & Accuracy & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Jaccard of \\ \textit{stratum corneum}\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Jaccard of \\ living epidermis\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Mean distance to \\ GT contour\end{tabular}} \\ \hline
Standard U-Net  & 98.33\%                       & 91.46\%                                                                                & 94.24\%                                                                                & 18 pixels                                                                                \\ \hline
Proposed method & 99.49\%                       & 97.43\%                                                                                & 94.82\%                                                                                & 4 pixels                                                                                \\ \hline
\end{tabular}
\label{tab_res}
\end{table}

\begin{figure}[htb]

  \begin{minipage}[b]{0.48\linewidth}
  \centering
\centerline{\includegraphics[width=5cm]{error1.png}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
  \centering
\centerline{\includegraphics[width=5cm]{error2.png}}

\end{minipage}
\caption{Zoom-in onto the two more significant errors found on the 23 images of the test database.}
\label{fig:errors}
\end{figure}

Processing times are as follows. The standard U-Net takes 171 seconds to process the full 23 test images on a conventional laptop with a NVidia GeForce GTX 980M graphics card. The improved method, including the geodesic reconstruction, takes 407 seconds. We think that the pre-processing could be optimized, but the current version is already fast enough for the application at hand.

\section{Conclusion}

A novel method to utilize global information within a convolutional neural network has been introduced. Based on the morphological reconstruction by dilation, it allows the network to take advantage of geodesic information.

This method has been successfully applied to the segmentation of histological images of reconstructed skin using a U-Net architecture. We believe that a similar improvement should be obtained with other fully convolutional neural networks, such as SegNet.

The method is being integrated in a complete software in order to use it in routine practice.

As a perspective, it would be interesting to explore other ways to use global information within convolutional neural networks, and compare them.

\bibliographystyle{splncs03}


\section{Introduction}
In recent years it has become obvious that the nature of
UltraLuminous  X-ray sources (ULXs) in nearby galaxies
cannot be elucidated from X-ray observations alone. In
particular, the question whether or not  ULXs harbour
specimen of the well-advertised intermediate mass black
holes (IMBH) lying in between the stellar and the AGN
variety has stimulated  numerous investigations. Other
ideas put forward to overcome  the classical Eddington
luminosity limit  ($10^{39.1} \;
\mathrm{M}/(10\mathrm{M}_{\odot})$ erg/s)  of a stellar mass black hole
with M$\sim10$M$_{\odot}$ include beamed emission into our
line of sight - either geometrically or
relativistically - and highly Super-Eddington accretion
rates, presumably coupled with the occurrence of massive
outflows from the systems.

Optical follow-up observations of nearby ULXs have in many
cases revealed  the presence of extended ionized
nebul\ae\, \citep{pam02,pam03} and of stellar
associations \citep{grpm06,gr08} that both can be used to derive important
constraints on possible formation scenarios, ages and on
the photon- and  mechanical luminosities involved. X-ray
photoionisation arguments have  previously been used to
exclude strongly-beamed X-ray emission from  the very
luminous ULX in Holmberg~II. In this contribution we will
mainly focus on the effects of mechanical power that is released
by ULXs and is injected into the ISM.

\section{Photoionisation versus Shock Excitation}
Galactic massive X-ray binaries are generally not located in
regions of  high interstellar density. That is why observable
X-ray photoionized  nebul\ae\ (XIN), as, e.g. N\,159F around the
black hole candidate LMC X-1  {\citep{pang86} are rare. The key
observation which indicated that X-ray  emission is largely
responsible for the high ionization structure of N\,159F came
from the detection of extended nebular He\,II$\lambda4686$
recombination radiation. This line is not present in normal
H\,II regions because even  the hottest O stars do not emit
significant amounts of He$^+$ Lyman continuum photons that
could ionize observable He\,III regions (exception:  hot
Wolf-Rayet stars with unusually weak winds).
An even more powerful XIN is embedded in the star forming
region  HSK~70 located in the M81 group dwarf galaxy
Holmberg\,II. This He\,III  region is excited by the bright
ULX in that galaxy \citep{pam02} and allows, through Zanstra-type
$\lambda4686$ photon
counting arguments, to establish the flux of the ionizing
He$^+$ Lyman continuum to be compatible with the
10$^{40}$ erg/s isotropic luminosity measured with X-ray
telescopes. In other words, optical observations have been employed to
exclude the possibility that the source is beamed towards us.
Possibly, a further XIN example is provided by the $\lambda4686$
emitting 30 pc diameter supernova remnant candidate MF16 around
the ULX NGC~6946 X-1 \citep{abol}. However, in this case, the
observed  expansion velocity is quite high, $\sim230$ km/s
\citep{du00},  which implies that shock-ionization must play a
key role here.

We have previously commented on the similarity of optical spectra
displayed by XIN on the one hand and by shock excited gas on the
other hand.  The situation is analogous to the
ambiguity in the interpretation of spectra of the {\it LINER}
subgroup of AGN and of extended emission line regions {\it (EELR)}, where
both shocks and ionisation by hard non-stellar continua have
been put forward to explain their characteristics.

\section{Structure and Energetics of Large ULX Bubbles}

It came as quite a surprise when it was realized that a
significant fraction  of ULXs are surrounded by large bubble-like
nebul\ae\ (hereafter ULXB) of  some 200-500 pc diameter.
Their optical spectra are reminiscent of the
generally more than 10  times smaller supernova remnants,  i.e.
shock-ionized gas combining strong emission from low-ionisation
species  like [OI]$\lambda6300$ and [SII]$\lambda6716,31$ with
emission  from more highly ionized gas, as e.g.
[OIII]$\lambda5007$.
The latter forbidden transition is excited only at sufficiently high shock
speeds, v$_s$, and the diagnostic $\lambda5007$/H$\alpha$ ratio varies by a
factor $\geq10$ in the very narrow critical range of 80-100 km/s
(somewhat dependent on the pre-shock conditions) before leveling off at about
unity at higher velocities \citep{dop84}.

Although the ULX counterparts often appear to be part of
moderately  rich stellar associations \citep{grpm06,gr08}, the
clusters  are mostly too old ($\geq$20-40 Myrs) to photoionize
observable H\,II regions. Naturally, even v$_s\sim100$ km/s
shocks become unobservable once the IS density drops below, say,\,
$\sim0.3$ cm$^{-3}$, when  the H$\alpha$ recombination line
intensity behind the shock -- which is  proportional to the
preshock density $n$ (not \mbox{$\sim n^2$ !)} -- drops below a threshold
where fluctuations of the  diffuse galactic background emission
become important.

Optical observations of ULX bubbles show an amazing variety of
filamentary and shell morphologies, as well as varying nebular
excitation displayed by diagnostic emission line ratios. This is
well illustrated by the huge, 500 pc diameter shell-like
ULXB MH~9-11 around Holmberg~IX X-1 (Fig. 1) first described by \citet{mi95},
and which also displays expansion velocities in the critical range.
The image underlines the markedly different OIII vs H$_{\alpha}$  morphology
in the bubble and shows a green-coded $\lambda5007$ dominated
extension towards the South-East (MH\,11). A possible explanation is
in terms of shocks
that propagate into a pre-ionized medium and which at the same time are
back-illuminated by the ionizing radiation of the ULX; see, e.g. \citet{du89}.
This prevents the formation of a full recombination region, and the emission
is characteristic of an uncomplete shock
\citep{ray88}.

Age and energetics of ULXB can readily be estimated from the
diameters and expansion velocities of these structures, depending
on whether we assume a SNR like nature or a continuously inflated
wind/jet-driven bubble, see, e.g. \citet{pagm06}. It turns out that
the 400 pc diameter, v$_s\sim100$ km/s, prototype ULXBs around
Holmberg~IX X-1 and NGC~1313 X-2 are typically $\sim1$ million
years old, have an energy content of \mbox{$\sim10^{53}$ erg}, or
equivalently, have experienced an average inflating wind/jet power of
\mbox{$\sim10^{40}$ erg/s} over  the age of the ULXB.

\begin{figure}
  \begin{minipage}[b]{0.5\linewidth}
    \centering \includegraphics[totalheight=2.5in]{pakull_fig1.ps}
    \caption{Multi-colour image of the ULXB MH9-11 around
    Holmberg~IX X-1. Note the change of excitation [OIII]/H$\alpha$
    across the nebula and the green-coded OIII dominated emission to the
    SE indicating the presence of X-ray illuminated uncomplete shocks.}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering \includegraphics[totalheight=2.5in]{pakull_fig2.ps}
    \caption{The Chandra triple X-ray source in the nearby galaxy NGC~7793
    superimposed on the outer H$\alpha$ intensity contour
    of the very large SNR candidate S26. The external sources
    (beambags?) are $15\arcsec$ $\cong 250$ pc apart and match the
    endpoints of the elongated bubble.}
  \end{minipage}
\end{figure}

\section{What ULX Bubbles can tell us about the Central Sources and their Jets}

An obvious question to ask is how ULX bubbles are being formed.  In
principle they could be superbubbles (SBs) blown by the combined
action of  many O stars and supernova remnants. However, two
considerations  argue against this interpretation:  known SBs don't
expand fast enough to create the fast shocks that are seen
in the SNR-like spectra of ULXBs. And perhaps more convincingly, we point
out that a mechanical energy input of \mbox{10$^{40}$ erg/s,} equivalent to 10-50
SN explosions within 10$^6$ years, requires a massive cluster of at
least $\sim10^6$ M$_{\odot}$. However, what we see in the case of
NGC1313 X-2 and Holmberg~IX X-1 are stellar associations with masses
below $10^4$ M$_{\odot}$ \citep{gr08}; in other words, the expected
mechanical power of the observed population falls short by two
orders of magnitude, or more.

Alternatively, there is the possibility of a rare energetic
hypernova injecting \mbox{$>10^{52}$ erg} into the ISM \citep{no03,rob03};
i.e. most probably the event that created the BH in the ULX system.
Although such a scenario has again recently been proposed by
\citet{lm07} to account for a large non-thermal radio bubble in the
Local Group galaxy IC~10, we think that the alternative scenario
of a ULX wind/jet blown bubble is more likely, not least for the
following reason:

After the SN explosion and the formation of a BH in the binary
system, mass transfer from the companion commences only
after a certain time span in which the new elliptical orbit
circularizes
by  tidal forces, and during which the star evolves and expands on
a nuclear  time scale (few 10$^6$ yrs) until it fills its Roche
lobe. The aforementioned ULXB MF16 in NGC~6946  is the
smallest, most rapidly expanding, and  therefore youngest object in
this group with a (Sedov) age of 2.5 10$^4$ yrs \citep{du00}.
Indeed, this period  would by far be too short for the onset of
mass transfer in this system.
We therefore believe that MF 16 provides ample evidence that
hypernov\ae\ are not the events that create ULX bubbles.
Instead, the view that ULXBs are being inflated by
quasi-relativistic winds or jets with mechanical power of some
10$^{39-40}$ erg/s (i.e. not unlike the famous SS433/W50  "beambag"
system \citep{beg80}) begins to materialize. Support for this
idea also comes from the discovery of a small (by ULXB standards)
$\sim5$ pc diameter radio and optical bubble around the famous black hole
X-ray binary Cyg X-1 \citep{gal05} which we now interpret as a
downscaled version (in jet power) of the bubbles described here, i.e. as
\mbox{a '$\mu$ULXB'.}

\section{More Tests with ULXBs}

Considering that extragalactic ultraluminous X-ray sources are
often  (we estimate in $>30\%$ of the cases) surrounded by large
ULXB  we might test the hypothesis of beamed emission by searching
for  (isotropically emitting) ULXB-like nebul\ae\ without
associated ULX. To be more specific: assuming a mild beaming factor
of  10 as often proposed \citep{king} we would expect,
for each visible ULX, to see about ten such systems that
are pointed away from
us. To that end, we have looked for large shocked ionized nebul\ae\
in our  VLT images of the galaxy NGC~1313 which harbours two
ultraluminous sources, X-1 and X-2, both of which are embedded in
ULXBs \citep{pam02}.  Although we find one possible such case, we
certainly don't see  20 such nebul\ae\ without associated
compact X-ray sources in  that galaxy.  Therefore, we conclude
that ULXs emit largely isotropically, as previously shown  for
Holmberg~II X-1 on the basis of optical Zanstra-method arguments.

Large shock-ionized bubbles without detectable X-ray point source
content may well represent systems that once contained ULX, but which
are no longer active; consider a typical 1 million years old standard
ULXB "beambag" of 400 pc diameter and 100 km/s expansion velocity which
is suddenly being deprived of wind/jet mechanical power input. What
would happen? The answer is that it  would still be visible for another
3$\times$10$^6$ years. Here we have assumed that the decreasing expansion
velocity must still be significantly supersonic (i.e. $\geq40$ km/s) in
order to form an observable radiative shock.
In view of some ULXBs that had previously been listed in SNR
catalogues of nearby galaxies -- sometimes as overlapping remnants
that turned out to be part of one bigger structure \citep{pam03} -- we
have extended our search by looking for point-like X-ray sources at
the position of unusually large ($\geq100$ pc) SNRs. This exercise
has already revealed some interesting results:

\begin{itemize}
 \item The well-known variable ULX NGC\,2403 X-1 which is located near
the galactic nucleus lies in between the SNR candidates MFBL\,14 and 15
listed by \citet{metal97}. This has lead \citet{sp03} to effectively
discard a likely association. However, archival HST H$\alpha$ images show
that the individual SNRs are part of a larger structure of
some 300 pc diameter, i.e., here is a new textbook ULXB.
 \item In their survey for SNR in the 7.0 Mpc distant spiral NGC~5885,
\cite{mf97} pointed out the enormous dimension of their candidate \#1
(we measured $200\times300$ pc). It has the form of a regular
elongated bubble, not unlike the ULXB around NGC~1313 X-2 \citep{pagm06}.
Unfortunately, an earlier proposal for XMM/Newton  observations of this
galaxy was turned down, but a recent public Chandra ACIS image now
reveals a bright point source with a luminosity of about
\mbox{10$^{39}$ erg/s} right in the middle of the 'remnant':
another beautiful example of a ULXB.
\item Finally, we mention the Chandra X-ray counterpart of the
well-known  huge SNR candidate S26 in the Sculptor group galaxy NGC\,7793
\citep{bl97}, which is  bright at both radio and optical wavelengths
\citep{pan02}. Deep H$\alpha$ images show an elongated bubble
with some filamentary structure and large extent of 300 pc.
Overlaying the outer H$\alpha$ contour on a deep archival Chandra image
reveals a very interesting configuration shown in Fig. 2:
Here the X-ray source is resolved into a linear triplet, with the outer
very soft sources being coincident with
the endpoints of the large axis of the optical bubble. Similarities
with X-ray images of SS433 and its X-ray lobes \citep{wat83} are striking,
and we believe that this system is indeed the long-searched (for nearly 30
years) extragalactic analogue of this famous microquasar. We finally
mention that the observed X-ray output \mbox{(L$_x\sim10^{37}$ erg/s)} of the
central source is far from 'ultraluminous', a property it shares with
the X-ray weak SS433. Future work on S26 will certainly shed more light on the
controversial relations between SS433 and ULXs on the one hand, and
between W50 and ULXB beambags on the other hand.
\end{itemize}

\acknowledgements MWP wishes to thank the organizers for an inspiring
conference and the editors for their patience with the delivery of this
manuscript.


\section{Introduction}
When $X$ is a smooth projective variety, the famous Kodaira-Nakano Vanishing theorem says that sufficiently high cohomologies vanish when $\Omega^p_X$ is twisted by any ample line bundle. Saito proved a more general vanishing theorem \cite{Sai90} using his theory of Hodge modules.
\begin{theorem}[\textbf{Kodaira-Saito Vanishing Theorem}]\label{KST}
Let $X$ be a complex projective variety with an ample line bundle $ L$, and $M$ a mixed Hodge module on $X$. Then
\[H^i(X, {\rm Gr}^{F}_{k}{\rm DR(M)}\otimes  L)=0,    \ i>0\]
\end{theorem}
A detailed discussion of the proof of this theorem can be found in \cite{Pop}, another proof following the approach of Esnault-Viehweg in \cite{Sch14a}. If X is smooth, taking $M=\mathbb Q_X^H:=(\omega_X, F_{\bullet}, \mathbb{Q}_X)$, the pure Hodge module corresponding to the trivial variation of Hodge structure on $X$, we have ${\rm Gr}^F_{-p}{\rm DR}(M)=\Omega^p_X[n-p]$, and so Saito's result implies Kodaira-Nakano vanishing. For arbitrary $M$, let $S(M)$ be the lowest graded piece of the Hodge filtration. In particular, \[H^i(X,S(M)\otimes  L)=0, \  i>0,\] which specializes to Kodaira vanishing as well.\\
\indent On the other hand, Kodaira vanishing can be generalized by replacing ample divisors by nef and big divisors (or even $\mathbb{Q}$-divisors). This generalization is the so-called Kawamata-Viehweg vanishing theorem. In \cite{Pop}, M. Popa proved a version of Kawamata-Viehweg vanishing for some special Hodge modules, and he suggested that a better result would be true. In this paper I remove the extra hypothesis in \cite{Pop} and prove a Kawamata-Viehweg type statement for pure Hodge modules in full generality.
\begin{theorem} \label{kvt1}
Let X be a complex projective variety, $ L$ a nef and big line bundle, and $M$ a polarizable pure Hodge module with strict support $X$. Then
\[H^i(X, S(M)\otimes  L)=0,    \ i>\ 0\]
\end{theorem}
This can be approached in two different ways. I first provide an inductive approach, a strategy similar to Kawamata's original method (see also \cite[\S 4]{Laz}), based in this setting on an adjunction-type formula that involves the nearby cycle functor and mixed Hodge modules. This is presented in Section 5.\par
At the same time as this proof was completed, J. Suh proved a Nakano type vanishing for the Deligne canonical extension of a polarizable variation of Hodge structure in \cite{Suh}, which in particular implies the same Kawamata-Viehweg type result. (Note that Suh also proves other types of vanishing statements as well, that apply to all graded quotients in the Hodge filtration of the de Rham complex.) His idea is based on the Esnault-Viehweg \cite{EV} approach to vanishing theorems. Following J. Suh's proof and C. Schnell's Esnault-Viehweg type proof of Theorem \ref{KST} in \cite{Sch14a}, I extend this to a version of the injectivity theorem of Koll$\acute{\text a}$r and Esnault-Viehweg for the Deligne canonical extensions of certain polarizable variations of Hodge structures. This is presented as a surjectivity statement below and proved in Section 6.
\begin{theorem} \label{main}
Let $X$ be a smooth projective variety with a line bundle $ L$, $D$ a reduced simple normal crossings divisor, and $\mathbb V=(\mathcal V, F^\bullet, \mathbb V_\mathbb Q)$ a variation of Hodge structure defined on $U=X\setminus D$. Assume  \[ L^N=\mathcal{O}_X(D')\]
for some $N\gg0$ and an effective divisor $D'$ supported on $D$. If $E$ is an effective divisor supported on ${\rm Supp}(D')$. Then for all $i$, the natural map induced by $E$
\[H^i(X, {\rm Gr}^{\textup{first}}_F {\rm DR}_{(X, D)}(\tilde{\mathcal V})\otimes  L^{-1}(-E))\longrightarrow H^i(X, {\rm Gr}^{\textup{first}}_F {\rm DR}_{(X, D)}(\tilde{\mathcal V})\otimes  L^{-1})\]
is surjective, where $\tilde\mathcal V$ is the Deligne canonical extension of $\mathcal V$, and ${\rm Gr}^{\textup{first}}_F {\rm DR}_{(X, D)}$ is the first non-zero graded piece of the logarithmic de Rham complex for $\tilde\mathcal V$ (see Section 5).
\end{theorem}
A injectivity theorem for pure Hodge modules with strict support $X$ follows from it.
\begin{theorem} \label{injectivity}
Let X be a complex projective variety, $E$ an effective divisor, and $M$ a polarizable pure Hodge module with strict support $X$. If a line bundle $ L$ is either nef and big or semi-ample and satisfying $\text{H}^0(X,  L^m(-E))\neq0$ for some $m>0$,  then the natural map
\[H^i(X, S(M)\otimes  L)\longrightarrow H^i(X, S(M)\otimes  L(E))\]
is injective for all $i$.
\end{theorem}
Replacing $E$ by a sufficiently large multiple of an ample divisor, Theorem \ref{kvt1} follows by Serre vanishing, which provides the second proof. See also Corollary \ref{kv}. We also obtain a version of Kawamata-Viehweg vanishing for $\mathbb Q$-divisors which contains the original version for $\omega_X$, as explained in Remark \ref{qdiv}.  \par
It is worth mentioning that, just like in Theorem \ref{KST}, the statement given here works on arbitrary (not necessary smooth) projective varieties $X$, due to the theory of Hodge modules on singular spaces; see Section 3.\par
In Sections 2, 3 and 4, we briefly review $\mathscr D$-modules and Hodge modules. Some necessary theorems are presented for later use. In Section 5, I present the inductive proof of the Kawamata-Viehweg vanishing for pure Hodge modules using nearby cycles. Section 6 is dedicated to the proof of the injectivity theorem in the normal crossings case (Theorem \ref{main}). Section 7 deals with applications.
\subsection*{Acknowledgements}
I would like to thank my advisor, Mihnea Popa, for suggesting the problem and for many helpful discussions. I am also grateful to Christian Schnell for answering my questions. Finally, I thank Junecue Suh for correspondence and for sharing his preprint \cite{Suh}, which helped crucially with the second part of this work.
\section{$\mathscr D$-modules and the de Rham complex}
In this section, I will recall some terminologies of $\mathscr D$-modules and the Riemann-Hilbert correspondence between regular holonomic $\mathscr D$-modules and perverse sheaves, which will be essentially used in the theory of Hodge modules. For a much detailed exposition of $\mathscr D$-modules, we refer to \cite{HTT} and \cite{MS}. The former focuses more on algebraic $\mathscr D$-modules, and the latter is mainly about the analytic story.
\subsection{The side-change operator of $\mathscr D$-modules}
Let $X$ be a complex manifold and let $\mathscr D_X$ be the sheaf of differential operators. It is well-known that the category of left $\mathscr D$-modules and the category of right $\mathscr D$-modules are equivalent via the so called side-change operation,
\[\mathcal N\longrightarrow\mathcal M=\mathcal N\otimes\omega_X, \] and its quasi-inverse
\[\mathcal M\longrightarrow\mathcal N=\mathcal{H}om_{\mathscr O_X}(\omega_X, \mathcal M),\]
where $\omega_X$ is the canonical sheaf.
When $\mathscr D$-modules are filtered, the correspondence of filtrations under the equivalence is
\[F_p(\mathcal M)=F_{p+n}(\mathcal N)\otimes\omega_X, \ \text{and}\  F_p(\mathcal N)=F_{p-n}(\mathcal M)\otimes\omega^{-1}_X,\] where $n=\text{dim}X$.
\subsection{The de Rham functor and the Riemann-Hilbert correspondence} The de Rham functor is defined to be
\[{\rm DR}(\mathcal N):=[\mathcal N\longrightarrow\Omega^1_X\otimes\mathcal N\longrightarrow\cdot\cdot\cdot\longrightarrow\Omega^n_X\otimes\mathcal N][n],\]
for left $\mathscr D$-modules, where $\Omega^k_X$ is the sheaf of holomorphic $k$-forms on $X$. Also
\[{\rm DR}(\mathcal M):=[\bigwedge^n\mathscr T_X\otimes\mathcal M\longrightarrow\bigwedge^{n-1}\mathscr T_X\otimes\mathcal M\longrightarrow\cdot\cdot\cdot\longrightarrow\mathcal M][n],\]
for right $\mathscr D$-modules, where $\mathscr T_X$ is the sheaf of vector fields on $X$. Both of the above complexes are concentrated in degree $-n,...,-1,0$. ${\rm DR}(\mathcal N)$ (${\rm DR}(\mathcal M$) respectively) is called the de Rham complex of $\mathcal N$ ($\mathcal M$ respectively). Clearly the de Rham functor is compatible with the side-change operation, i.e.
${\rm DR}(\mathcal N)$ is canonically isomorphic to ${\rm DR}(\mathcal M)$ provided $\mathcal M=\mathcal N\otimes \omega_X$, because of the canonical isomorphism
\[\bigwedge^{n-k}\mathscr T_X\otimes\omega_X\simeq\Omega^k.\]
\par If the right $\mathscr D$-module $\mathcal M$ is filtered, then ${\rm DR}(\mathcal M)$ is filtered naturally by
\[F_p{\rm DR}(\mathcal M):=[\bigwedge^n\mathscr T_X\otimes F_{p-n}\mathcal M\longrightarrow\bigwedge^{n-1}\mathscr T_X\otimes F_{p-n+1}\mathcal M\longrightarrow\cdot\cdot\cdot\longrightarrow F_p\mathcal M][n].\]
Similarly, for filtered left $\mathscr D$-module $\mathcal N$,
\[F_p{\rm DR}(\mathcal N):=[F_p\mathcal N\longrightarrow\Omega^1_X\otimes F_{p+1}\mathcal N\longrightarrow\cdot\cdot\cdot\longrightarrow\Omega^n_X\otimes F_{p+n}\mathcal N][n].\]
Obviously, the filtered de Rham functor is also compatible with the side-change operation. Hence, it is not necessary to distinguish the right version and the left version of the (filtered) de Rham functor and (filtered) de Rham complexes.
The associated graded complexes for the filtration above are
\[{\rm Gr}^F_p{\rm DR}(\mathcal M):=[\bigwedge^n\mathscr T_X\otimes {\rm Gr}^F_{p-n}\mathcal M\longrightarrow\bigwedge^{n-1}\mathscr T_X\otimes {\rm Gr}^F_{p-n+1}\mathcal M\longrightarrow\cdot\cdot\cdot\longrightarrow {\rm Gr}^F_p\mathcal M][n],\]which are complexes of $\mathscr O_X$-modules.

\par The Riemann-Hilbert correspondence says that the de Rham functor induces an equivalence between the category of regular holonomic $\mathscr D$-modules and the category of perverse sheaves. In particular, holomorphic vector bundles with flat connections correspond to local systems under this equivalence. This is just part of the general Riemann-Hilbert correspondence. See for instance \cite[\S 7 ]{HTT} for the whole statement. \par
Convention: We denote increasing filtrations (decreasing filtrations respectively) by $F_{\bullet}$ ($F^{\bullet}$ respectively), and the associated graded objects by ${\rm Gr}^F$(${\rm Gr}_F$ respectively).
\section{Hodge modules}
This section will be devoted to briefly recall Morihiko Saito's theory of Hodge modules. I will only mention basic information of Hodge modules and important theorems that will be needed later on. The main two references are Saito's original paper \cite{Sai88} and \cite{Sai90}. Another useful reference is the recent survey \cite{Sch14}. Since Hodge modules are originally defined for complex manifolds, when we say Hodge modules on some complex algebraic variety, it means on the underlying analytic space. Since we are about to work on projective varieties, all the ingredients of Hodge modules will be algebraic by the GAGA principle. Varieties always mean reduced and irreducible complex algebraic schemes or analytic schemes in this paper.
\subsection{Pure Hodge modules}
Let $X$ be a smooth complex variety or complex manifold.  A variation of Hodge structure $\mathbb V$ of weight $l$ is
\[\mathbb V=(\mathcal V, F^{\bullet}, \mathbb V_{\mathbb Q}),\]
consisting of
\begin{itemize}
\item a $\mathbb Q$-local system $\mathbb V_{\mathbb Q}$;
\item a finite decreasing filtration $F^{\bullet}$ of the vector bundle $\mathcal V=\mathbb V_{\mathbb Q}\otimes\mathscr O_X$ by subbundles,
satisfying
\item $\forall x\in X$, $\mathbb{V}_x=(\mathcal{V}_x, F^{\bullet}_x, \mathbb{V}_{\mathbb Q, x})$ is a  Hodge structure of weight $l$;
\item the filtration $F^{\bullet}$ satisfies the Griffiths transversality condition, namely for the induced connection $\nabla$,
\[\nabla(F^p)\subset \Omega^1\otimes F^{p-1}.\]
\end{itemize}
Additionally, a polarization of a variation Hodge structure $\mathbb V$ of weight $l$ is a morphism
\[Q: \mathbb V_{\mathbb Q}\otimes \mathbb V_{\mathbb Q}\longrightarrow\mathbb Q(-l),\]
such that $Q$ induces a polarization of $\mathbb V_x$ for every $x\in X$, where $\mathbb Q(-l)=(2\pi i)^{-l}\mathbb Q$\\\\
Note: If we let $F_p\mathcal V= F^{-p}\mathcal V$, then the last requirement means exactly that the new filtration $F_{\bullet}$ is good for left $\mathscr D$-module $\mathcal V$. From now on, $F_{\bullet}\mathcal V$ of a variation of Hodge structure always means this induced increasing filtration.\par
Saito's theory generalizes variations of Hodge structure by allowing filtered holonomic $\mathscr D$-modules with $\mathbb Q$-structure instead.
Here a filtered holonomic $\mathscr D$-modules with $\mathbb Q$-structure is a triple
\[M=(\mathcal M, F_{\bullet}\mathcal M, K)\]
where $(\mathcal M, F_{\bullet}\mathcal M)$ is a holonomic right $\mathscr D$-module with a good filtration, and $K$ is a $\mathbb Q$-perverse sheaf, satisfying
\[{\rm DR}(\mathcal M)\simeq \mathbb C\otimes_{\mathbb Q} K.\]
We use right $\mathscr D$-modules because it is more natural to define the direct image funtor for right $\mathscr D$-modules. See Section \ref{directimage}. From now on, all $\mathscr D$-modules will mean right $\mathscr D$-modules unless stated explicitly. I will also use the expression "a filtered left $\mathscr D$-module underlying a Hodge module (pure or mixed)" if it is so after side-change. \par
Saito constructed an abelian category $\text{HM}(X,l)^p$ of polarizable pure Hodge modules of weight $l$ for smooth complex variety $X$ (or complex manifold) in \cite{Sai88} which is a fully faithful subcategory of category of filtered holonomic $\mathscr D$-modules with $\mathbb Q$-structures. (M. Saito constructed pure Hodge modules first, but we are only interested in polarizable ones.) To be more precise, $\text{HM}(X,l)^p$ is semi-simple, i.e.
\[\text{HM}(X, l)^p=\bigoplus_{Z\subseteq X} \text{HM}^Z(X, l)^p,\]
direct sum over all irreducible subvariety of X. Here $\text{HM}^Z(X, l)^p$ is of objects extended from polarizable variations of Hodge structures of weight $l-\text{dim}Z$ on Zariski open subset of the smooth locus of $Z$. From this, it is clear that when $X$ is a point, $\textup{HM}(X, l)^p$ is just the category of polarizable Hodge structures of weight $l$. \par
If $Y$ is an irreducible subvariety of $X$, Hodge modules on $Y$ can be defined as,
\begin{definition}[Pure Hodge Modules on Singular Spaces] \label{singhm}
\[\textup{HM}(Y, l)^p:=\textup{HM}_Y(X, l)^p,\]
where the right-hand side is the category of polarizable pure Hodge modules of weight $l$ on $X$ supported on $Y$.
\end{definition}
In this way, the underlying $\mathscr D$-module of $M\in\textup{HM}(Y, l)^p$ makes sense. So do ${\rm DR}(\mathcal M)$, $F_{\bullet} {\rm DR}(M)$ and ${\rm Gr}^F_{\bullet}{\rm DR}(\mathcal M)$. In fact, by the filtered version of Kashiwara's equivalence, $\textup{HM}(Y, l)^p$ doesn't depend on embeddings of $Y$ into ambient space. Moreover, ${\rm Gr}^F_k{\rm DR}(\mathcal M)$ are well-defined complexes of coherent sheaves on $Y$, independent of the choice of an embedding. Namely, $\textup{HM}(Y, l)^p$ and ${\rm Gr}^F_k{\rm DR}(\mathcal M)$ are intrinsically defined for $Y$. See \cite[\S14]{Sch14} for details. Indeed, only local embeddings are needed. Then gluing local data together
gives rise to the definition of $\textup{HM}(Y, l)^p$ \cite{Sai88}.
\par
A polarizable pure Hodge module $M$ of weight $l$ is of strict support $Z$, or strictly supported on $Z$, if $M\in \text{HM}^Z(X, l)^p$. Indeed, it is called strict because its underlying perverse sheaf is the intersection complex of the local system. Saito also proved the following equivalence, which will play an important role later on.
\begin{theorem}[{\cite[Theorem 3.21]{Sai90}}] \label{geneq}
Let $X$ be a complex variety of dimension $n$. Then the following two categories are equivalent:
\[\textup{HM}^X(X, l)^p\simeq\textup{VHS} ^p_{gen}(X, l-n),\]
where $\textup{VHS}^p_{gen}(X, l-n)$ is the inductive limit of $\textup{VHS}^p(U, l-n)$ the categories of polarizable variations of Hodge structures of weight $l-n$ on smooth dense Zariski open subsets $U$.
\end{theorem}

\subsection{Mixed Hodge modules}
In \cite{Sai90}, Saito constructed another abelian category, the category of mixed Hodge modules. Similar to mixed Hodge structures generalizing pure Hodge structures, the basic input of a mixed Hodge module on a smooth variety $X$ is a filtered holonomic $\mathscr D$-module with $\mathbb Q$-structure, $(\mathcal M, F_{\bullet}\mathcal M, K)$ plus a finite increasing filtration $W_{\bullet}$ on $K$, which in turn induces a finite filtration $W_{\bullet}\mathcal M$ for $\mathcal M$ by the Riemann-Hilbert correspondence. This $W$-filtration is called the weight filtration, while the $F$-filtration is called the Hodge filtration.
\par Saito first constructed the category of weakly mixed Hodge modules (graded polarizable), $\text{MHW}(X)^p$. An $M=(\mathcal M, F_{\bullet}\mathcal M, K, W_{\bullet})$ is a weakly mixed Hodge module (graded polarizable) if \[ ({\rm Gr}^W_l\mathcal M, F_{\bullet}, {\rm Gr}^W_lK)\in\text{HM}(X, l)^p,\]
where $F_{\bullet}$ is the induced filtration.
If in addition, such an $M$ also satisfies some "admissibility" condition (originally defined in \cite{SZ85} for admissible variations of mixed Hodge structures), then $M$ is a mixed Hodge module. See \cite[\S20]{Sch14} or originally \cite[\S2]{Sai90} for a detailed discussion of the definition of mixed Hodge modules. \par
Denote the category of graded polarizable mixed Hodge modules by $\text{MHM}(X)^p$. Mixed Hodge modules have many good properties, one of which is that $\text{MHM}(X)^p$ is stable by $j_*j^{-1}$ (see \cite[Proposition 2.11]{Sai90}), where $j$ is the open embedding
\[j:X\setminus D\longrightarrow X\]
for some effective divisor $D$. This is called localizations of mixed Hodge modules in \cite{Pop}. At the level of perverse sheaves, this functor is precisely
\[Rj_*j^{-1}(K).\]
If $M\in \text{MHM}(X)^p$, it is hard to compute $j_*j^{-1}M$ explicitly in general, even for $j_*j^{-1}\mathbb Q_X^H[n]$. But in some special situation like for instance $D$ is a smooth irreducible divisor, $j_*j^{-1}\mathbb Q_X^H[n]$ is understood very well (\cite[\S 5]{Pop}). I will calculate some special examples of mixed Hodge modules of this type in the next section.
\subsection{Direct image functor and the Stability Theorem}\label{directimage}
Let $f: X\longrightarrow Y$ be a projective morphism of smooth complex varieties (or complex manifolds), and let $M$ be a Hodge module (pure or mixed) on $X$. The direct image of $M$ is \[f_*M:=(f_+(\mathcal M, F_{\bullet}\mathcal M), Rf_*K),\]
i.e. combining the direct image of the filtered $\mathscr D$-module and the direct image of the perverse sheaf.
Detailed discussion about direct image functors can be found in \cite[\S26, \S28]{Sch14}. One of the main results of M. Saito's theory is the following theorem about the direct image functor.
\begin{theorem}[Stability Theorem and Decomposition Theorem \cite{Sai88}] \label{stability}
Let $f:X\longrightarrow Y$ is a projective morphism between complex manifolds, and let \[M=(\mathcal M, F_{\bullet}\mathcal M, K)\in\textup{HM}^Z(X, l)^p,\] for some subvariety Z of $X$. Then,\\
(1) $f_+(\mathcal M, F_{\bullet}\mathcal M)$ is strict, and $\mathcal H^if_*M\in\textup{HM}(Y, n+i)^p$;\\
(2) $f_*M\simeq \bigoplus\mathcal H^if_*M[-i]$ (non-canonical).
\end{theorem}
Statement (1) is called the Stability Theorem. The statement in \cite{Sai88} is more general, but this is all I need in this article. The second statement is called the Decomposition Theorem, and is easily deduced from the first because $f_*M$ is a complex of pure weights (see \cite[$\S$14.1.2]{PS}). This is also the reason why the quasi-isomorphism in (2) isn't canonical (\cite[Corollary 14.4]{PS}).
\begin{remark}
When $(\mathcal M, F_{\bullet}\mathcal M)$ underlies a graded-polarizable weakly mixed Hodge module $M$, Stability Theorem is still true (see \cite[Theorem 2.14]{Sai90}). But since $f_*M$ may not be of pure weights, Decomposition Theorem doesn't hold for weak mixed Hodge modules or even mixed Hodge modules.
\end{remark}
As an easy consequence of the Stability Theorem, one has the following useful corollary:
\begin{corol}[{\cite[Example 27.1]{Sch14}}] \label{degen}
Let $f: X\longrightarrow \bullet$ be the constant morphism from a smooth projective variety, with $(\mathcal M, F_{\bullet}\mathcal M)$ a filtered holonomic $\mathscr D$-module underlying some graded-polarizable weakly mixed Hodge module. Then the Hodge-de Rham spectral sequence
\[E^{p, q}_1=H^{p+q}(X, {\rm Gr}^F_{-p}{\rm DR}(\mathcal M))\implies H^{p+q}(X, {\rm DR}(\mathcal M))\]
degenerates at $E_1$.
\end{corol}
Since the filtration is bounded below, it is useful to emphasize on where the filtration starts.
\begin{definition}
Suppose $M=(\mathcal M, F_{\bullet}\mathcal M, K)$ is a pure Hodge module on a smooth complex variety $X$. Set
\[p(M):=\text{min}\{p\ |\ F_{p}\mathcal M\neq 0\},\]
and
\[S(M):=F_{p(M)}\mathcal M= {\rm Gr}^F_{p(M)}{\rm DR}(\mathcal M).\]
\end{definition}
$S(M)$ also makes sense when $X$ is singular, because $S(M)= {\rm Gr}^F_{p(M)}{\rm DR}(\mathcal M)$ and the latter one is intrinsic for $M$ as explained in Definition \ref{singhm}. $S(M)$ will play a similar role as the canonical sheaf does in vanishing theorems.
From the Decomposition Theorem, Saito also proved,
\begin{prop}[{\cite[Theorem 28.1]{Sch14}}] \label{kol}
Under the setting of Theorem \ref{stability},
\[Rf_*{\rm Gr}^F_p{\rm DR}(\mathcal M)\simeq\bigoplus_i{\rm Gr}^F_p{\rm DR}(\mathcal M_i)[-i].\] In particular,
\[Rf_*S(M)\simeq \bigoplus_iF_{p(M)}\mathcal M_i[-i],\]
where $(\mathcal M_i, F_{\bullet}\mathcal M_i):=\mathcal H^if_+(\mathcal M, F_{\bullet}\mathcal M)$.
\end{prop}
\subsection{$V$-filtrations and the torsion-freeness of $S(M)$}
The $V$-filtration is fundamentally important in Saito's theory of Hodge modules. It is used in this section to prove the torsion-freeness of $S(M)$ that due to Saito because of completeness.  \par
Let $X$ be a smooth complex variety or complex manifold of dimension $n$ and let $X_0$ be a smooth divisor of $X$ with local defining equation $t$. $\mathscr D_X$ is filtered by
\[V_{i}\mathscr D_X=\{P\in\mathscr D_X| P\cdot\mathcal{I}^j_{X_0}\subseteq\mathcal{I}^{j-i}_{X_0}\},\]
where $\mathcal{I}_{X_0}$ is the ideal sheaf of $X_0$ with the convention that $\mathcal{I}^j_{X_0}=\mathscr O_X$ for $j\leq 0$.
\begin{definition}[$V$-filtration]
A $V$-filtration of a coherent right $\mathscr D$-module $\mathcal M$ along $X_0$ is an increasing filtration $V_{\alpha}\mathcal M$ indexed by rational number $\alpha$ satisfying,
\begin{itemize}
\item The filtration is exhaustive, and each $V_{\alpha}\mathcal M$ is a coherent $V_0\mathscr D_X$-module. \\
\item $V_{\alpha}\mathcal M\cdot V_i\mathscr D_X\subseteq V_{\alpha+i}\mathcal M$ for every $\alpha\in\mathbb Q$ and $i\in\mathbb Z$; and
\[V_{\alpha}\mathcal M\cdot t=V_{\alpha-1}\mathcal M, \  \text{for}\  \alpha<0\]
\item The action of $t\partial_t-\alpha$ on $Gr^V_{\alpha}\mathcal M$ is nilpotent for any $\alpha$, where $Gr^V_{\alpha}\mathcal M:=V_{\alpha}\mathcal M/V_{<\alpha}\mathcal M$ and $V_{<\alpha}\mathcal M=\cup_{\beta<\alpha}V_\alpha\mathcal M$.
\end{itemize}
\end{definition}
In fact, the morphism of the action $t$ in the second requirement of the above definition is an isomorphism (\cite[Lemme 3.1.4]{Sai88}), i.e.
\[t: V_{\alpha}\mathcal M\xrightarrow{\simeq} V_{\alpha-1}\mathcal M\]
is an isomorphism for all $\alpha<0$. \par
The $V$-filtration is a refinement of the Kashiwara-Malgrange filtration. It is known that the Kashiwara-Malgrange filtration exists and is unique for any regular holonomic $\mathscr D$-module. It can be refined to a $V$-filtration if the monodromy action is quasi-unipotent. Hence in particular, the $V$-filtration exists for any Hodge module. See \cite[\S8]{Sch14} for a detailed exposition.  \par
For filtered holonomic $\mathscr D$-modules, one needs to consider the compatibility of the $V$-filtration and the $F_{\bullet}$ filtration. Such compatibility condition is called "quasi-unipotent and regular along $X_0$ or some locally defined holomorphic function $f$" (see \cite[\S3.2]{Sai88}). Pure Hodge modules are quasi-unipotent and regular along any locally defined holomorphic function $f$. This condition makes the Hodge filtration of any Hodge module only depends on its restriction to the open subset complementary to the divisor defined by $f$. To be precise, if $M=(\mathcal M, F_{\bullet}\mathcal M, K)$ is a pure Hodge module of strict support $Z$ on $X$, then for holomorphic function $f:X\longrightarrow \mathbb C$ whose restriction to $Z$ is not constant,
\begin{equation} \label{filtr}
F_{p}\mathcal M_f=\sum_{i=0}^\infty (V_{<0}\mathcal M_f\cap j_*j^*F_{p-i}\mathcal M_f)\partial_t^i,
\end{equation}
where $(\mathcal M_f, F_{\bullet}\mathcal M_f)=i_{f,+}(\mathcal M, F_{\bullet}\mathcal M)$, $i_f$ is the graph embedding of $f$, and $t$ is the coordinate of $\mathbb C$.
\begin{prop}\label{otorsionfree}
Let $X$ be a complex variety (irreducible), and $M$ a pure Hodge module strictly supported on $X$. Then $S(M)$ is torsion-free.
\end{prop}
\begin{proof}
Since torsion-freeness is local, we can assume $X\hookrightarrow Y$ embedded into a complex manifold Y, and the underlying filtered $\mathscr D$-module is $(\mathcal M, F_{\bullet}\mathcal M)$ on $Y$. We know
\[S(M)=F_{p(M)}\mathcal M\]
is a coherent $\mathscr O_X$-module. Suppose $\mathcal T$ is the torsion submodule of $S(M)$, and $\mathcal T$ is annihilated by $\tilde f$ which can be lifted to a holomorphic function $f:Y\longrightarrow \mathbb C$ on Y. By equation \eqref{filtr},
\[S(M)=F_{p(M)}\mathcal M_f=V_{<0}\mathcal M_f\cap j_*j^*F_{p(M)}\mathcal M_f.\]
And the action of $\tilde f$ on $S(M)$ is exactly the action of $t$ on $F_{p(M)}\mathcal M_f$. But since  $t$ acts on $V_\alpha$ injectively for $\alpha<0$, $\mathcal T$ must be 0.
\end{proof}
Because of the compatibility of the $V$-filtration and the Hodge filtration, Saito also proved the following theorem about $p(M)$.
\begin{prop}[{\cite[Proposition 2.6]{Sai91}}] \label{p(M)}
Suppose $f:X\longrightarrow Y$ is a projective morphism of complex manifolds with $M\in{HM^Z(X, l)^p}$ for some subvariety $Z$. If $M'$ is a direct summand of $\mathcal H^jf_*M$ and its strict support  $Z'\neq f(Z)$, then
\[p(M')> p(M).\]
\end{prop}
Combining Proposition \ref{otorsionfree} and Proposition \ref{p(M)}, one obtains,
\begin{corol}\label{torsionfree}
Let $f:X\longrightarrow Y$ be a surjective projective morphism of complex varieties, and let $M$ be a polarizable pure Hodge module strictly supported on $X$. Then
$R^if_*S(M)$ is torsion-free for all $i$.
\end{corol}
This corollary was first conjectured by Koll\'{a}r as generalizations of his theorem about higher direct images of dualizing sheaves \cite{Kol}, and was proved by M. Saito in \cite{Sai91}.
\begin{corol}\label{hdim}
Assume $f:X\longrightarrow Y$ is a birational morphism of complex projective varieties, and $M\in HM^X(X, l)^p$. Then
\[R^if_*S(M) = \left\{ \begin{array}{ll}
         S(M') & \mbox{ $i = 0$}\\
        0 & \mbox{ $i<0$}\end{array} \right., \]
where $M'$ is the direct summand of $\mathcal H^0f_*M$ strictly supported on Y.
\end{corol}
\begin{proof}
The fact that $R^0f_*S(M)=S(M')$ follows from Proposition \ref{p(M)}. Now $R^if_*S(M)=0$ for $i>0$ because $R^if_*S(M)$ is both torsion and torsion-free.
\end{proof}
\subsection{Non-characteristic inverse image} \label{nonchar}
Let $f:X\longrightarrow Y$ be a morphism of complex manifolds. We have the following two morphisms of cotangent bundles,
\[p_2: X\times_Y T^*Y\longrightarrow T^*Y\] and
\[df^*: X\times_Y T^*Y\longrightarrow T^*X\] given by differential, $df^*(x, \omega)=df^*_x(\omega)$.

\begin{definition}[Non-Characteristic morphism {\cite[\S3.5]{Sai88}}]
Let $(\mathcal M, F_{\bullet}\mathcal M)$ be a filtered coherent $\mathscr D_Y$-module. $f:X\longrightarrow Y$ is said to be non-characteristic for $(\mathcal M, F_{\bullet}\mathcal M)$ if the following two conditions are satisfied:
\begin{itemize}
\item $L^if^*{\rm Gr}^F_p(\mathcal M)=0$ for all $i\neq0$ and all $k$;\\
\item $df^*|_{p_2^{-1}\text{Ch}(\mathcal M)}$ is finite.
\end{itemize}
If $f$ is a closed embedding, one says that $X$ is non-characteristic for $(\mathcal M, F_{\bullet}\mathcal M)$ if $f$ is so.\par
The first condition means that the pull-back of the filtration is still a filtration of the pull-back of $\mathcal M$;
the second says the pull-back filtration is also good. Hence one gets the filtered pull-back $f^*(\mathcal M, F_{\bullet}\mathcal M)=(\tilde\mathcal M, F_{\bullet}\tilde\mathcal M)$ when $f$ is non-characteristic for $(\mathcal M, F_{\bullet}\mathcal M)$ by
\[\tilde\mathcal M=f^{-1}\mathcal M\otimes_{f^{-1}\mathscr O_Y}\omega_{Y/X}\]
and
\[F_p\tilde\mathcal M=f^{-1}F_{p+d}\mathcal M\otimes_{f^{-1}\mathscr O_Y}\omega_{Y/X},\]
where $d=\text{dim}X-\text{dim}Y$.
Here the degree of the filtration of $\tilde\mathcal M$ is shifted because $\mathcal M$ is a right $\mathscr D$-module. $\tilde\mathcal M$ is holonomic if $\mathcal M$ is so (\cite[Lemme 3.5.5]{Sai88}). If $(\mathcal M, F_{\bullet}\mathcal M)$ underlies a polarizable pure Hodge module $M$ of weight $l$ on $Y$, then $f^*(\mathcal M, F_{\bullet}\mathcal M)$ underlies a polarizable pure Hodge module of weight $l+d$, denoted by $f^*M$ (\cite[Theorem 9.3]{Sch14a}).
\end{definition}
\section{Hodge modules and the Deligne extension}\label{snc}
Let $X$ be a complex smooth projective variety, and let $D=\sum^r_{i=1} D_i$ be a reduced simple normal crossings divisor with irreducible components $D_i$. Set $U=X\setminus D$, and
\[j:U\hookrightarrow X,\] the open embedding. \par
Assume that $\mathbb V=(\mathcal V, F^{\bullet}\mathcal V, \mathbb{V}_\mathbb Q)$ is a polarizable variation of Hodge structure on $U$. Then it is well-known that $\mathcal V$ extends to vector bundles with flat connections with logarithmic poles along $D$. Such extension is unique if the eigenvalues of the residue along each $D_i$ are required to be in a fixed strip of $\mathbb C$ of length $1$ (\cite[Theorem 5.2.17]{HTT}). Because of the monodromy theorem (see \cite{Sch}), the eigenvalues are in fact rational numbers in this setting. The Deligne canonical extension is the extension with eigenvalues in $[0, 1)$, denoted by $\mathcal{\tilde V}$. $\mathcal{\tilde V}$ is filtered by
\begin{equation}\label{filtrDe}
F_p\mathcal{\tilde V}:=\mathcal{\tilde V}\cap j_*(F_p\mathcal V),
\end{equation}
where $F_p\mathcal V=F^{-p}\mathcal V$. Because of Schmid's theorem on nilpotent orbits, filtrations of all Deligne extensions are locally free (see \cite[\S3.b]{Sai90}).
Hence we have the filtered logarithmic de Rham complex for $\tilde{\mathcal V}$, similar to the filtered de Rham complex,
\[{\rm DR}_{(X,D)}(\tilde{\mathcal V}, F_{\bullet}):=\lbrack\tilde{\mathcal V}\longrightarrow \Omega^1({\rm log}~D)\otimes\tilde{\mathcal V}\longrightarrow\cdots\longrightarrow\Omega^n({\rm log}~D)\otimes\tilde{\mathcal V}\rbrack [n]. \] with filtration
\[F_p {\rm DR}_{(X,D)}(\tilde{\mathcal V}, F_{\bullet}):=\lbrack F_p\tilde{\mathcal V}\longrightarrow \Omega^1({\rm log}~D)\otimes F_{p+1}\tilde{\mathcal V}\longrightarrow\cdots\longrightarrow\Omega^n({\rm log}~D)\otimes F_{p+n}\tilde{\mathcal V}\rbrack [n], \]
where $n=\text{dim}X$.
\par
Then $\tilde{\mathcal V}(*D)=\tilde{\mathcal V}\otimes\mathscr O_X(*D)$ is the regular meromorphic connection extending $\mathcal V$, which is a left regular holonomic $\mathscr D$-module a priori. According to \cite[S3.b]{Sai90}, $\tilde{\mathcal V}(*D)$ can be filtered by
\[F_p\tilde{\mathcal V}(*D)=\sum F_{p-i}\mathscr D_X\cdot F_i\tilde{\mathcal V}_1,\]
where $\tilde{\mathcal V}_1$ is another extension of $\mathcal V$ with eigenvalues in $[-1, 0)$. \par
By Theorem \ref{geneq}, one denotes the pure Hodge module of strict support X corresponding to $\mathbb V$ by $M$. Since $\mathbb V$ is defined outside of a normal crossings divisor, $M$ can be described explicitly in terms of a Deligne extension  (see \cite[Theorem 3.20]{Sai90}). In particular,
\[S(M)=\omega_X\otimes F_{p(M)+n}\mathcal V_2,\]
where $\mathcal V_2$ is another Deligne extension with eigenvalues in $(-1, 0]$. Hence $S(M)$ is locally free. Saito proved that the filtered left $\mathscr D$-module $(\tilde{\mathcal V}(*D), F_{\bullet}\tilde{\mathcal V}(*D))$ underlies $j_*j^{-1}M$. This point is clear at least for $\tilde{\mathcal V}(*D)$ without filtration. It is the underlying left $\mathscr D$-module of $j_*j^{-1}M$ because ${\rm DR}(\tilde{\mathcal V}(*D))\simeq Rj_*\mathbb V_\mathbb C$ (see \cite[Theorem 5.2.24]{HTT}).
Furthermore, by \cite[Proposition 3.11 (3.11.4)]{Sai90}, there is a filtered quasi-isomorphism
\begin{equation} \label{qi}
{\rm DR}_{(X,D)}(\tilde{\mathcal V}, F_{\bullet}\tilde{\mathcal V})\simeq {\rm DR}(\tilde{\mathcal V}(*D), F_{\bullet}\tilde{\mathcal V}(*D)).
\end{equation} This filtered quasi-isomorphism will be used repeatedly. \par
Since $\mathbb V$ is finitely filtered by $F_{\bullet}\mathbb V$, it is also useful to define (following \cite{Sai91})
\[q(M)=q(\mathbb V):=\text{max}\{p\  |\ {\rm Gr}_p^F(\mathbb V)\neq 0\}.\]
For any pure Hodge module N strictly supported on $X$, besides $S(N)$ we define (following \cite{Sai91})
\begin{equation} \label{Q(M)}
Q_X(N):=\mathbb{D}(S_X(N^*))=R\mathcal{H}om_{\mathscr O_X}(S_X(N^*), \omega_X[n]),
\end{equation}
where $N^*$ is the extension of the dual of the variation of Hodge structure defined on a Zariski open dense subset in the sense of Theorem \ref{geneq}, and $\mathbb{D}$ is the Grothendieck duality functor.  When $X$ is a singular projective variety, $Q_X(N)$ is defined to be
\[Q_X(N)=R\mathcal{H}om_{\mathscr O_Y}(S_X(N^*), \omega_Y[\text{dim}Y])\] for some smooth $Y$ containing $X$ as a subvariety. This definition is independent with $Y$ because of the well-known fact that the pushforward functor and the Grothendieck duality functor commute.   \par
By \cite[Lemma 2.4]{sai92}, under this normal crossings assumption,
\[Q_X(M)\simeq\frac{\tilde{\mathcal V}}{F_{q(M)-1}\tilde {\mathcal V}}[n].\] Hence one obtains,
\begin{equation} \label{dualiso}
Q_X(M)\simeq\footnote{ I learned this point from J. Suh.}\ {\rm Gr}^F_{q(M)}{\rm DR}_{(X,D)}(\tilde{\mathcal V}, F_{\bullet})\simeq {\rm Gr}^F_{q(M)}{\rm DR}(\tilde{\mathcal V}(*D), F_{\bullet}).
\end{equation}
 The second one is the quasi-isomorphism induced by the filtered quasi-isomorphism \eqref{qi}.
\section{An inductive proof of Kawamata-Viehweg type vanishing theorem}
In this section, I prove the Kawamata-Viehweg type vanishing theorem inductively by the method similar to Kawamata's original approach (see \cite[the proof of Theorem 4.3.1]{Laz}). It will be proved alternatively by the injectivity theorem in Section 7. See the first half of Corollary \ref{kv}.
\begin{theorem}\label{KVV}
Let $X$ be a complex projective variety with polarizable pure Hodge module $M$ strictly supported on $X$ and let $ L$ be a nef and big line bundle on $X$. Then
\[H^i(X, S(M)\otimes L)=0, \ \ i>0.\]
\end{theorem}
\begin{proof}
The proof will be divided into two steps. Step 1 is a Norimatsu-type statement (see \cite[Lemma 4.3.5] {Laz}); Step 2 is to reduce the general statement to the case of Step 1.  \\\\
$Step$ 1. In this step, we make the following assumption:
\begin{itemize}
\item X is smooth;\\
\item M is extended from a polarizable variation of Hodge structure $\mathbb V=(\mathcal V, F^{\bullet}\mathcal V, \mathbb{V}_\mathbb Q)$ on $U$, with $D=X\setminus U$ a normal crossings divisor; \\
\item The monodromy of $\mathbb V$ along each irreducible component of $D$  is unipotent;\\
\item $ L\simeq\mathscr O_X(A+E)$ with $A$ an ample divisor and $E=\sum_{i=1}^tE_i$ a reduced simple normal crossings divisor, and ${\rm Supp}(E)\subset D$.
\end{itemize}
\par
Denote the Deligne canonical extension of $\mathcal V$ by $\tilde{\mathcal V}$, as in section \ref{snc}. Then the third assumption means the residue of $\tilde\mathcal V$ along each irreducible component of $D$ is nilpotent. As explained in Section \ref{snc},
\[Q(M)=\dfrac{\tilde\mathcal V}{F_{q(M)-1}\tilde{\mathcal V}}[\text{dim}X]={\rm Gr}^F_{q(M)}\tilde\mathcal V[\text{dim}X]. \]
Set $Q'=Q(M)[-\text{dim}X];$ it is a locally free sheaf. \par
Under these assumptions, we show that
\[H^i(X, Q'(-A-E))=0, \ i<\text{dim}X. \]
We use induction on $t$, the number of components of $E$, and on dimension. The case $t=0$ is Kodaira-Saito vanishing because of the filtered quasi-isomorphism \eqref{qi}. Assuming the result known for $E$ with $\leq k$ components, consider the short exact sequence,
\[0\longrightarrow \mathscr O_X(-A-\sum_{i=1}^{k+1}E_i)\longrightarrow \mathscr O_X(-A-\sum_{i=1}^kE_i)\longrightarrow \mathscr O_{E_{k+1}}(-A-\sum_{i=1}^kE_i)\longrightarrow 0.  \] Tensoring with $Q'$ gives,
\begin{equation}\label{ses}
0\longrightarrow Q'(-A-\sum_{i=1}^{k+1}E_i)\longrightarrow Q'(-A-\sum_{i=1}^kE_i)\longrightarrow Q'|_{E_{k+1}}\otimes\mathscr O_{E_{k+1}}(-A-\sum_{i=1}^kE_i)\longrightarrow 0.
\end{equation}
By the inductive assumption,
\begin{equation}\label{ind1}
H^i(X, Q'(-A-\sum_{i=1}^kE_i))=0,\ i<\text{dim}X.
\end{equation}
Now $\tilde{\mathcal V}|_{E_{k+1}}$ is not a Deligne canonical extension of some variation of Hodge structure defined on some open set of $E_{k+1}$. However, the residue along $E_{k+1}$ gives rise to a finite increasing filtration $W_{\bullet}$ for $\tilde{\mathcal V}|_{E_{k+1}}$, the monodromy weight filtration. Moreover, $Gr^{W}_l(\tilde{\mathcal V}|_{E_{k+1}})$ for any $l$, is the Deligne canonical extension of some polarizable variation of Hodge structure defined on $E_{k+1}\setminus D$, following from the multi-valued SL$_2$-orbit theorem \cite{CKS}. The filtration on $Gr^{W}_l(\tilde{\mathcal V}|_{E_{k+1}})$ induced from the variation of Hodge structure is precisely the filtration induced from $\tilde\mathcal V$, i.e.
\[F_p{\rm Gr}^{W}_l(\tilde{\mathcal V}|_{E_{k+1}})=\dfrac{F_p\tilde\mathcal V|_{E_{k+1}}\cap W_l}{F_p\tilde\mathcal V|_{E_{k+1}}\cap W_{l-1}}.\]
Namely $\tilde{\mathcal V}|_{E_{k+1}}$ is the Deligne canonical extension of an admissible variation of mixed Hodge structure because of the unipotency assumption of the monodromy. See \cite[Theorem 3.20]{Bru},\footnote{This process is called the graded nearby-cycle functor in \cite{Bru}.} or \cite[Proposition 4.3]{FFS}.\par
By the inductive assumption again,
\[H^i(E_{k+1}, {\rm Gr}^F_{q(M)}{\rm Gr}^W_l\tilde{\mathcal V}|_{E_{k+1}}\otimes\mathscr O_{E_{k+1}}(-A-\sum_{i=1}^kE_i))=0,\] for $i<\text{dim}X-1$ and all $l$. By a simple calculation, there are short exact sequences for any $p$ and $l$,
\[0\longrightarrow\dfrac{F_p\tilde\mathcal V|_{E_{k+1}}\cap W_{l-1}}{F_{p-1}\tilde\mathcal V|_{E_{k+1}}\cap W_{l-1}}\longrightarrow\dfrac{F_p\tilde\mathcal V|_{E_{k+1}}\cap W_l}{F_{p-1}\tilde\mathcal V|_{E_{k+1}}\cap W_{l}}\longrightarrow {\rm Gr}^F_{p}{\rm Gr}^W_l\tilde{\mathcal V}|_{E_{k+1}}\longrightarrow 0.\] Using the long exact sequences of cohomology associated to the above short exact sequences, one sees that
\begin{equation}\label{ind2}
H^i(E_{k+1}, Q'|_{E_{k+1}}\otimes\mathscr O_{E_{k+1}}(-A-\sum_{i=1}^kE_i))=0, \ i<\text{dim}X-1 .
\end{equation}
Using the long exact sequence of cohomology associated to the short exact sequence \eqref{ses}, together with \eqref{ind1} and \eqref{ind2}, we get
\[H^i(X, Q'(-A-\sum_{i=1}^{k+1}E_i))=0, \ i<\text{dim}X. \]
$Step$ 2. Since the Leray spectral sequence degenerates at $E_1$ because of Corollary \ref{hdim}, by passing to a log-resolution (see also the proof of Theorem \ref{semiample}), it suffices to assume that $X$ is smooth, $M$ is extended from a polarizable variation of Hodge structure $\mathbb V$ defined on $U$ with $D=X\setminus U=\sum D_i$ a reduced simple normal crossings divisor, and $ L^m=\mathscr O(A+E)$ with $A$ an ample divisor and $E$ an effective divisor with support contained in $D$. Denote
\[E=\sum^t_{i=1} \alpha_iD_i,\ \alpha=\alpha_1\cdot...\cdot \alpha_t, \ \text{and}\  \alpha_i'=\alpha/\alpha_i,\] with $\alpha_i>0$.
Using a Kawamata covering \cite{Kaw},  we can construct a finite flat covering
\[f:Y\longrightarrow X,\]
such that $f^*D$ still has simple normal crossings support,
\[f^*D_i=m\alpha_i'D_i',\] for $i=1,... ,t$
and $E'=\sum^t_{i=1}D'_i$ is a reduced simple normal crossings divisor. Furthermore, the monodromy along each component of $f^*D$ of $\mathbb V':=f_1^*\mathbb V$ is unipotent, since $m$ may be as divisible as needed. Here $f_1=f|_{f^{-1}U}$ and the pull-back is just the non-characteristic pull-back for filtered left $\mathscr D$-modules (see Section \ref{nonchar}). Hence $\mathbb V'$ is also a polarizable variation of Hodge structure on $f^{-1}U$. \par
Now put $ L'=f^* L$ and $A'=f^*(A)$, so that
\[( L')^m\simeq\mathscr O_Y(A'+m\alpha E').\]
Hence
\[( L'(-E'))^{m\alpha}\simeq( L')^{m(\alpha-1)}(A').\]
Since $ L'$ is nef, one knows that the line bundle on the right-hand side is also ample. Therefore
\[ L'\simeq\mathscr O_Y(H+E'),\]
for some ample divisor $H$ on $Y$. \par
It is obvious that $\mathbb V$ is a direct summand of $f_{1, *}\mathbb V'$.\footnote{Here the push-forward is the Hodge module direct image of $\mathbb V$ after side-change.} Hence so is $M$ of $\mathcal H^0f_*M'$ where $M'$ is the pure Hodge module uniquely determined by $\mathbb V'$. Therefore, $S(M)$ is also a direct summand of $f_*(S(M'))$. Since $f$ is finite, it is sufficient to prove vanishing for $S(M')$ and $ L'$ on $Y$. By Serre duality (see \eqref{Q(M)}), it is equivalent to prove
\[H^{i}(Y, Q(M'^*)[-n]\otimes ( L')^{-1})=0, \ i<n,\]
where $M'^*$ is the pure Hodge module extended from the dual of $\mathbb V'$ and  $n=\text{dim}Y$. Consequently, the proof is done by step 1.
\end{proof}
\section{Injectivity in the Normal Crossings Case}
Let $X$ be a smooth complex projective variety, and $D=\sum^r_{i=1} D_i$ a reduced simple normal crossings divisor with irreducible components $D_i$. Assume $M$ is the polarizable pure Hodge module extending a polarizable variation of Hodge structure $\mathbb V=(\mathcal V, F_{\bullet}\mathcal V, \mathbb V_\mathbb Q)$ defined on $U=X\setminus D$.
Then one uses the notations introduced in Section \ref{snc}.\par
The result in this section owes a lot to the approach of J. Suh to the Kawamata-Viehweg vanishing theorem for Hodge modules in \cite{Suh}. The proof follows closely his argument and a similar argument of C. Schnell in \cite{Sch14a} with a slight refinement that leads to a more general injectivity statement. All of these arguments follow in turn the general strategy of Esnault-Viehweg \cite{EV} towards vanishing and injectivity theorems. \par
Let $ L$ be a line bundle on $X$. Assume  \[ L^N\simeq\mathcal{O}_X(D')\]
for some $N$ large enough, such that \[D'=\sum^r_{i=1} \alpha_i D_i,  \ \ \ 0\leq\alpha_i \ll N,\] and $D'\neq0$.
\begin{theorem} \label {thm1}
Let $E=\sum^r_{i=1}\mu_i D_i$ be an effective divisor such that ${\rm Supp}(E)\subset {\rm Supp}(D')$. Then for all $i$, the natural map induced by $E$
\[H^i(X, {\rm Gr}^F_{q(M)} {\rm DR}(\tilde{\mathcal V}(*D))\otimes  L^{-1}(-E))\longrightarrow H^i(X, {\rm Gr}^F_{q(M)} {\rm DR}(\tilde{\mathcal V}(*D))\otimes  L^{-1})\]
is surjective.
\end{theorem}
\begin{proof}
Set
\[ L^{(i)^{-1}}:= L^{\otimes -i}(\lfloor\dfrac{iD'}{N}\rfloor).\] By \cite[Theorem 3.2]{EV}, one knows each $ L^{(i)^{-1}}$ has a flat logarithmic connection and its residue along $D_i$ is just multiplication by $\{\dfrac{i\alpha_i}{N}\}$ (the fractional part of $\dfrac{i\alpha_i}{N}$).
\par Let $\pi: Y\longrightarrow X$ be the cyclic cover obtained by taking the $N$-th root of $D'$. See \cite[\S3]{EV} for the construction of cyclic covers. Set $\pi'=\pi|_{U'=\pi^{-1}(X\setminus D)}$. So $\pi'$ is $\acute{\text{e}}$tale. Since the relative de Rham complex is trivial, we get a variation of Hodge structure (the Gauss-Manin connection of $\pi'$, see \cite[\S10]{PS}),
\[\mathbb V_1:=(\pi'_{*}\mathscr O_{U'}, F_{\bullet}\pi'_{ *}\mathscr O_{U'}, \pi'_{*}\mathbb Q_{U'}),\] with trivial filtration. \par
Since $\pi'_{*}\mathscr O_{U'}=\bigoplus^{N-1}_{i=0} L^{(i)^{-1}}|_{U'}$, we have a new variation of Hodge structure
\[\mathbb{V}\otimes\mathbb V_1=\left(\mathcal V\otimes\bigoplus^{N-1}_{i=0}\mathcal{L}^{(i)^{-1}}|_U, F^\bullet, \mathbb V_{\mathbb Q}\otimes \pi'_*\mathbb Q_{U'}\right),\] with
\[F^p\left(\mathcal V\otimes\bigoplus^{N-1}_{i=0}\mathcal{L}^{(i)^{-1}}|_U\right)=F^p\mathcal V\otimes\bigoplus^{N-1}_{i=0}\mathcal{L}^{(i)^{-1}}|_U.\]
Then we obtain a mixed Hodge module $j_*j^{-1}M_1$, where $M_1$ is the pure Hodge module corresponding to $\mathbb V\otimes \mathbb V_1$ by Theorem \ref{geneq}.
The filtered left $\mathscr D$-module $\mathcal N$,
\[\mathcal N:=\tilde{{\mathcal V}}(*D)\otimes\bigoplus^{N-1}_{i=0} L^{(i)^{-1}}(*D),\] underlies $j_*j^{-1}M_1$. Hence the filtered left $\mathscr D$-module $\tilde{{\mathcal V}}(*D)\otimes  L^{(1)^{-1}}(*D)$ is a direct summand of $\mathcal N$. We also know $\tilde {\mathcal V}\otimes  L^{(1)^{-1}}$ is the Deligne canonical extension of $\mathcal V\otimes  L^{(1)^{-1}}|_U$ because of the assumption $\alpha_i \ll N$.
Therefore, by filtered quasi-isomorphism \eqref{qi} we have a filtered quasi-isomorphism
\begin{equation}\label{qi1}
{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{(1)^{-1}}, F_{\bullet})\simeq {\rm DR}(\tilde{\mathcal V}(*D)\otimes  L^{(1)^{-1}}(*D), F_{\bullet}).
\end{equation}
\par By Corollary \ref{degen}, we see that the spectral sequence
\[E^{p,q}_1=H^{p+q}(X, {\rm Gr}^F_{-p}{\rm DR}(\mathcal N,F_{\bullet}))\implies H^{p+q}(X, {\rm DR}(\mathcal N))\]
degenerates at $E_1$. Hence as a direct summand, we know that the spectral sequence (since $ L^{(1)^{-1}}= L^{-1}$)
\[E^{p,q}_1=H^{p+q}(X, {\rm Gr}^F_{-p}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}, F_{\bullet}))\implies H^{p+q}(X, {\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}, F_{\bullet}))\]
degenerates at $E_1$. From the filtration of $\mathbb V\otimes \mathbb V_1$ and the definition of the filtration on the Deligne canonical extension (equation \eqref{filtrDe}), it is clear that
\begin{equation}\label{iso}
{\rm Gr}^F_{p}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}, F_{\bullet}))={\rm Gr}^F_{p}{\rm DR}_{(X,D)}(\tilde{\mathcal V}, F_{\bullet}))\otimes  L^{-1}.
\end{equation}
For any integer $a\leq 0$, all the eigenvalues of residue of $\tilde {\mathcal V}\otimes  L^{-1}(aD_i)$ along $D_i$ (irreducible components of $E$) are strictly positive (\cite[Lemma 2.7]{EV}). Therefore, by repeatedly applying \cite[Lemma 2.10]{EV} for the components of $E$, we see that the natural map
\begin{equation}\label{qwf}
 {\rm DR}_{(X,D)}(\tilde {\mathcal V}\otimes  L^{-1}(-E))\longrightarrow {\rm DR}_{(X,D)}(\tilde {\mathcal V}\otimes  L^{-1})
\end{equation}
is a quasi-isomorphism (without filtrations).
If one puts the trivial filtration on $ L^{-1}(-E)$, we get another spectral sequence
\[E^{p,q}_1=H^{p+q}(X, {\rm Gr}^F_{-p}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}(-E), F_{\bullet}))\implies H^{p+q}(X, {\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}(-E), F_{\bullet}))\]
Clearly as before,
\begin{equation}\label{iso1}
{\rm Gr}^F_{p}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}(-E), F_{\bullet}))={\rm Gr}^F_{p}{\rm DR}_{(X,D)}(\tilde{\mathcal V}, F_{\bullet}))\otimes  L^{-1}(-E).
\end{equation}
Now this spectral sequence may not degenerate at $E_1$. However, one has the following diagram,
\begin{diagram}
 H^{p-q(M)}(X, \mathcal A)  &\lTo{\alpha}      &H^{p-q(M)}(X, \mathcal C) \\
\uTo_{\beta}   &                   &\uTo_{\gamma}\\
H^{p-q(M)}(X, \mathcal B) &\lTo         &H^{p-q(M)}(X, \mathcal D),
\end{diagram}
where $\mathcal A={\rm Gr}^F_{q(M)}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}, F_{\bullet}))$, $\mathcal B={\rm Gr}^F_{q(M)}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}(-E), F_{\bullet}))$, $\mathcal C={\rm DR}_{(X,D)}(\tilde {\mathcal V}\otimes  L^{-1})$ and $\mathcal D={\rm DR}_{(X,D)}(\tilde {\mathcal V}\otimes  L^{-1}(-E))$.\par
Observe that $\alpha$ is surjective because of $E_1$ degeneracy. Also, $\gamma$ is an isomorphism because of the quasi-isomorphism \eqref{qwf}. Hence,
\[\beta:H^{i}(X, {\rm Gr}^F_{q(M)}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}(-E), F_{\bullet})))\longrightarrow H^{i}(X, {\rm Gr}^F_{q(M)}{\rm DR}_{(X,D)}(\tilde{\mathcal V}\otimes  L^{-1}, F_{\bullet}))) \] is surjective for all $i$. Combining this with the filtered quasi-isomorphism \eqref{qi1} and the isomorphisms \eqref{iso} and \eqref{iso1}, the proof is finished.

\end{proof}
\begin{remark} \label{re1}
In the above proof, the condition $0<\alpha_i\ll N$ is needed because it ensures that  $\tilde {\mathcal V}\otimes  L^{-1}$ is the Deligne canonical extension. Therefore, Theorem \ref{thm1} is still true if we require that
\[\alpha_i+\lambda_{D_i}<1\] instead, where $\lambda_{D_i}$ is the maximal eigenvalue of the residue of $\tilde {\mathcal V}$ along $D_i$.
\end{remark}
An injectivity statement for $S(M)$ follows from the above theorem as follows. By \eqref{dualiso},
\[H^b(X, {\rm Gr}^F_{q(M)}{\rm DR}(\tilde{\mathcal V}(*D), F_{\bullet})\otimes  L^{-1})=H^b(X, Q_X(M)\otimes  L^{-1}).\]
By Grothendieck-Serre Duality,
\[H^i(X, Q_X(M)\otimes  L^{-1})=H^{-i}(X, S_X(M^*)\otimes  L)^*.\] Here $M^*$ is the pure Hodge module extended from the dual of the variation of Hodge structure $\mathbb V$. Therefore, the following corollary has been proved by replacing $M$ by $M^*$ in Theorem \ref{thm1}.
\begin{corol} \label{inj}
Under the assumption of Theorem \ref{thm1},
\[H^i(X, S_X(M)\otimes  L)\longrightarrow H^{i}(X, S_X(M)\otimes  L(E))\] is injective for all $i$.
\end{corol}
\section{Injectivity in the general case}
In this section I prove Theorem \ref{injectivity} from the introduction, and further extensions.
From now on, divisors always mean Cartier divisors. When X is smooth, Cartier and Weil divisors will not be distinguished.
\begin{theorem} \label{semiample}
Let $X$ be a complex projective variety, and let $M$ be a pure Hodge module with strict support $X$. If $ L$ is a semi-ample line bundle and $E$ an effective divisor with $H^{0}(X,  L^{v}(-E))\neq 0$ for some $v>0$, then the natural map
\[H^i(X, S_X(M)\otimes  L)\longrightarrow H^i(X, S_X(M)\otimes  L(E))\]
is injective for all $i$.
\end{theorem}
\begin{proof}
By assumption, write
\[ L^v\simeq\mathcal{O}_X(E+C)\]
for an effective divisor $C$. Take a log resolution of $E+C+\text{singular locus of }M$. (Since $M$ is extended from a polarizable variation of Hodge structure defined on a smooth Zariski open set $U$, the singular locus means $X\setminus U.$\footnote{This is not a good definition, because $U$ can be shrunk by subtracting any proper Zariski-closed set as needed.}):
\[f: X_1\longrightarrow X\]
Set $E_1=f^*E$, $C_1=f^*C$. Since $f$ is birational, the variation of Hodge structure also extends to a polarizable pure Hodge module strictly supported on $Y$, called $M_1$, and the singular locus of $M_1$ is just the exceptional divisor of $f$, which is a simple normal crossing divisor.
Clearly, $ L_1:=f^*( L)$ is still semi-ample. Hence, $ L_1^{u}$ is base-point free for some $u$ large enough. Pick a general divisor $D_1$ in $\abs{ L_1^{u}}$ so that it is transversal to the exceptional divisor of $f$. Then \[ L_1^{u+v}\simeq\mathcal{O}_X(D_1+E_1+C_1),\] and $D_1+E_1+C_1$ is a divisor with simple normal crossings support. So the assumption of Corollary \ref{inj} has been fulfilled. \par
By Corollary \ref{hdim},
\[R^if_*S(M_1)=
\begin{cases}
S(M)& i=0\\
0& i>0
\end{cases}.\] Hence by the degeneracy of the Leray spectral sequence and the projection formula,
\[H^b(X_1, S(M_1)\otimes f^* L)=H^b(X, S(M)\otimes  L).\]
The proof is done by Corollary \ref{inj}.
\end{proof}
Recall that a line bundle $ L$ on $X$ is $f$-semi-ample for a proper morphism $f:X\longrightarrow Y$ of varieties if the natural map \[ f^*f_* L^n\longrightarrow  L^n\] is surjective for some $n>0$.
As a corollary, we obtain the following torsion-freeness statement, generalizing Corollary \ref{torsionfree}.
\begin{corol}
Let $f:X\longrightarrow Y$ be a surjective projective morphism of complex algebraic varieties, and let M be a pure Hodge module strictly supported on $X$. If $ L$ is an $f$-semi-ample line bundle on $X$, then for $i\geq0$
\[R^if_*(S(M)\otimes  L) \]
is torsion-free.
\begin{proof}
By a standard reduction process (see for instance \cite[proof of Theorem 1-2-3]{KMM}), one can assume $X$ and $Y$ are projective and $ L$ is semi-ample. Suppose $\mathcal T$ is a torsion sub-sheaf of $R^if_*(S(M)\otimes  L) $ for some $i$. Pick a sufficient ample line bundle $A$ on Y such that $\mathcal T\otimes A$ is globally generated and
\[H^i(X, S(M)\otimes L\otimes f^*A)=H^0(Y, A\otimes R^if_*(S(M)\otimes L)).\]
Hence, by Theorem \ref{inj}, for any effective divisor $D$ on Y, the natural map
\[H^0(Y, A\otimes R^if_*(S(M)\otimes  L))\longrightarrow H^0(Y, A(D)\otimes R^if_*(S(M)\otimes  L))\]
is injective.
Choose $D$ so that multiplication by the equation of $D$ kills $\mathcal T$. Then $H^0(Y, A\otimes \mathcal T)=0$, which implies $\mathcal T=0$.
\end{proof}
\end{corol}
Similarly, we also get an injectivity theorem for nef and big line bundles.
\begin{theorem}\label{nefbig}
Let X be a complex projective variety, and let $M$ be a pure Hodge module with strict support X. If $ L$ is a nef and big line bundle and $E$ an effective divisor, then
\[H^i(X, S(M)\otimes  L)\longrightarrow H^i(X, S(M)\otimes  L(E))\]
is injective for all $i$.
\end{theorem}
\begin{proof}
Since $ L$ is nef and big, we can write
\[ L^v\simeq\mathcal{O}_X(A+D+E)\]
for an ample divisor $A$, an effective divisor $D$ and some $v>0$.
By taking a log resolution for $D+E+\text{singular locus of }M$, it is enough to assume that $X$ is smooth and the singular locus of $M$ is a simple normal crossings divisor containing ${\rm Supp}(D+E)$. Choosing $N$ large enough, then
\[ L^N\simeq L^{N-v}(A)\otimes \mathcal{O}_X(D+E).\]
Since $ L$ is nef, $H:= L^{N-v}(A)$ is ample. Hence
\[ L^{mN}\simeq H^m\otimes \mathcal{O}_X(mD+mE)\simeq\mathcal{O}_X(D_1+mD+mE)\] for some sufficiently general $D_1\in \abs{H^m}$, so that all the transversality conditions are satisfied. The statement of the theorem follows from Corollary \ref{inj}.
\end{proof}
\begin{remark}{(Injectivity for nef and abundant line bundles.)} After a series of blowing-ups, nef and abundant ($\kappa( L)=v( L)$) line bundles can be reduced to semi-ample ones. Therefore, Theorem \ref{nefbig} is also true if $ L$ is only nef and abundant. (More details can be found in \cite[\S5] {EV}.)
\end{remark}
Choosing $E$ as a multiple of a very ample divisor, by Serre vanishing we obtain another proof of Theorem \ref{KVV}.
\begin{corol} \label{kv}
If X is a complex projective variety, $ L$ a nef and big line bundle, and M a pure Hodge module with strict support X, then
\[H^i(X, S(M)\otimes  L)= 0, \ \ \ i>0.\]
More generally, if $ L$ is nef only, then
\[H^i(X, S(M)\otimes  L)= 0, \ \ \ i>n-\kappa( L).\]
\end{corol}
\begin{proof}
First, $\kappa( L)\leq\kappa(f^* L)$ for any birational morphism $f: Y\longrightarrow X .$
Hence, it is enough to assume $X$ is smooth, $S_X(M)$ is locally free and $\kappa( L)<n$. Choose a general hyperplane section $H$ of a very ample line bundle, such that
\[H^b(X, S(M)\otimes  L(H))=0, \ \ \ b>0,\]
and $H$ is non-characteristic for $M$.
Therefore, we get a short exact sequence,
\[0\longrightarrow S(M)\otimes  L\longrightarrow S_X(M)\otimes  L(H)\longrightarrow S(M)|_H\otimes \mathcal O_H(H)\otimes  L|_H\longrightarrow 0.\]
Hence, by passing to the long exact sequence of cohomology,
\[H^{i-1}(H, S(M)|_H\otimes \mathcal O_H(H)\otimes  L|_H)=H^i(X, S(M)\otimes  L).\]
Since $\kappa( L|_H)\geq \kappa( L)$ and $ S(M)|_H\otimes \mathcal O_H(H)\simeq S(i^*(M))$, both groups vanish for $i>n-\kappa( L)$ by induction on dimension.
\end{proof}

Since the eigenvalues of the Deligne canonical extension lie in $[0, 1)$, the injectivity theorem is still true if the nef and big line bundle is perturbed by an effective $\mathbb Q$-divisor of sufficiently small coefficients in the following sense.
\begin{theorem}
Let X be a projective variety with a polarizable pure Hodge module $M$ strictly supported on X, and let $N$ be a nef and big $\mathbb Q$-divisor, $D$ and $B$ two effective divisors. There exists an $\varepsilon=\varepsilon(M, N, D, B)>0$ such that if a line bundle $ L\sim_{\mathbb Q}N+ \varepsilon_1 D$ for $0<\varepsilon_1<\varepsilon$, then the natural map
\[H^i(X, S(M)\otimes  L)\longrightarrow H^i(X, S(M)\otimes  L(B)) \]
is injective for all $i$.
\end{theorem}
\begin{proof}
Since $N$ is nef and big, by Kodaira's lemma,
\[nN\sim A+C+B\]
for some $0<n\in \mathbb Z$, $A$ an ample divisor and $C$ an effective divisor.
Take a log-resolution for $C+B+D+\text{singular locus of }M$,
\[f: Y\longrightarrow X.\]
Then $A_1:=f^*pA-E$ is ample for some $p\ll0$ and $E$ an effective divisor with simple normal crossings support. Hence,
\[f^*npN\sim A_1+E+pf^*C+pf^*B.\]
Write $f^*D=\sum d_iD'_i$.
Then we can assume $U$ is Zariski open in $Y$ and $Y\setminus U$ is a simple normal crossings divisor containing the support of all effective divisors that appear, and there is a variation of Hodge structure $\mathbb V$ on $U$ which corresponds to $M$ by the equivalence of Theorem \ref{geneq}. Take $\varepsilon$ to be
\[\varepsilon=\min\{\dfrac{1-\lambda_{D'_i}}{d_i}\},\] where $\lambda_{D'_i}$ is the maximal eigenvalue of the residue of the Deligne canonical extension of $\mathbb V$ along $D'_i$.
If a line bundle $ L\sim_{\mathbb Q}N+ \varepsilon_1 D$ for $0<\varepsilon_1<\varepsilon$, then
\[f^* L^k\sim \dfrac{k}{np}(A_1+E+pf^*C+pf^*B)+k\varepsilon_1 f^*D,\] where $k$ is some big enough multiple of $np$. Since $f^* L^l(-\varepsilon_1 lD)$ is a nef line bundle for $l\ll0$ and sufficiently divisible,
\[f^* L^{k+l}\sim A_2+\dfrac{k}{np}(E+pf^*C+pf^*B)+(k+l)\varepsilon_1 f^*D,\] for some other ample divisor $A_2$. Hence
\[f^* L^{(k+l)m}\sim H+\dfrac{km}{np}(E+pf^*C+pf^*B)+(k+l)m\varepsilon_1 f^*D,\] where $H\in |\mathscr O(mA_2)|$ is sufficiently general.
The statement then follows from Corollary \ref{inj} and Remark \ref{re1}.

\end{proof}
Note that $\varepsilon$ in the statement depends on the choice of a resolution, but can be made effective once one has been fixed; see also Remark \ref{qdiv} below.
A similar argument works for the following Kawamata-Viehweg type vanishing for $\mathbb Q$-divisors.
\begin{theorem}
Let $X$ be a projective variety with a pure Hodge module $M$ strictly supported on X, and let $N$ be a nef and big $\mathbb Q$-divisor and $D$ an effective divisor. There exists an $\varepsilon=\varepsilon(M, N, D)>0$ such that if a line bundle $ L\sim_{\mathbb Q}N+\varepsilon_1 D$ for $0<\varepsilon_1<\varepsilon$, then
\[H^i(X, S(M)\otimes  L)=0,\ \ \ i>0.\]
\end{theorem}
\begin{remark}\label{qdiv}
If X is a smooth projective variety, $M=\mathbb Q^H_X$ (or more generally for any smooth $M$, i.e. $M$ corresponding to a polarizable variation of Hodge structure defined on $X$), and $D$ is a reduced simple normal crossings divisor, then since there is no residue under these assumptions, $\varepsilon=1$ by \cite[Theorem 9.4.17(i)]{Laz}. Hence the above theorem reduces to the original Kawamata-Viehweg vanishing for $\mathbb Q$-divisors \cite[Theorem 9.1.18]{Laz}, or a special case of \cite[Theorem 11.1]{Pop}.
\end{remark}

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