# Good wild harmonic bundles and good filtered Higgs bundles

Takuro Mochizuki

###### Abstract

We prove the Kobayashi-Hitchin correspondence between good wild harmonic
bundles and polystable good filtered $\lambda$-flat bundles satisfying a
vanishing condition. We also study the correspondence for good wild harmonic
bundles with the homogeneity with respect to a group action, which is expected
to provide another way to construct Frobenius manifolds.

MSC: 53C07, 58E15, 14D21, 81T13

##  1 Introduction

Let $X$ be a smooth projective variety with a simple normal crossing
hypersurface $H$. Let $L$ be an ample line bundle on $X$. We shall prove the
following theorem, that is the Kobayashi-Hitchin correspondence for good wild
harmonic bundles and good filtered $\lambda$-flat bundles.

######  Theorem 1.1 (Corollary 2.24)

The following objects are equivalent.

  * •

Good wild harmonic bundles on $(X,H)$.

  * •

$\mu_{L}$-Polystable filtered $\lambda$-flat bundles
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ on $(X,H)$ satisfying
$\int c_{1}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-1}=0$ and
$\int\mathop{\rm ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim
X-2}=0$.

We shall recall the precise definitions of the objects in §2.

In [50], we have already proved that good wild harmonic bundles on $(X,H)$
induce $\mu_{L}$-polystable good filtered $\lambda$-flat bundles satisfying
the vanishing condition. Note that $0$-flat bundles are equivalent to Higgs
bundles, and $1$-flat bundles are flat bundles in the ordinary sense.
Moreover, we studied an analogue of Theorem 1.1 in the case $\lambda=1$, i.e.,
the correspondence between good wild harmonic bundles and $\mu_{L}$-polystable
good filtered flat bundles satisfying a similar vanishing condition [50,
Theorem 16.1]. It was applied to the study of the correspondence between
semisimple algebraic holonomic $D$-modules and pure twistor $D$-modules.

In this paper, as a complement, we shall explain the proof for all $\lambda$.
There is no new essential difficulty to prove Theorem 1.1 after our studies
[46, 47, 48, 50] on the basis of [61, 62]. Moreover, in some parts of the
proof, the arguments can be simplified in the Higgs case. However, because the
Higgs case is also particularly important, it would be useful to explain a
rather detailed proof. We shall also explain the correspondences in
homogeneous cases which would be useful in a generalized Hodge theory.

###  1.1 Kobayashi-Hitchin correspondences

####  1.1.1 Kobayashi-Hitchin correspondence for vector bundles

We briefly recall a part of the history of this type of correspondences. (See
also [25, 35, 41].) For a holomorphic vector bundle $E$ on a compact Riemann
surface $C$, we set $\mu(E):=\deg(E)/\mathop{\rm rank}\nolimits(E)$, which is
called the slope of $E$. A holomorphic bundle $E$ is called stable (resp.
semistable) if $\mu(E^{\prime})<\mu(E)$ (resp. $\mu(E^{\prime})\leq\mu(E)$)
holds for any holomorphic subbundle $E^{\prime}\subset E$ such that
$0<\mathop{\rm rank}\nolimits(E^{\prime})<\mathop{\rm rank}\nolimits(E)$. It
is called polystable if it is a direct sum of stable subbundles with the same
slope. This stability, semistability and polystability conditions were
introduced by Mumford [55] for the construction of the moduli spaces of vector
bundles with reasonable properties. Narasimhan and Seshadri [57] established
the equivalence between unitary flat bundles and polystable bundles of degree
$0$ on compact Riemann surfaces.

Let $(X,\omega)$ be a compact connected Kähler manifold. For any torsion-free
$\mathcal{O}_{X}$-module $\mathcal{F}$, the slope of $\mathcal{F}$ with
respect to $\omega$ is defined as

$\mu_{\omega}(\mathcal{F}):=\frac{\int_{X}c_{1}(\mathcal{F})\omega^{\dim
X-1}}{\mathop{\rm rank}\nolimits\mathcal{F}}.$

If the cohomology class of $\omega$ is the first Chern class of an ample line
bundle $L$, then $\mu_{\omega}(\mathcal{F})$ is also denoted by
$\mu_{L}(\mathcal{F})$. Then, a torsion-free $\mathcal{O}_{X}$-module
$\mathcal{F}$ is called $\mu_{\omega}$-stable if
$\mu_{\omega}(\mathcal{F}^{\prime})<\mu_{\omega}(\mathcal{F})$ holds for any
saturated coherent subsheaf $\mathcal{F}^{\prime}\subset\mathcal{F}$ such that
$0<\mathop{\rm rank}\nolimits(\mathcal{F}^{\prime})<\mathop{\rm
rank}\nolimits(\mathcal{F})$. This condition was first studied by Takemoto
[70, 71]. It is also called $\mu_{\omega}$-stability, or slope stability.
Slope semistability and slope polystability are naturally defined.

Bogomolov [4] introduced a different stability condition for torsion-free
sheaves on connected projective surfaces, and he proved the inequality of the
Chern classes $c_{2}(E)-(r-1)c_{1}^{2}/2\mathop{\rm rank}\nolimits(E)<0$ for
any unstable bundle $E$ in his sense. Gieseker [19] proved the inequality for
slope semistable bundles. The inequality is called Bogomolov-Gieseker
inequality.

Inspired by these works, Kobayashi [32] introduced the concept of Hermitian-
Einstein condition for metrics of holomorphic vector bundles. Let
$(E,\overline{\partial}_{E})$ be a holomorphic vector bundle on a Kähler
manifold $(X,\omega)$. Let $h$ be a Hermitian metric of $E$. Let $R(h)$ denote
the curvature of the Chern connection
$\nabla_{h}=\overline{\partial}_{E}+\partial_{E,h}$, associated to $h$ and
$\overline{\partial}_{E}$. Then, $h$ is called Hermitian-Einstein if $\Lambda
R(h)^{\bot}=0$, where $R(h)^{\bot}$ denote the trace-free part of $R(h)$. In
[32], he particularly studied the case where the tangent bundle of a compact
Kähler manifold has a Hermitian-Einstein metric, and he proved that such
bundles are not unstable in the sense of Bogomolov. Kobayashi [33, 34] and
Lübke [40] proved that a holomorphic vector bundle on a compact connected
Kähler manifold satisfies the slope polystability condition if it has a
Hermitian-Einstein metric. Moreover, Lübke [39] established the so called
Kobayashi-Lübke inequality for the first and the second Chern forms associated
to Hermitian-Einstein metrics, which is reduced to the inequality $\mathop{\rm
Tr}\nolimits\bigl{(}(R(h)^{\bot})^{2}\bigr{)}\omega^{\dim X-2}\geq 0$ in the
form level. It particularly implies the Bogomolov-Gieseker inequality for
holomorphic vector bundles $(E,\overline{\partial}_{E})$ with a Hermitian-
Einstein metric $h$ on compact Kähler manifolds $(X,\omega)$. Moreover, if
$c_{1}(E)=0$ and $\int_{X}\mathop{\rm ch}\nolimits_{2}(E)\omega^{\dim X-2}=0$
are satisfied for such $(E,\overline{\partial}_{E},h)$, and if we impose
$\det(h)$ is flat, then the Kobayashi-Lübke inequality implies that $R(h)=0$,
i.e., $\nabla_{h}$ is flat.

Independently, in [36], Hitchin proposed a problem to ask an equivalence of
the stability condition and the existence of a metric $h$ such that $\Lambda
R(h)=0$, under the vanishing of the first Chern class of the bundle. (See [25]
for more precise explanation.) It clearly contains the most important essence.
He also suggested possible applications on the vanishings. His problem
stimulated Donaldson whose work on this topic brought several breakthroughs to
whole geometry.

In [14], Donaldson introduced the method of global analysis to reprove the
theorem of Narasimhan-Seshadri. In [15], by using the method of the heat flow
associated to the Hermitian-Einstein condition, he established the equivalence
of the slope polystability condition and the existence of a Hermitian-Einstein
metric for holomorphic vector bundles on any complex projective surface. The
important concept of Donaldson functional was also introduced in [15].

Eventually, Donaldson [16] and Uhlenbeck-Yau [73] established the equivalence
on any dimensional complex projective manifolds. Note that Uhlenbeck-Yau
proved it for any compact Kähler manifolds, more generally. The correspondence
is called with various names; Kobayashi-Hitchin correspondence, Hitchin-
Kobayashi correspondence, Donaldson-Hitchin-Uhlenbeck-Yau correspondence, etc.
In this paper, we adopt Kobayashi-Hitchin correspondence.

As a consequence of the Kobayashi-Hitchin correspondence and the Kobayashi-
Lübke inequality, we also obtain an equivalence between unitary flat bundles
and slope polystable holomorphic vector bundles $E$ satisfying
$\mu_{\omega}(E)=0$ and $\int_{X}\mathop{\rm ch}\nolimits_{2}(E)\omega^{\dim
X-2}=0$. Note that Mehta and Ramanathan [43, 44] deduced the equivalence on
complex projective manifolds directly from the equivalence in the surface case
due to Donaldson [15].

####  1.1.2 Higgs bundles and $\lambda$-flat bundles

Such correspondences have been also studied for vector bundles equipped with
something additional, which are also called Kobayashi-Hitchin correspondences
in this paper. One of the most rich and influential is the case of Higgs
bundles, pioneered by Hitchin and Simpson.

Let $(E,\overline{\partial}_{E})$ be a holomorphic vector bundle on a compact
Riemann surface $C$. A Higgs field of $(E,\overline{\partial}_{E})$ is a
holomorphic section $\theta$ of $\mathop{\rm
End}\nolimits(E)\otimes\Omega^{1}_{C}$. Let $h$ be a Hermitian metric of $E$.
We obtain the Chern connection $\overline{\partial}_{E}+\partial_{E,h}$ and
its curvature $R(h)$. Let $\theta^{\dagger}_{h}$ denote the adjoint of
$\theta$. In [24], Hitchin introduced the following equation, called the
Hitchin equation:

$R(h)+[\theta,\theta_{h}^{\dagger}]=0.$ (1)

Such $(E,\overline{\partial}_{E},\theta,h)$ is called a harmonic bundle. He
particularly studied the case $\mathop{\rm rank}\nolimits E=2$. Among many
deep results in [24], he proved that a Higgs bundle
$(E,\overline{\partial}_{E},\theta)$ has a Hermitian metric $h$ satisfying (1)
if and only if it is polystable of degree $0$. Here, a Higgs bundle
$(E,\overline{\partial}_{E},\theta)$ is called stable (resp. semistable) if
$\mu(E^{\prime})<\mu(E)$ (resp. $\mu(E^{\prime})\leq\mu^{\prime}(E)$) holds
for any holomorphic subbundle $E^{\prime}\subset E$ such that
$\theta(E^{\prime})\subset E^{\prime}\otimes\Omega^{1}_{C}$ and that
$0<\mathop{\rm rank}\nolimits(E^{\prime})<\mathop{\rm rank}\nolimits(E)$, and
a Higgs bundle is called polystable if it is a direct sum of stable Higgs
subbundles with the same slope. By this equivalence and another equivalence
due to Donaldson [17] between irreducible flat bundles and twisted harmonic
maps, Hitchin obtained that the moduli space of polystable Higgs bundles of
degree $0$ and the moduli space of semisimple flat bundles are isomorphic. His
work revealed that the moduli spaces of Higgs bundles and flat bundles have
extremely rich structures.

The higher dimensional case was studied by Simpson [61]. Note that Simpson
started his study independently motivated by a new way to construct variations
of Hodge structure, which we shall mention later in §1.2.1. For a holomorphic
vector bundle $(E,\overline{\partial}_{E})$ on a complex manifold $X$ with
arbitrary dimension, a Higgs field $\theta$ is defined to be a holomorphic
section of $\mathop{\rm End}\nolimits(E)\otimes\Omega^{1}_{X}$ satisfying the
additional condition $\theta\wedge\theta=0$. Suppose that $X$ has a Kähler
form. Let $h$ be a Hermitian metric of $E$. Let $F(h)$ denote the curvature of
the connection $\nabla_{h}+\theta+\theta_{h}^{\dagger}$. A Hermitian metric
$h$ of a Higgs bundle $(E,\overline{\partial}_{E},\theta)$ is called
Hermitian-Einstein if $\Lambda F(h)^{\bot}=0$. When $X$ is compact, the slope
stability, semistability and polystability conditions for Higgs bundles are
naturally defined in terms of the slopes of Higgs subsheaves. Simpson
established that a Higgs bundle $(E,\overline{\partial}_{E},\theta)$ on a
compact Kähler manifold $(X,\omega)$ has a Hermitian-Einstein metric if and
only if it is slope polystable. Moreover, he generalized the Kobayashi-Lübke
inequality for the Chern forms to the context of Higgs bundles, which is
reduced to the inequality $\mathop{\rm
Tr}\nolimits\bigl{(}(F(h)^{\bot})^{2}\bigr{)}\omega^{\dim X-2}\geq 0$ in the
form level for any Hermitian-Einstein metric $h$ of
$(E,\overline{\partial}_{E},\theta)$. Here, the condition
$\theta\wedge\theta=0$ is essential. It particularly implies that if
$(E,\overline{\partial}_{E},\theta)$ on a compact Kähler manifold $(X,\omega)$
satisfies $\mu_{\omega}(E)=0$ and $\int_{X}\mathop{\rm
ch}\nolimits_{2}(E)\omega^{\dim X-2}=0$, then a Hermitian-Einstein metric $h$
of $(E,\overline{\partial}_{E},\theta)$ is a pluri-harmonic metric, i.e., the
connection $\nabla_{h}+\theta+\theta_{h}^{\dagger}$ is flat. It is equivalent
to the following:

$\partial_{E,h}\theta=0,\quad\overline{\partial}\theta_{h}^{\dagger}=0,\quad
R(h)+[\theta,\theta_{h}^{\dagger}]=0.$

A Higgs bundle $(E,\overline{\partial}_{E},\theta)$ with a pluri-harmonic
metric is called a harmonic bundle. This equivalence and another important
equivalence due to Corlette [10] induce an equivalence between semisimple flat
bundles and polystable Higgs bundles $(E,\overline{\partial}_{E},\theta)$
satisfying $\mu_{\omega}(E)=0$ and $\int_{X}\mathop{\rm
ch}\nolimits_{2}(E)\omega^{\dim X-2}=0$ on any connected compact Kähler
manifold.

After the work of Corlette, Donaldson, Hitchin and Simpson, it turned out that
the moduli space $\mathcal{M}(X)$ of flat bundles on a complex projective
manifold $X$ has a hyper-Kähler metric. It particularly induces the twistor
space of the moduli space $\mathop{\rm TW}\nolimits(\mathcal{M}(X))$, which is
a complex analytic space with a fibration $\mathop{\rm
TW}\nolimits(\mathcal{M}(X))\longrightarrow\mathbb{P}^{1}$, such that the
fiber over $1$ is the moduli space of flat bundles, and that the fiber over
$0$ is the moduli space of Higgs bundles. The notion of $\lambda$-connections
was introduced and developed by Deligne and Simpson [64, 65] for a more
complex analytic construction of the twistor space $\mathop{\rm
TW}\nolimits(\mathcal{M}(X))$. They obtain the family of the moduli spaces
$\mathcal{M}^{\lambda}(X)$ of $\lambda$-flat bundles on $X$, and the family of
the moduli spaces $\mathcal{M}^{\mu}(X^{\dagger})$ of $\mu$-flat bundles on
the conjugate $X^{\dagger}$ of $X$. They proved that the twistor space
$\mathop{\rm TW}\nolimits(\mathcal{M}(X))$ can be obtained as the gluing of
the two families $\coprod_{\lambda}\mathcal{M}^{\lambda}(X)$ and
$\coprod_{\mu}\mathcal{M}^{\mu}(X^{\dagger})$ by the natural identification of
$\mathcal{M}^{\lambda}(X)=\mathcal{M}^{\mu}(X^{\dagger})$ for $\lambda\mu=1$.

These correspondences are not only really interesting, but also provide a
starting point of the further investigations. Simpson pursuit the comparison
of flat bundles, Higgs bundles and more generally $\lambda$-flat bundles in
deeper levels [63, 64], and developed the non-abelian Hodge theory [65]. In
particular, he explained that the Kobayashi-Hitchin correspondences for
$\lambda$-flat bundles can be studied in a unified way [63]. For more recent
study on the moduli spaces of $\lambda$-connections, see [12, 26, 27, 66], for
example.

####  1.1.3 Filtered case

It is interesting to generalize such correspondences for objects on complex
quasi-projective manifolds. We need to impose a kind of boundary condition,
that is parabolic structure.

Mehta and Seshadri [45] introduced the concept of parabolic structure of
vector bundles on compact Riemann surfaces. Let $C$ be a compact Riemann
surface with a finite subset $D\subset C$. Let $E$ be a holomorphic vector
bundle on $C$. A parabolic structure of $E$ at $D$ is a tuple of filtrations
$F_{\bullet}(E_{|P})$ $(P\in D)$ indexed by $]-1,0]$ satisfying
$F_{a}(E_{|P})=\bigcap_{b>a}F_{b}(E_{|P})$. Set $\mathop{\rm
Gr}\nolimits^{F}_{a}(E_{|P}):=F_{a}(E)/F_{<a}(E)$, and

$\deg(E,F):=\deg(E)-\sum_{P\in D}\sum_{-1<a\leq 0}a\dim\mathop{\rm
Gr}\nolimits^{F}_{a}(E_{|P}).$

We set $\mu(E,F):=\deg(E,F)/\mathop{\rm rank}\nolimits(E)$. For any subbundle
$E^{\prime}\subset E$, filtrations $F(E^{\prime}_{|P})$ on $E^{\prime}_{|P}$
are induced as $F_{a}(E^{\prime}_{|P}):=F_{a}(E_{|P})\cap E^{\prime}_{|P}$.
Then, $(E,F)$ is called stable if $\mu(E^{\prime},F)<\mu(E,F)$ for any
subbundle $E^{\prime}\subset E$ with $0<\mathop{\rm
rank}\nolimits(E^{\prime})<\mathop{\rm rank}\nolimits(E)$. Semistability and
polystability conditions are also defined naturally. Then, Mehta and Seshadri
proved an equivalence of unitary flat bundles on $C\setminus D$ and parabolic
vector bundles $(E,F)$ with $\mu(E,F)=0$ on $(C,D)$.

For some purposes, it is more convenient to replace parabolic bundles with
filtered bundles introduced by Simpson [61, 62]. Let $\mathcal{O}_{C}(\ast D)$
denote the sheaf of meromorphic functions on $C$ which may have poles along
$D$. Let $\mathcal{V}$ be a locally free $\mathcal{O}_{C}(\ast D)$-module. A
filtered bundle $\mathcal{P}_{\ast}\mathcal{V}$ over $\mathcal{V}$ is a tuple
of lattices $\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}$
$({\boldsymbol{a}}=(a_{P})_{P\in D}\in{\mathbb{R}}^{D})$ such that (i)
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}(\ast D)=\mathcal{V}$, (ii) the
restriction of $\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}$ to a neighbourhood
of $P\in D$ depends only on $a_{P}$, (iii)
$\mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{n}}}\mathcal{V}=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}(\sum
n_{P}P)$ for any ${\boldsymbol{a}}\in{\mathbb{R}}^{D}$ and
${\boldsymbol{n}}\in{\mathbb{Z}}^{D}$, (iv) for any
${\boldsymbol{a}}\in{\mathbb{R}}^{D}$, there exists
${\boldsymbol{\epsilon}}\in{\mathbb{R}}_{>0}^{D}$ such that
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}=\mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{\epsilon}}}\mathcal{V}$.
Let ${\boldsymbol{0}}$ denote $(0,\ldots,0)\in{\mathbb{R}}^{D}$. Then,
$\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V}$ is equipped with the parabolic
structure $F$ induced by the images of
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|P}\longrightarrow\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V}_{|P}$
$(P\in D)$. It is easy to observe that filtered bundles are equivalent to
parabolic bundles. We set
$\mu(\mathcal{P}_{\ast}\mathcal{V}):=\mu(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},F)$
for filtered bundles $\mathcal{P}_{\ast}\mathcal{V}$.

Simpson [61, 62] generalized the theorem of Mehta-Seshadri to the
correspondences of tame harmonic bundles and regular filtered $\lambda$-flat
bundles and regular filtered flat bundles on compact Riemann surfaces. A
harmonic bundle $(E,\overline{\partial}_{E},\theta,h)$ on $C\setminus D$ is
called tame on $(C,D)$ if the closure of the spectral curve of $\theta$ in
$T^{\ast}C(\log D)$ is proper over $C$. A regular filtered $\lambda$-flat
bundle consists of a filtered bundle $\mathcal{P}_{\ast}\mathcal{V}$ equipped
with a flat $\lambda$-connection
$\mathbb{D}^{\lambda}:\mathcal{V}\longrightarrow\mathcal{V}\otimes\Omega^{1}_{C}$
such that
$\mathbb{D}^{\lambda}\cdot\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\subset\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\otimes\Omega^{1}_{C}(\log
D)$ for any ${\boldsymbol{a}}\in{\mathbb{R}}^{D}$. Stability, semistable and
polystable conditions are naturally defined in terms of the slope. Then,
Simpson established the equivalence of tame harmonic bundles on $(C,D)$ and
polystable regular filtered $\lambda$-flat bundles
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ satisfying
$\mu(\mathcal{P}_{\ast}\mathcal{V})=0$. Note that filtered bundles express the
growth order of the norms of holomorphic sections with respect to the metrics.
We should mention that the study of the asymptotic behaviour of tame harmonic
bundles is much harder than that of the asymptotic behaviour of unitary flat
bundles. Hence, it is already hard to prove that tame harmonic bundles induce
regular filtered $\lambda$-flat bundles.

There are several directions to generalize. One is a generalization in the
context of tame harmonic bundles on higher dimensional varieties. Let $X$ be a
smooth connected projective variety with a simple normal crossing hypersurface
$H$ and an ample line bundle $L$. Then, there should be equivalences of tame
harmonic bundles on $(X,H)$ and $\mu_{L}$-polystable regular filtered
$\lambda$-flat bundles $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ on $(X,H)$
satisfying $\int_{X}c_{1}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-1}=0$
and $\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-2}=0$ for each
$\lambda\in{\mathbb{C}}$. In [2], Biquard studied the case where $H$ is
smooth. In [37, 38, 69], Li, Narasimhan, Steer and Wren studied the
correspondence for parabolic bundles without flat $\lambda$-connections. In
[30], Jost and Zuo studied the correspondence between semisimple flat bundles
and tame harmonic bundles. In [46, 47, 48], the author obtained the
satisfactory equivalences for tame harmonic bundles. Note that Donagi and
Pantev recently proposed an attractive application of the Kobayashi-Hitchin
correspondence for tame harmonic bundles to the study of geometric Langlands
theory [13].

In another natural direction of generalization, we should consider more
singular objects than regular filtered Higgs or flat bundles. A harmonic
bundle $(E,\overline{\partial}_{E},\theta,h)$ on $X\setminus H$ is called wild
if the closure of the spectral variety of $\theta$ in the projective
completion of $T^{\ast}X$ is complex analytic. For the analysis, we should
impose that the spectral variety of the harmonic bundle satisfies some non-
degeneracy condition along $H$. (See §2.7.1.) This is not essential because
the condition is always satisfied once we replace $X$ by its appropriate blow
up. The notion of regular filtered $\lambda$-flat bundle is appropriately
generalized to the notion of good filtered $\lambda$-flat bundle. The results
of Simpson should be generalized to equivalences of good wild harmonic bundles
and $\mu_{L}$-polystable good filtered $\lambda$-flat bundles
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ satisfying
$\int_{X}c_{1}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-1}=0$ and
$\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{2}(L)^{\dim X-2}=0$. Sabbah
[58] studied the correspondence between semisimple meromorphic flat bundles
and wild harmonic bundles in the one dimensional case. Biquard and Boalch [3]
obtained generalization for wild harmonic bundles in the one dimensional case.
Boalch informed the author that wild generalization in the context of the
Higgs case was not expected in those days.

As mentioned, the author studied the wild harmonic bundles on any dimensional
varieties in [50]. We obtained that good wild harmonic bundles induce
$\mu_{L}$-polystable good filtered $\lambda$-flat bundles satisfying the
vanishing conditions. Moreover, we proved that the construction induces an
equivalence of good wild harmonic bundles and slope polystable good filtered
flat bundles satisfying the vanishing condition. Such an equivalence for
meromorphic flat bundles is particularly interesting because we may apply it
to prove a conjecture of Kashiwara [31] on semisimple algebraic holonomic
$\mathcal{D}$-modules. See a survey paper [52] for more details on this
application.

In [50], we did not give a proof of the equivalence for wild harmonic bundles
in the case $\lambda\neq 1$ because it is rather obvious that a similar
argument can work after [46, 47, 48, 50] on the basis of [61, 62]. But,
because the Higgs case is also important, it would be better to have a
reference in which a rather detailed proof is explained. It is one reason why
the author writes this manuscript. As another reason, in the next subsection,
we shall explain an application to the correspondence for good wild harmonic
bundles with homogeneity, which is expected to be useful in the generalized
Hodge theory.

###  1.2 Homogeneity with respect to group actions

####  1.2.1 Variation of Hodge structure

As mentioned, Simpson [61] was motivated by the construction of polarized
variation of Hodge structure. Let us recall the definition of polarized
complex variation of Hodge structure given in [61], instead of the original
definition of polarized variation of Hodge structure due to Griffiths. A
complex variation of Hodge structure of weight $w$ is a graded
$C^{\infty}$-vector bundle $V=\bigoplus_{p+q=w}V^{p,q}$ equipped with a flat
connection $\nabla$ satisfying the Griffiths transversality condition, i.e.,
$\nabla^{0,1}(V^{p,q})\subset\Omega^{0,1}\otimes\bigl{(}V^{p+1,q-1}\oplus
V^{p,q}\bigr{)}$ and
$\nabla^{1,0}(V^{p,q})\subset\Omega^{1,0}\otimes\bigl{(}V^{p-1,q+1}\oplus
V^{p,q}\bigr{)}$, where $\nabla^{p,q}$ denote the $(p,q)$-part of $\nabla$. A
polarization of a complex variation of Hodge structure is a flat Hermitian
pairing $\langle\cdot,\cdot\rangle$ satisfying the following conditions: (i)
the decomposition $V=\bigoplus V^{p,q}$ is orthogonal with respect to
$\langle\cdot,\cdot\rangle$, (ii) $(\sqrt{-1})^{p-q}\langle\cdot,\cdot\rangle$
is positive definite on $V^{p,q}$.

A polarization of pure Hodge structure typically appears when we consider the
Gauss-Manin connection associated to a smooth projective morphism
$f:\mathcal{X}\longrightarrow\mathcal{Y}$. Namely, the family of vector spaces
$H^{w}(f^{-1}(y))$ $(y\in\mathcal{Y})$ naturally induces a flat bundle on
$\mathcal{Y}$. With the Hodge decomposition, it is a variation of Hodge
structure of weight $w$. A relatively ample line bundle induces a polarization
on the variation of Hodge structure.

Simpson discovered a completely different way to construct a polarized
variation of Hodge structure. Let $(V=\bigoplus V^{p,q},\nabla)$ be a complex
variation of Hodge structure. Note that $\nabla^{0,1}$ induces holomorphic
structures $\overline{\partial}_{V^{p,q}}:V^{p,q}\longrightarrow
V^{p,q}\otimes\Omega^{0,1}$ of $V^{p,q}$. We set
$\overline{\partial}_{V}:=\bigoplus\overline{\partial}_{V^{p,q}}$. Then,
$(V=\bigoplus V^{p,q},\overline{\partial}_{V})$ is a graded holomorphic vector
bundle. We also note that $\nabla^{1,0}$ induces linear maps
$V^{p,q}\longrightarrow V^{p-1,q+1}\otimes\Omega^{1,0}$, and hence
$\theta:V\longrightarrow V\otimes\Omega^{1,0}$. It is easy to check that
$\theta$ is a Higgs field of $(V,\overline{\partial}_{V})$. Such a graded
holomorphic bundle $V=\bigoplus_{p+q=w}V^{p,q}$ with a Higgs field $\theta$
such that $\theta\cdot V^{p,q}\subset V^{p-1,q+1}\otimes\Omega^{1,0}$ is
called a Hodge bundle of weight $w$. In general, we cannot construct a complex
variation of Hodge structure from a Hodge bundle. However, Simpson discovered
that if a Hodge bundle $(V=\bigoplus V^{p,q},\theta)$ on a compact Kähler
manifold satisfies the stability condition and the vanishing condition, then
there exists a flat connection $\nabla$ and a flat Hermitian pairing
$\langle\cdot,\cdot\rangle$ such that (i) $(V=\bigoplus V^{p,q},\nabla)$ is a
complex variation of Hodge structure which induces the Hodge bundle, (ii)
$\langle\cdot,\cdot\rangle$ is a polarization of $(V=\bigoplus
V^{p,q},\nabla)$. Indeed, according to the equivalence of Simpson between
Higgs bundles and harmonic bundles, there exists a pluri-harmonic metric $h$
of $(V,\theta)$. It turns out that the flat connection
$\nabla_{h}+\theta+\theta^{\dagger}_{h}$ satisfies the Griffiths
transversality. Moreover, the decomposition $V=\bigoplus V^{p,q}$ is
orthogonal with respect to $h$, and flat Hermitian paring
$\langle\cdot,\cdot\rangle$ is constructed by the relation
$(\sqrt{-1})^{p-q}\langle\cdot,\cdot\rangle_{V^{p,q}}=h_{|V^{p,q}}$.

Note that a Hodge bundle is regarded as a Higgs bundle
$(V,\overline{\partial}_{V},\theta)$ with an $S^{1}$-homogeneity, i.e.,
$(V,\overline{\partial}_{V})$ is equipped with an $S^{1}$-action such that
$t\circ\theta\circ t^{-1}=t\cdot\theta$ for any $t\in S^{1}$. It roughly means
that Hodge bundles correspond to the fixed points in the moduli space of Higgs
bundles with respect to the natural $S^{1}$-action induced by
$t(E,\overline{\partial}_{E},\theta)=(E,\overline{\partial}_{E},t\theta)$.

By the deformation $(E,\overline{\partial}_{E},\alpha\theta)$
$(\alpha\in{\mathbb{C}}^{\ast})$, any Higgs bundles is deformed to an
$S^{1}$-fixed point in the moduli space, i.e., a Hodge bundle as $\alpha\to
0$. Note that the Higgs field of the limit is not necessarily $0$. Hence, by
the equivalence between Higgs bundles and flat bundles, it turns out that any
flat bundle is deformed to a flat bundle underlying a polarized variation of
Hodge structure.

Simpson [61] particularly applied these ideas to construct uniformizations of
some types of projective manifolds. He also applied it to prove that some type
of discrete groups cannot be the fundamental group of any projective manifolds
in [63].

####  1.2.2 TE-structure

We recall that a complex variation of Hodge structure on $X$ induces a TE-
structure in the sense of Hertling [21], i.e., a holomorphic vector bundle
$\mathcal{V}$ on $\mathcal{X}:={\mathbb{C}}_{\lambda}\times X$ with a
meromorphic flat connection

$\widetilde{\nabla}:\mathcal{V}\longrightarrow\mathcal{V}\otimes\mathcal{O}_{\mathcal{X}}(\mathcal{X}^{0})\otimes\Omega_{\mathcal{X}}^{1}(\log\mathcal{X}^{0}),$

where $\mathcal{X}^{0}:=\\{0\\}\times X$. Indeed, for a complex variation of
Hodge structure $(V=\bigoplus V^{p,q},\nabla)$,
$F^{p}(V):=\bigoplus_{p_{1}\geq p}V^{p_{1},q_{1}}$ are holomorphic subbundles
with respect to $\nabla^{0,1}$. Thus, we obtain a decreasing filtration of
holomorphic subbundles $F^{p}(V)$ $(p\in{\mathbb{Z}})$ satisfying the
Griffiths transversality $\nabla^{1,0}F^{p}(V)\subset
F^{p-1}(V)\otimes\Omega^{1,0}$. Let $p:{\mathbb{C}}^{\ast}_{\lambda}\times
X\longrightarrow X$ denote the projection. We obtain the induced flat bundle
$(p^{\ast}V,p^{\ast}\nabla)$. By the Rees construction, $p^{\ast}V$ extends to
a locally free $\mathcal{O}_{\mathcal{X}}$-module $\mathcal{V}$, on which
$\widetilde{\nabla}:=p^{\ast}\nabla$ is a meromorphic flat connection
satisfying the condition
$\widetilde{\nabla}\mathcal{V}\subset\mathcal{V}\otimes\mathcal{O}_{\mathcal{X}}(\mathcal{X}^{0})\otimes\Omega^{1}_{\mathcal{X}}(\log\mathcal{X}^{0})$.

It is recognized that a TE-structure appears as a fundamental piece of
interesting structures in various fields of mathematics. For instance, TE-
structure is an ingredient of Frobenius manifold, which is important in the
theory of primitive forms and flat structures due to K. Saito [60], the
topological field theory of Dubrovin [18], the $tt^{\ast}$-geometry of
Cecotti-Vafa [7, 8], the Gromov-Witten theory, the theory of Landau-Ginzburg
models, etc. For the construction of Frobenius manifolds, it is an important
step to obtain TE-structures. Abstractly, TE-structure is also an important
ingredient of semi-infinite variation of Hodge structure [1, 9, 28], TERP
structure [21, 22, 23], integrable variation of twistor structure [59], etc.
(See also [49, 51].)

####  1.2.3 Homogeneous harmonic bundles

As Simpson applied his Kobayashi-Hitchin correspondence to construct complex
variations of Hodge structure, we may apply Theorem 1.1 to construct TE-
structures with something additional. It is done through harmonic bundles with
homogeneity as in the Hodge case.

Let $X$ be a complex manifold equipped with an $S^{1}$-action. Let
$(E,\overline{\partial}_{E})$ be an $S^{1}$-equivariant holomorphic vector
bundle. Let $\theta$ be a Higgs field of $(E,\overline{\partial}_{E})$, which
is homogeneous with respect to the $S^{1}$-action, i.e.,
$t^{\ast}\theta=t^{m}\theta$ for some $m\neq 0$. Let $h$ be an
$S^{1}$-invariant pluri-harmonic metric of
$(E,\overline{\partial}_{E},\theta)$. Then, as studied in [51, §3], we
naturally obtain a TE-structure. More strongly, it is equipped with a grading
in the sense of [9, 28], and it also underlies a polarized integrable
variation of pure twistor structure of weight $0$ [59]. Moreover, if there
exists an $S^{1}$-equivariant isomorphism between
$(E,\overline{\partial}_{E},\theta,h)$ and its dual, the TE-structure is
enhanced to a semi-infinite variation of Hodge structure with a grading [1, 9,
28]. If the $S^{1}$-action on $X$ is trivial, this is the same as the
construction of a variation of Hodge structure from a Hodge bundle with a
pluri-harmonic metric for which the Hodge decomposition is orthogonal.

Let $H$ be a simple normal crossing hypersurface of $X$. From an
$S^{1}$-homogeneous good wild harmonic bundle
$(E,\overline{\partial}_{E},\theta,h)$ on $(X,H)$, as mentioned above, we
obtain a TE-structure with a grading on $X\setminus H$. Moreover, it extends
to a meromorphic TE-structure on $(X,H)$ as studied in [51, §3]. We obtain the
mixed Hodge structure as the limit objects at the boundary, which is useful
for the study of more detailed properties of the TE-structure.

####  1.2.4 An equivalence

Let $X$ be a complex projective manifold with a simple normal crossing
hypersurface $H$ and an ample line bundle $L$, equipped with a
${\mathbb{C}}^{\ast}$-action. We may define a good filtered Higgs bundle
$(\mathcal{P}_{\ast}\mathcal{V},\theta)$ is called
${\mathbb{C}}^{\ast}$-homogeneous if $\mathcal{P}_{\ast}\mathcal{V}$ is
${\mathbb{C}}^{\ast}$-equivariant and $t^{\ast}\theta=t^{m}\cdot\theta$ for
some $m\neq 0$. Then, we obtain the following theorem by using Theorem 1.1.
(See §8.1.2 for the precise definition of the stability condition in this
context.)

######  Theorem 1.2 (Corollary 8.11)

There exists an equivalence between the following objects.

  * •

$\mu_{L}$-polystable ${\mathbb{C}}^{\ast}$-homogeneous good filtered Higgs
bundles $(\mathcal{P}_{\ast}\mathcal{V},\theta)$ on $(X,H)$ satisfying

$\int_{X}c_{1}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim
X-1}=\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-2}=0.$

  * •

$S^{1}$-homogeneous good wild harmonic bundles on $(X,H)$.

As mentioned in §1.2.3, Theorem 1.2 allows us to obtain a meromorphic TE-
structure on $(X,H)$ with a grading from a $\mu_{L}$-polystable
${\mathbb{C}}^{\ast}$-equivariant good filtered Higgs bundle satisfying the
vanishing condition. We already applied it to a classification of solutions of
the Toda equations on ${\mathbb{C}}^{\ast}$ [54]. It seems natural to expect
that this construction would be another way to obtain Frobenius manifolds.

Although we explained the homogeneity with respect to an $S^{1}$-action,
Theorem 1.2 is generalized for $K$-homogeneous good wild harmonic bundles as
explained in §8, where $K$ is any compact Lie group.

##### Acknowledgement

I thank Carlos Simpson for his fundamental works on harmonic bundles which are
most important in this study. I thank Philip Boalch and Andy Neitzke for their
kind comments to a preliminary note on the proof in the one dimensional case.
I thank Claude Sabbah for discussions on many occasions and for his kindness.
I thank François Labourie for his comment on the definition of wild harmonic
bundles. I thank Pengfei Huang for discussions. I am grateful to Ya Deng for
asking questions related to Mehta-Ramanathan type theorems in §3.4 and tensor
products in §4.3. I thank Akira Ishii and Yoshifumi Tsuchimoto for their
constant encouragement. A part of this manuscript was prepared for lectures in
the Oka symposium and ICTS program “Quantum Fields, Geometry and
Representation Theory”. I thank the organizers for the opportunities.

I am partially supported by the Grant-in-Aid for Scientific Research (S) (No.
17H06127), the Grant-in-Aid for Scientific Research (S) (No. 16H06335), the
Grant-in-Aid for Scientific Research (C) (No. 15K04843) and the Grant-in-Aid
for Scientific Research (C) (No. 20K03609) Japan Society for the Promotion of
Science.

##  2 Good filtered $\lambda$-flat bundles and wild harmonic bundles

###  2.1 Filtered sheaves and filtered $\lambda$-flat sheaves

####  2.1.1 Filtered sheaves

Let $X$ denote a complex manifold with a simple normal crossing hypersurface
$H$. Let $H=\bigcup_{i\in\Lambda}H_{i}$ denote a decomposition such that each
$H_{i}$ is smooth. Note that $H_{i}$ are not necessarily connected. For any
$P\in H$, a holomorphic coordinate neighbourhood $(X_{P},z_{1},\ldots,z_{n})$
around $P$ is called admissible if $H_{P}:=H\cap
X_{P}=\bigcup_{i=1}^{\ell(P)}\\{z_{i}=0\\}$. For such an admissible coordinate
neighbourhood, there exists the map
$\rho_{P}:\\{1,\ldots,\ell(P)\\}\longrightarrow\Lambda$ determined by
$H_{\rho_{P}(i)}\cap X_{P}=\\{z_{i}=0\\}$. We obtain the map
$\kappa_{P}:{\mathbb{R}}^{\Lambda}\longrightarrow{\mathbb{R}}^{\ell(P)}$ by
$\kappa_{P}({\boldsymbol{a}})=(a_{\rho(1)},\ldots,a_{\rho(\ell(P))})$.

Let $\mathcal{O}_{X}(\ast H)$ denote the sheaf of meromorphic functions which
may have poles along $H$. Let $\mathcal{E}$ be any coherent torsion free
$\mathcal{O}_{X}(\ast H)$-module. A filtered sheaf over $\mathcal{E}$ is
defined to be a tuple of coherent $\mathcal{O}_{X}$-submodules
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\subset\mathcal{E}$
$({\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda})$ satisfying the following
conditions.

  * •

$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\subset\mathcal{P}_{{\boldsymbol{b}}}\mathcal{E}$
if ${\boldsymbol{a}}\leq{\boldsymbol{b}}$, i.e., $a_{i}\leq b_{i}$ for any
$i\in\Lambda$.

  * •

$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}(\ast H)=\mathcal{E}$ for any
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$.

  * •

$\mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{n}}}\mathcal{E}=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\Bigl{(}\sum_{i\in\Lambda}n_{i}H_{i}\Bigr{)}$
for any ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$ and
${\boldsymbol{n}}\in{\mathbb{Z}}^{\Lambda}$.

  * •

For any ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$ there exists
${\boldsymbol{\epsilon}}\in{\mathbb{R}}_{>0}^{\Lambda}$ such that
$\mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{\epsilon}}}\mathcal{E}=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}$.

  * •

For any $P\in H$, we take an admissible coordinate neighbourhood
$(X_{P},z_{1},\ldots,z_{n})$ around $P$. Then, for any
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$,
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}_{|X_{P}}$ depends only on
$\kappa_{P}({\boldsymbol{a}})$.

For any coherent $\mathcal{O}_{X}(\ast H)$-submodule
$\mathcal{E}^{\prime}\subset\mathcal{E}$, we obtain a filtered sheaf
$\mathcal{P}_{\ast}\mathcal{E}^{\prime}$ over $\mathcal{E}^{\prime}$ by
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}^{\prime}:=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\cap\mathcal{E}^{\prime}$.
If $\mathcal{E}^{\prime}$ is saturated, i.e.,
$\mathcal{E}^{\prime\prime}:=\mathcal{E}/\mathcal{E}^{\prime}$ is torsion-
free, we obtain a filtered sheaf
$\mathcal{P}_{\ast}\mathcal{E}^{\prime\prime}$ over
$\mathcal{E}^{\prime\prime}$ by
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}^{\prime\prime}:=\mathop{\rm
Im}\nolimits\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}\longrightarrow\mathcal{E}^{\prime\prime}\bigr{)}$.

A morphism of filtered sheaves
$f:\mathcal{P}_{\ast}\mathcal{E}_{1}\longrightarrow\mathcal{P}_{\ast}\mathcal{E}_{2}$
is defined to be a morphism $f:\mathcal{E}_{1}\longrightarrow\mathcal{E}_{2}$
of $\mathcal{O}_{X}(\ast H)$-modules such that
$f(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}_{1})\subset\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}_{2}$
for any ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$.

###### Remark 2.1

The concept of filtered bundles on curves was introduced by Mehta and Seshadri
[45] and Simpson [61, 62]. A higher dimensional version was first studied by
Maruyama and Yokogawa [42] for the purpose of the construction of the moduli
spaces.

####  2.1.2 Restriction and gluing

Let $U\subset X$ be any open subset. We set $H_{U}=H\cap U$. Let
$H_{U}=\bigcup_{j\in\Lambda_{U}}H_{U,j}$ be the irreducible decomposition. For
any $j\in\Lambda_{U}$, we have $i(j)\in\Lambda$ such that $H_{U,j}$ is a
connected component of $H_{i(j)}\cap U$. For any $P\in H_{U}$, we set
$\Lambda_{U}(P):=\\{j\in\Lambda_{U}\,|\,P\in H_{U,j}\\}$.

Let $\mathcal{P}_{\ast}\mathcal{E}$ be a filtered sheaf over $\mathcal{E}$.
Let ${\boldsymbol{b}}\in{\mathbb{R}}^{\Lambda_{U}}$. For any $P\in H_{U}$, we
choose ${\boldsymbol{a}}(P,{\boldsymbol{b}})\in{\mathbb{R}}^{\Lambda}$ such
that $a(P,{\boldsymbol{b}})_{i(j)}=b_{j}$ for any $j\in\Lambda_{U}(P)$, and we
obtain the following $\mathcal{O}_{U,P}$-submodule of the stalk
$(\mathcal{E}_{|U})_{P}$:

$\mathcal{P}_{{\boldsymbol{b}}}(\mathcal{E}_{|U})_{P}:=\mathcal{P}_{{\boldsymbol{a}}(P,{\boldsymbol{b}})}(\mathcal{E})_{P}.$

It is independent of the choice of ${\boldsymbol{a}}(P,{\boldsymbol{b}})$ as
above. There exists a coherent $\mathcal{O}_{U}$-submodule
$\mathcal{P}_{{\boldsymbol{b}}}(\mathcal{E}_{|U})$ of $\mathcal{E}_{|U}$ whose
stalk at $P$ is $\mathcal{P}_{{\boldsymbol{b}}}(\mathcal{E}_{|U})_{P}$. Thus,
we obtain a filtered sheaf $\mathcal{P}_{\ast}(\mathcal{E}_{|U})$ over
$\mathcal{E}_{|U}$, which is denoted as $\mathcal{P}_{\ast}\mathcal{E}_{|U}$.

Let $X=\bigcup_{k\in\Gamma}X^{(k)}$ be an open covering. We set $H^{(k)}=H\cap
X^{(k)}$. For any a filtered sheaf $\mathcal{P}_{\ast}\mathcal{E}$ over
$\mathcal{E}$, we obtain filtered sheaves
$\mathcal{P}_{\ast}\mathcal{E}_{|X^{(k)}}$ over $\mathcal{E}_{|X^{(k)}}$ as
the restriction. Conversely, let $\mathcal{P}_{\ast}(\mathcal{E}_{|X^{(k)}})$
be filtered sheaves over $\mathcal{E}_{|X^{(k)}}$ such that
$\mathcal{P}_{\ast}(\mathcal{E}_{X^{(k)}})_{|X^{(k)}\cap
X^{(\ell)}}=\mathcal{P}_{\ast}(\mathcal{E}_{X^{(\ell)}})_{|X^{(k)}\cap
X^{(\ell)}}$ for any $k,\ell\in\Gamma$.

###### Lemma 2.2

There uniquely exists a filtered sheaf $\mathcal{P}_{\ast}\mathcal{E}$ over
$\mathcal{E}$ such that
$\mathcal{P}_{\ast}\mathcal{E}_{|X^{(k)}}=\mathcal{P}_{\ast}(\mathcal{E}_{|X^{(k)}})$
for any $k\in\Gamma$.

Proof Let ${\boldsymbol{a}}\in\Lambda$. For any $P\in H$, there exists
$k\in\Gamma$ such that $P\in X^{(k)}$. Let
$H^{(k)}=\bigcup_{j\in\Lambda^{(k)}}H^{(k)}_{j}$ be the irreducible
decomposition. For any $j\in\Lambda^{(k)}$, we have $i(k,j)\in\Lambda$ such
that $H^{(k)}_{j}$ is a connected component of $H_{i(k,j)}\cap X^{(k)}$. Thus,
we obtain a map $\Lambda^{(k)}\longrightarrow\Lambda$. For any
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$, let ${\boldsymbol{a}}^{(k)}$ be
the image of ${\boldsymbol{a}}$ by the induced map
${\mathbb{R}}^{\Lambda}\longrightarrow{\mathbb{R}}^{\Lambda^{(k)}}$, and we
obtain the following $\mathcal{O}_{X,P}$-submodule of $\mathcal{E}_{P}$:

$\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{E})_{P}:=\mathcal{P}_{{\boldsymbol{a}}^{(k)}}(\mathcal{E}_{|X^{(k)}})_{P}.$

There exists a coherent $\mathcal{O}_{X}$-submodule
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}$ whose stalk at $P$ is
$\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{E})_{P}$. Thus, we obtain a filtered
sheaf $\mathcal{P}_{\ast}\mathcal{E}$ over $\mathcal{E}$ with the desired
property. The uniqueness is also clear.

####  2.1.3 Reflexive filtered sheaves

A filtered sheaf $\mathcal{P}_{\ast}\mathcal{E}$ on $(X,H)$ is called
reflexive if each $\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}$ is a reflexive
$\mathcal{O}_{X}$-module. Note that it is equivalent to the “reflexive and
saturated” condition in [46, Definition 3.17] by the following lemma.

###### Lemma 2.3

Suppose that $\mathcal{P}_{\ast}\mathcal{E}$ is reflexive. Let
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$. We take $a_{i}-1<b\leq a_{i}$,
and let ${\boldsymbol{a}}^{\prime}\in{\mathbb{R}}^{\Lambda}$ be determined by
$a_{j}^{\prime}=a_{j}$ $(j\neq i)$ and $a^{\prime}_{i}=b$. Then,
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}/\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{E}$
is a torsion-free $\mathcal{O}_{H_{i}}$-module.

Proof Let $s$ be a section of
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}/\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{E}$
on an open set $U\subset D_{i}$. There exists an open subset
$\widetilde{U}\subset X$ and a section $\widetilde{s}$ of
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}$ on $\widetilde{U}$ such that
$\widetilde{U}\cap D_{i}=U$ and that $\widetilde{s}$ induces $s$. Note that
there exists $Z\subset\widetilde{U}$ of codimension $2$ such that
$\widetilde{s}_{|\widetilde{U}\setminus Z}$ is a section of
$\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{E}_{|\widetilde{U}\setminus
Z}$. Because $\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{E}$ is
reflexive, there exists a section $\widetilde{s}^{\prime}$ of
$\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{E}$ on $\widetilde{U}$ such
that $\widetilde{s}^{\prime}_{|\widetilde{U}\setminus
Z}=\widetilde{s}_{|\widetilde{U}\setminus Z}$. Hence, we obtain that
$\widetilde{s}$ is a section of
$\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{E}$, i.e., $s=0$.

The following lemma is clear.

###### Lemma 2.4

Let $\mathcal{P}_{\ast}\mathcal{E}$ be a reflexive filtered sheaf on $(X,H)$.
Then a coherent $\mathcal{O}_{X}(\ast H)$-submodule
$\mathcal{E}^{\prime}\subset\mathcal{E}$ is saturated if and only if the
induced filtered sheaf $\mathcal{P}_{\ast}\mathcal{E}^{\prime}$ is reflexive.

####  2.1.4 Filtered $\lambda$-flat sheaves

Let $\lambda$ be any complex number. Let $\mathcal{E}$ be a coherent torsion-
free $\mathcal{O}_{X}(\ast H)$-module. A $\lambda$-connection
$\mathbb{D}^{\lambda}:\mathcal{E}\longrightarrow\Omega_{X}^{1}\otimes\mathcal{E}$
is a ${\mathbb{C}}$-linear morphism of sheaves such that
$\mathbb{D}^{\lambda}(fs)=f\mathbb{D}^{\lambda}(s)+\lambda df\otimes s$ for
any local sections $f$ and $s$ of $\mathcal{O}_{X}$ and $\mathcal{E}$,
respectively. Note that an $\mathcal{O}_{X}$-morphism
$\mathbb{D}^{\lambda}\circ\mathbb{D}^{\lambda}:\mathcal{E}\longrightarrow\Omega_{X}^{2}\otimes\mathcal{E}$
is induced. If $\mathbb{D}^{\lambda}\circ\mathbb{D}^{\lambda}=0$, it is called
a flat $\lambda$-connection. When $\mathcal{E}$ is equipped with a flat
$\lambda$-connection, a $\lambda$-flat subsheaf of $\mathcal{E}$ means a
coherent $\mathcal{O}_{X}$-submodule $\mathcal{E}^{\prime}\subset\mathcal{E}$
such that
$\mathbb{D}^{\lambda}(\mathcal{E}^{\prime})\subset\Omega_{X}^{1}\otimes\mathcal{E}^{\prime}$.
A pair of a filtered sheaf $\mathcal{P}_{\ast}\mathcal{E}$ over $\mathcal{E}$
and a flat $\lambda$-connection $\mathbb{D}^{\lambda}$ of $\mathcal{E}$ is
called a filtered $\lambda$-flat connection. It is called reflexive if
$\mathcal{P}_{\ast}\mathcal{E}$ is reflexive.

###  2.2 $\mu_{L}$-Stability condition for filtered $\lambda$-flat sheaves

Let $X$ be a connected projective manifold with a simple normal crossing
hypersurface $H=\bigcup_{i\in\Lambda}H_{i}$. Let $L$ be an ample line bundle.

####  2.2.1 Slope of filtered sheaves

Let $\mathcal{P}_{\ast}\mathcal{E}$ be a filtered sheaf on $(X,H)$. Recall the
definition of the parabolic first Chern class
$c_{1}(\mathcal{P}_{\ast}\mathcal{E})$. Let $\eta_{i}$ be the generic point of
$H_{i}$. Note that $\mathcal{O}_{X,\eta_{i}}$-modules
$(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E})_{\eta_{i}}$ depends only on
$a_{i}$, which is denoted by $\mathcal{P}_{a_{i}}(\mathcal{E}_{\eta_{i}})$. We
obtain $\mathcal{O}_{H_{i},\eta_{i}}$-modules $\mathop{\rm
Gr}\nolimits^{\mathcal{P}}_{a}(\mathcal{E}_{\eta_{i}}):=\mathcal{P}_{a}(\mathcal{E}_{\eta_{i}})\big{/}\mathcal{P}_{<a}(\mathcal{E}_{\eta_{i}})$.
Then, we set

$c_{1}(\mathcal{P}_{\ast}\mathcal{E}):=c_{1}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E})-\sum_{i\in\Lambda}\sum_{a_{i}-1<a\leq
a_{i}}a\mathop{\rm rank}\nolimits\mathop{\rm
Gr}\nolimits^{\mathcal{P}}_{a}(\mathcal{E}_{\eta_{i}})[H_{i}]\in
H^{2}(X,{\mathbb{R}}).$ (2)

Here, $[H_{i}]$ denote the cohomology class induced by $H_{i}$. It is easy to
see that $c_{1}(\mathcal{P}_{\ast}\mathcal{V})$ is independent of a choice of
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$. We set

$\mu_{L}(\mathcal{P}_{\ast}\mathcal{E}):=\frac{1}{\mathop{\rm
rank}\nolimits\mathcal{E}}\int_{X}c_{1}(\mathcal{P}_{\ast}\mathcal{E})\cdot
c_{1}(L)^{n-1}.$

It is called the slope of $\mathcal{P}_{\ast}\mathcal{E}$ with respect to $L$.
The following is proved in [46, Lemma 3.7].

###### Lemma 2.5

Let
$f:\mathcal{P}_{\ast}\mathcal{E}^{(1)}\longrightarrow\mathcal{P}_{\ast}\mathcal{E}^{(2)}$
be a morphism of filtered sheaves which is generically an isomorphism, i.e.,
the induced morphism
$\mathcal{E}^{(1)}_{\eta(X)}\longrightarrow\mathcal{E}^{(2)}_{\eta(X)}$ at the
generic point of $X$ is an isomorphism. Then,
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{E}^{(1)})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{E}^{(2)})$
holds. If the equality holds, $f$ is an isomorphism in codimension one, i.e.,
there exists an algebraic subset $Z\subset X$ such that (i) the codimension of
$Z$ is larger than $2$, (ii) $f_{|X\setminus
Z}:\mathcal{P}_{\ast}\mathcal{E}^{(1)}_{|X\setminus
Z}\longrightarrow\mathcal{P}_{\ast}\mathcal{E}^{(2)}_{|X\setminus Z}$ is an
isomorphism.

####  2.2.2 $\mu_{L}$-Stability condition

A filtered $\lambda$-flat sheaf
$(\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda})$ on $(X,H)$ is called
$\mu_{L}$-stable (resp. $\mu_{L}$-semistable) if the following holds.

  * •

Let $\mathcal{E}^{\prime}\subset\mathcal{E}$ be any $\lambda$-flat
$\mathcal{O}_{X}(\ast H)$-submodule such that $0<\mathop{\rm
rank}\nolimits(\mathcal{E}^{\prime})<\mathop{\rm rank}\nolimits(\mathcal{E})$.
Then,
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{E}^{\prime})<\mu_{L}(\mathcal{P}_{\ast}\mathcal{E})$
(resp.
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{E}^{\prime})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{E})$)
holds.

A filtered $\lambda$-flat sheaf
$(\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda})$ is called
$\mu_{L}$-polystable if the following holds.

  * •

$(\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda})$ is
$\mu_{L}$-semistable.

  * •

$(\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda})=\bigoplus(\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i})$,
where each $(\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i})$ is
$\mu_{L}$-stable.

The following is standard. (See [46, §3.1.3] and [48, §2.1.4].)

###### Lemma 2.6

Suppose that $(\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda})$ is a
$\mu_{L}$-polystable reflexive filtered $\lambda$-flat sheaf. Then, there
exists a unique decomposition
$(\mathcal{P}_{\ast}\mathcal{E},\mathbb{D}^{\lambda})=\bigoplus_{i=1}^{N}(\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i})\otimes{\mathbb{C}}^{m(i)}$
such that (i) $(\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i})$
are $\mu_{L}$-stable, (ii)
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{E}_{i})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{E})$,
(iii)
$(\mathcal{P}_{\ast}\mathcal{E}_{i},\mathbb{D}^{\lambda}_{i})\not\simeq(\mathcal{P}_{\ast}\mathcal{E}_{j},\mathbb{D}^{\lambda}_{j})$
$(i\neq j)$.

###### Remark 2.7

In [46, §3.1.3], “the inequality $\mathop{\rm
par\textrm{-}deg}\nolimits_{L}(\mathcal{E}^{\prime}_{\ast})<\mathop{\rm
par\textrm{-}deg}\nolimits_{L}(\mathcal{E}_{\ast})$” should be corrected to
“the inequality
$\mu_{L}(\mathcal{E}^{\prime}_{\ast})<\mu_{L}(\mathcal{E}_{\ast})$”.

###  2.3 Filtered bundles

####  2.3.1 Filtered bundles in the local case

We recall the notion of filtered bundle in the local case. We shall explain it
in the global case in §2.3.3. Let $U$ be a neighbourhood of $(0,\ldots,0)$ in
${\mathbb{C}}^{n}$. We set $H_{U,i}:=U\cap\\{z_{i}=0\\}$, and
$H_{U}:=\bigcup_{i=1}^{\ell}H_{U,i}$ for some $0\leq\ell\leq n$. Let
$\mathcal{V}$ be a locally free $\mathcal{O}_{U}(\ast H_{U})$-module. A
filtered bundle $\mathcal{P}_{\ast}\mathcal{V}$ over $\mathcal{V}$ is a tuple
of locally free $\mathcal{O}_{U}$-submodules
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}$
$({\boldsymbol{a}}\in{\mathbb{R}}^{\ell})$ such that the following holds.

  * •

$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\subset\mathcal{P}_{{\boldsymbol{b}}}\mathcal{V}$
if ${\boldsymbol{a}}\leq{\boldsymbol{b}}$, i.e., $a_{i}\leq b_{i}$ for any
$i=1,\ldots,\ell$.

  * •

There exit a frame ${\boldsymbol{v}}=(v_{1},\ldots,v_{r})$ of $\mathcal{V}$
and tuples ${\boldsymbol{a}}(v_{j})\in{\mathbb{R}}^{\ell}$ $(j=1,\ldots,r)$
such that

$\mathcal{P}_{{\boldsymbol{b}}}\mathcal{V}=\bigoplus_{j=1}^{r}\mathcal{O}_{U}\Bigl{(}\sum_{i}\bigl{[}b_{i}-a_{i}(v_{j})\bigr{]}H_{U,i}\Bigr{)}\cdot
v_{j},$

where we set $[c]:=\max\\{p\in{\mathbb{Z}}\,|\,p\leq c\\}$ for any
$c\in{\mathbb{R}}$.

Clearly, a filtered bundle over $\mathcal{V}$ is a filtered sheaf over
$\mathcal{V}$.

###### Remark 2.8

We set $\mathcal{R}:={\mathbb{C}}[\\![z_{1},\ldots,z_{n}]\\!]$ and
$\widetilde{\mathcal{R}}:=\mathcal{R}[z_{1}^{-1},\ldots,z_{\ell}^{-1}]$. For a
free $\widetilde{\mathcal{R}}$-module $\widehat{\mathcal{V}}$, a filtered
bundle over $\widehat{\mathcal{V}}$ is defined to be a tuple
$\mathcal{P}_{\ast}\widehat{\mathcal{V}}:=\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}\widehat{\mathcal{V}}\,\big{|}\,{\boldsymbol{a}}\in{\mathbb{R}}^{\ell(P)}\bigr{)}$
of free $\mathcal{R}$-submodules satisfying similar conditions as above.

####  2.3.2 Pull back, push-forward and descent with respect to ramified
coverings in the local case

Let $\varphi:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}^{n}$ be given by
$\varphi(\zeta_{1},\ldots,\zeta_{n})=(\zeta_{1}^{m_{1}},\ldots,\zeta_{\ell}^{m_{\ell}},\zeta_{\ell+1},\ldots,\zeta_{n})$.
We set $U^{\prime}:=\varphi^{-1}(U)$,
$H_{U^{\prime},i}:=\varphi^{-1}(H_{U,i})$ and
$H_{U^{\prime}}:=\varphi^{-1}(H_{U})$. The induced ramified covering
$U^{\prime}\longrightarrow U$ is also denoted by $\varphi$.

For any ${\boldsymbol{b}}\in{\mathbb{R}}^{\ell}$, we set
$\varphi^{\ast}({\boldsymbol{b}})=(m_{i}b_{i})\in{\mathbb{R}}^{\ell}$. For any
filtered bundle $\mathcal{P}_{\ast}\mathcal{V}_{1}$ on $(U,H_{U})$, we define
a filtered bundle $\mathcal{P}_{\ast}\mathcal{V}^{\prime}_{1}$ on
$(U^{\prime},H_{U^{\prime}})$ as follows:

$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}^{\prime}_{1}=\sum_{\varphi^{\ast}({\boldsymbol{b}})+{\boldsymbol{n}}\leq{\boldsymbol{a}}}\varphi^{\ast}\bigl{(}\mathcal{P}_{{\boldsymbol{b}}}\mathcal{V}_{1}\bigr{)}\Bigl{(}\sum
n_{i}H_{U^{\prime},i}\Bigr{)}.$

We set
$\varphi^{\ast}(\mathcal{P}_{\ast}\mathcal{V}_{1}):=\mathcal{P}_{\ast}\mathcal{V}^{\prime}_{1}$.
Thus, we obtain the pull back functor $\varphi^{\ast}$ from the category of
filtered bundles on $(U,H_{U})$ to the category of filtered bundles on
$(U^{\prime},H_{U^{\prime}})$.

For any ${\boldsymbol{b}}\in{\mathbb{R}}^{\ell}$, we set
$\varphi_{\ast}({\boldsymbol{b}})=(m_{i}^{-1}b_{i})$. For any filtered bundle
$\mathcal{P}_{\ast}\mathcal{V}_{2}$ on $(U^{\prime},H_{U^{\prime}})$, we
obtain the following filtered bundle

$\mathcal{P}_{{\boldsymbol{b}}}\varphi_{\ast}(\mathcal{V}_{2}):=\varphi_{\ast}\mathcal{P}_{\varphi_{\ast}({\boldsymbol{b}})}\mathcal{V}_{2}.$

In this way, we obtain a functor $\varphi_{\ast}$ from the category of
filtered bundles on $(U^{\prime},H_{U^{\prime}})$ to the category of filtered
bundles on $(U,H_{U})$.

We set
$G:=\prod_{i=1}^{\ell}\\{\mu_{i}\in{\mathbb{C}}^{\ast}\,|\,\mu_{i}^{m_{i}}=1\\}$.
We define the action of $G$ on $U^{\prime}$ by
$(\mu_{1},\ldots,\mu_{\ell})(\zeta_{1},\ldots,\zeta_{n})=(\mu_{1}\zeta_{1},\ldots,\mu_{\ell}\zeta_{\ell},\zeta_{\ell+1},\ldots,\zeta_{n})$.
We identify $G$ as the Galois group of the ramified covering
$U^{\prime}\longrightarrow U$. Let $\mathcal{P}_{\ast}\mathcal{V}_{3}$ be a
$G$-equivariant filtered bundles on $(U^{\prime},H_{U^{\prime}})$. Then,
$\mathcal{P}_{\ast}\varphi_{\ast}\mathcal{V}_{3}$ is equipped with an induced
$G$-action. We obtain a filtered bundle
$(\mathcal{P}_{\ast}\varphi_{\ast}\mathcal{V}_{3})^{G}$ on $(U,H_{U})$ as the
$G$-invariant part of $\mathcal{P}_{\ast}\varphi_{\ast}\mathcal{V}_{3}$, which
is called the descent of $\mathcal{P}_{\ast}\mathcal{V}_{3}$ with respect to
the $G$-action. In this way, we obtain a functor from the category of
$G$-equivariant filtered bundles on $(U^{\prime},H_{U^{\prime}})$ to the
category of filtered bundles on $(U,H_{U})$.

For a filtered bundle $\mathcal{P}_{\ast}\mathcal{V}_{1}$ on $(U,H_{U})$, the
pull back $\varphi^{\ast}(\mathcal{P}_{\ast}\mathcal{V}_{1})$ is a
$G$-equivariant filtered bundle on $(U^{\prime},H_{U^{\prime}})$, and it
descent is naturally isomorphic to $\mathcal{P}_{\ast}\mathcal{V}_{1}$.

####  2.3.3 Filtered bundles in the global case

We use the notation in §2.1.1. Let $\mathcal{V}$ be a locally free
$\mathcal{O}_{X}(\ast H)$-module. A filtered bundle
$\mathcal{P}_{\ast}\mathcal{V}=\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\,\big{|}\,{\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}\bigr{)}$
over $\mathcal{V}$ be a sequence of locally free $\mathcal{O}_{X}$-submodules
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}$ of $\mathcal{V}$ such that the
following holds.

  * •

For any $P\in H$, we take an admissible coordinate neighbourhood
$(X_{P},z_{1},\ldots,z_{n})$ around $P$. Then, for any
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$,
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|X_{P}}$ depends only on
$\kappa_{P}({\boldsymbol{a}})$, denoted as
$\mathcal{P}^{(P)}_{\kappa_{P}({\boldsymbol{a}})}(\mathcal{V}_{|X_{P}})$.

  * •

The sequence
$(\mathcal{P}^{(P)}_{{\boldsymbol{b}}}(\mathcal{V}_{|X_{P}})\,|\,{\boldsymbol{b}}\in{\mathbb{R}}^{\ell(P)})$
is a filtered bundle over $\mathcal{V}_{|X_{P}}$ in the sense of §2.3.1.

In other words, a filtered bundle is a filtered sheaf (see §2.1.1) satisfying
the condition in §2.3.1 locally around any point of $H$.

###### Remark 2.9

The higher dimensional version of filtered bundles was introduced in [47] with
a different formulation. See also [5, 6]. In this paper, we essentially follow
Iyer and Simpson [29].

####  2.3.4 The induced bundles and filtrations

For any $I\subset\Lambda$, let
${\boldsymbol{\delta}}_{I}\in{\mathbb{R}}^{\Lambda}$ be the element whose
$j$-th component is $0$ $(j\in\Lambda\setminus I)$ or $1$ $(j\in I)$. We also
set $H_{I}:=\bigcap_{i\in I}H_{i}$ and $\partial
H_{I}:=H_{I}\cap\bigl{(}\bigcup_{j\in\Lambda\setminus I}H_{j}\bigr{)}$.

Let $\mathcal{P}_{\ast}\mathcal{V}$ be a filtered bundle on $(X,H)$. Take
$i\in\Lambda$. Let ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$. For any
$a_{i}-1\leq b\leq a_{i}$, we set
${\boldsymbol{a}}(b,i):={\boldsymbol{a}}+(b-a_{i}){\boldsymbol{\delta}}_{i}$.
We set

${}^{i}\\!F_{b}\bigl{(}\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{V})_{|H_{i}}\bigr{)}:=\mathcal{P}_{{\boldsymbol{a}}(b,i)}\mathcal{V}/\mathcal{P}_{{\boldsymbol{a}}(a_{i}-1,i)}\mathcal{V}.$

It is naturally regarded as a locally free $\mathcal{O}_{H_{i}}$-module.
Moreover, it is a subbundle of
$\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{V})_{|H_{i}}$. In this way, we obtain
a filtration ${}^{i}\\!F$ of
$\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{V})_{|H_{i}}$ indexed by
$]a_{i}-1,a_{i}]$. We shall also denote it as just $F$ if there is no risk of
confusion.

We obtain the induced filtrations ${}^{i}\\!F$ of
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{I}}$ if $i\in I$. Let
${\boldsymbol{a}}_{I}\in{\mathbb{R}}^{I}$ denote the image of
${\boldsymbol{a}}$ by the projection
${\mathbb{R}}^{\Lambda}\longrightarrow{\mathbb{R}}^{I}$. Set
$]{\boldsymbol{a}}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}_{I}]:=\prod_{i\in
I}]a_{i}-1,a_{i}]$. For any
${\boldsymbol{b}}\in]{\boldsymbol{a}}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}_{I}]$,
we set

${}^{I}\\!F_{{\boldsymbol{b}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{I}}):=\bigcap_{i\in
I}{}^{i}\\!F_{b_{i}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{I}}).$

By the condition of filtered bundles, the following compatibility condition
holds.

  * •

Let $P$ be any point of $H_{I}$. There exist a neighbourhood $X_{P}$ of $P$ in
$X$ and a decomposition

$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|X_{P}\cap
H_{I}}=\bigoplus_{{\boldsymbol{b}}\in]{\boldsymbol{a}}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}_{I}]}\mathcal{G}_{P,{\boldsymbol{b}}}$

such that the following holds for any
${\boldsymbol{c}}\in]{\boldsymbol{a}}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}_{I}]$:

${}^{I}\\!F_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{I}\cap
X_{P}})=\bigoplus_{{\boldsymbol{b}}\leq{\boldsymbol{c}}}\mathcal{G}_{P,{\boldsymbol{b}}}.$

For any
${\boldsymbol{c}}\in]{\boldsymbol{a}}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}_{I}]$,
we obtain the following locally free $\mathcal{O}_{H_{I}}$-modules:

${}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}):=\frac{{}^{I}\\!F_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{I}})}{\sum_{{\boldsymbol{b}}\lneq{\boldsymbol{c}}}{}^{I}\\!F_{{\boldsymbol{b}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{I}})}.$

Here, ${\boldsymbol{b}}=(b_{i})\lneq{\boldsymbol{c}}=(c_{i})$ means that
$b_{i}\leq c_{i}$ for any $i$ and that ${\boldsymbol{b}}\neq{\boldsymbol{c}}$.
Clearly, if
${\boldsymbol{c}}\in]{\boldsymbol{a}}^{\prime}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}^{\prime}_{I}]$
and ${\boldsymbol{a}}^{\prime}_{\Lambda\setminus
I}={\boldsymbol{a}}_{\Lambda\setminus I}$, we obtain ${}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})={}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}^{\prime}}\mathcal{V})$.

####  2.3.5 The induced filtered bundles

For ${\boldsymbol{c}}\in{\mathbb{R}}^{I}$, we choose
${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$ such that
${\boldsymbol{c}}\in]{\boldsymbol{a}}_{I}-{\boldsymbol{\delta}}_{I},{\boldsymbol{a}}_{I}]$,
and we obtain the following $\mathcal{O}_{H_{I}}(\ast\partial H_{I})$-module:

${}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V}):={}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})(\ast\partial
H_{I}).$ (3)

It is independent of the choice of ${\boldsymbol{a}}$ as above. We obtain the
irreducible decomposition $\partial H_{I}=\bigcup_{i\in\Lambda(I)}H_{I,i}$.
For any $i\in\Lambda(I)$, there exists $j(i)\in\Lambda\setminus I$ such that
$H_{I,i}$ is a connected component of $H_{I}\cap H_{j(i)}$. Let
${\boldsymbol{d}}\in{\mathbb{R}}^{\Lambda(I)}$. For $P\in\partial H_{I}$,
there exists
${\boldsymbol{a}}({\boldsymbol{c}},{\boldsymbol{d}},P)\in{\mathbb{R}}^{\Lambda}$
such that (i) $a({\boldsymbol{c}},{\boldsymbol{d}},P)_{j}=c_{j}$ $(j\in I)$,
(ii) $a({\boldsymbol{c}},{\boldsymbol{d}},P)_{j(i)}=d_{i}$
$(i\in\Lambda(I),\,P\in H_{I,i})$. We obtain an $\mathcal{O}_{X,P}$-submodule

$\mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}_{P}:={}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}({\boldsymbol{c}},{\boldsymbol{d}},P)}\mathcal{V})_{P}\subset{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})_{P}.$ (4)

Note that $\mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}_{P}$ is independent
of the choice of ${\boldsymbol{a}}(P)$. There uniquely exists an
$\mathcal{O}_{H_{I}}$-submodule
$\mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}\subset{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})$ whose stalk at $P$
$(P\in\partial H_{I})$ are equal to
$\mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}_{P}$. Thus, we obtain
the following filtered bundle over ${}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})$ on $(H_{I},\partial H_{I})$:

${}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{\ast}\mathcal{V}):=\Bigl{(}\mathcal{P}_{{\boldsymbol{d}}}\bigl{(}{}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{V})\bigr{)}\,\Big{|}\,{\boldsymbol{d}}\in{\mathbb{R}}^{\Lambda(I)}\Bigr{)}.$

####  2.3.6 First and second Chern characters for filtered bundles

Let $\mathcal{P}_{\ast}\mathcal{V}$ be a filtered bundle over $(X,H)$. Take
any ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$. As recalled in §2.2.1, we
obtain the parabolic first Chern class:

$c_{1}(\mathcal{P}_{\ast}\mathcal{V})=c_{1}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})-\sum_{i\in\Lambda}\sum_{a_{i}-1<b\leq
a_{i}}a_{i}\mathop{\rm rank}\nolimits{}^{i}\\!\mathop{\rm
Gr}\nolimits^{F}_{b}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{E}_{|H_{i}})\cdot[H_{i}]\in
H^{2}(X,{\mathbb{R}}).$ (5)

To explain the second parabolic Chern character in $H^{4}(X,{\mathbb{R}})$,
let us introduce some notation. Let $\mathop{\rm Irr}\nolimits(H_{i}\cap
H_{j})$ be the set of the irreducible components of $H_{i}\cap H_{j}$. For
$C\in\mathop{\rm Irr}\nolimits(H_{I})$, let $[C]\in H^{4}(X,{\mathbb{R}})$
denotes the induced cohomology class, and let ${}^{C}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})$
denote the restriction of ${}^{I}\\!\mathop{\rm
Gr}\nolimits^{F}_{{\boldsymbol{c}}}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})$
to $C$. Moreover, $\iota_{i\ast}:H^{2}(H_{i},{\mathbb{R}})\longrightarrow
H^{4}(X,{\mathbb{R}})$ denotes the Gysin map induced by
$\iota_{i}:H_{i}\longrightarrow X$. Then, the second parabolic Chern character
is given as follows.

$\mathop{\rm ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V}):=\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})-\sum_{i\in\Lambda}\sum_{a_{i}-1<b\leq
a_{i}}b\cdot\iota_{i\ast}\Bigl{(}c_{1}\bigl{(}{}^{i}\\!\mathop{\rm
Gr}\nolimits^{F}_{b}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|H_{i}})\bigr{)}\Bigr{)}\\\
+\frac{1}{2}\sum_{i\in\Lambda}\sum_{a_{i}-1<b\leq a_{i}}b^{2}\mathop{\rm
rank}\nolimits\left({}^{i}\\!\mathop{\rm
Gr}\nolimits^{F}_{b}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})\right)\cdot[H_{i}]^{2}\\\
+\frac{1}{2}\sum_{\begin{subarray}{c}(i,j)\in\Lambda^{2}\\\ i\neq
j\end{subarray}}\sum_{C\in\mathop{\rm Irr}\nolimits(H_{i}\cap
H_{j})}\sum_{\begin{subarray}{c}a_{i}-1<c_{i}\leq a_{i}\\\ a_{j}-1<c_{j}\leq
a_{j}\end{subarray}}c_{i}\cdot c_{j}\mathop{\rm
rank}\nolimits{}^{C}\\!\mathop{\rm
Gr}\nolimits^{F}_{(c_{i},c_{j})}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})\cdot[C].$
(6)

###### Remark 2.10

The higher Chern character for filtered sheaves was defined by Iyer and
Simpson [29] in a systematic way. In this paper, we adopt the definition of
$\mathop{\rm ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})$ in [46].

###  2.4 Good filtered $\lambda$-flat bundles

Let $X$ be a complex manifold with a simple normal crossing hypersurface
$H=\bigcup_{i\in\Lambda}H_{i}$.

####  2.4.1 Good set of irregular values at $P$

Let $P$ be any point of $H$. We take an admissible holomorphic coordinate
neighbourhood $(X_{P},z_{1},\ldots,z_{n})$ around $P$. Let
$f\in\mathcal{O}_{X}(\ast H)_{P}$. If $f\in\mathcal{O}_{X,P}$, we set
$\mathop{\rm ord}\nolimits(f):=(0,\ldots,0)\in{\mathbb{R}}^{\ell(P)}$. If
there exists ${\boldsymbol{n}}\in{\mathbb{Z}}_{\leq
0}^{\ell(P)}\setminus\\{(0,\ldots,0)\\}$ such that (i) $g:=f\prod
z_{i}^{-n_{i}}\in\mathcal{O}_{X,P}$, (ii) $g(P)\neq 0$, then we set
$\mathop{\rm ord}\nolimits(f):={\boldsymbol{n}}$. Otherwise, $\mathop{\rm
ord}\nolimits(f)$ is not defined.

For any $\mathfrak{a}\in\mathcal{O}_{X}(\ast H)_{P}/\mathcal{O}_{X,P}$, we
take a lift $\widetilde{\mathfrak{a}}\in\mathcal{O}_{X}(\ast H)_{P}$. If
$\mathop{\rm ord}\nolimits(\widetilde{\mathfrak{a}})$ is defined, we set
$\mathop{\rm ord}\nolimits(\mathfrak{a}):=\mathop{\rm
ord}\nolimits(\widetilde{\mathfrak{a}})$. Otherwise, $\mathop{\rm
ord}\nolimits(\mathfrak{a})$ is not defined. Note that it is independent of
the choice of a lift $\widetilde{\mathfrak{a}}$.

Let $\mathcal{I}_{P}\subset\mathcal{O}_{X}(\ast H)_{P}/\mathcal{O}_{X,P}$ be a
finite subset. We say that $\mathcal{I}_{P}$ is a good set of irregular values
if the following conditions are satisfied.

  * •

$\mathop{\rm ord}\nolimits(\mathfrak{a})$ is defined for any
$\mathfrak{a}\in\mathcal{I}_{P}$.

  * •

$\mathop{\rm ord}\nolimits(\mathfrak{a}-\mathfrak{b})$ is defined for any
$\mathfrak{a},\mathfrak{b}\in\mathcal{I}_{P}$.

  * •

$\\{\mathop{\rm
ord}\nolimits(\mathfrak{a}-\mathfrak{b})\,|\,\mathfrak{a},\mathfrak{b}\in\mathcal{I}_{P}\\}$
is totally ordered with respect to the order $\leq_{{\mathbb{Z}}^{\ell(P)}}$.
Here, we define
${\boldsymbol{n}}\leq_{{\mathbb{Z}}^{\ell(P)}}{\boldsymbol{n}}^{\prime}$ if
$n_{i}\leq n_{i}^{\prime}$ for any $i$.

####  2.4.2 Good filtered $\lambda$-flat bundles

Let $\mathcal{V}$ be a locally free $\mathcal{O}_{X}(\ast H)$-module with a
flat $\lambda$-connection. Let $\mathcal{P}_{\ast}\mathcal{V}$ be a filtered
bundle over $\mathcal{V}$. For any $P\in X$, let $\mathcal{O}_{X,\widehat{P}}$
denote the completion of the local ring $\mathcal{O}_{X,P}$ with respect to
the maximal ideal. We say that
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is unramifiedly good at
$P$ if the following holds.

  * •

There exist a good set of irregular values
$\mathcal{I}_{P}\subset\mathcal{O}_{X}(\ast H)_{P}/\mathcal{O}_{X,P}$ and a
decomposition

$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})\otimes\mathcal{O}_{X,\widehat{P}}=\bigoplus_{\mathfrak{a}\in\mathcal{I}_{P}}(\mathcal{P}_{\ast}\mathcal{V}_{\mathfrak{a}},\mathbb{D}^{\lambda}_{\mathfrak{a}})$
(7)

such that
$\mathbb{D}^{\lambda}_{\mathfrak{a}}-d\widetilde{\mathfrak{a}}\mathop{\rm
id}\nolimits_{\mathcal{V}_{\mathfrak{a}}}$ are logarithmic with respect to the
lattices $\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{\mathfrak{a}}$ for any
${\boldsymbol{a}}\in{\mathbb{R}}^{\ell(P)}$ and
$\mathfrak{a}\in\mathcal{I}_{P}$, i.e.,

$(\mathbb{D}^{\lambda}_{\mathfrak{a}}-d\widetilde{\mathfrak{a}}\mathop{\rm
id}\nolimits_{\mathcal{V}_{\mathfrak{a}}})\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{\mathfrak{a}}\subset\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{\mathfrak{a}}\otimes\Omega^{1}_{X}(\log
H).$ (8)

Here, $\widetilde{\mathfrak{a}}$ denote lifts of $\mathfrak{a}$ to
$\mathcal{O}_{X}(\ast H)_{P}$.

We say that $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is good at
$P$ if the following holds.

  * •

There exist a neighbourhood $X_{P}$ of $P$ in $X$ and a covering map
$\varphi_{P}:X_{P}^{\prime}\longrightarrow X_{P}$ ramified over $H_{P}=H\cap
X_{P}$ such that
$\varphi_{P}^{\ast}(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is
unramifiedly good at $\varphi_{P}^{-1}(P)$. (See §2.3.2 for the pull back of
filtered bundles.)

We say that $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is good
(resp. unramifiedly good) if it is good (resp. unramifiedly good) at any point
of $H$.

###  2.5 Prolongation of holomorphic vector bundles with a Hermitian metric

Let $X$ be any complex manifold with a simple normal crossing hypersurface
$H=\bigcup_{i\in\Lambda}H_{i}$. Let $(E,\overline{\partial}_{E})$ be a
holomorphic vector bundle on $X\setminus H$ with a Hermitian metric $h$. Let
us recall the construction of $\mathcal{O}_{X}(\ast H)$-module
$\mathcal{P}^{h}E$ and $\mathcal{O}_{X}$-modules
$\mathcal{P}^{h}_{{\boldsymbol{a}}}E$
$({\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda})$.

Let ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$. For any open subset
$\mathcal{U}\subset X$, let $\mathcal{P}_{{\boldsymbol{a}}}^{h}E(\mathcal{U})$
be the space of holomorphic sections $s$ of $E_{|\mathcal{U}\setminus H}$
satisfying the following condition.

  * •

For any point $P$ of $\mathcal{U}\cap H$, let $(X_{P},z_{1},\ldots,z_{n})$ be
an admissible holomorphic coordinate neighbourhood around $P$ such that
$X_{P}$ is relatively compact in $\mathcal{U}$. Set
${\boldsymbol{c}}=\kappa_{P}({\boldsymbol{a}})$. (See §2.1.1.) Then,

$\bigl{|}s\bigr{|}_{h}=O\Bigl{(}\prod_{i=1}^{\ell(P)}|z_{i}|^{-c_{i}-\epsilon}\Bigr{)}$

holds on $X_{P}\setminus H$ for any $\epsilon>0$.

We obtain an $\mathcal{O}_{X}$-module $\mathcal{P}^{h}_{{\boldsymbol{a}}}E$.
We set
$\mathcal{P}^{h}E:=\bigcup_{{\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}}\mathcal{P}^{h}_{{\boldsymbol{a}}}E$
which is an $\mathcal{O}_{X}(\ast H)$-module. Note that in general,
$\mathcal{P}^{h}_{{\boldsymbol{a}}}E$ are not necessarily coherent
$\mathcal{O}_{X}$-modules.

###### Definition 2.11

Let $\mathcal{P}_{\ast}\mathcal{V}$ be a filtered bundle over $(X,H)$. Let
$(E,\overline{\partial}_{E})$ be the holomorphic vector bundle obtained as the
restriction of $\mathcal{V}$ to $X\setminus H$. A Hermitian metric $h$ is
called adapted to $\mathcal{P}_{\ast}\mathcal{V}$ if
$\mathcal{P}^{h}_{\ast}E=\mathcal{P}_{\ast}\mathcal{V}$ in
$\iota_{\ast}(E)=\iota_{\ast}(\mathcal{V}_{|X\setminus H})$, where
$\iota:X\setminus H\longrightarrow X$ denotes the inclusion.

####  2.5.1 A sufficient condition

We mention a useful sufficient condition for $\mathcal{P}^{h}_{\ast}E$ to be a
filtered bundle, although we do not use it in this paper. Let $g_{X\setminus
H}$ be a Kähler metric satisfying the following condition [11]:

  * •

For any $P\in H$, we take an admissible holomorphic coordinate neighbourhood
$(X_{P},z_{1},\ldots,z_{n})$ around $P$ such that $X_{P}$ is isomorphic to
$\prod_{i=1}^{n}\\{|z_{i}|<1\\}$ by the coordinate system. Set
$X_{P}^{\prime}:=\prod_{i=1}^{n}\\{|z_{i}|<1/2\\}$. Then,
$g_{|X_{P}^{\prime}\setminus H}$ is mutually bounded with the restriction of
the Poincaré metric

$\sum_{i=1}^{\ell(P)}\frac{dz_{i}\,d\overline{z}_{i}}{|z_{i}|^{2}(\log|z_{i}|^{2})^{2}}+\sum_{i=\ell(P)+1}^{n}dz_{i}\,d\overline{z}_{i}.$

A Hermitian metric $h$ of $(E,\overline{\partial}_{E})$ is called acceptable
if the curvature of the Chern connection is bounded with respect to $h$ and
$g_{X\setminus H}$. The following theorem is proved in [50, Theorem 21.3.1].

###### Theorem 2.12

If $h$ is acceptable, then $\mathcal{P}^{h}_{\ast}E$ is a filtered bundle, and
$\mathcal{P}^{h}E$ is a locally free $\mathcal{O}_{X}(\ast H)$-module.

###  2.6 Harmonic bundles

####  2.6.1 Pluri-harmonic metrics for $\lambda$-flat bundles

Let $Y$ be any complex manifold. Let $E$ be a $C^{\infty}$-vector bundle on
$Y$. Let $A^{p,q}(E)$ denote the space of $C^{\infty}$-sections of
$\Omega^{p,q}\otimes E$. We set $A^{\ell}(E):=\bigoplus_{p+q=\ell}A^{p,q}(E)$.
In this context, a $\lambda$-connection of $E$ is a differential operator
$\mathbb{D}^{\lambda}:A^{0}(E)\longrightarrow A^{1}(E)$ such that
$\mathbb{D}^{\lambda}(fs)=f\mathbb{D}^{\lambda}(s)+(\lambda\partial_{Y}+\overline{\partial}_{Y})f\otimes
s$ for any $f\in C^{\infty}(Y)$ and $s\in A^{0}(E)$. We obtain a section
$\mathbb{D}^{\lambda}\circ\mathbb{D}^{\lambda}\in A^{2}(\mathop{\rm
End}\nolimits(E))$. A $\lambda$-connection is called flat if
$\mathbb{D}^{\lambda}\circ\mathbb{D}^{\lambda}=0$.

Let $(E,\mathbb{D}^{\lambda})$ be a $\lambda$-flat bundle on $Y$. We decompose
$\mathbb{D}^{\lambda}=d_{E}^{\prime\prime}+d_{E}^{\prime}$ into the
$(0,1)$-part and the $(1,0)$-part. Then, $(E,d_{E}^{\prime\prime})$ is a
holomorphic vector bundle. Let $h$ be a Hermitian metric of $E$. From $h$ and
$d^{\prime\prime}_{E}$, we obtain the differential operator
$\delta^{\prime}_{E,h}$ such that $d^{\prime\prime}_{E}+\delta^{\prime}_{E,h}$
is a Chern connection. From $d^{\prime}_{E}$ and $h$, we obtain the
$(0,1)$-operator $\delta^{\prime\prime}_{E,h}$ determined by $\lambda\partial
h(u,v)=h(d^{\prime}_{E}u,v)+h(u,\delta^{\prime\prime}_{E,h}v)$. As in [48,
§2.2.1], we obtain the operators

$\overline{\partial}_{E,h}:=\frac{1}{1+|\lambda|^{2}}(d^{\prime\prime}_{E}+\lambda\delta^{\prime}_{E,h}),\quad\partial_{E,h}:=\frac{1}{1+|\lambda|^{2}}(\overline{\lambda}d^{\prime}_{E}+\delta^{\prime}_{E,h})$
$\theta^{\dagger}_{E,h}:=\frac{1}{1+|\lambda|^{2}}(\overline{\lambda}d^{\prime\prime}_{E}-\delta^{\prime\prime}_{E,h}),\quad\theta_{E,h}:=\frac{1}{1+|\lambda|^{2}}(d^{\prime}_{E}-\lambda\partial^{\prime}_{E,h}).$

Note that
$\mathbb{D}^{\lambda}=\overline{\partial}_{E,h}+\theta_{E,h}+\lambda(\partial_{E,h}+\theta^{\dagger}_{E,h})$.
We set
$\mathbb{D}^{\lambda\star}_{E,h}:=\delta^{\prime}_{E,h}-\delta^{\prime\prime}_{E,h}=\partial_{E,h}+\theta_{E,h}^{\dagger}-\overline{\lambda}(\overline{\partial}_{E,h}+\theta_{E,h})$,
and
$G(h):=\bigl{[}\mathbb{D}^{\lambda},\mathbb{D}^{\lambda\star}_{E,h}\bigr{]}$.

###### Definition 2.13

$h$ is called a pluri-harmonic metric of $(E,\mathbb{D}^{\lambda})$ if
$G(h)=0$. Such a tuple $(E,\mathbb{D}^{\lambda},h)$ is called a harmonic
bundle.

If $\lambda\neq 0$, Because
$(1+|\lambda|^{2})(\overline{\partial}_{E,h}+\theta_{E,h})=\mathbb{D}^{\lambda}-\lambda\mathbb{D}_{E,h}^{\lambda\star}$,
and $(\mathbb{D}^{\lambda})^{2}=(\mathbb{D}_{E,h}^{\lambda\star})^{2}=0$, we
obtain

$G(h)=-\frac{(1+|\lambda|^{2})^{2}}{\lambda}(\overline{\partial}_{E,h}+\theta_{E,h})^{2}=-\frac{(1+|\lambda|^{2})^{2}}{\lambda}\bigl{(}\overline{\partial}_{E,h}^{2}+\overline{\partial}_{E,h}\theta_{E,h}+\theta_{E,h}^{2}\bigr{)}.$
(9)

Hence, $G(h)=0$ implies that $(E,\overline{\partial}_{E,h},\theta_{E,h})$ is a
Higgs bundle. The metric $h$ is a pluri-harmonic metric for
$(E,\overline{\partial}_{E,h},\theta_{E,h})$. Conversely, if $h$ is a pluri-
harmonic metric for a Higgs bundle $(E,\overline{\partial}_{E},\theta)$, we
obtain the Chern connection $\overline{\partial}_{E}+\partial_{E,h}$
associated with $\overline{\partial}_{E}$ and $h$, and the adjoint
$\theta^{\dagger}_{h}$ of $\theta$ with respect to $h$. We obtain a flat
$\lambda$-connection
$\mathbb{D}^{\lambda}_{h}=\overline{\partial}_{E}+\lambda\theta_{h}+\lambda\partial_{E,h}+\theta$.
The metric $h$ is a pluri-harmonic metric for $(E,\mathbb{D}^{\lambda}_{h})$.

###### Remark 2.14

If $\lambda=0$, a flat $0$-connection is equivalent to a Higgs bundle
$(E,\overline{\partial}_{E},\theta)$ by the relation
$\mathbb{D}^{1}=\overline{\partial}_{E}+\theta$. In this case, we obtain
$G(h)=\bigl{[}\overline{\partial}_{E}+\theta,\partial_{E,h}+\theta_{h}^{\dagger}\bigr{]}$,
and hence $2G(h)$ is equal to the curvature $F(h)$ of the connection
$\mathbb{D}^{1}_{h}=\overline{\partial}_{E}+\theta_{h}^{\dagger}+\partial_{E,h}+\theta$.

####  2.6.2 The case $\lambda\neq 0$

Let $G(h)=G(h)^{2,0}+G(h)^{1,1}+G(h)^{0,2}$ denote the decomposition into
$(p,q)$-parts. If $G(h)=0$, we clearly obtain $G(h)^{1,1}=0$. If $\lambda\neq
0$, we obtain the converse.

###### Proposition 2.15

Suppose $\lambda\neq 0$. If $G(h)^{1,1}=0$, we obtain $G(h)=0$, i.e., $h$ is a
pluri-harmonic metric of $(E,\mathbb{D}^{\lambda})$.

Proof As in [48, Lemma 2.28], the following holds:

$\overline{\lambda}^{-1}\partial_{E,h}^{2}+\lambda^{-1}\theta_{E,h}^{2}=0,\quad\lambda^{-1}\overline{\partial}_{E,h}^{2}+\overline{\lambda}^{-1}(\theta_{E,h}^{\dagger})^{2}=0.$
(10)

It is easy to check that
$\overline{\partial}_{E,h}^{2}=-(\partial_{E,h}^{2})^{\dagger}$,
$(\theta_{E,h}^{\dagger})^{2}=-(\theta_{E,h}^{2})^{\dagger}$ and
$(\overline{\partial}_{E,h}\theta_{E,h})^{\dagger}=\partial_{E,h}\theta^{\dagger}_{E,h}$.

From the flatness $\mathbb{D}^{\lambda}\circ\mathbb{D}^{\lambda}=0$, we obtain

$(\lambda\partial_{E,h}+\theta_{E,h})^{2}=\lambda^{2}\partial_{E,h}^{2}+\lambda\partial_{E,h}\theta_{E,h}+\theta_{E,h}^{2}=0,$
(11)
$\bigl{[}\overline{\partial}_{E,h}+\lambda\theta^{\dagger}_{E,h},\lambda\partial_{E,h}+\theta_{E,h}\bigr{]}=\lambda\bigl{(}[\overline{\partial}_{E,h},\partial_{E,h}]+[\theta_{E,h}\theta^{\dagger}_{E,h}]\bigr{)}+\overline{\partial}_{E,h}\theta_{E,h}+\lambda^{2}\partial_{E,h}\theta^{\dagger}_{E,h}=0.$
(12)

From (10) and (11), we obtain

$\partial_{E,h}\theta_{E,h}=-\lambda^{-1}(1-|\lambda|^{2})\theta_{E,h}^{2}.$
(13)

Because
$(\overline{\partial}_{E,h}\theta^{\dagger}_{E,h})^{\dagger}=\partial_{E,h}\theta_{E,h}$,
we obtain

$\overline{\partial}_{E,h}\theta_{E,h}^{\dagger}=\overline{\lambda}^{-1}(1-|\lambda|^{2})(\theta^{\dagger}_{E,h})^{2}.$
(14)

Note that $G(h)^{1,1}=0$ is equivalent to
$\overline{\partial}_{E,h}\theta_{E,h}=0$ and
$\partial_{E,h}\theta_{E,h}^{\dagger}=0$. To obtain $G(h)=0$, it is enough to
prove $\mathop{\rm
Tr}\nolimits(\theta_{E,h}^{2}(\theta_{E,h}^{\dagger})^{2})=0$. Indeed, there
exists $C\neq 0$ depending on $\dim Y$ such that for any Kähler form $\omega$
of $Y$ we obtain $\mathop{\rm
Tr}\nolimits(\theta_{E,h}^{2}(\theta_{E,h}^{\dagger})^{2})\omega^{\dim
Y-2}=C\bigl{|}\theta_{E,h}^{2}\bigr{|}^{2}_{h,\omega}\omega^{\dim Y}$. Hence,
the vanishing $\mathop{\rm
Tr}\nolimits(\theta_{E,h}^{2}(\theta_{E,h}^{\dagger})^{2})=0$ implies
$\theta_{E,h}^{2}=(\theta_{E,h}^{\dagger})^{2}=0$ and
$\partial_{E,h}^{2}=\overline{\partial}_{E,h}^{2}=0$.

From (14) and $\partial_{E,h}\theta^{\dagger}_{E,h}=0$, we obtain
$\partial_{E,h}\overline{\partial}_{E,h}\theta_{E,h}^{\dagger}=\overline{\lambda}^{-1}(1-|\lambda|^{2})\partial_{E,h}\bigl{(}(\theta^{\dagger}_{E,h})^{2}\bigr{)}=0$.
We also have
$\overline{\partial}_{E,h}\partial_{E,h}\theta_{E,h}^{\dagger}=0$. Hence, we
obtain the following equality:

$0=\mathop{\rm
Tr}\nolimits(\theta_{E,h}\partial_{E,h}\overline{\partial}_{E,h}\theta^{\dagger}_{E,h})=\mathop{\rm
Tr}\nolimits\bigl{(}\theta_{E,h}\cdot[\overline{\partial}_{E,h}\partial_{E,h}+\partial_{E,h}\overline{\partial}_{E,h},\theta_{E,h}^{\dagger}]\bigr{)}.$
(15)

From (12), $\overline{\partial}_{E,h}\theta_{E,h}=0$ and
$\partial_{E,h}\theta_{E,h}^{\dagger}=0$, we obtain

$[\overline{\partial}_{E,h},\partial_{E,h}]+[\theta_{E,h},\theta_{E,h}^{\dagger}]=0.$
(16)

Hence, we obtain the following:

$\mathop{\rm
Tr}\nolimits\bigl{(}\theta_{E,h}\cdot[\overline{\partial}_{E,h}\partial_{E,h}+\partial_{E,h}\overline{\partial}_{E,h},\theta_{E,h}^{\dagger}]\bigr{)}=\mathop{\rm
Tr}\nolimits\bigl{(}\theta_{E,h}\cdot\bigl{[}-[\theta_{E,h},\theta_{E,h}^{\dagger}],\theta_{E,h}^{\dagger}\bigr{]}\bigr{)}\\\
=-2\mathop{\rm
Tr}\nolimits\bigl{(}\theta_{E,h}^{2}(\theta^{\dagger}_{E,h})^{2}\bigr{)}.$
(17)

We obtain the claim of the proposition from (15) and (17).

By using Proposition 2.15 we can improve [48, Corollary 2.30] as follows.

###### Corollary 2.16

If $\lambda\neq 0$, the pluri-harmonicity of the metric $h$ is equivalent to
the vanishing $G(h)^{1,1}=0$, i.e., $\overline{\partial}_{E,h}\theta_{E,h}=0$.

###### Remark 2.17

In [48, Lemma 2.29], the claim
$[\overline{\partial}_{V,h},\partial_{V,h}]+[\theta_{V,h},\theta_{V,h}^{\dagger}]=0$
is incorrect, in general. The author thanks Pengfei Huang for pointing out it.

###### Remark 2.18

If $\lambda=1$, the claim of Proposition 2.15 also follows from a Bochner type
formula [47, Proposition 21.39], which originally goes back to the study of
Simpson [63] in the context of harmonic bundles, the study of Corlette [10] in
the context of harmonic metrics for flat bundles on Riemannian manifolds, and
the study of Siu [68] in the context of harmonic maps.

###  2.7 Wild harmonic bundles

####  2.7.1 Higgs case

Let $X$ be a complex manifold with a simple normal crossing hypersurface
$H=\bigcup_{i\in\Lambda}H_{i}$. Let $(E,\overline{\partial}_{E},\theta,h)$ be
a harmonic bundle on $X\setminus H$. It is called wild on $(X,H)$ if the
following holds.

  * •

Let $\Sigma_{\theta}\subset T^{\ast}(X\setminus H)$ denote the spectral cover
of $\theta$, i.e., $\Sigma_{\theta}$ denotes the support of the coherent
$\mathcal{O}_{T^{\ast}(X\setminus H)}$-module induced by
$(E,\overline{\partial}_{E},\theta)$. Then, the closure of $\Sigma_{\theta}$
in the projective completion of $T^{\ast}X$ is complex analytic.

A wild harmonic bundle $(E,\overline{\partial}_{E},\theta,h)$ is called
unramifiedly good at $P\in H$ if the following holds.

  * •

There exists a good set of irregular values
$\mathcal{I}_{P}\subset\mathcal{O}_{X}(\ast H)_{P}/\mathcal{O}_{X,P}$, a
neighbourhood $X_{P}$, and a decomposition

$(E,\overline{\partial}_{E},\theta)_{|X_{P}\setminus
H}=\bigoplus_{\mathfrak{a}\in\mathcal{I}_{P}}(E_{\mathfrak{a}},\overline{\partial}_{E_{\mathfrak{a}}},\theta_{\mathfrak{a}})$

such that the closure of the spectral cover $\Sigma_{\mathfrak{a}}$ of
$\theta_{\mathfrak{a}}-d\widetilde{\mathfrak{a}}\mathop{\rm
id}\nolimits_{E_{\mathfrak{a}}}$ in $T^{\ast}X_{P}(\log(X_{P}\cap H))$ is
proper over $X_{P}$, where $\widetilde{\mathfrak{a}}$ denote lifts of
$\mathfrak{a}$ to $\mathcal{O}_{X}(\ast H)_{P}$.

A wild harmonic bundle $(E,\overline{\partial}_{E},\theta,h)$ is called good
at $P\in H$ if the following holds.

  * •

There exist a neighbourhood $X_{P}$ and a covering
$\varphi_{P}:X_{P}^{\prime}\longrightarrow X_{P}$ ramified along
$H_{P}^{\prime}$ such that the pull back
$\varphi_{P}^{-1}(E,\overline{\partial}_{E},\theta,h)_{|X_{P}}$ is
unramifiedly good wild at any point of $\varphi_{P}^{-1}(H)$.

We say that $(E,\overline{\partial}_{E},\theta,h)$ is good wild (resp.
unramifiedly good wild) on $(X,H)$ if it is good wild (resp. unramifiedly good
wild) at any point of $H$.

Note that every wild harmonic bundle on $(X,H)$ is not necessarily good on
$(X,H)$. But, the following is known [53, Corollary 15.2.8].

###### Theorem 2.19

Let $(E,\overline{\partial}_{E},\theta,h)$ be a wild harmonic bundle on
$(X,H)$. Then, there exists a proper birational morphism
$\varphi:X^{\prime}\longrightarrow X$ of complex manifolds such that (i)
$H^{\prime}:=\varphi^{-1}(H)$ is simple normal crossing, (ii)
$X^{\prime}\setminus H^{\prime}\simeq X\setminus H$, (iii)
$\varphi^{-1}(E,\overline{\partial}_{E},\theta,h)$ is good wild on
$(X^{\prime},H^{\prime})$.

####  2.7.2 The case of $\lambda$-flat bundles

A $\lambda$-flat bundle $(E,\mathbb{D}^{\lambda})$ with a pluri-harmonic
metric $h$ on $X\setminus H$ is called (good, unramifiedly good) wild if the
associated Higgs bundle with a pluri-harmonic metric
$(E,\overline{\partial}_{E,h},\theta_{E,h},h)$ is a (good, unramifiedly good)
wild harmonic bundle.

####  2.7.3 Prolongation of good wild harmonic bundles to good filtered
$\lambda$-flat bundles

The following is one of the fundamental theorems in the study of wild harmonic
bundles [50, Theorem 7.4.3].

###### Theorem 2.20

If $(E,\mathbb{D}^{\lambda},h)$ is a good wild harmonic bundle on $(X,H)$,
then $(\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda})$ is a good filtered
$\lambda$-flat bundle on $(X,H)$.

The following is a consequence of the norm estimate for good wild harmonic
bundles [50, Theorem 11.7.2].

###### Theorem 2.21

Let $(E,\mathbb{D}^{\lambda},h_{i})$ $(i=1,2)$ be good wild harmonic bundles
on $X\setminus H$ such that
$\mathcal{P}^{h_{1}}_{\ast}E=\mathcal{P}^{h_{2}}_{\ast}E$. Then, $h_{i}$ are
mutually bounded around any point of $H$.

####  2.7.4 Prolongation of good wild harmonic bundles in the projective case

Suppose that $X$ is projective and connected. Let $L$ be any ample line bundle
on $X$. The following is proved in [50, Proposition 13.6.1, Proposition
13.6.4].

###### Proposition 2.22

Let $(E,\mathbb{D}^{\lambda},h)$ be a good wild harmonic bundle on $(X,H)$.

  * •

$(\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda})$ is $\mu_{L}$-polystable with
$\mu_{L}(\mathcal{P}^{h}_{\ast}E)=0$.

  * •

We obtain $c_{1}(\mathcal{P}_{\ast}E)=0$ and $\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}E)c_{1}(L)^{\dim X-2}=0$.

  * •

Let $h^{\prime}$ be another pluri-harmonic metric of
$(E,\mathbb{D}^{\lambda},h)$ such that
$\mathcal{P}^{h^{\prime}}_{\ast}E=\mathcal{P}^{h}_{\ast}E$. Then, there exists
a decomposition of the $\lambda$-flat bundle
$(E,\mathbb{D}^{\lambda})=\bigoplus(E_{j},\mathbb{D}^{\lambda}_{j})$ such that
(i) the decomposition is orthogonal with respect to both $h$ and $h^{\prime}$,
(ii) $h_{|E_{i}}=a_{i}\cdot h^{\prime}_{|E_{i}}$ for some $a_{i}>0$.

  * •

Let $(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1})$ be any
direct summand of $(\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda})$. Let
$(E_{1},\mathbb{D}^{\lambda}_{1})$ be the $\lambda$-flat bundle on $X\setminus
H$ obtained as the restriction of
$(\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1})$, and let $h_{1}$ be the metric of
$E_{1}$ induced by $h$. Then, $(E_{1},\mathbb{D}^{\lambda}_{1},h_{1})$ is a
harmonic bundle. In particular, we obtain
$c_{1}(\mathcal{P}_{\ast}\mathcal{V}_{1})=0$ and $\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V}_{1})c_{1}(L)^{\dim X-2}=0$.

###  2.8 Main existence theorem in this paper

Let $X$ be a smooth connected projective complex manifold with a simple normal
crossing hypersurface $H$. Let $L$ be any ample line bundle on $X$. Let
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ be a good filtered
$\lambda$-flat bundle on $(X,H)$. Let
$(E,\overline{\partial}_{E},\mathbb{D}^{\lambda})$ be the $\lambda$-flat
bundle obtained as the restriction of
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ to $X\setminus H$.

###### Theorem 2.23

Suppose that $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is
$\mu_{L}$-polystable, and the following vanishing:

$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})=0,\quad\int_{X}\mathop{\rm
ch}\nolimits_{2}(\mathcal{P}_{\ast}\mathcal{V})c_{1}(L)^{\dim X-2}=0.$ (18)

Then, there exists a pluri-harmonic metric $h$ of
$(E,\overline{\partial}_{E},\mathbb{D}^{\lambda})$ such that the isomorphism
$(\mathcal{V},\mathbb{D}^{\lambda})_{|X\setminus
H}\simeq(E,\mathbb{D}^{\lambda})$ extends to
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})\simeq(\mathcal{P}^{h}_{\ast}E,\mathbb{D}^{\lambda})$.

We proved the claim of the theorem in the case $\lambda=1$ in [50, Theorem
16.1.1]. We shall explain the proof in §3–7. Note that the one dimensional
case is due to Biquard-Boalch [3].

###### Corollary 2.24

There exists the equivalence of the following objects for each $\lambda$.

  * •

Good wild harmonic bundles on $(X,H)$.

  * •

$\mu_{L}$-polystable good filtered $\lambda$-flat bundles
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ satisfying the
vanishing condition (18).

##  3 Preliminaries

###  3.1 Hermitian-Einstein metrics of $\lambda$-flat bundles

Let $Y$ be a Kähler manifold with a Kähler form $\omega$. Let
$(E,\mathbb{D}^{\lambda})$ be a $\lambda$-flat bundle on $Y$ with a Hermitian
metric. Recall that $h$ is called a Hermitian-Einstein metric of the
$\lambda$-flat bundle if $\Lambda_{\omega}G(h)^{\bot}=0$, where $G(h)^{\bot}$
denote the trace-free part of $G(h)$, and $\Lambda_{\omega}$ denote the
adjoint of the multiplication of $\omega$ (see [35, §3.2]). The following is a
generalization of Kobayashi-Lübke inequality to the context of $\lambda$-flat
bundles due to Simpson [61, Proposition 3.4].

######  Proposition 3.1 (Simpson)

If $h$ is a Hermitian-Einstein metric, there exists $C>0$ depending only on
$n=\dim Y$ such that the following holds:

$\mathop{\rm
Tr}\nolimits\Bigl{(}\bigl{(}G(h)^{\bot}\bigr{)}^{2}\Bigr{)}\omega^{n-2}=C\bigl{|}G(h)^{\bot}\bigr{|}_{h,\omega}^{2}\omega^{n}.$

As a result, if $\mathop{\rm
Tr}\nolimits\Bigl{(}\bigl{(}G(h)^{\bot}\bigr{)}^{2}\Bigr{)}\omega^{n-2}=0$,
then we obtain $G(h)^{\bot}=0$.

###  3.2 Rank one case

Let $X$ be an $n$ dimensional smooth connected projective variety with a
simple normal crossing hypersurface $H$. Let $\omega$ be a Kähler form. Let
$H=\bigcup_{i\in\Lambda}H_{i}$ be the irreducible decomposition. Let $g_{i}$
be a $C^{\infty}$-Hermitian metric of the line bundle $\mathcal{O}(H_{i})$.
Let $\sigma_{i}$ denote the section of $\mathcal{O}_{X}(H_{i})$ induced by the
inclusion $\mathcal{O}_{X}\longrightarrow\mathcal{O}_{X}(H_{i})$.

Let $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ be a good filtered
$\lambda$-flat bundle on $(X,H)$ of rank one. For each $i\in\Lambda$, there
uniquely exists $a_{i}\in]-1,0]$ such that ${}^{i}\\!\mathop{\rm
Gr}\nolimits^{F}_{a_{i}}(\mathcal{P}_{\ast}\mathcal{V})\neq 0$. Let $A$ be the
constant determined by

$A\int_{X}\omega^{n}=2\pi
n(1+|\lambda|^{2})\int_{X}c_{1}(\mathcal{P}_{\ast}\mathcal{V})\omega^{n-1}.$

The following proposition is standard.

###### Proposition 3.2

There exists a Hermitian metric $h$ of the line bundle
$E:=\mathcal{V}_{|X\setminus D}$ such that (i)
$\sqrt{-1}\Lambda_{\omega}G(h)=A$, (ii)
$h\prod_{i\in\Lambda}|\sigma_{i}|_{g_{i}}^{2a_{i}}$ is a Hermitian metric of
$\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{V})$ of $C^{\infty}$-class. Such a
metric is unique up to the multiplication of a positive constant. Moreover, if
$c_{1}(\mathcal{P}_{\ast}\mathcal{V})=0$, then $R(h)=0$ holds, and hence $h$
is a pluri-harmonic metric of $(E,\mathbb{D}^{\lambda})$.

Proof Note that $G(h)=(1+|\lambda|^{2})R(h)$ holds in the rank one case. (See
[48, Lemma 2.31].) Let $h_{0}^{\prime}$ be a $C^{\infty}$-metric of
$\mathcal{P}_{{\boldsymbol{a}}}E$. We obtain the metric
$h_{0}:=h_{0}^{\prime}\cdot\prod_{i\in\Lambda}|\sigma_{i}|^{-2a_{i}}_{g_{i}}$
of $E$ on $X\setminus H$. It is well known that
$\frac{\sqrt{-1}}{2\pi}R(h_{0})$ naturally extends to a closed $(1,1)$-form on
$X$ of $C^{\infty}$-class which represents $c_{1}(\mathcal{P}_{\ast}E)$. By
the condition of $A$, we obtain
$\int_{X}\bigl{(}\sqrt{-1}\Lambda_{\omega}R(h_{0})-(1+|\lambda|^{2})^{-1}A\bigr{)}\omega^{n}=0$.
Note that
$\sqrt{-1}\Lambda_{\omega}R(h_{0}e^{\varphi})=\sqrt{-1}\Lambda_{\omega}R(h_{0})+\sqrt{-1}\Lambda_{\omega}\overline{\partial}\partial\varphi$.
Hence, there exists an ${\mathbb{R}}$-valued $C^{\infty}$-function
$\varphi_{0}$ such that
$\sqrt{-1}\Lambda_{\omega}R(h_{0}e^{\varphi_{0}})-(1+|\lambda|^{2})^{-1}A=0$.
The metric $h=h_{0}e^{\varphi_{0}}$ has the desired property. The uniqueness
is clear.

Suppose that $c_{1}(\mathcal{P}_{\ast}E)=0$. In the rank one case, a Hermitian
metric of $E$ is a pluri-harmonic metric of $(E,\mathbb{D}^{\lambda})$, if and
only if $R(h)=0$. Because the cohomology class of $R(h_{0})$ is $0$, there
exists an ${\mathbb{R}}$-valued $C^{\infty}$-function $\varphi_{0}$ such that
$R(h_{0}e^{\varphi_{0}})=0$ by the standard
$\partial\overline{\partial}$-lemma. By the uniqueness, we obtain the second
claim of the lemma.

For the metric $h$ in Proposition 3.2, $\frac{\sqrt{-1}}{2\pi}R(h)$ induces a
closed $(1,1)$-form on $X$ of $C^{\infty}$-class which represents
$c_{1}(\mathcal{P}_{\ast}E)$.

###  3.3 $\beta$-subobject and socle for reflexive filtered $\lambda$-flat
sheaves

Let $X$ and $H$ be as in §3.2. Let $L$ be an ample line bundle $L$ on $X$. For
any coherent $\mathcal{O}_{X}$-module $\mathcal{M}$, we set
$\deg_{L}(\mathcal{M}):=\int_{X}c_{1}(\mathcal{M})c_{1}(L)^{\dim X-1}$.

####  3.3.1 $\beta$-subobjects

Let $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ be a reflexive
filtered $\lambda$-flat sheaf on $(X,H)$. For any $A\in{\mathbb{R}}$, let
$\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)$ denote the family
of saturated coherent subsheaves $\mathcal{F}$ of
$\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V}$ such that
$\deg_{L}(\mathcal{F})\geq-A$ and that $\mathcal{F}(\ast H)$ is a
$\lambda$-flat subsheaf of $\mathcal{V}$. Any
$\mathcal{F}\in\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)$
induces a reflexive filtered sheaf $\mathcal{P}_{\ast}(\mathcal{F}(\ast H))$
by $\mathcal{P}_{{\boldsymbol{c}}}(\mathcal{F}(\ast
H)):=\mathcal{P}_{{\boldsymbol{c}}}\mathcal{V}\cap\mathcal{F}(\ast H)$ for any
${\boldsymbol{c}}\in{\mathbb{R}}^{\Lambda}$. We set
$f_{A}(\mathcal{F}):=\mu_{L}(\mathcal{P}_{\ast}(\mathcal{F}(\ast H)))$. Thus,
we obtain a function $f_{A}$ on
$\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)$.

###### Lemma 3.3

The image
$f_{A}\bigl{(}\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)\bigr{)}$
is a finite subset of ${\mathbb{R}}$. In particular, $f_{A}$ has the maximum.

Proof According to [20, Lemma 2.5], the family
$\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)$ is bounded. Hence,
by using the flattening stratifications [56, §8], it is easy to see that there
exists a finite decomposition
$\mathcal{S}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)=\coprod_{i=1}^{N}\mathcal{S}_{i}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)$
such that $f_{A}$ is constant on each
$\mathcal{S}_{i}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V},A)$.

It is standard that any reflexive filtered $\lambda$-flat sheaf has a
$\beta$-subobject, i.e., the following holds.

###### Proposition 3.4

For any reflexive filtered $\lambda$-flat sheaf
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$, there uniquely exists
a non-zero $\lambda$-flat subsheaf $\mathcal{V}_{0}\subset\mathcal{V}$ such
that the following holds for any non-zero reflexive $\lambda$-flat subsheaf
$\mathcal{V}^{\prime}\subset\mathcal{V}$.

  * •

$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{0})$
holds.

  * •

If
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{0})$
holds, then we obtain $\mathcal{V}^{\prime}\subset\mathcal{V}_{0}$.

Proof By the formula (2), there exists $N>0$ such that the following holds for
any saturated subsheaf
$\mathcal{F}\subset\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V}$:

$\left|\deg_{L}(\mathcal{F})-\mathop{\rm
rank}\nolimits(\mathcal{F})\mu_{L}(\mathcal{P}_{\ast}\mathcal{F}(\ast
H))\right|<N.$

We set $A_{0}:=|\deg_{L}(\mathcal{P}_{{\boldsymbol{0}}}\mathcal{V})|+10N$. Let
$B_{0}$ denote the maximum of $f_{A_{0}}$. Then, it is easy to see that
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})\leq B_{0}$ for any saturated
$\lambda$-flat subsheaf $\mathcal{V}^{\prime}\subset\mathcal{V}$. Moreover, if
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})=B_{0}$, then
$(\mathcal{P}_{\ast}\mathcal{V}^{\prime},\mathbb{D}^{\lambda}_{\mathcal{V}^{\prime}})$
is $\mu_{L}$-semistable, where $\mathbb{D}^{\lambda}_{\mathcal{V}^{\prime}}$
denote the flat $\lambda$-connection induced by $\mathbb{D}^{\lambda}$.

Suppose that the $\lambda$-flat subsheaves $\mathcal{V}_{i}\subset\mathcal{V}$
$(i=1,2)$ satisfy $\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{i})=B_{0}$. We
obtain the subsheaf $\mathcal{V}_{1}+\mathcal{V}_{2}\subset\mathcal{V}$.
Because $\mathcal{V}_{1}+\mathcal{V}_{2}$ is a quotient of
$\mathcal{V}_{1}\oplus\mathcal{V}_{2}$, we obtain a filtered sheaf
$\mathcal{P}_{\ast}(\mathcal{V}_{1}+\mathcal{V}_{2})$ over
$\mathcal{V}_{1}+\mathcal{V}_{2}$. induced by
$\mathcal{P}_{\ast}\mathcal{V}_{1}\oplus\mathcal{P}_{\ast}\mathcal{V}_{2}$.
Then, by the $\mu_{L}$-semistability of
$(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i})$, we obtain that
$B_{0}=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1}\oplus\mathcal{P}_{\ast}\mathcal{V}_{2})\leq\mu_{L}(\mathcal{P}_{\ast}(\mathcal{V}_{1}+\mathcal{V}_{2}))$.
Let $\mathcal{V}_{3}$ denote the saturated subsheaf of $\mathcal{V}$ generated
by $\mathcal{V}_{1}+\mathcal{V}_{2}$. We obtain a filtered sheaf
$\mathcal{P}_{\ast}\mathcal{V}_{3}$ by
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{3}=\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{V})\cap\mathcal{V}_{3}$.
Because the natural morphism
$\mathcal{P}_{\ast}(\mathcal{V}_{1}+\mathcal{V}_{2})\longrightarrow\mathcal{P}_{\ast}\mathcal{V}_{3}$
is generically an isomorphism, we obtain
$\mu_{L}(\mathcal{P}_{\ast}(\mathcal{V}_{1}+\mathcal{V}_{2}))\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{3})\leq
B_{0}$ by Lemma 2.5. Hence, we obtain
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{3})=B_{0}$. Then, the claim of the
lemma is clear.

####  3.3.2 Socle

Let $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ be a
$\mu_{L}$-semistable reflexive filtered $\lambda$-flat sheaf on $(X,H)$. Let
$\mathcal{T}$ denote the family of saturated $\lambda$-flat subsheaves
$\mathcal{V}^{\prime}\subset\mathcal{V}$ such that the induced filtered
$\lambda$-flat sheaf
$(\mathcal{P}_{\ast}\mathcal{V}^{\prime},\mathbb{D}^{\lambda}_{\mathcal{V}^{\prime}})$
is $\mu_{L}$-stable with
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{\prime})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$.
Let $\mathcal{V}_{1}$ be the saturated $\mathcal{O}_{X}(\ast H)$-submodule of
$\mathcal{V}$ generated by
$\sum_{\mathcal{V}^{\prime}\in\mathcal{T}}\mathcal{V}^{\prime}$. It is a
$\lambda$-flat subsheaf of $\mathcal{V}$.

###### Proposition 3.5

$(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{\mathcal{V}_{1}})$
is equal to the direct sum
$\bigoplus_{k=1}^{\ell}(\mathcal{P}_{\ast}\mathcal{V}^{(k)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(k)}})$
of a tuple of $\mu_{L}$-stable filtered $\lambda$-flat subsheaves of
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$. In particular,
$(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1})$ is
$\mu_{L}$-polystable. The filtered $\lambda$-flat subsheaf
$(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{\mathcal{V}_{1}})$
is called the socle of $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$.

Proof Let $\mathcal{V}^{(i)}$ $(i=1,2)$ be saturated $\lambda$-flat subsheaves
of $\mathcal{V}$ such that (i)
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{(i)})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$,
(ii)
$(\mathcal{P}_{\ast}\mathcal{V}^{(1)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(1)}})$
is $\mu_{L}$-semistable, (iii)
$(\mathcal{P}_{\ast}\mathcal{V}^{(2)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(2)}})$
is $\mu_{L}$-stable.

###### Lemma 3.6

Either $\mathcal{V}^{(2)}\subset\mathcal{V}^{(1)}$ or
$\mathcal{V}^{(1)}\cap\mathcal{V}^{(2)}=0$ holds.

Proof Let us consider the morphism
$\iota_{1}-\iota_{2}:\mathcal{V}^{(1)}\oplus\mathcal{V}^{(2)}\longrightarrow\mathcal{V}$,
where $\iota_{i}:\mathcal{V}^{(i)}\longrightarrow\mathcal{V}$ denote the
inclusions. Let $\mathcal{K}$ denote the kernel. We obtain a filtered sheaf
$\mathcal{P}_{\ast}\mathcal{K}$ over $\mathcal{K}$ by
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{K}:=\mathcal{K}\cap\mathcal{P}_{{\boldsymbol{a}}}(\mathcal{V}_{1}\oplus\mathcal{V}_{2})$.
The projection
$\mathcal{V}^{(1)}\oplus\mathcal{V}^{(2)}\longrightarrow\mathcal{V}^{(2)}$
induces
$\mathcal{K}\simeq\mathcal{V}^{(1)}\cap\mathcal{V}^{(2)}=:\mathcal{I}$. It
induces a morphism of filtered $\lambda$-flat sheaves
$g:(\mathcal{P}_{\ast}\mathcal{K},\mathbb{D}^{\lambda}_{\mathcal{K}})\longrightarrow(\mathcal{P}_{\ast}\mathcal{V}^{(2)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(2)}})$.
We set $\mu_{0}:=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$. Because
$\bigoplus_{i=1,2}(\mathcal{P}_{\ast}\mathcal{V}^{(i)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(i)}})$
and $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ are
$\mu_{L}$-semistable with the same slope $\mu_{0}$, we obtain that
$(\mathcal{P}_{\ast}\mathcal{K},\mathbb{D}^{\lambda}_{\mathcal{K}})$ is also
$\mu_{L}$-semistable with $\mu_{L}(\mathcal{P}_{\ast}\mathcal{K})=\mu_{0}$.

Suppose that $\mathcal{K}\neq 0$, i.e., $\mathcal{I}\neq 0$. Because
$\mathcal{I}$ is a subsheaf of $\mathcal{V}^{(2)}$, we also obtain a filtered
sheaf $\mathcal{P}_{\ast}\mathcal{I}$ induced by
$\mathcal{P}_{\ast}\mathcal{V}^{(2)}$. Because $\mathcal{I}\simeq\mathcal{K}$,
we obtain a filtered sheaf $\mathcal{P}^{\prime}_{\ast}\mathcal{I}$ over
$\mathcal{I}$ induced by $\mathcal{P}_{\ast}\mathcal{K}$. Then, we obtain

$\mu_{0}=\mu_{L}(\mathcal{P}_{\ast}\mathcal{K})=\mu_{L}(\mathcal{P}^{\prime}_{\ast}\mathcal{I})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{I})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{(2)})=\mu_{0}.$

Because
$(\mathcal{P}_{\ast}\mathcal{V}^{(2)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(2)}})$
is $\mu_{L}$-stable and because $\mathcal{I}\neq 0$, we obtain that
$\mathop{\rm rank}\nolimits(\mathcal{I})=\mathop{\rm
rank}\nolimits\mathcal{V}^{(2)}$, i.e., $\mathcal{I}$ and $\mathcal{V}^{(2)}$
are generically isomorphic. Because
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{I})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{(2)})$,
Lemma 2.5 implies that
$\mathcal{P}_{\ast}\mathcal{I}\longrightarrow\mathcal{P}_{\ast}\mathcal{V}^{(2)}$
is an isomorphism in codimension $1$. Hence, there exists a closed algebraic
subset $Z\subset X$ such that (i) the codimension of $Z$ is larger than $2$,
(ii) $\mathcal{V}^{(2)}_{|X\setminus Z}\subset\mathcal{V}^{(1)}_{|X\setminus
Z}$. Because $\mathcal{V}^{(1)}$ is reflexive we obtain that
$\mathcal{V}^{(2)}\subset\mathcal{V}^{(1)}$.

Let us study the case where $\mathcal{V}^{(1)}\cap\mathcal{V}^{(2)}=0$. Let
$\mathcal{V}^{(3)}$ denote the saturated $\lambda$-flat subsheaf of
$\mathcal{V}$ generated by $\mathcal{V}^{(1)}+\mathcal{V}^{(2)}$. Let
$\mathcal{P}_{\ast}\mathcal{V}^{(3)}$ denote the filtered sheaf over
$\mathcal{V}^{(3)}$ induced by $\mathcal{P}_{\ast}\mathcal{V}$.

###### Lemma 3.7

$(\mathcal{P}_{\ast}\mathcal{V}^{(3)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(3)}})$
is $\mu_{L}$-semistable, and the induced morphism
$g:\mathcal{P}_{\ast}\mathcal{V}^{(1)}\oplus\mathcal{P}_{\ast}\mathcal{V}^{(2)}\longrightarrow\mathcal{P}_{\ast}\mathcal{V}^{(3)}$
is an isomorphism in codimension one.

Proof We obtain
$\mu_{0}=\mu_{L}(\mathcal{P}_{\ast}(\mathcal{V}^{(1)}\oplus\mathcal{V}^{(2)}))\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{(3)})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})=\mu_{0}$.
Hence, we obtain that $\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}^{(3)})=\mu_{0}$
and that
$(\mathcal{P}_{\ast}\mathcal{V}^{(3)},\mathbb{D}^{\lambda}_{\mathcal{V}^{(3)}})$
is $\mu_{L}$-semistable. Because
$g:\mathcal{P}_{\ast}\mathcal{V}^{(1)}\oplus\mathcal{P}_{\ast}\mathcal{V}^{(2)}\longrightarrow\mathcal{P}_{\ast}\mathcal{V}^{(3)}$
is generically an isomorphism, and because they have the same slope, $g$ is an
isomorphism in codimension one by Lemma 2.5.

By Lemma 3.7, it is easy to observe that there exists a finite sequence of
reflexive $\lambda$-flat subsheaves $\mathcal{V}^{\prime}_{j}$
$(j=1,\ldots,m)$ such that (i) the induced filtered $\lambda$-flat sheaves
$(\mathcal{P}_{\ast}\mathcal{V}^{\prime}_{j},\mathbb{D}^{\lambda}_{\mathcal{V}_{j}^{\prime}})$
are $\mu_{L}$-stable, (ii) the image of the induced morphism
$g:\widetilde{\mathcal{V}}:=\bigoplus\mathcal{V}^{\prime}_{j}\longrightarrow\mathcal{V}_{1}$
is generically an isomorphism. Because
$\mu_{0}=\mu_{L}\bigl{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}\bigr{)}\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})\leq\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})=\mu_{0}$,
we obtain that
$\mu_{L}\bigl{(}\mathcal{P}_{\ast}\widetilde{\mathcal{V}}\bigr{)}=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{1})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V})$.
Hence, $g$ is an isomorphism in codimension one by Lemma 2.5. Because both
$\mathcal{P}_{\ast}\widetilde{\mathcal{V}}$ and
$\mathcal{P}_{\ast}\mathcal{V}_{1}$ are reflexive, we obtain that
$\mathcal{P}_{\ast}\widetilde{\mathcal{V}}\simeq\mathcal{P}_{\ast}\mathcal{V}_{1}$.
Thus, we obtain Proposition 3.5.

###  3.4 Mehta-Ramanathan type theorems

Let $X$ be a smooth connected $n$-dimensional projective variety with a simple
normal crossing hypersurface $H$. Let $H=\bigcup_{i\in\Lambda}H_{i}$ be the
irreducible decomposition. Let $L$ be an ample line bundle on $X$.

####  3.4.1 Restriction to general curves

Let $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ be a reflexive
filtered $\lambda$-flat sheaf on $(X,H)$. There exists a Zariski closed subset
$W\subset H$ with $\dim W<\dim H$ such that (i) the singular locus of $H$ is
contained in $W$, (ii) $\mathcal{P}_{\ast}\mathcal{V}_{|X\setminus W}$ is a
filtered bundle on $(X\setminus W,H\setminus W)$.

Let $Y$ be a smooth curve in $X$ such that (i) $Y\cap W=\emptyset$, (ii) $Y$
intersects with the smooth part of $H$ transversally. Set $H_{Y}:=H\cap Y$. We
obtain a locally free $\mathcal{O}_{Y}(\ast H_{Y})$-module $\mathcal{V}_{|Y}$.
It is equipped with the induced flat $\lambda$-connection
$\mathbb{D}^{\lambda}_{|Y}$. Let ${\boldsymbol{b}}\in{\mathbb{R}}^{H_{Y}}$.
For any $P\in H_{Y}$, there exists $i\in\Lambda$ such that $P\in H_{i}$. For
any $b\in{\mathbb{R}}$, we choose
${\boldsymbol{a}}(P,b)\in{\mathbb{R}}^{\Lambda}$ such that $a(P,b)_{i}=b$, and
we obtain an $\mathcal{O}_{Y,P}$-submodule
$\mathcal{P}_{{\boldsymbol{b}}}(\mathcal{V}_{|Y})_{P}:=\mathcal{P}_{{\boldsymbol{a}}(P,b)}(\mathcal{V})_{P}$
of $(\mathcal{V}_{|Y})_{P}$, which is independent of the choice of
${\boldsymbol{a}}(P,b)$ as above. There exists a locally free
$\mathcal{O}_{Y}$-module
$\mathcal{P}_{{\boldsymbol{b}}}(\mathcal{V}_{|Y})\subset\mathcal{V}_{|Y}$
whose stalk at $P$ is $\mathcal{P}_{{\boldsymbol{b}}}(\mathcal{V}_{|Y})_{P}$.
Thus, we obtain a filtered $\lambda$-flat bundle
$(\mathcal{P}_{\ast}(\mathcal{V}_{|Y}),\mathbb{D}^{\lambda}_{|Y})$ which is
denoted by $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}$.

####  3.4.2 The stability condition

###### Proposition 3.8

A reflexive filtered $\lambda$-flat sheaf
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ on $(X,H)$ is
$\mu_{L}$-stable (resp. $\mu_{L}$-semistable) if and only if the following
holds.

  * •

For any $m_{1}>0$, there exists $m>m_{1}$ such that
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}$ is
$\mu_{L}$-stable (resp. $\mu_{L}$-semistable) where $Y$ denotes a generic
$1$-dimensional complete intersection of hypersurfaces of $L^{\otimes m}$.

Proof The case $\lambda=1$ is already studied in [50, §13.2]. The case
$\lambda\neq 0$ is reduced to the case $\lambda=1$. As for the case
$\lambda=0$, we can prove the claim of the proposition by the argument in [46,
§3.4], which closely follows the arguments of Mehta-Ramanathan [43, 44] and
Simpson [63]. We use $\mathcal{W}=\Omega^{1}(ND)$ for a large $N$ instead of
$\Omega^{1}(\log D)$ in [46, §3.4]. (See also [50, §13.2].)

####  3.4.3 Restrictions of morphisms and the polystability condition

Let us give a complement on the restriction of morphisms of reflexive filtered
$\lambda$-flat sheaves to generic complete intersection curves, which is a
variant of [63, Lemma 3.9]. Let
$(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda})$ $(i=1,2)$ be
reflexive filtered $\lambda$-flat sheaves on $(X,H)$. Let $\mathop{\rm
Hom}\nolimits\bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}),(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})\bigr{)}$
denote the vector space of morphisms of filtered $\lambda$-flat sheaves
$(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1})\longrightarrow(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})$.
We shall prove a refined claim (Proposition 3.16) of the following proposition
in §3.4.4–3.4.6.

###### Proposition 3.9

There exists a positive integer $m_{0}>0$ such that the restriction

$\mathop{\rm
Hom}\nolimits\bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}),(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})\bigr{)}\longrightarrow\mathop{\rm
Hom}\nolimits\bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1})_{|Y},(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})_{|Y}\bigr{)}$

is an isomorphism for a generic $1$-dimensional complete intersection $Y$ of
hypersurfaces of $L^{\otimes m}$ $(m\geq m_{0})$.

Before the proof of Proposition 3.9, we state a variant of Proposition 3.8 on
the $\mu_{L}$-polystability condition.

###### Corollary 3.10

A reflexive filtered $\lambda$-flat sheaf
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ on $(X,H)$ is
$\mu_{L}$-polystable if and only if the following holds.

  * •

For any $m_{1}>0$, there exists $m>m_{1}$ such that
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}$ is
$\mu_{L}$-polystable where $Y$ denotes the $1$-dimensional complete
intersection of generic hypersurfaces of $L^{\otimes m}$.

Proof If $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is
$\mu_{L}$-polystable, we obtain a decomposition
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})=\bigoplus(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i})$
into $\mu_{L}$-stable filtered $\lambda$-flat sheaves. Applying Proposition
3.8 to each stable component, we obtain the “only if” claim.

Let $m_{1}$ be an integer larger than $m_{0}$ in Proposition 3.9 for
$\mathop{\rm
Hom}\nolimits\bigl{(}(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda}),(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})\bigr{)}$.
Suppose that there exists $m>m_{1}$ such that
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}$ is
$\mu_{L}$-polystable for a generic $1$-dimensional complete intersection $Y$
of hypersurfaces of $L^{\otimes m}$. We obtain the decomposition

$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}=\bigoplus_{i=1}^{\ell}(\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i})$
(19)

into stable filtered Higgs bundles. Let $\pi_{Y,i}$ denote the endomorphisms
of $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}$ obtained as the
projection
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}\longrightarrow(\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i})$
with respect to the decomposition (19), and the inclusion
$(\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i})\longrightarrow(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|Y}$.
Note that they satisfy $\pi_{Y,i}\circ\pi_{Y,i}=\pi_{Y,i}$,
$\pi_{Y,i}\circ\pi_{Y,j}=0$ $(i\neq j)$ and $\sum\pi_{Y,i}=\mathop{\rm
id}\nolimits$. By Proposition 3.9, there uniquely exist the endomorphisms
$\pi_{i}$ of $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ such that
$\pi_{i|Y}=\pi_{Y,i}$. By Proposition 3.9 again, they satisfy
$\pi_{i}\circ\pi_{i}=\pi_{i}$, $\pi_{i}\circ\pi_{j}=0$ $(i\neq j)$ and
$\sum\pi_{i}=\mathop{\rm id}\nolimits$. Let
$\mathcal{V}_{i}\subset\mathcal{V}$ denote the image of $\pi_{i}$. We define
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{i}=\mathcal{V}_{i}\cap\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}$
for any ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$. Because $\pi_{i}$ are
compatible with $\mathbb{D}^{\lambda}$ and the filtration
$\mathcal{P}_{\ast}\mathcal{V}$, we obtain the decomposition
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})=\bigoplus(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i})$.
By the construction,
$(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i})_{|Y}=(\mathcal{P}_{\ast}\mathcal{V}_{Y,i},\mathbb{D}^{\lambda}_{Y,i})$
are stable. Hence,
$(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i})$ are
$\mu_{L}$-stable with
$\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{i},\mathbb{D}^{\lambda}_{i})=\mu_{L}(\mathcal{P}_{\ast}\mathcal{V}_{j},\mathbb{D}^{\lambda}_{j})$
$(i\neq j)$, i.e., $(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ is
$\mu_{L}$-polystable.

####  3.4.4 General Enriques-Severi lemma due to Mehta-Ramanathan

To prove Proposition 3.9, we recall the general Enriques-Severi lemma in [43].
Recall $n=\dim X$. For a positive integer $m$, let $S_{m}$ denote the
projective space of lines in $H^{0}(X,L^{\otimes m})$. For sequences
${\boldsymbol{m}}=(m_{1},\ldots,m_{t})\in{\mathbb{Z}}_{>0}^{t}$ with $t\leq
n-1$, we set $S_{{\boldsymbol{m}}}:=\prod S_{m_{i}}$. There exists the
correspondence variety $Z_{{\boldsymbol{m}}}\subset X\times
S_{{\boldsymbol{m}}}$, i.e.,
$Z_{{\boldsymbol{m}}}=\bigl{\\{}(x,s_{1},\ldots,s_{t})\in X\times
S_{{\boldsymbol{m}}}\,\big{|}\,s_{i}(x)=0,\,\,1\leq i\leq t\bigr{\\}}$. For
any $s\in S_{{\boldsymbol{m}}}$, we set
$X_{s}:=Z_{{\boldsymbol{m}}}\times_{S_{{\boldsymbol{m}}}}\times\\{s\\}$.

Let $F$ be a coherent reflexive $\mathcal{O}_{X}$-module on $X$. For any
${\boldsymbol{m}}=(m_{1},\ldots,m_{t})\in{\mathbb{Z}}_{>0}^{t}$ with $t\leq
n-1$, and for any $s\in S_{{\boldsymbol{m}}}$, we set
$F_{s}:=F\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X_{s}}$. For any integer $m$,
let $F_{s}(m)=F_{s}\otimes_{\mathcal{O}_{X}}L^{\otimes m}$.

According to [43, Proposition 1.5]. there exists a non-empty Zariski open
subset $\widetilde{S}_{{\boldsymbol{m}}}\subset S_{{\boldsymbol{m}}}$ such
that the following holds.

  * •

$\widetilde{S}_{{\boldsymbol{m}}}\times_{S_{{\boldsymbol{m}}}}Z_{{\boldsymbol{m}}}\longrightarrow\widetilde{S}_{{\boldsymbol{m}}}$
is smooth.

  * •

For any $s\in S_{{\boldsymbol{m}}}$, $F_{s}$ is a reflexive
$\mathcal{O}_{X_{s}}$-module.

In the proof of [43, Proposition 3.2], the following proposition is proved.

###### Proposition 3.11

Let $t\leq n-2$. There exists a positive integer $m_{0}$ depending only on $F$
such that the following holds.

  * •

For any ${\boldsymbol{m}}=(m_{1},\ldots,m_{t})\in{\mathbb{Z}}_{>0}^{t}$ with
$m_{i}\geq m_{0}$, there exists a non-empty Zariski open subset
$U\subset\widetilde{S}_{{\boldsymbol{m}}}$ such that
$H^{1}(X_{s},F_{s}(-\ell))=0$ for any $s\in U$ and any $\ell\geq m_{0}$.

###### Corollary 3.12

Let $t\leq n-1$. There exists a positive integer $m_{0}$ depending only on $F$
such that the following holds.

  * •

For any ${\boldsymbol{m}}=(m_{1},\ldots,m_{t})\in{\mathbb{Z}}_{>0}^{t}$ with
$m_{i}\geq m_{0}$, there exists a non-empty Zariski open subset
$U\subset\widetilde{S}_{{\boldsymbol{m}}}$ such that
$H^{0}(X_{s},F_{s}(-\ell))=0$ for any $s\in U$ and any $\ell\geq m_{0}$.

Proof Let $m_{0}$ be a positive integer as in Proposition 3.11. We also assume
that $H^{0}(X,F(-\ell))=0$ for any $\ell\geq m_{0}$. We use an induction on
$t$. For ${\boldsymbol{m}}=(m_{1},\ldots,m_{t})$ with $m_{i}\geq m_{0}$, we
set ${\boldsymbol{m}}^{\prime}=(m_{1},\ldots,m_{t-1})$. By the assumption of
the induction and Proposition 3.11, there exists a non-empty Zariski open
subset $U_{1}^{\prime}\subset\widetilde{S}_{{\boldsymbol{m}}^{\prime}}$ such
that $H^{i}(X_{s^{\prime}},F_{s^{\prime}}(-\ell))=0$ $(i=0,1)$ for any
$s^{\prime}\in U_{1}^{\prime}$ and any $\ell\geq m_{0}$.

For any $s\in S_{{\boldsymbol{m}}}$, let $s^{\prime}$ denote the image $s$ in
$S_{{\boldsymbol{m}}^{\prime}}$ by the projection
$S_{{\boldsymbol{m}}}\longrightarrow S_{{\boldsymbol{m}}^{\prime}}$. There
exists the exact sequence

$0\longrightarrow\mathcal{O}_{X_{s^{\prime}}}(-m_{t})\longrightarrow\mathcal{O}_{X_{s^{\prime}}}\longrightarrow\mathcal{O}_{X_{s}}\longrightarrow
0.$ (20)

By [43, Proposition 1.5], there exists a Zariski open subset
$U_{1}\subset\widetilde{S}_{{\boldsymbol{m}}}$ such that if $s\in U_{1}$ then
we obtain the following exact sequence from (20) by taking the tensor product
with $F$:

$0\longrightarrow F_{s^{\prime}}(-m_{t})\longrightarrow
F_{s^{\prime}}\longrightarrow F_{s}\longrightarrow 0.$

We shrink $U_{1}$ so that $U_{1}^{\prime}$ contains the image of $U_{1}$ by
the projection $S_{{\boldsymbol{m}}}\longrightarrow
S_{{\boldsymbol{m}}^{\prime}}$. Let $\ell\geq m_{0}$. For any $s\in U_{1}$, we
obtain $H^{0}(X_{s^{\prime}},F_{s^{\prime}}(-\ell))=0$ and
$H^{1}(X_{s^{\prime}},F_{s^{\prime}}(-\ell-m_{t}))=0$ because $s^{\prime}\in
U_{1}^{\prime}$. Hence, we obtain $H^{0}(X_{s},F_{s}(-\ell))=0$.

###### Corollary 3.13

Let $t\leq n-1$. There exists $m_{0}$ depending only on $F$ such that the
following holds.

  * •

For any ${\boldsymbol{m}}=(m_{1},\ldots,m_{t})\in{\mathbb{Z}}_{>0}^{t}$ with
$m_{i}\geq m_{0}$, there exists a non-empty Zariski open subset
$U\subset\widetilde{S}_{{\boldsymbol{m}}}$ such that the natural morphism
$H^{0}(X,F)\longrightarrow H^{0}(X_{s},F_{s})$ is an isomorphism for any $s\in
U$.

Proof It is enough to apply the argument in the first paragraph of the proof
of [43, Proposition 3.2] with Proposition 3.11 and Corollary 3.12. (This is
essentially pointed out in [63, Lemma 3.9].)

Let $T^{\ast}_{X_{s}}X$ denote the conormal bundle of $X_{s}$ in $X$ for any
$s\in\widetilde{S}_{{\boldsymbol{m}}}$.

###### Corollary 3.14

Let $t\leq n-1$. There exists $m_{0}$ depending only on $F$ such that the
following holds.

  * •

For any ${\boldsymbol{m}}=(m_{1},\ldots,m_{t})\in{\mathbb{Z}}_{>0}^{t}$ with
$m_{i}\geq m_{0}$, there exists a non-empty Zariski open subset
$U\subset\widetilde{S}_{{\boldsymbol{m}}}$ such that
$H^{0}(X_{s},T^{\ast}_{X_{s}}X\otimes F_{s})=0$ for any $s\in U$.

Proof Because
$T^{\ast}_{X_{s}}X\simeq\bigoplus_{j=1}^{t}\mathcal{O}_{X_{s}}(-m_{j})$, the
claim follows from Corollary 3.12.

####  3.4.5 Flat sections of reflexive filtered $\lambda$-flat sheaves

Let $H$ be a simple normal crossing hypersurface of $X$ with the irreducible
decomposition $H=\bigcup_{i\in\Lambda}H_{i}$. Let
$(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})$ be a reflexive filtered
$\lambda$-flat sheaf on $(X,H)$. Note that there exists a Zariski closed
subset $W\subset H$ with $\dim W<\dim H$ such that (i) $W$ contains the
singular locus of $H$, (ii) $\mathcal{P}_{\ast}\mathcal{V}_{|X\setminus W}$ is
a filtered bundle on $(X\setminus W,H\setminus W)$. For any
${\boldsymbol{m}}\in{\mathbb{Z}}^{n-1}$ and for any $s\in
S_{{\boldsymbol{m}}}$, we set $H_{s}:=X_{s}\times_{X}H$.

According to [43, Proposition 1.5], there exists a non-empty Zariski open
subset $S^{\circ}_{{\boldsymbol{m}}}\subset S_{{\boldsymbol{m}}}$ such that
the following holds.

  * •

$Z_{{\boldsymbol{m}}}\times_{S_{{\boldsymbol{m}}}}S^{\circ}_{{\boldsymbol{m}}}\longrightarrow
S^{\circ}_{{\boldsymbol{m}}}$ is smooth.

  * •

For any $s\in S^{\circ}_{{\boldsymbol{m}}}$, $X_{s}$ intersects with $H$ in
$H\setminus W$ transversally. Moreover,
$\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{s}:=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}_{|X_{s}}$
$({\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda})$ are locally free
$\mathcal{O}_{X_{s}}$-modules.

There exists a non-negative integer $N$ such that $\mathbb{D}^{\lambda}$
induces a morphism of sheaves
$\mathbb{D}^{\lambda}:\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\longrightarrow\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\otimes\Omega^{1}_{X}(\log
H)\otimes\mathcal{O}_{X}(NH)$. For $j=0,1,\ldots,n$, we set

$\mathcal{C}^{j}_{N}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})=\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\otimes\Omega^{j}_{X}(\log
H)\otimes\mathcal{O}_{X}(jNH).$

The flat $\lambda$-connection $\mathbb{D}^{\lambda}$ induces
$\mathbb{D}^{\lambda}:\mathcal{C}^{j}_{N}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})\longrightarrow\mathcal{C}^{j+1}_{N}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})$
such that $\mathbb{D}^{\lambda}\wedge\mathbb{D}^{\lambda}=0$. Thus, we obtain
a complex of sheaves
$\mathcal{C}^{\bullet}_{N}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V},\mathbb{D}^{\lambda})$
on $X$. Clearly, the following holds:

$\mathbb{H}^{0}\bigl{(}X,\mathcal{C}^{\bullet}_{N}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V},\mathbb{D}^{\lambda})\bigr{)}=\mathop{\rm
Ker}\nolimits\Bigl{(}H^{0}(X,\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})\stackrel{{\scriptstyle\mathbb{D}^{\lambda}}}{{\longrightarrow}}H^{0}\bigl{(}X,\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\otimes\Omega_{X}^{1}(\log
H)\otimes\mathcal{O}_{X}(NH)\bigr{)}\Bigr{)}.$

For any $s\in S^{\circ}_{{\boldsymbol{m}}}$, we obtain the filtered
$\lambda$-flat bundle
$(\mathcal{P}_{\ast}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s}):=(\mathcal{P}_{\ast}\mathcal{V},\mathbb{D}^{\lambda})_{|X_{s}}$.
Let ${\boldsymbol{a}}(s)$ denote the image of
${\mathbb{R}}^{\Lambda}\longrightarrow{\mathbb{R}}^{H_{s}}$ induced by the
natural map $H_{s}\longrightarrow\Lambda$. Let $\iota_{s}:X_{s}\longrightarrow
X$ denote the inclusion. We obtain the natural morphism of complexes of
sheaves
$\mathcal{C}_{N}^{\bullet}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V},\mathbb{D}^{\lambda})\longrightarrow\iota_{s\ast}\mathcal{C}_{N}^{\bullet}(\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s})$,
which induces

$\mathbb{H}^{0}(X,\mathcal{C}^{\bullet}_{N}(\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V},\mathbb{D}^{\lambda}))\longrightarrow\mathbb{H}^{0}(X_{s},\mathcal{C}^{\bullet}_{N}(\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s},\mathbb{D}^{\lambda}_{s})).$
(21)

The following proposition is essentially [63, Lemma 3.9].

###### Proposition 3.15

There exists a positive integer $m_{0}>0$ such that the following claim holds
for any ${\boldsymbol{m}}=(m_{1},\ldots,m_{n-1})$ with $m_{i}\geq m_{0}$ and a
non-empty Zariski open subset $U\subset S^{\circ}_{{\boldsymbol{m}}}$.

  * •

For any $s\in U$, the natural morphism (21) is an isomorphism.

Proof According to Corollary 3.13, if $m_{0}$ is sufficiently large, there
exists a non-empty Zariski open subset $U_{1}\subset
S^{\circ}_{{\boldsymbol{m}}}$ such that the following natural morphisms are
isomorphisms for any $s\in U$:

$H^{0}(X,\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V})\longrightarrow
H^{0}(X_{s},\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}),$ (22)
$H^{0}\bigl{(}X,\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\otimes_{\mathcal{O}_{X}}\Omega^{1}_{X}(\log
H)\otimes\mathcal{O}_{X}(NH)\bigr{)}\longrightarrow
H^{0}\Bigl{(}X_{s},\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\otimes_{\mathcal{O}_{X_{s}}}\bigl{(}\Omega^{1}_{X}(\log
H)\otimes\mathcal{O}_{X_{s}}(NH_{s})\bigr{)}\Bigr{)}.$ (23)

There exists the following exact sequence:

$0\longrightarrow
T_{X_{s}}^{\ast}X\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\otimes\mathcal{O}_{X_{s}}(NH_{s})\longrightarrow\bigl{(}\Omega^{1}_{X}(\log
H)\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X_{s}}(NH_{s})\bigr{)}\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\longrightarrow\\\
\Omega^{1}_{X_{s}}(\log
H_{s})\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\otimes\mathcal{O}_{X_{s}}(NH_{s})\longrightarrow
0.$ (24)

According to Corollary 3.14, if $m_{0}$ is sufficiently large, there exists a
non-empty Zariski open subset $U_{2}\subset U_{1}$ such that the following
holds for any $s\in U_{2}$:

$H^{0}\bigl{(}X_{s},T_{X_{s}}^{\ast}X\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\otimes\mathcal{O}_{X_{s}}(NH_{s})\bigr{)}=0.$

Hence, the natural morphism

$H^{0}\bigl{(}X_{s},(\Omega^{1}_{X}(\log
H)\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X_{s}}(NH_{s}))\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\bigr{)}\longrightarrow\\\
H^{0}\bigl{(}X_{s},\Omega^{1}_{X_{s}}(\log
H_{s})\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\otimes\mathcal{O}_{X_{s}}(NH_{s})\bigr{)}$
(25)

is injective for any $s\in U_{2}$. We obtain the injectivity of the following
natural morphism for any $s\in U_{2}$:

$H^{0}\bigl{(}X,\Omega^{1}_{X}(\log
H)\otimes\mathcal{P}_{{\boldsymbol{a}}}\mathcal{V}\otimes\mathcal{O}_{X}(NH)\bigr{)}\longrightarrow
H^{0}\bigl{(}X_{s},\Omega^{1}_{X_{s}}(\log
H_{s})\otimes\mathcal{P}_{{\boldsymbol{a}}(s)}\mathcal{V}_{s}\otimes\mathcal{O}_{X_{s}}(NH_{s})\bigr{)}.$
(26)

Then, we obtain the claim of the proposition.

####  3.4.6 Morphisms of reflexive filtered $\lambda$-flat sheaves

Let $\mathcal{P}_{\ast}\mathcal{V}_{i}$ $(i=1,2)$ be reflexive filtered
sheaves with meromorphic flat $\lambda$-connection $\mathbb{D}^{\lambda}_{i}$
on $(X,H)$. Let $\mathop{\rm
Hom}\nolimits\bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}),(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})\bigr{)}$
denote the vector space of morphisms of filtered $\lambda$-flat sheaves
$(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1})\longrightarrow(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})$.

###### Proposition 3.16

There exists a positive integer $m_{0}>0$ such that the following claim holds
for any ${\boldsymbol{m}}=(m_{1},\ldots,m_{n-1})$ with $m_{i}\geq m_{0}$ and
for a non-empty Zariski open subset $U\subset S_{{\boldsymbol{m}}}$.

  * •

For any $s\in U$, let
$(\mathcal{P}_{\ast}\mathcal{V}_{i,s},\mathbb{D}^{\lambda}_{i,s})$ denote the
induced filtered $\lambda$-flat bundles on $(X_{s},H_{s})$. Then, the natural
morphism

$\mathop{\rm
Hom}\nolimits\Bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}),(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})\Bigr{)}\longrightarrow\mathop{\rm
Hom}\nolimits\Bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1,s},\mathbb{D}^{\lambda}_{1,s}),(\mathcal{P}_{\ast}\mathcal{V}_{2,s},\mathbb{D}^{\lambda}_{2,s})\Bigr{)}$

is an isomorphism.

Proof For ${\boldsymbol{a}}\in{\mathbb{R}}^{\Lambda}$, let
$\mathcal{P}_{{\boldsymbol{a}}}{\mathcal{H}om}(\mathcal{V}_{1},\mathcal{V}_{2})$
be the subsheaf of the $\mathcal{O}_{X}(\ast H)$-module
${\mathcal{H}om}_{\mathcal{O}_{X}(\ast H)}(\mathcal{V}_{1},\mathcal{V}_{2})$
determined as follows for any open subset $U\subset X$:

$H^{0}\Bigl{(}U,\mathcal{P}_{{\boldsymbol{a}}}({\mathcal{H}om}(\mathcal{V}_{1},\mathcal{V}_{2}))\Bigr{)}=\\\
\Bigl{\\{}f\in
H^{0}(U,{\mathcal{H}om}(\mathcal{V}_{1},\mathcal{V}_{2}))\,\Big{|}\,f(\mathcal{P}_{{\boldsymbol{b}}}\mathcal{V}_{1|U})\subset\mathcal{P}_{{\boldsymbol{a}}+{\boldsymbol{b}}}(\mathcal{V}_{2|U})\,\,\forall{\boldsymbol{b}}\in{\mathbb{R}}^{\Lambda}\Bigr{\\}}.$

It is easy to see that
$\mathcal{P}_{{\boldsymbol{a}}}{\mathcal{H}om}(\mathcal{V}_{1},\mathcal{V}_{2})$
are reflexive $\mathcal{O}_{X}$-modules. Thus, we obtain a reflexive filtered
sheaf $\mathcal{P}_{\ast}{\mathcal{H}om}(\mathcal{V}_{1},\mathcal{V}_{2})$
with the induced flat $\lambda$-connection $\widetilde{\mathbb{D}}^{\lambda}$.
We can easily observe that

$\mathop{\rm
Hom}\nolimits\Bigl{(}(\mathcal{P}_{\ast}\mathcal{V}_{1},\mathbb{D}^{\lambda}_{1}),(\mathcal{P}_{\ast}\mathcal{V}_{2},\mathbb{D}^{\lambda}_{2})\Bigr{)}=\mathbb{H}^{0}\Bigl{(}X,\mathcal{C}^{\bullet}_{N}\bigl{(}\mathcal{P}_{{\boldsymbol{0}}}{\mathcal{H}om}(\mathcal{V}_{1},\mathcal{V}_{2}),\widetilde{\mathbb{D}}^{\lambda}\bigr{)}\Bigr{)}$

for any large $N$, where