Good Wild Harmonic Bundles and Good Filtered Higgs Bundles

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.


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 Higgs bundles. • Good wild harmonic bundles on (X, H).
We shall recall the precise definitions of the objects in §2.
In [47], we have already proved that good wild harmonic bundles on (X, H) induce µ L -polystable good filtered Higgs bundles satisfying the vanishing condition. Indeed, more generally, for any complex number λ, good wild harmonic bundles induce µ L -polystable good filtered λ-flat bundles satisfying a similar vanishing condition. Note that 0-flat bundles are equivalent to Higgs bundles, and 1-flat bundles are equivalent to flat bundles in the ordinary sense. Moreover, we studied an analogue of Theorem 1.1 in the case λ = 1, i.e., the correspondence between good wild harmonic bundles and µ L -polystable good filtered flat bundles satisfying a similar vanishing condition [47,Theorem 16.1]. It was applied to the study of the correspondence between semisimple algebraic holonomic D-modules and pure twistor D-modules.
There is no new essential difficulty to prove Theorem 1.1 after our studies [43,44,45,47] on the basis of [57,58]. 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 of the correspondence.

Kobayashi-Hitchin correspondence for vector bundles
We briefly recall a part of the history of this type of correspondences. (See also [24,32,38].) For a holomorphic vector bundle E on a compact Riemann surface C, we set µ(E) := deg(E)/ rank(E), which is called the slope of E. A holomorphic bundle E is called stable (resp. semistable) if µ(E ′ ) < µ(E) (resp. µ(E ′ ) ≤ µ(E)) holds for any holomorphic subbundle E ′ ⊂ E such that 0 < rank(E ′ ) < rank(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 [52] for the construction of the moduli spaces of vector bundles with reasonable properties. Narasimhan and Seshadri [53] established the equivalence between unitary flat bundles and polystable bundles of degree 0 on compact Riemann surfaces.
Let (X, ω) be a compact connected Kähler manifold. For any torsion-free O X -module F , the slope of F with respect to ω is defined as µ ω (F ) := X c 1 (F )ω dim X−1 rank F .
If the cohomology class of ω is the first Chern class of an ample line bundle L, then µ ω (F ) is also denoted by µ L (F ). Then, a torsion-free O X -module F is called µ ω -stable if µ ω (F ′ ) < µ ω (F ) holds for any saturated subsheaf F ′ ⊂ F such that 0 < rank(F ′ ) < rank(F ). This condition was first studied by Takemoto [64,65]. It is also called µ ω -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 2 1 /2r < 0 for any unstable bundle E of rank r in his sense. Gieseker [18] proved the inequality for slope semistable bundles. The inequality is called Bogomolov-Gieseker inequality.
Inspired by these works, Kobayashi [29] introduced the concept of Hermitian-Einstein condition for metrics of holomorphic vector bundles. Let (E, ∂ E ) be a holomorphic vector bundle on a Kähler manifold (X, ω). Let h be a Hermitian metric of E. Let R(h) denote the curvature of the Chern connection ∇ h = ∂ E + ∂ E,h , associated to h and ∂ E . Then, h is called Hermitian-Einstein if ΛR(h) ⊥ = 0, where R(h) ⊥ denote the trace-free part of R(h). In [29], 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 [30,31] and Lübke [37] 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 [36] 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 Tr (R(h) ⊥ ) 2 ω dim X−2 ≥ 0 in the form level. It particularly implies the Bogomolov-Gieseker inequality for holomorphic vector bundles (E, ∂ E ) with a Hermitian-Einstein metric h on compact Kähler manifolds (X, ω). Moreover, if c 1 (E) = 0 and X ch 2 (E)ω dim X−2 = 0 are satisfied for such (E, ∂ E , h), and if we impose det(h) is flat, then the Kobayashi-Lübke inequality implies that R(h) = 0, i.e., ∇ h is flat.
Independently, in [33], Hitchin proposed a problem to ask an equivalence of the stability condition and the existence of a metric h such that ΛR(h) = 0, under the vanishing of the first Chern class of the bundle. (See [24] for more precise.) 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 [13], Donaldson introduced the method of global analysis to reprove the theorem of Narasimhan-Seshadri. In [14], 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 [14].
Eventually, Donaldson [15] and Uhlenbeck-Yau [66] 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 µ ω (E) = 0 and X ch 2 (E)ω dim X−2 = 0. Note that Mehta and Ramanathan [40,41] deduced the equivalence on complex projective manifolds directly from the equivalence in the surface case due to Donaldson [14].

Higgs 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, ∂ E ) be a holomorphic vector bundle on a compact Riemann surface C. A Higgs field of (E, ∂ E ) is a holomorphic section θ of End(E) ⊗ Ω 1 C . Let h be a Hermitian metric of E. We obtain the Chern connection ∂ E + ∂ E,h and its curvature R(h). Let θ † h denote the adjoint of θ. In [23], Hitchin introduced the following equation, called the Hitchin equation: Such (E, ∂ E , θ, h) is called a harmonic bundle. He particularly studied the case rank E = 2. Among many deep results in [23], he proved that a Higgs bundle (E, ∂ E , θ) has a Hermitian metric h satisfying (1) if and only if it is polystable of degree 0. Here, a Higgs bundle (E, ∂ E , θ) is called stable (resp. semistable) if µ(E ′ ) < µ(E) (resp. µ(E ′ ) ≤ µ ′ E) holds for any holomorphic subbundle E ′ ⊂ E such that θ(E ′ ) ⊂ E ′ ⊗ Ω 1 C and that 0 < rank(E ′ ) < rank(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 [16] 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 bundle are isomorphic. Together with another equivalence due to Donaldson [16] between irreducible flat bundles and twisted harmonic maps, Hitchin's work showed that the moduli spaces of Higgs bundles and flat bundles have extremely rich structures.
The higher dimensional case was studied by Simpson [57]. 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, ∂ E ) on a complex manifold X with arbitrary dimension, a Higgs field θ is defined to be a holomorphic section of End(E) ⊗ Ω 1 X satisfying the additional condition θ ∧ θ = 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 ∇ h + θ + θ † h . A Hermitian metric h of a Higgs bundle (E, ∂ E , θ) is called Hermitian-Einstein if ΛF (h) ⊥ = 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, ∂ E , θ) on a compact Kähler manifold (X, ω) 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 Tr (F (h) ⊥ ) 2 ω dim X−2 ≥ 0 in the form level for any Hermitian-Einstein metric h of (E, ∂ E , θ). Here, the condition θ ∧ θ = 0 is essential. It particularly implies that if (E, ∂ E , θ) on a compact Kähler manifold (X, ω) satisfies µ ω (E) = 0 and X ch 2 (E)ω dim X−2 = 0, then a Hermitian-Einstein metric h of (E, ∂ E , θ) is a pluri-harmonic metric, i.e., i.e., the connection ∇ h + θ + θ † h is flat. It is equivalent to the following: A Higgs bundle (E, ∂ E , θ) with a pluri-harmonic metric is called a harmonic bundle. The equivalence and another equivalence due to Corlette [10] induces an equivalence between semisimple flat bundles and polystable Higgs bundles (E, ∂ E , θ) satisfying µ ω (E) = 0 and X ch 2 (E)ω dim X−2 = 0 on any connected compact Kähler manifold. This correspondence is not only really interesting, but also a starting point of the further investigations. Simpson pursuit the comparison of flat bundles and Higgs bundles in deeper levels [59], and developed the non-abelian Hodge theory [61].

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 [42] 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 ⊂ C. Let E be a holomorphic vector bundle on C. A parabolic structure of E at D is a tuple of filtrations We set µ(E, F ) := deg(E, F )/ rank(E). For any subbundle . Semistability and polystability conditions are also defined naturally. Then, Mehta and Seshadri proved an equivalence of unitary flat bundles on C \ D and parabolic vector bundles (E, F ) with µ(E, F ) = 0 on (C, D). For some purposes, it is more convenient to replace parabolic bundles with filtered bundles introduced by Simpson [57,58]. Let V be a locally free O C ( * D)-module. A filtered bundle P * V over V is a tuple of lattices P a V (a = (a P ) P ∈D ∈ R D ) such that (i) P a V( * D) = V, (ii) the restriction of P a V to a neighbourhood of P ∈ D depends only on a P , (iii) P a+n V = P a V( n P P ) for any a ∈ R D and n ∈ Z D , (iv) for any a ∈ R D , there exists ǫ ∈ R D >0 such that P a V = P a+ǫ V. Let 0 denote (0, . . . , 0) ∈ R D . Then, P 0 V is equipped with the parabolic structure F induced by the images of P a V |P −→ P 0 V |P . It is easy to observe that filtered bundles are equivalent to parabolic bundles. We set µ(P * V) := µ(P 0 V, F ) for filtered bundles P * V.
Simpson [57,58] generalized the theorem of Mehta-Seshadri to the correspondences of tame harmonic bundles, regular filtered Higgs bundles and regular filtered Higgs bundles on compact Riemann surfaces. A harmonic bundle (E, ∂ E , θ, h) on C \ D is called tame on (C, D) if the closure of the spectral curve of θ in T * C(log D) is proper over C. A regular filtered Higgs bundle consists of a filtered bundle P * V equipped with a Higgs field θ : Similarly, a regular filtered flat bundle consists of a filtered bundle P * V equipped with a connection ∇ : for any a ∈ 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), polystable regular filtered Higgs bundles (P * V, θ) satisfying µ(P * V) = 0, and polystable regular filtered flat bundles (P * V, θ) satisfying µ(P * 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 Higgs bundles and regular filtered 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, D), µ L -polystable regular filtered Higgs bundles (P * V, θ) on (X, D) satisfying X par-c 1 (P * V)c 1 (L) dim X−1 = 0 and X par-ch 2 (P * V)c 1 (L) dim X−2 = 0, and µ L -polystable regular filtered flat bundles (P * V, θ) on (X, D) satisfying a similar vanishing condition. In [2], Biquard studied the case where D is smooth. In [34,35,63], Li, Narasimhan, Steer and Wren studied the correspondence for parabolic bundles without Higgs field nor flat connection. In [27], Jost and Zuo studied the correspondence between semisimple flat bundles and tame harmonic bundles. Eventually, in [43,44,45], the author obtained the satisfactory equivalences for tame harmonic bundles. Note that Donagi and Pantev proposed an attractive application of the Kobayashi-Hitchin correspondence for tame harmonic bundles to the study of geometric Langlands theory [12].
In another natural direction of generalization, we should consider more singular objects than regular filtered Higgs or flat bundles. A harmonic bundle (E, ∂ E , θ) on X \ D is called wild if the closure of the spectral variety of θ in the projective completion of T * X is complex analytic. For the analysis, we should impose that the spectral variety of harmonic bundles satisfy some non-degeneracy condition along D. (See §2.6.2.) This is not essential because the condition is always satisfied once we replace X by its appropriate blow up. The notion of regular filtered Higgs (resp. flat) bundle is appropriately generalized to the notion of good filtered Higgs (resp. flat) bundle. The results of Simpson should be generalized to equivalences of good wild harmonic bundles, µ L -polystable good filtered Higgs bundles (P * V, θ) satisfying X par-c 1 (P * V)c 1 (L) dim X−1 = 0 and X par-c 2 (P * V)c 2 (L) dim X−2 = 0, and µ L -polystable good filtered flat bundles satisfying a similar vanishing condition. Sabbah [54] 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, in [47], the author studied the wild harmonic bundles on any dimensional varieties. We obtained that good wild harmonic bundles induce µ L -polystable good filtered Higgs bundles and µ L -polystable good filtered 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 [28] on semisimple algebraic holonomic D-modules. See [49] for more details on this application.
In [47], we did not give a proof of the equivalence for wild harmonic bundles on the Higgs side because it is rather obvious that a similar argument can work even in the Higgs case after [43,44,45,47] on the basis of [57,58]. 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

Variation of Hodge structure
As mentioned, Simpson [57] 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 [57], 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 ∞ -vector bundle V = p+q=w V p,q equipped with a flat connection ∇ satisfying the Griffiths transversality condition, i.e., ∇ 0, where ∇ p,q denote the (p, q)-part of ∇. A polarization of a complex variation of Hodge structure is a flat Hermitian pairing ·, · satisfying the following conditions: (i) the decomposition V = V p,q is orthogonal with respect to ·, · , (ii) ( √ −1) p−q ·, · 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 : X −→ Y. Namely, the family of vector spaces H w (f −1 (y)) (y ∈ Y) naturally induces a flat bundle on 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 polarized variation of Hodge structure. Let (V = V p,q , ∇) be a complex variation of Hodge structure. Note that ∇ 0,1 induces holomorphic structures is a graded holomorphic vector bundle. We also note that ∇ 1,0 induces linear maps V p,q −→ V p−1,q+1 ⊗ Ω 1,0 , and hence θ : V −→ V ⊗ Ω 1,0 . It is easy to check that θ is a Higgs field of (V, ∂ V ). Such a graded holomorphic bundle V = p+q=w V p,q with a Higgs field θ such that θ · V p,q ⊂ V p−1,q+1 ⊗ Ω 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 = V p,q , θ) on a compact Kähler manifold satisfies the stability condition and the vanishing condition, then there exists a flat connection ∇ and a flat Hermitian pairing ·, · such that (i) (V = V p,q , ∇) is a complex variation of Hodge structure which induces the Hodge bundle, (ii) ·, · is a polarization of (V = V p,q , ∇). Indeed, according to the equivalence of Simpson between Higgs bundles and harmonic bundles, there exists a pluri-harmonic metric h of (V, θ). It turns out that the flat connection ∇ h + θ + θ † h satisfies the Griffiths transversality. Moreover, the decomposition V = V p,q is orthogonal with respect to h, and flat Hermitian paring ·, · is constructed by the relation ( √ −1) p−q ·, · V p,q = h |V p,q . Note that a Hodge bundle is regarded as a Higgs bundle (V, ∂ V , θ) with an S 1 -homogeneity, i.e., (V, ∂ V ) is equipped with an S 1 -action such that t • θ • t −1 = t · θ for any t ∈ 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, ∂ E , θ) = (E, ∂ E , tθ).
By the deformation (E, ∂ E , αθ) (α ∈ C * ), any Higgs bundles is deformed to an S 1 -fixed point in the moduli space, i.e., a Hodge bundle as α → 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 flat bundle underlying a polarized variation of Hodge structure.
Simpson [57] 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 [59].

TE-structure
We recall that a complex variation of Hodge structure on X induces a TE-structure in the sense of Hertling [20], i.e., a holomorphic vector bundle V on X := C λ × X with a meromorphic flat connection where X 0 := {0} × X. Indeed, for a complex variation of Hodge structure (V = V p,q , ∇), F p (V ) := p1≥p V p1,q1 are holomorphic subbundles with respect to ∇ 0,1 . Thus, we obtain a decreasing filtration of holomorphic subbundles F p (V ) (p ∈ Z) satisfying the Griffiths transversality Let p : C * λ × X −→ X denote the projection. We obtain the induced flat bundle (p * V, p * ∇). By the Rees construction, p * V is extended to a locally free O X -module V, on which ∇ := p * ∇ is a meromorphic flat connection satisfying the condition ∇V ⊂ V ⊗ O X (X 0 ) ⊗ Ω 1 X (log 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 due to K. Saito [56], the topological field theory of Dubrovin [17], the tt *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, TEstructure is also an important ingredient of semi-infinite variation of Hodge structure [1,9,25], TERP structure [20,21,22], integrable variation of twistor structure [55], etc. (See also [46,48].)

Homogeneous harmonic bundles
As Simpson applied his Kobayashi-Hitchin correspondence to construct complex variation of Hodge structure, we may apply Theorem 1.1 to construct TE-structure 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, ∂ E ) be an S 1 -equivariant holomorphic vector bundle. Let θ be a Higgs field of (E, ∂ E ), which is homogeneous with respect to the S 1 -action, i.e., t * θ = t m θ for some m = 0. Let h be an S 1 -invariant pluri-harmonic metric of (E, ∂ E , θ). Then, as studied in [48, §3], we naturally obtain a TE-structure. More strongly, it is equipped with a grading in the sense of [9,25], and it also underlies a polarized integrable variation of pure twistor structure of weight 0 [55]. Moreover, if there exists an S 1 -equivariant isomorphism between (E, ∂ E , θ, h) and its dual, the TE-structure is enhanced to a semi-infinite variation of Hodge structure with a grading [1,9,25]. 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. If we are given an S 1 -homogeneous good wild harmonic bundle (E, ∂ E , θ, h) on (X, H), as mentioned above, we obtain a TE-structure with a grading on X \ H. Moreover, it is extended to a meromorphic TE-structure on (X, H) as studied in [48, §3]. We obtain the mixed Hodge structure as the limit objects at the boundary, which is useful for the study of more detailed property of the TE-structure.

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 C * -action. We may define a good filtered Higgs bundle (P * V, θ) is called C * -homogeneous if P * V is C * -equivariant and t * θ = t m · θ for some m = 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 8.10) There exists an equivalence between the following objects.
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 µ L -polystable C * -equivariant good filtered Higgs bundle satisfying the vanishing condition. We already applied it to a classification of solutions of the Toda equations on C * [51]. 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 Ghomogeneous good wild harmonic bundles as explained in §8, where G is any compact Lie group.
Acknowledgement I thank Carlos Simpson for his fundamental works on harmonic bundles which are most fundamental 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 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.  Let X denote a complex manifold with a simple normal crossing hypersurface H. Let H = i∈Λ H i denote the irreducible decomposition. For any P ∈ H, a holomorphic coordinate neighbourhood (X P , z 1 , . . . , z n ) around P is called admissible if For such an admissible coordinate neighbourhood, there exists the map ρ P : {1, . . . , ℓ(P )} −→ Λ determined by H ρP (i) ∩ X P = {z i = 0}. We obtain the map κ P : R Λ −→ R ℓ(P ) by κ P (a) = (a ρ(1) , . . . , a ρ(ℓ(P )) ).
Let E be any coherent torsion free O X ( * H)-module. A filtered sheaf over E is defined to be a tuple of coherent O X -submodules P a E ⊂ E (a ∈ R Λ ) satisfying the following conditions.
• P a E( * H) = E for any a ∈ R Λ .
• P a+n E = P a E i∈Λ n i H i for any a ∈ R Λ and n ∈ Z Λ .
• For any a ∈ R Λ there exists ǫ ∈ R Λ >0 such that P a+ǫ E = P a E.
• For any P ∈ H, we take an admissible coordinate neighbourhood (X P , z 1 , . . . , z n ) around P . Then, for any a ∈ R Λ , P a E |XP depends only on κ P (a).
For any coherent O X ( * H)-submodule E ′ ⊂ E, we obtain a filtered sheaf P * E ′ over E ′ by P a E ′ := P a E ∩E ′ . If E ′ is saturated, i.e., E ′′ := E/E ′ is torsion-free, we obtain a filtered sheaf P * E ′′ over E ′′ by P a E ′′ := Im P a E −→ E ′′ . A morphism of filtered sheaves f : P * E 1 −→ P * E 2 is defined to be a morphism f : Remark 2.1 The concept of filtered bundles on curves was introduced by Mehta and Seshadri [42] and Simpson [57,58]. A higher dimensional version was first studied by Maruyama and Yokogawa [39] for the purpose of the construction of the moduli spaces.

Reflexive filtered sheaves
A filtered sheaf P * E on (X, H) is called reflexive if each P a E is a reflexive O X -module. Note that it is equivalent to the "reflexive and saturated" condition in [43,Definition 3.17] by the following lemma.
Lemma 2.2 Suppose that P * E is reflexive. Let a ∈ R Λ . We take a i −1 < b ≤ a i , and let a ′ ∈ R Λ be determined by a ′ j = a j (j = i) and a ′ i = b. Then, P a E/P a ′ E is a torsion-free O Hi -module.
Proof Let s be a section of P a E/P a ′ E on an open set U ⊂ D i . There exists an open subset U ⊂ X and a section s of P a E on U such that U ∩ D i = U and that s induces s. Note that there exists Z ⊂ U of codimension 2 such that s | U\Z is a section of P a ′ E | U\Z . Because P a ′ E is reflexive, there exists a section s ′ of P a ′ E on U such that s ′ | U\Z = s | U\Z . Hence, we obtain that s is a section of P a ′ E, i.e., s = 0. The following lemma is clear.

Lemma 2.3
Let P * E be a reflexive filtered sheaf on (X, H). Then a coherent O X ( * H)-submodule E ′ ⊂ E is saturated if and only if the induced filtered sheaf P * E ′ is reflexive.

Filtered Higgs sheaves
X is induced by the composition of morphisms and the wedge product. If θ ∧ θ = 0 is satisfied, θ is called a Higgs field of E. When a Higgs field θ is given, a Higgs subsheaf of E means a coherent A pair of a filtered sheaf P * E over E and a Higgs field θ of E is called a filtered Higgs sheaf. It is called reflexive if P * E is reflexive.

µ L -Stability condition for filtered Higgs sheaves
Let X be a connected projective manifold with a simple normal crossing hypersurface H = i∈Λ H i . Let L be an ample line bundle.

Slope of filtered sheaves
Let P * E be a filtered sheaf on (X, H) which is not necessarily a filtered bundle. Recall that par-c 1 (P * E) is defined as follows. Let η i be the generic point of H i . Note that O X,ηi -modules (P a E) ηi depends only on a i , which is denoted by P ai (E ηi ). We obtain O Hi,ηi -modules Gr P a (E ηi ) := P a (E ηi ) P <a (E ηi ). Then, we have We set It is called the slope of P * E with respect to L. The following is proved in [43,Lemma 3.7].
be a morphism of filtered sheaves which is generically an isomorphism, i.e., the induced morphism E η(X) at the generic point of X is an isomorphism. Then, µ L (P * E (1) ) ≤ µ L (P * E (2) ) holds. If the equality holds, f is an isomorphism in codimension one, i.e., there exists an algebraic subset Z ⊂ X such that (i) the codimension of Z is larger than 2, (ii) f |X\Z : |X\Z is an isomorphism.
A filtered Higgs sheaf (P * E, θ) is called µ L -polystable if the following holds.

Filtered bundles in the local case
We explain the notion of filtered bundle in the local case. We shall explain it in the global case in §2.

Pull back, push-forward and descent with respect to ramified coverings in the local case
Let ϕ : C n −→ C n be given by We set ϕ * (P * V 1 ) := P * V ′ 1 . Thus, we obtain the pull back functor ϕ * from the category of filtered bundles on (U, H U ) to the category of filtered bundles on (U ′ , H U ′ ). For , we obtain the following filtered bundle In this way, we obtain a functor ϕ * from the category of filtered bundles on (U ′ , H U ′ ) to the category of filtered bundles on (U, H U ).

Filtered bundles in the global case
We use the notation in §2.1.1. Let V be a locally free O X ( * H)-module. A filtered bundle P * V = P a V a ∈ R Λ be a sequence of locally free O X -submodules P a V of V such that the following holds.
• For any P ∈ H, we take an admissible coordinate neighbourhood (X P , z 1 , . . . , z n ) around P . Then, for any a ∈ R Λ , P a V |XP depends only on κ P (a). We denote P • The sequence (P ) is a filtered bundle over V |XP in the sense of §2.3.1. Clearly, a filtered bundle is a special type of filtered sheaf in §2.1.1.

Remark 2.7
The higher dimensional version of filtered bundles was introduced in [44] with a different formulation. See also [5,6]. In this paper, we follow Iyer and Simpson [26].

The induced bundles and filtrations
For any I ⊂ Λ, let δ I ∈ R Λ be the element whose j-th component is 0 (j ∈ I) or 1 (j ∈ I). We also set It is naturally regarded as a locally free O Hi -module. Moreover, it is a subbundle of P a (V) |Hi . In this way, we obtain a filtration i F of P a (V) |Hi indexed by ]a i − 1, a i ].
We obtain the induced filtrations i F of P a V |HI if i ∈ I. Let a I ∈ R I denote the image of a by the projection R Λ −→ R I . Set ]a I − δ I , a I ] := i∈I ]a i − 1, a i ]. For any b ∈]a I − δ I , a I ], we set By the condition of filtered bundles, the following compatibility condition holds.
• Let P be any point of H I . There exists a neighbourhood X P of P in X and a decomposition such that the following holds for any c ∈]a I − δ I , a I ]: For any c ∈]a I − δ I , a I ], we obtain the following locally free O HI -modules: Here, b c means "b ≤ c and b = c". We introduce some notation. We set Par(P * V,

First and second Chern characters for filtered bundles
Let P * V be a filtered bundle over (X, H). Take any a ∈ R Λ . We set Here, [H i ] denote the cohomology class induced by H i . It is easy to see that par-c 1 (P * V) is independent of a choice of a ∈ R Λ . We also obtain the following element in H 4 (X, R): Here, ι i * : denote the cohomology class induced by C.

Remark 2.8
The higher Chern character for filtered sheaves was defined by Iyer and Simpson [26] in a systematic way. In this paper, we adopt the definition of par-ch 2 (P * V) in [43].

Good filtered Higgs bundles
Let X be a complex manifold with a simple normal crossing hypersurface H = i∈Λ H i .

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 , . . . , z n ) around ∈ O X,P , (ii) g(P ) = 0, then we set ord(f ) := n. Otherwise, ord(f ) is not defined. For any a ∈ O X ( * H) P /O X,P , we take a lift a ∈ O X ( * H) P . If ord( a) is defined, we set ord(a) := ord( a). Otherwise, ord(a) is not defined. Note that it is independent of the choice of a lift a.
Let I P ⊂ O X ( * H) P /O X,P be a finite subset. We say that I P is a good set of irregular values if the following holds.
• ord(a) is defined for any a ∈ I P .
• ord(a − b) is defined for any a, b ∈ I P .

Good filtered Higgs bundles
Let V be a locally free O X ( * H)-module with a Higgs field θ. Let P * V be a filtered bundle over V. We say that (P * V, θ) is unramifiedly good at P if the following holds.
• There exist a good set of irregular values I P ⊂ O X ( * H) P /O X,P , an admissible holomorphic coordinate neighbourhood (X P , z 1 , . . . , z n ) around P , and a decomposition such that θ a − d a id Va are logarithmic with respect to the lattice P a V a for any a ∈ R ℓ(P ) and a ∈ I P , i.e., Here a denote lifts of a to O X ( * H) P .
We say that (P * V, θ) is good at P if the following holds.
• There exist a neighbourhood X P of P in X and a covering map ϕ P : X ′ P −→ X P ramified over H P = H ∩ X P such that ϕ * P (P * V, θ) is unramifiedly good at any point of ϕ −1 P (H P ). We say that (P * V, θ) is good (resp. unramifiedly good) if it is good (resp. unramifiedly good) at any point of H.

Prolongation of holomorphic vector bundles with a Hermitian metric
Let X be any complex manifold with a simple normal crossing hypersurface For any open subset U ⊂ X, let P h a E(U) be the space of holomorphic sections of E |U \H satisfying the following condition.
• For any point P of U ∩H, take an small admissible holomorphic coordinate neighbourhood (X P , z 1 , . . . , z n ) around P such that X P is relatively compact in U. Set c = κ P (a). Then,

A sufficient condition to be filtered bundles
We mention a useful sufficient condition for P h * E to be a filtered bundle, although we do not use it in this paper. Let g X\H be a Kähler metric satisfying the following condition [11]: • For any P ∈ H, we take an admissible holomorphic coordinate neighbourhood (X P , z 1 , . . . , z n ) around P such that X P is isomorphic to A Hermitian metric h of (E, ∂ E ) is called acceptable if the curvature of the Chern connection is bounded with respect to h and g X\H . The following theorem is proved in [ A Higgs bundle (E, ∂ E , θ) with a pluri-harmonic metric h is called a harmonic bundle.

Wild harmonic bundles
Let X be a complex manifold with a simple normal crossing hypersurface H = i∈Λ H i . Let (E, ∂ E , θ, h) be a harmonic bundle on X \ H. It is called wild on (X, H) if the following holds.
• Let Σ θ ⊂ T * (X \ H) denote the spectral cover of θ, i.e., Σ θ denotes the support of the coherent O T * (X\H)module induced by (E, ∂ E , θ). Then, the closure of Σ θ in the projective completion of T * X is complex analytic.
A wild harmonic bundle (E, ∂ E , θ, h) is called unramifiedly good at P ∈ H if the following holds.
• There exists a good set of irregular values I P ⊂ O X ( * H) P /O X,P , a neighbourhood X P , and a decompo- A wild harmonic bundle (E, ∂ E , θ, h) is called good at P ∈ H if the following holds.
• There exist a neighbourhood X P and a covering ϕ P : X ′ P −→ X P ramified along H ′ P such that the pull back ϕ −1 P (E, ∂ E , θ, h) |XP is unramifiedly good wild at any point of ϕ −1 P (H).
We say that (E, ∂ E , θ, 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 not every wild harmonic bundle on (X, H) is good on (X, H). But, the following is known [50, Corollary 15.2.8].
Theorem 2.12 Let (E, ∂ E , θ, h) be a wild harmonic bundle on (X, H). Then, there exists a proper birational morphism ϕ : The following is one of the fundamental theorem in the study of wild harmonic bundles [47,Theorem 7.4.3].
The following is a consequence of the norm estimate for good wild harmonic bundles [47, Theorem 11.7.2].
Theorem 2.14 If h 1 is another pluri-harmonic metric of (E, ∂ E , θ) such that P h1 * E = P h * E. Then, h and h 1 are mutually bounded.

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 [47, Proposition 13.6.1, Proposition 13.6.4].

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 (P * V, θ) be a good filtered Higgs bundle on (X, H). Let (E, ∂ E , θ) be the Higgs bundle obtained as the restriction of (P * V, θ) to X \ H.
Theorem 2.16 Suppose that (P * V, θ) is µ L -polystable, and the following vanishing: Then, there exists a pluri-harmonic metric h of (E, We proved a similar theorem for good filtered flat bundles in [47, Theorem 16.1.1]. Theorem 2.16 can be proved similarly and more easily on the basis of the fundamental theorem of Simpson [57] after [43,45]. We shall explain a proof in §3-7. Note that the one dimensional case is due to Biquard-Boalch [3].

Corollary 2.17
We have the equivalence of the following objects.
• Good wild harmonic bundles on (X, H).

Hermitian-Einstein metrics of Higgs bundles
Let Y be a Kähler manifold with a Kähler form ω. Let (E, ∂ E , θ) be a Higgs bundle on Y with a Hermitian metric. We set D 1 : Recall that h is called a Hermitian-Einstein metric of the Higgs bundle if Λ ω F (h) ⊥ = 0, where F (h) ⊥ denote the trace-free part of F (h), and Λ ω denote the adjoint of the multiplication of ω (see [32, §3.2]). The following is a generalization of Kobayashi-Lübke inequality to the context of Higgs bundles due to Simpson [57,Proposition 3.4].

Proposition 3.1 (Simpson)
If h is a Hermitian-Einstein metric, there exists C(n) > 0 depending only on n = dim Y such that the following holds: Corollary 3.2 (Simpson) If Y is compact, and if a Higgs bundle (E, ∂ E , θ) on Y has a Hermitian-Einstein metric h, then the Bogomolov-Gieseker type inequality holds:

Rank one case
Let X be an n-dimensional smooth connected projective variety with a simple normal crossing hypersurface H. Let ω be a Kähler form. Let Λ ω denote the adjoint of the multiplication of ω. We have the irreducible The following proposition is standard.

Proposition 3.3 There exists a Hermitian metric h of the line bundle
gi is a Hermitian metric of P a (V) of C ∞ -class. Such a metric is unique up to the multiplication of positive constants. Moreover, if c 1 (P * E) = 0, then R(h) = 0 holds, and hence h is a pluri-harmonic metric of (E, θ).
Proof Note that F (h) = R(h) holds in the rank one case. Let h ′ 0 be a C ∞ -metric of P a E. We obtain the metric The metric h = h 0 e ϕ0 has the desired property. The uniqueness is clear. Suppose that c 1 (P * E) = 0. In the rank one case, a Hermitian metric of E is a pluri-harmonic metric of (E, ∂ E , θ), if and only if R(h) = 0. Because the cohomology class of R(h 0 ) is 0, there exists an R-valued C ∞ -function ϕ 0 such that R(h 0 e ϕ0 ) = 0 by the standard ∂∂-lemma.

β-subobject and socle for reflexive filtered Higgs sheaves
Let X be a complex projective connected manifold with a simple normal crossing hypersurface H = i∈Λ H i and an ample line bundle L.

β-subobjects
Let (P * V, θ) be a reflexive filtered Higgs sheaf on (X, H). For any A ∈ R, let S(P 0 V, A) denote the family of saturated subsheaves F of P 0 V such that deg L (F ) ≥ −A and that F ( * H) is a Higgs subsheaf of V. Any F ∈ S(P 0 V, A) induces a reflexive filtered Higgs sheaf P * (F ( * H)) by P c (F ( * H)) := P c V ∩ F ( * H) for any c ∈ R Λ . We set f A (F ) := µ L (P * (F ( * H))). Thus, we obtain a function f A on S(P 0 V, A).
In particular, f A has the maximum.
Proof According to [19,Lemma 2.5], S(P 0 V, A) is bounded. Hence, it is easy to see that there exists a finite decomposition S(P 0 V, . It is standard that any reflexive filtered Higgs sheaf has a β-subobject, i.e., the following holds. Proposition 3.5 For any reflexive filtered Higgs sheaf (P * V, θ), there uniquely exists a non-zero Higgs subsheaf V 0 ⊂ V such that the following holds for any non-zero reflexive Higgs subsheaf V ′ ⊂ V.
Proof There exists N > 0 such that the following holds for any saturated subsheaf F ⊂ P 0 V: where θ ′ denote the Higgs field induced by θ.
Suppose that the Higgs subsheaves V i ⊂ V (i = 1, 2) satisfy µ L (P * V i ) = B 0 . We obtain the subsheaf Then, the claim of the lemma is clear.
Suppose that K = 0, i.e., I = 0. Because I is a subsheaf of V (2) , we also obtain a filtered sheaf P * I induced by P * V (2) . Because I ≃ K, we obtain a filtered sheaf P ′ * I over I induced by P * K. Then, we obtain Because (P * V (2) , θ) is µ L -stable and because I = 0, we obtain that rank(I) = rank V (2) , i.e., I and V (2) are generically isomorphic. Because µ L (P * I) = P * V (2) , Lemma 2.4 implies that P * I −→ P * V (2) is an isomorphism in codimension 1. Hence, there exists a closed algebraic subset Z ⊂ X such that (i) the codimension of Z is larger than 2, (ii) V Let us study the case where V (1) ∩ V (2) = 0. Let V (3) denote the saturated Higgs subsheaf of V generated by V (1) + V (2) . Let P * V (3) denote the filtered sheaf over V (3) induced by P * V. Lemma 3.8 (P * V (3) , θ (3) ) is µ L -semistable, and the induced morphism g : is generically an isomorphism, and because they have the same slope, g is an isomorphism in codimension one by Lemma 2.4.
By Lemma 3.8, it is easy to observe that there exists a finite sequence of reflexive Higgs subsheaves V ′ j (j = 1, . . . , m) such that (i) the induced filtered Higgs sheaves (P * V ′ j , θ ′ j ) are µ L -stable, (ii) the image of the induced morphism g : Hence, g is an isomorphism in codimension one by Lemma 2.4. Because both P * V and P * V 1 are reflexive, we obtain that P * V ≃ P * V 1 . Thus, we obtain Proposition 3.6.

Mehta-Ramanathan type theorem
Let X be a smooth connected projective variety with a simple normal crossing hypersurface H. Let L be an ample line bundle on X.
Proposition 3.9 Let P * V be a filtered sheaf on (X, H) with a meromorphic Higgs field θ. Suppose that (P * V, θ) is µ L -stable. Then, it is µ L -stable (resp. µ L -semistable) if and only if the following holds.
• For any m 1 > 0, there exists m > m 1 such that (P * V, θ) |Y is µ L -stable (resp. µ L -semistable) where Y denotes the 1-dimensional complete intersection of generic hypersurfaces of L ⊗m .
Proof We can prove this proposition by the argument in [43, §3.4], which closely follows the arguments of Mehta-Ramanathan [40,41] and Simpson [59].
such that θ a − da id Va are logarithmic with respect to the lattices P a V a . We obtain the endomorphism . By taking the direct sum, we obtain the following endomorphism of i Gr F b (P a V): Note that Res i (θ) |HI preserves the induced filtrations j F (j ∈ I \ {i}) of i Gr F b (P a V) |HI . We set ∂H I := j ∈I (H j ∩ H I ). Let π I : R ℓ −→ R I be the projection. We obtain the following filtered bundle on (H I , ∂H I ): Note that Res i (θ) (i ∈ I) are endomorphisms of the filtered bundle I Gr F b (P * V). Let ϕ : C n −→ C n be given by ϕ(ζ 1 , . . . , ζ n ) = (ζ m1 1 , . . . , ζ m ℓ ℓ , ζ ℓ+1 , . . . , ζ n ). Let U ′ := ϕ −1 (U ). The induced map U ′ −→ U is also denoted by ϕ. Set H U ′ := ϕ −1 (H U ). We obtain the good filtered Higgs bundle (P * V 1 , θ 1 ) := ϕ * (P * V, θ) on (U ′ , H U ′ ) obtained as the pull back. We obtain the endomorphisms Res i (θ 1 ) (i ∈ I) of the filtered bundles I Gr F b1 (P * V 1 ).
. We obtain endomorphisms Res i (θ 1 ) ′ (i ∈ I) of I Gr F ϕ * (b) (P * V 1 ) obtained as the descent of Res i (θ 1 ). By the relation dζ i /ζ i = m i ϕ * (dz i /z i ), we obtain the following relation:

Residue in the local and ramified case
Let (P * V, θ) be a good filtered Higgs bundle on (U, H U ). There exists a ramified covering ϕ : has a decomposition as in (5). For any I ⊂ {1, . . . , ℓ}, we obtain the endomorphisms Res i (θ 1 ) (i ∈ I) of the filtered bundles I Gr F b1 (P * V 1 ) on (H ′ I , ∂H ′ I ). We obtain the endomorphism as the descent of Res i (θ 1 ). We set It is easy to check that Res i (θ) are independent of the choice of a ramified covering U ′ −→ U . In particular, we obtain endomorphisms Res i (θ) of I Gr F b (P a V) for any a ∈ π −1 I (b). The above construction is independent of the choice of a holomorphic coordinate system.

Global case
Let (P * V, θ) be a good filtered Higgs bundle on (X, H). Then, by gluing the residues locally obtained in §3.5.2 for any I ⊂ Λ, we obtain the endomorphisms Res i (θ) (i ∈ I) of I Gr F b (P a V) for any b ∈ π −1 I (a).

Gap of filtered bundles
Let X be a complex manifold with a simple normal crossing hypersurface H = i∈Λ H i . For simplicity, we assume that Λ is finite. Let (P * V, θ) be a good filtered Higgs bundle on (X, H).
We take a ∈ R Λ such that a i ∈ Par(P * V, i). We set Par(P * V, a, i) := Par(P * V, i)∩]a i − 1, a i [. We also set We set gap(P * V, a) := min i∈Λ gap(P * V, a, i). Recall that Λ is assumed to be finite.

Curve case
Let C be a complex curve with a finite subset D ⊂ C. Let (P * V, θ) be a good filtered Higgs bundle on (C, D).
For any (k, b) ∈ Z×Par(P * V, a, P ), we obtain the subspace W k F b (P a V |P ) as the pull back of W k Gr F b (P a V |P ) by the projection F b (P a V |P ) −→ Gr F b (P a (V |P )). We define the filtration F (ǫ) on P a (V) |P indexed by ]a(P ) − 1, a(P )] as follows: F (ǫ) c P a (V) |P := (k,b)∈Z×Par(P * V,a,P ) ϕǫ,P (k,b)≤c We have the corresponding good filtered Higgs bundle (P (ǫ) * V, θ). We clearly have lim ǫ→0 par-c 1 (P (ǫ) * V) = par-c 1 (P * V). The following is standard. Lemma 3.10 Suppose that C is compact and that (P * V, θ) is stable (resp. polystable). Then, if ǫ is sufficiently small, (P (ǫ) * V, θ) is also stable (resp. polystable).
Proof See [43, Proposition 3.28] for the stability. If (P * V, θ) = P * (V i , θ i ), then we have (P Hence, we obtain the claim for the polystability.

Surface case
Let X be a complex projective surface with a simple normal crossing hypersurface H = i∈Λ H i . Let (P * V, θ) be a good filtered Higgs bundle on (X, H). We shall explain a similar perturbation of good filtered Higgs bundles. We take a ∈ R Λ such that a i ∈ Par(P * V, i) for any i ∈ Λ. We choose η > 0 such that 0 < 10 rank(V)η < gap(P * V, a).
For any 0 < ǫ < η, let ψ ǫ,i be a map Par( Note that the eigenvalues of the endomorphism Res i (θ) on Gr F b (P a V |Hi ) are constant on H i because H i are compact. Hence, we have the well defined nilpotent part N i,b of Res i (θ). Note that there exists a finite subset Z i ⊂ H i such that the conjugacy classes of the nilpotent part of N i,b|Q (Q ∈ H i \ Z i ) are constant. We obtain the filtration W of Gr F b (P a V Hi\Zi ) by algebraic vector subbundles whose restriction to Q ∈ H i \ Z i are the weight filtration of N i,b|Q . By the valuative criterion, it is uniquely extended to a filtration of Gr F b (P a V |Hi ) by holomorphic subbundles, which is also denoted by W . For . We define the filtration F (ǫ) on P a (V) |Hi indexed by ]a i − 1, a i ] as follows: We have the corresponding good filtered Higgs bundle (P (ǫ) * V, θ). We clearly have lim ǫ→0 par-c 1 (P (ǫ) * V) = par-c 1 (P * V) and lim ǫ→0 par-ch 2 (P (ǫ) * V) = par-ch 2 (P * V). The following is standard, and similar to Lemma 3.10. (See also [43,Proposition 3.28].) Lemma 3.11 Suppose that (P * V, θ) is stable (resp. polystable). Then, if ǫ is sufficiently small, (P (ǫ) * V, θ) is also stable (polystable).
Let h ǫ be the C ∞ -metric of E given by Lemma 3.14 (E, ∂ E , θ, h ǫ ) are harmonic bundles.
Proof Let H ǫ be the matrix valued function on X \ D determined by (H ǫ ) i,j := h ǫ (v i , v j ). Then, the following holds: Let Θ be the matrix valued function representing θ with respect to the frame ( Let θ † ǫ denote the adjoint of θ with respect to h ǫ . Let Θ † ǫ denote the matrix valued function representing θ † ǫ . The following holds: Hence, we obtain It implies that ∂ H −1 ǫ ∂H ǫ + Θ, Θ † ǫ = 0. It is exactly the Hitchin equation for (E, ∂ E , θ, h ǫ ).
Let (P (ǫ) * E, θ) denote the associated filtered Higgs bundle. We have Let s ǫ be determined by h ǫ = h 0 s ǫ .

Families of equivariant harmonic bundles with nilpotent Higgs fields
Let G := {µ ∈ C * | µ ℓ = 1} for some ℓ. Let V be a finite dimensional C-vector space equipped with a G-action and a G-invariant nilpotent endomorphism N . We set V = V ⊗ O X ( * D) with the Higgs field θ = N dz/z. Let W denote the weight filtration of N on V . We fix 0 < η such that 10 rank(V)η < 1. Take c ∈ R and c(ǫ) ∈ R (0 ≤ ǫ ≤ η) such that |c(ǫ) − c| ≤ 2ǫ. We set We consider the G-action on X by the multiplication on the coordinate. Then, (P (ǫ,c(ǫ)) * V, θ) is naturally G-equivariant.
• lim ǫ→0 h ǫ,c(ǫ) = h 0,c in the C ∞ -sense locally on X \ D. Moreover, there exists C > 1 such that Proof We set G ∨ := Hom(G, C * ). For each χ ∈ G ∨ , let C χ denote the irreducible G-representation corresponding to χ. There exists the canonical decomposition (V, N ) = (V χ , N χ ) ⊗ C χ , where (V χ , N χ ) denote finite dimensional C-vector spaces with a nilpotent endomorphism. We For any finite dimensional vector space U , let Sym ℓ (U ) denote the ℓ-th symmetric tensor product of U . For any nilpotent endomorphism N U on U , let Sym ℓ (N U ) denote the endomorphism of Sym ℓ (U ) induced by the Leibniz rule.

Example of family of equivariant unramifiedly good wild harmonic bundles
Let X, D and G be as in §3.7.3. Note that G acts on z −1 C[z −1 ] by the pull back. Let a ∈ z −1 C[z −1 ]. We set G · a := {µ * a | µ ∈ G}. Let V be a finite dimensional C-vector space equipped with a nilpotent endomorphism N , a grading and a G-action such that µ • N = N • µ for any µ ∈ G, and µV b = V µ * b for any µ ∈ G and b ∈ G · a. We set V := V ⊗ O X ( * D) and V b := V b ⊗ O X ( * D). We have the decomposition V = b∈G·a V b . Let α ∈ C. Let θ be the Higgs field of V given by We have the decomposition (V, θ) = b∈G·a (V b , θ b ). Let W denote the weight filtration on V with respect to N . Take η > 0 such that 10 rank(V)η < 1. Take c ∈ R and c(ǫ) ∈ R (0 ≤ ǫ ≤ η) such that |c(ǫ) − c| ≤ 2ǫ. For a ∈ R, we set P (ǫ,c(ǫ)) Lemma 3.17 There exists a family of harmonic metrics h ǫ,c(ǫ) (0 ≤ ǫ ≤ η) of (V, θ) |X\D such that the following holds • h ǫ,c(ǫ) is adapted to P (c,ǫ) * V.
• lim ǫ→0 h ǫ,c(ǫ) = h 0,c in the C ∞ -sense locally on X \ D. Moreover, there exists C > 1 such that N a ) is naturally G a -equivariant. Let h a,ǫ,c(ǫ) be a family of G a -invariant harmonic metrics of (V a , θ a ) as in Lemma 3.16. By the isomorphisms µ * V a ≃ V µ * a , we obtain harmonic metrics h b,ǫ,c(ǫ) for (V b , θ b ). We set h ǫ,c(ǫ) := b h b,ǫ,c(ǫ) . Then, the family of the harmonic metrics has the desired property.

Proof of Proposition 3.12
Let ϕ : C −→ C be the map determined by ϕ(ζ) = ζ ℓ for some ℓ. We set X ′ := ϕ −1 (X). We may assume to have I ⊂ ζ −1 C[ζ −1 ] and a decomposition where θ a,α − (da + αdz/z) id are logarithmic, and the eigenvalues of the residues are 0. There exists the natural action of G := µ ∈ C * | µ ℓ = 1 on X ′ given by the multiplication on the coordinate. Because ϕ * (P * V, θ) is naturally G-equivariant, there exists the natural G-action on I. We obtain the orbit decomposition It is naturally G-equivariant.
equipped with the induced G-action such that µ * Gr F b (P a V a,α ) = Gr F b (P a V µ * a,α ), and the G-invariant nilpotent endomorphism N which is compatible with the grading. We set b(ǫ) := ψ ǫ (b). By applying Lemma 3.17, we obtain a family of G-equivariant unramifiedly good filtered Higgs fields equipped with a family of harmonic metrics h (ǫ,b(ǫ)) i,b,α,ǫ satisfying the conditions in Lemma 3.17. We set

Lemma 3.18
There exists a G-equivariant isomorphism of filtered bundles f : P for any c ∈ R.
Proof Let F (0) denote the filtration on P (0) a V ′ ai,b,α induced by the filtered bundle P (0) * V ′ . By the construction, we have the isomorphisms f ai,b,α : Gr There exists a G ai -equivariant isomorphism f ai,α : We set f := i,α f i,α . Then, f has the desired property by the construction.
We also remark that . Hence, we obtain the claim of the proposition.  We give an outline of the proof in §4.2 based on the fundamental theorem of Simpson [57,Theorem 1] because we obtain a consequence on the Donaldson functional from the proof, which will be useful in the proof of Proposition 4.2.

Convergence of some families of Hermitian metrics
For each P ∈ D, we take a holomorphic coordinate neighbourhood (C P , z P ) around P such that z P (P ) = 0. Set C * P := C P \ {P }. Fix N > 10. Let g ǫ be a sequence of C ∞ -metrics of C \ D, such that The following proposition is a variant of [45, Proposition 5.1].
• Let b (i) be the automorphism E which is self-adjoint with respect to h (ǫi) and determined by h Then, b (i) and (b (i) ) −1 are bounded with respect to h (ǫi) on C \D. We do not assume the uniform estimate.
Proof We have only to apply the argument in the proof of [45, Proposition 5.1] by replacing G(h) and D λ with F (h) and ∂ + θ, respectively.

Proof of Theorem 4.1
Take a ∈ R D such that a P ∈ Par(P * V, P ) for any P ∈ D. Let (C P , z P ) be a holomorphic coordinate neighbourhood around P such that z P (P ) = 0. Set C * P := C P \ {P }. Take η > 0 such that 10 rank(V)η < gap(P * , a). We take a Kähler metric g C\D,η of C \ D satisfying the following condition.

Lemma 4.4
There exists a Hermitian metric h 0 of E such that the following holds.
(a) (E, ∂ E , h 0 ) is acceptable, and P h0 Proof By applying Proposition 3.12 only in the case ǫ = 0, we obtain a Hermitian metric h ′ 0 of E satisfying (a) and (b). We define the function ϕ : Then, ϕ induces a C ∞ -function on C. We set h 0 := h ′ 0 e ϕ/ rank(E) . Then, the metric h 0 has the desired property. For any holomorphic Higgs subbundle E ′ ⊂ E, let h ′ 0 denote the Hermitian metric of E ′ induced by h 0 . Let θ ′ denote the Higgs field of E ′ obtained as the restriction of θ. We have the Chern connection ∇ h ′ 0 and the adjoint θ ′ † Let Λ g C\D,η denote the adjoint of the multiplication of the Kähler form associated to g C\D,η . Because F (h 0 ) is bounded with respect to h 0 and g C\D,η , deg(E ′ , h 0 ) is well defined in R ∪ {−∞} by the Chern-Weil formula [57,Lemma 3.2]: Here, π E ′ denotes the orthogonal projection E −→ E ′ with respect to h 0 .
Hence, we obtain that (E, ∂ E , θ, h 0 ) is analytically stable. According to the existence theorem of Simpson [57,Theorem 1], there exists a harmonic metric h of (E, ∂ E , θ) such that det(h) = det(h 0 ) and that h and h 0 are mutually bounded. Thus, we obtain Theorem 4.1.

Complement on the Donaldson functional
Let P(h 0 ) be the space of C ∞ -Hermitian metrics h 1 of E satisfying the following condition.
Here, we consider the L p -norms induced by h 0 and g C\D,η .

Proof of Proposition 4.2
For 0 ≤ ǫ ≤ η 2 , let g C\D,ǫ,η1 be the Kähler metric on C \ D such that the following holds on C * P for any P ∈ D: Let Λ ω,ǫ denote the adjoint of the multiplication of the Kähler form ω C\D,ǫ,η1 associated to g C\D,ǫ,η1 . By using families of Hermitian metrics as in Proposition 3.12, we construct a family of metrics h (ǫ) in (0 ≤ ǫ ≤ η 2 ) of E such that the following holds: in locally on C \ D in the C ∞ -sense as ǫ −→ 0.
where the L p -norms are taken with respect to h (ǫi) in and g C\D,ǫi,η1 . We do not assume that the estimate is uniform in i.
Then, there exists C 3 , C 4 > 0 such that the following holds for any ǫ i 1 ) = 1. Take any sequence ǫ i → 0. By Proposition 4.6 and Lemma 4.7, there exists a constant C 10 > 0 such that the following holds for any i: in and g C\D,ǫi,η1 , we obtain that Λ ǫi,η1 ∂∂ Tr(b (ǫi) 1 ) is L 1 with respect to g C\D,ǫi,η1 . Because (∂ + θ)b (ǫi) 1 is L 2 with respect to g C\D,ǫi,η1 and h By [57, Lemma 3.1], the following holds: Therefore, there exists C 12 > 0 such that the following holds for any i: Take Q ∈ C \ D. Let (C Q , z Q ) be a holomorphic coordinate neighbourhood around Q which is rela- . Hence, by (7), there exists C 13 (Q) > 0 such that the following holds for any i: According to a variant of Simpson's main estimate (for example, see [43, Proposition 2.10]), there exists C 14 (Q) > 0 such that the following holds on C Q for any i: Because R(h (ǫi) ) + [θ, θ † h (ǫ i ) ] = 0, we obtain the following estimate on C Q for any i: , there exists C 15 (Q) > 0 such that for any i. By a bootstrapping argument, for any p ≥ 2, there exists C 16 (Q, p) > 0 such that p) for any i. There exists a subsequence ǫ ′ j such that the sequence b is weakly convergent locally on in b ′ ∞ is a harmonic metric of (E, ∂ E , θ) such that (i) h ′(0) and h (0) are mutually bounded on C \ D, (ii) det(h ′ (0) ) = h det E . Then, by the uniqueness, we obtain that b ∞ = id E . Namely, h (ǫ ′ j ) is weakly convergent to h (0) locally in L p 2 for any p. By a bootstrapping argument, we obtain that h (ǫ ′ j ) is convergent to h (0) locally in the C ∞ -sense. Then, the claim of the proposition follows.

Continuity of family of harmonic metrics
Let π : C −→ ∆ be a smooth projective family of complex curves. Let D ⊂ C be a smooth hypersurface such that the induced map D −→ ∆ is proper and locally bi-holomorphic. For each t ∈ ∆, we set C t := π −1 (t) and D t := C t ∩ D.
Let (P * V, θ) be a good filtered Higgs bundle on (C, D). The induced good filtered Higgs bundles (P * V, θ) |Ct are denoted by (P * V t , θ t ).
Let (E, ∂ E , θ) be the Higgs bundle on C \ D obtained as the restriction of (P * V, θ) to C \ D. Let (E t , ∂ Et , θ t ) be the Higgs bundle on C t \ D t obtained as the restriction of (E, ∂ E , θ). Suppose the following.
• There exists a Hermitian metric h det E of E such that (i) R(h det E ) = 0, (ii) h det E is adapted to P * (det V).
• Each (P * V t , θ t ) is stable of degree 0.
According to Theorem 4.1, there exists harmonic metrics h t of (E t , ∂ Et , θ t ) adapted to P * V t such that det(h t ) = h det(E)|Ct\Dt . We obtain the Hermitian metric h of E determined by h |Ct\Dt = h t . We obtain the following proposition by using Proposition 4.6 and an argument similar to the proof of Proposition 4.2. (See also [45,Proposition 4.2].) Proposition 4.9 h is continuous. Moreover, any derivatives of h in the fiber direction are continuous.

Kähler metrics
Let X be a smooth projective surface with a simply normal crossing hypersurface H = i∈Λ H i . Let L be an ample line bundle on X. Let g X be the Kähler metric of X such that the associated Kähler form ω X represents c 1 (L).
We take Hermitian metrics g i of O(H i ). Let σ i : O X −→ O X (H i ) denote the canonical section. Take N > 10. There exists C > 0 such that the following form defines a Kähler form on X \ H for any 0 ≤ ǫ < 1/10: It is easy to observe that X ω 2 ǫ = X ω 2 X and that X ω ǫ τ = X ω X τ for any closed C ∞ -(1, 1)-form τ on X.

Condition for good filtered Higgs bundles and initial metrics
Let (P * V, θ) be a good filtered Higgs bundle on (X, H) satisfying the following condition.

Condition 5.1
• There exists c ∈ R Λ and m ∈ Z >0 such that Par(P * V, i) = {c i + n/m | n ∈ Z} for each i ∈ Λ.
• The nilpotent part of Res i (θ) on i Gr F b (P a V) are 0 for any i ∈ Λ, a ∈ R Λ and b ∈]a i − 1, a i [.
Definition 5.4 A Hermitian metric h of E is called strongly adapted to P * V if the following holds.
• For any P ∈ H, there exists a small neighbourhood X P of P such that h |XP \H is strongly adapted to P * V |XP in the sense of Definition 5.2 and Definition 5.3.

Lemma 5.5
Let h be a Hermitian metric of E strongly adapted to P * V. Then, the following holds: Proof It is the equality (36) in the proof of [43,Proposition 4.18].
For each i ∈ Λ, we choose b i ∈ Par(P * det V, i). Set b = (b i ) ∈ R Λ . We take a Hermitian metric h det(E) of det(E) such that h det(E) i∈Λ |σ i | 2bi gi induces a Hermitian metric of P b det V of C ∞ -class. We shall prove the following proposition in §5.4 after preliminaries in §5.2-5.3.

Proposition 5.6
There exists a Hermitian metric h in of E such that the following holds.
• h in is strongly adapted to P * V.
• F (h in ) is bounded with respect to h in and ω ǫ , where ǫ := m −1 .
Such a Hermitian metric h in is called an initial metric of (P * V, θ).

Preliminary existence theorem for Hermitian-Einstein metrics
Let (P * V, θ) be a good filtered Higgs bundle satisfying Condition 5.1. Let h in be an initial metric for (P * V, θ) as in Proposition 5.6.
(i) h HE and h in are mutually bounded.
(iv) The following equalities hold:

Unramified case
Suppose that (P * V, θ) satisfies the following condition.
• There exists a decomposition of good filtered Higgs bundles such that θ a,α − (da + α i dz i /z i ) induce holomorphic Higgs fields of P c V a,α .
We take any holomorphic frame v = (v j ) of P c V compatible with the decomposition. For j = 1, . . . , r, . We have ∂v = v − k=1,2 c k dz k /z k I, where I denotes the identity matrix. We have the description θv = v Λ 0 + Λ 1 such that the following holds.
Note that there exists a C ∞ -function u on X 0 such that det(h 0 ) = e u h det(E) . We set h in := h 0 e −u/ rank E .

Around smooth points
We set X 0 := (z 1 , z 2 ) ∈ C 2 |z i | < 1 and H := {z 1 = 0}. Let ν : X 0 \ H −→ R >0 be a C ∞ -function such that ν|z 1 | −1 induces a nowhere vanishing function on X 0 of C ∞ -class. Let (P * V, θ) be a good filtered Higgs bundle on (X 0 , H). Let (E, ∂ E , θ) be the Higgs bundle obtained as the restriction of (P * V, θ) to X 0 \ H. We choose b ∈ Par(P * ) and a Hermitian metric h det(E) of det(E) such that h det(E) ν 2c induces a C ∞ metric of P c (det V).
We take C ∞ -metrics h a,α of P c V a,α , and we set h 0 := ν −2c h a,α . We may assume that det(h 0 ) = h det(E) . Let v = (v 1 , . . . , v r ) be any holomorphic frame of P c V compatible with the decomposition. For each i, a i and α i are determined by the condition that v i is a section of P c V ai,αi . There exist matrix valued C ∞ -(1, 0)-forms A a,α such that where I denotes the identity matrix, and (A a,α ) i,j = 0 unless (a i , α i ) = (a j , α j ) = (a, α). Let Λ denote the matrix valued holomorphic 1-form determined by θv = vΛ. There exists the decomposition Λ = Λ 0 + Λ 1 such that the following holds.

Proof of Proposition 5.6
Let X, H and L be as in §5. Then, we obtain (8) from Lemma 5.11 and Lemma 5.14. Thus, we obtain Proposition 5.6.

Proof of Theorem 5.7
Let E ′ ⊂ E be any coherent Higgs O X\H -subsheaf. We assume that E ′ is saturated, i.e., E/E ′ is torsion-free.
holds. As a result, (E, ∂ E , θ, h in ) is analytically stable in the sense of [57].
As studied in [34,35] It is equal to 2 X par-ch 2 (P * V) by Lemma 5.5 and Proposition 5.6. Thus, Theorem 5.7 is proved.

Bogomolov-Gieseker inequality
Let X be any dimensional smooth connected projective variety with a simple normal crossing hypersurface H = i∈Λ H i . Let L be any ample line bundle on X.
Theorem 6.1 Let (P * V, θ) be a µ L -polystable good filtered Higgs bundle on (X, H). Then, the Bogomolov-Gieseker inequality holds: Proof By the Mehta-Ramanathan type theorem (Proposition 3.9), it is enough to study the case dim X = 2, which we shall assume in the rest of the proof. We use the notation in §3.6.3. We take a ∈ R Λ such that a i ∈ Par(P * V, i) for any i ∈ Λ. We choose η > 0 such that 0 < 10 rank(V)η < gap(P * V, a).
Applying the construction in §3.6.3, we obtain a good filtered Higgs bundle (P (ǫ) * V, θ) on (X, H). By the construction, it satisfies Condition 5.1. There exists m 0 such that (P be the Higgs bundle obtained as the restriction of (P * V, θ) to X \ H. We use the Kähler metric g ǫ of X \ H as in §5.1.1. There exists a Hermitian-Einstein metric h (ǫ) HE of the Higgs bundle (E, ∂ E , θ) as in Theorem 5.7 for the good filtered Higgs bundle (P (ǫ) * V, θ). By Proposition 3.1, the equality (9), and the equality By taking the limit as m → ∞, i.e., ǫ → 0, we obtain the desired inequality.
Let (E, ∂ E , θ) be the Higgs bundle obtained as the restriction (P * V, θ) |X\H . Let h det(E) denote the pluriharmonic metric of (det(E), ∂ det(E) , Tr θ) strongly adapted to P * (det(E)). For the proof of Theorem 2.16, it is enough to prove the following theorem.
Theorem 7.1 There exists a unique pluri-harmonic metric h of the Higgs bundle (E, ∂ E , θ) such that P h * E = P * V and det(h) = h det(E) .

Local holomorphic coordinate systems
Let P ∈ X \ W M . We take s ∞ ∈ Z △ M such that P ∈ X s∞ . The following is clear because P 1 (X △ M ) ∩ P(T * P X) is dense in P(T * P X). Lemma 7.3 There exist s i ∈ Z △ M (i = 1, 2) and ǫ > 0 such that the following holds. • P ∈ X si (i = 1, 2).
• X s1 and X s2 are transversal at P .
• {s 1 + as ∞ | |a| < ǫ}, {s 2 + as ∞ | |a| < ǫ}, s 1 + s 2 + as ∞ |a| < ǫ and s 1 + √ −1s 2 + as ∞ |a| < ǫ are contained in Z △ M . We set x i := s i /s ∞ (i = 1, 2). There exists a neighbourhood U P of P in X \ H such that (x 1 , x 2 ) is a holomorphic coordinate system on U P . Note that P with respect to ∂ zi and ∂ zi (i = j) are continuous. Hence, we obtain that h P,∞ is C 1 . Thus, we obtain the claim of the lemma.
The curvature R(h ∞ ) of the Chern connection is defined as a current. We also obtain the adjoint of Higgs field θ † h∞ as a C 1 -section of End(E) ⊗ Ω 0,1 . We obtain F (h ∞ ) ( Because h ∞|{xi=a} is equal to h si+as∞ , we obtain F (h ∞ ) ii = 0 for i = 1, 2. By considering the holomorphic coordinate system (w 1 , w 2 ) = (x 1 +x 2 , x 1 −x 2 ) and the coefficient of Proof The first claim immediately follows from Lemma 7.6. We obtain the second claim by the elliptic regularity and a standard bootstrapping argument.
HE |Xs is weakly convergent locally on X s \ H in L p 2 for s ∈ X ♯ M , we obtain that ∂b (ǫi) is convergent to 0 almost everywhere. Hence, , we obtain the claim of the lemma.
Lemma 7.10 h ∞ induces a C ∞ -metric of E on X \ H, and hence it is a pluri-harmonic metric of (E, ∂ E , θ).
Proof Take P ∈ W M \ H. We take a holomorphic coordinate neighbourhood (X P , z 1 , z 2 ) around P in X \ H. We may assume that X P is bi-holomorphic with {(z 1 , z 2 ) | |z i | < 2} by the coordinate system, and that P corresponds to (0, 0). Let C i,a := {z i = a} ∩ X P . Let g P denote the metric dz i dz i . We have the expression θ = f 1 dz 1 + f 2 dz 2 . According to a variant of Simpson's main (for example, see [43, Proposition 2.10]), there exist A > 0 such that |f 1|C2,a | < A and |f 2|C1,a | < A for any {a ∈ C | 0 < |a| < 1}. Hence, we obtain that |f 1 | < A and |f 2 | < A on X P \ {P }. We obtain that R(h) |XP \{P } |gP ,h < BA 2 , for a constant B > 0 depending only on rank(E).
We take a C ∞ -metric h 0,P of E |XP . Let b P be the automorphism of E |XP which is self-adjoint with respect to both h ∞|XP and h 0,P and determined by h ∞|XP = h 0,P b P . By the norm estimate for tame harmonic bundles [44], we obtain that b P and b −1 P are bounded with respect to h 0,P .
Therefore, we obtain that |∂ E,z1 b P | h0,P is L 2 on X P . Similarly, we obtain that |∂ E,z2 b P | h0,P is L 2 on X P . Hence, we obtain that b −1 P ∂ h0,P b P is L 2 . Because ∂(b −1 P ∂ h0,P b P ) = R(h ∞|XP ) − R(h 0,P ) on X P \ {P }, we obtain that ∂(b −1 P ∂ h0,P b P ) is bounded on X P \ {P }. It is easy to observe that ∂(b −1 P ∂ h0,P b P ) = R(h ∞|XP ) − R(h 0,P ) holds X P as distributions. By the elliptic regularity, we obtain that b −1 P ∂ h0,P b P is L p 1 for any p > 1. By using and the elliptic regularity, we obtain that b P is L p 2 for any p > 1. Then, by using (15) and the standard bootstrapping argument, we obtain that b P is C ∞ on X P .
Because (E, ∂ E , θ, h ∞ ) is a good wild harmonic bundle on (X, H), we obtain that a good filtered Higgs bundle (P h∞ E, θ) on (X, H). We put H [2] = i =j (H i ∩ H j ). For any P ∈ H \ (W M ∪ H [2] ), there exists s ∈ Z △ M such that P ∈ X s . By the construction, h ∞|Xs\Hs = h s . Hence, we obtain P h∞ (E) |Xs = P * (V) |Xs . Let Y := (H ∩ W M ) ∩ H [2] , which is a finite subset of H. We obtain that P h∞ * (E) X\Y ≃ P * V |X\Y . By Hartogs theorem, we obtain that P h∞ * (E) ≃ P * V. Thus, the proof of Proposition 7.2 is completed.
There exists a pluri-harmonic metric h s of (E s , ∂ Es , θ s ) := (E, ∂ E , θ) |Xs\H adapted to P * V s such that det(h s ) = h det(E)|Xs\H . Take another s ′ ∈ Z △ M such that P ∈ X s ′ . There exists a pluri-harmonic metric h s ′ of (E s ′ , ∂ E s ′ , θ s ′ ) adapted to P * V s ′ such that det(h s ′ ) = h det(E)|X s ′ \H . Proof Suppose that X s ∪ X s ′ ∪ H is simply normal crossing. We set X s,s ′ := X s ∩ X s ′ . It is smooth and connected. We obtain a good filtered Higgs bundle (P * V, θ) |X s,s ′ , and h s|X s,s ′ and h s ′ |X s,s ′ are adapted to P * V |X s,s ′ . Let b s,s ′ be the automorphism of E |X s,s ′ which is self-adjoint with respect to both h s|X s,s ′ and h s ′ |X s,s ′ , and determined by h s ′ |X s,s ′ = h s|X s,s ′ · b s,s ′ . There exists a decomposition (P * V, θ) |X s,s ′ = (P * V i , θ i ), which is orthogonal with respect to both h s|X s,s ′ and h s ′ |X s,s ′ , and b s,s ′ = a i id Vi for some positive constants a i .
In general, there exists s 2 ∈ Z △ M such that (i) P ∈ X s2 , (ii) X s ∪ X s2 ∪ H and X s ′ ∪ X s2 ∪ H are simply normal crossing. By the above consideration, we obtain h s|P = h s2|P = h s ′ |P . Therefore, we obtain Hermitian metrics h P of E |P (P ∈ X \ (H ∪ W M )). By using the argument in Lemma 7.5, we can prove that they induce a Hermitian metric h of E |X\(H∪WM ) of C 1 -class. We obtain F (h) as a current. Because h |Xs (s ∈ U △ M ) are pluri-harmonic metrics of (E, ∂ E , θ) |Xs\H , we obtain that F (h) = 0. It also implies that h is a C ∞ on X \ (H ∪ W M ). By using the argument in the proof of Lemma 7.10, we obtain that h induces a pluri-harmonic metric of (E, ∂ E , θ) on X \ H. Then, as in the proof of Proposition 7.2, we can conclude that P h * (E) = P * V. Thus, we obtain Theorem 7.1.
8 Homogeneity with respect to group actions 8.1 Preliminary

Homogeneous harmonic bundles
Let Y be a complex manifold. Let K be a compact Lie group. Let ρ : K × Y −→ Y be a K-action on Y such that ρ k : Y −→ Y is holomorphic for any k ∈ K. Let κ : K −→ S 1 be a homomorphism of Lie groups. Let (E, ∂ E , θ, h) be a harmonic bundle on Y . It is called (K, ρ, κ)-homogeneous if (E, ∂ E , h) is K-equivariant and k * θ = κ(k)θ.
Remark 8.1 According to [60], harmonic bundles are equivalent to polarized variation of pure twistor structure of weight w, for any given integer w. As studied in [48, §3], by choosing a vector v in the Lie algebra of K such that dκ(v) = 0, we obtain the integrability of the variation of pure twistor structure from the homogeneity of harmonic bundles. G 0 −→ E is K-equivariant. Hence, K is a K-equivariant holomorphic vector bundle. Similarly H 0 (X, K⊗L ⊗m2 ) is a K-representation, and G 1 is K-equivariant holomorphic vector bundle, and G 1 −→ K 2 is K-equivariant.
The K-representations on H 0 (X, E ⊗ L ⊗ m1 ) and H 0 (X, K ⊗ L ⊗m2 ) naturally induce G-representations on H 0 (X, E ⊗ L ⊗ m1 ) and H 0 (X, K ⊗ L ⊗m2 ). Hence, G i are naturally algebraic G-equivariant vector bundles on X. Moreover, the morphism G 1 −→ G 0 is G-equivariant and algebraic. Hence, E is a G-equivariant algebraic vector bundle on X.

An equivalence
8.2.1 Good filtered Higgs bundles associated to homogeneous good wild Higgs bundles Let X be a connected complex projective manifold with a simple normal crossing hypersurface H. Let G be a complex reductive group acting on (X, H). Let K be a compact real form of G. The actions of G and K on X are denoted by ρ. Let κ : G −→ C * be a character. The induced homomorphism K −→ S 1 is also denoted by κ.
Let (E, ∂ E , θ, h) be a (K, ρ, κ)-homogeneous harmonic bundle on X \ H which is good wild on (X, H). We obtain a good filtered Higgs bundle (P h * E, θ) on (X, H). Because each P h a E is naturally a K-equivariant holomorphic vector bundle on X, P h * E is naturally G-equivariant by Lemma 8.6. Because k * θ = κ(k)θ for any k ∈ K, we obtain g * θ = κ(g)θ for any g ∈ G. Therefore, (P h * E, θ) is a (G, ρ, κ)-homogeneous good filtered Higgs bundle on (X, H).
Let L be a G-equivariant ample line bundle on X.

Uniqueness
Let (E, ∂ E , θ, h) be a (K, ρ, κ)-homogeneous harmonic bundle on X \ H which is good wild on (X, H). Let h ′ be another pluri-harmonic metric of (E, ∂ E , θ) such that (i) h ′ is K-invariant, (ii) P h ′ * E = P h * E. The following is clear from Proposition 2.15.
Proposition 8.8 There exists a decomposition (E, ∂ E , θ) = m i=1 (E i , ∂ Ei , θ i ) such that (i) the decomposition is orthogonal with respect to both h and h ′ , (ii) there exist a i > 0 (i = 1, . . . , m) such that h ′ |Ei = a i h Ei , (iii) the decomposition E = E i is preserved by the K-action.
Let (E, ∂ E , θ) be the Higgs bundle on X \ H obtained as the restriction of (P * V, θ).
Theorem 8.9 Suppose that (P * V, θ) is µ L -stable with respect to the G-action. Then, there exists a K-invariant pluri-harmonic metric h of (E, ∂ E , θ) such that P h * E = P * V. If h ′ is another K-invariant pluri-harmonic metric of (E, ∂ E , θ), there exists a positive constant a such that h ′ = ah.
We obtain a Hermitian metric h := k * h 1 dk of E. Then, h is K-invariant. Moreover, h = h E0 ⊗ K Ψ dk holds. Hence, h is a pluri-harmonic metric of (E, ∂ E , θ) such that P h * E = P * V. Hence, we obtain the claim of the theorem. The uniqueness is clear.
Corollary 8. 10 We obtain the equivalence between the isomorphism classes of the following objects.