Let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle on a projective manifold $(X,\omega)$ of dimension $n$, where $\omega$ is a Kähler form; the holomorphic structure of $E$ defines an operator $\bar{\partial}\_E:\Omega^0(E)\to\Omega^{0,1}(E)$ and, since $\varphi:\Omega^0(E)\to\Omega^{1,0}(E)$, one defines $D^{\prime\prime}=\varphi+\bar{\partial}\_E$; in particular **this is not a connection**.


Let $h$ be a Hermitian metric on $E$, one can define a *connection* $D\_h$ (the *Hitchin-Simpson connection*) on $E$ (with respect to $h$) as follow:


1. $\partial\_h+\bar{\partial}\_E=D$ is the *Chern connection* on $E$ with respect to $h$;
2. $\bar{\varphi}$ is the adjoint of $\varphi$ with respect to $h$;
3. $D\_h=D^{\prime}\_h+D^{\prime\prime}$, where $D^{\prime}\_h=\partial\_h+\bar{\varphi}$.


Simpson proved in *Higgs Bundles and Local Systems* (Pubblications Mathématiques de l'I.H.É.S., **75** (1992) 5-95):



> 
> **Corollary 3.10**: There is an equivalence of categories between the category of flat bundles on $X$ and the category of semistable Higgs bundles on $X$ with $ch\_1(\cdot)\cdot[\omega]^{n-1}=0$ and $ch\_2(\cdot)\cdot[\omega]^{n-2}=0$.
> 
> 
> 


that is: if $\mathfrak{E}$ is semistable, $ch\_1(E)\cdot[\omega]^{n-1}=0$ and $ch\_2(E)\cdot[\omega]^{n-2}=0$ then there exists a Hermitian metric $h$ on $E$ such that the relevant Chern connection $D$ is flat; **am I correct?**


Moreover:



> 
> **Theorem 2**: If $\mathfrak{E}$ is semistable, $ch\_1(E)\cdot[\omega]^{n-1}=0$ and $ch\_2(E)\cdot[\omega]^{n-2}=0$ then there exists a filtration $0=\mathfrak{E}\_0\subsetneqq\mathfrak{E}\_1\subsetneqq...\subsetneqq\mathfrak{E}\_{k-1}\subsetneqq\mathfrak{E}\_k=\mathfrak{E}$ of Higgs subbundles such that any $\mathfrak{E}\_i$ and $\mathfrak{E}\_{j\displaystyle/\mathfrak{E}\_{j-1}}$ are stable Higgs bundles.
> 
> 
> 


**Preserves $D$ this filtration?**


Thanks in advance.



Semistable Higgs bundles and flat connections
Let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle on a complex, projective manifold $(X,\omega)$ of dimension $n$, where $\omega$ is a Kähler form; by hypothesis: $ch\_1(E)\cdot\omega^{n-1}=0$, $ch\_2(E)\cdot\omega^{n-2}=0$ and $\mathfrak{E}$ is semistable.


By [S] theorem 2, $\mathfrak{E}$ is the extension of stable Higgs bundles
\begin{equation\*}
0\to\mathfrak{K}\to\mathfrak{E}\to\mathfrak{Q}\to0
\end{equation\*}
with vanishing Chern classes.


By [S] theorem 1.(2), $\mathfrak{K}=(K,\chi)$ and $\mathfrak{Q}=(Q,\psi)$ admit Hermitian Yang-Mills metrics $h\_K$ and $h\_Q$, wich are harmonic; that is the relevant Hitchin-Simpson connections $D\_{h\_K}\equiv D\_K$ and $D\_{h\_Q}\equiv D\_Q$ are flat; in particular $E$ is an extension of flat bundles.


**Remark 1.** In general a complex bundle extension of flat complex bundles is not flat; see [BH].


In [S] is proved that $\left(K\otimes Q^{\vee},D\_K\otimes D\_Q^{\vee}\right)$ is a harmonic bundle; by [S] lemma 2.2 for all $i\in\{0,...,n\}$ the complex vector spaces $H^i\_{DR}\left(X,K\otimes Q^{\vee}\right)\cong H^i\_{Dol}\left(X,K\otimes Q^{\vee}\right)$ are naturally isomorphic.


**Remark 2.** Because $E$ is a holomorphic bundle on a complex manifold, by a theorem of Koszul and Malgrange, $E$ is a complex bundle with a (natural) holomorphic connection $\nabla$ such that the $(0,1)$-component of relevant covariant derivative $d^{\nabla}$ is $\bar{\partial}\_E$.


For $i=1$, the natural isomorphism $H^1\_{DR}\left(X,K\otimes Q^{\vee}\right)\cong H^1\_{Dol}\left(X,K\otimes Q^{\vee}\right)$ assures that:


1. $E$ as extension of flat bundles is holomorphic;
2. $\bar{\partial}\_E+\varphi$ preserves $K$;
3. on $E$ there exists a Hermitian metric $h$ extension of $h\_K$ and $h\_Q$;


then the Hitchin-Simpson connection of $(\mathfrak{E},h)\equiv(E,\varphi,h)$ preserves $\mathfrak{K}$.




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[BH] Biswas, Heu - *Non-flat extension of flat vector bundles*, avalaible at [arxiv.org](https://arxiv.org/abs/1409.3178)


[S] Simpson - *Higgs Bundles and Local Systems*, Pubblications Mathématiques de l'I.H.É.S., 75 (1992) 5-95


