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 because $\varphi:\Omega^0(E)\to\Omega^{1,0}(E)$ one defines $D^{\prime\prime}=\varphi+\bar{\partial}_E$, **which 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:

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

flat connection$\nabla$ from a semistable Higgs bundle $\mathfrak{E}$ with $ch_1(E)\cdot\omega^{n-1}=0$ and $ch_2(E)\cdot\omega^{n-2}=0$? $\endgroup$ – Armando j18eos Dec 12 '16 at 11:11