Semistable Higgs bundles and flat connections

Question

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.

Answer 1

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

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