# Semistable Higgs bundles and flat connections

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:

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?

• If the equivalence is the obvious one you sketched out, then the connection should preserve the filtration if and only if the associated graded pieces also have vanishing first and second Chern class integrals. The reason is that the equivalence should preserve the property of being a sub-bundle, so a sub-vector bundle should be a preserved by the connection if and only if it is a sub-Higgs bundle satisfying the vanishing conditions. – Will Sawin Dec 11 '16 at 22:44
• Yes, it is all clear; indeed, in my problem, the $\mathfrak{E}_i$'s are also flat. But I can not understand how can I construct a flat connection $\nabla$ from a semistable Higgs bundle $\mathfrak{E}$ with $ch_1(E)\cdot\omega^{n-1}=0$ and $ch_2(E)\cdot\omega^{n-2}=0$? – Armando j18eos Dec 12 '16 at 11:11
• Have you read the paper of Simpson? Does that not clear it up? – Will Sawin Dec 12 '16 at 12:25
• I had read again the paper of Simpson, and I solved my doubts; or I think so. Thank you @WillSawin. ;) – Armando j18eos Dec 14 '16 at 17:56

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 $\mathfrak{K}$.

[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