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Let $X$ be a compact Riemann surface. We fix a complex vector bundle $E$ of rank $n$ and degree $d$ (unique up to diffeomorphism). From results coming originally (I think at least) by Simpson,Donaldson and Hitchin one gets an equivalence between:

  • Polystable Higgs field $(\bar{\partial_E},\phi )$ where $\bar{\partial_E}$ is a $(0,1)$ connection giving an holomorphic structure to $E$

-Projectively flat connections $\nabla$ over $E$ with semisimple monodromy.

The correspondence more or less is based on the following fact. We fix an Hermitian metric $h$ on $E$ and we decompose $$\nabla=\nabla^h+ \Phi $$ where $\nabla^h$ is an Hermitian connection and $\Phi$ an adjoint operator. Then we take the pair $((\nabla^h)^{(0,1)},\Phi^{(1,0)})$.In the semisimple monodromy situation this is a (poly)stable Higgs field.

However, I've not understood a fact: starting from a projectively flat $\nabla$ we can look at its $(0,1)$ part $\nabla^{(0,1)}$: this defines an holomorphic structure over $E$ as $ \nabla^{(0,1)}\nabla^{(0,1)}=0$. Is this holomorphic structure the same (or better said isomorphic) to the structure induced by the couple $((\nabla^h)^{(0,1)},\Phi^{(1,0)})$. I tried to prove this by computing everything by hand, but I'm not able to provide an answer, neither a possible counterexample.

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In general, the complex structures associated to $\nabla$ and $\nabla^h$ are different. One case where things can be made explicit is when the Higgs bundle arise from a (complex) variation of Hodge structure, c.f. Simpson, Higgs bundles and local systems. Then $(E,\nabla^{0,1})$ is a flat bundle associated to the local system underlying the VHS. While $(E,(\nabla^h)^{0,1})$ is isomorphic to the associated graded of $E$ with respect to the Hodge filtration, and this won't be flat except in special cases. One case worth noting is when flat connection $\nabla$ (or local system $\ker \nabla$) is unitary. Then it preserves a Hermitian metric $h$. So in this case $\nabla=\nabla^h$, and the Higgs field $\Phi=0$. When one specializes the nonabelian Hodge correspondence to this case, one recovers the theorem of Narasimhan-Seshadri.

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  • $\begingroup$ Can we at least say that for example the connection $\nabla$ is holomorphic with respect to the structure induced by $\nabla^h$ or this is not at all true? $\endgroup$ May 10, 2021 at 19:57
  • $\begingroup$ @TommasoScognamiglio perhaps I've misunderstood, but isn't this the same question? $\endgroup$ May 10, 2021 at 22:20
  • $\begingroup$ You are totally right. Sorry for the stupid mistake. $\endgroup$ May 11, 2021 at 7:43
  • $\begingroup$ Can we say something in some more specific cases? For example what happens what $\nabla^{(0,1)}$ induces the trivial holomorphic structure? $\endgroup$ May 12, 2021 at 6:17
  • $\begingroup$ OK, I have made an edit that hopefully answers your question. $\endgroup$ May 12, 2021 at 14:33

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