3
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.