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Let $X$ be a smooth connected quasiprojective variety over $\mathbb{C}$ and let $G$ be a complex reductive group. Let $$\iota: G\to GL_N$$ be a representation and let $$\rho: \pi_1(X(\mathbb{C}))\to G(\mathbb{C})$$ be a homomorphism. I am looking for a reference for the following fact:

$\rho$ is deformation-equivalent to a homomorphism $$\rho': \pi_1(X(\mathbb{C}))\to G(\mathbb{C})$$ such that the local system on $X$ associated to $\iota\circ \rho'$ underlies a polarizable complex variation of Hodge structure.

If $X$ is projective, this is due to Simpson [1, Theorem 3]; in the quasiprojective case some special cases have been written down by T. Mochizuki, e.g. if $G=GL_N$ [2, Theorem 10.5] or if $\rho$ is rigid with Zariski-dense image [2, Lemma 10.13].

I think this can probably be extracted with some difficulty from existing literature, but I'm hoping the statement has been written down explicitly somewhere.

EDIT: If it helps, I'm primarily interested in the case $G=PGL_n$.

[1] Simpson, Carlos T., Higgs bundles and local systems, Publ. Math., Inst. Hautes Étud. Sci. 75, 5-95 (1992). ZBL0814.32003.

[2] Mochizuki, Takuro, Kobayashi-Hitchin correspondence for tame harmonic bundles and an application, Astérisque 309. Paris: Société Mathématique de France (ISBN 978-2-85629-226-6/pbk). viii, 117 p. (2006). ZBL1119.14001.

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    $\begingroup$ I remember that Simpson proved this for projective varieties. I do not remember seeing a quasiprojective version. $\endgroup$
    – Moishe Kohan
    Commented Jan 7, 2022 at 4:39
  • $\begingroup$ @MoisheKohan: As I remark in the question, Mochizuki has proven a quasiprojective version, just not for arbitrary $G$... $\endgroup$
    – Daniel Litt
    Commented Jan 7, 2022 at 15:21
  • $\begingroup$ I see, I just did not read the OP carefully. $\endgroup$
    – Moishe Kohan
    Commented Jan 7, 2022 at 15:28
  • $\begingroup$ @MoisheKohan: No worries! $\endgroup$
    – Daniel Litt
    Commented Jan 7, 2022 at 15:54
  • $\begingroup$ Apologies for an off-topic comment, but do you know if the analogous statement (say, at least for $GL_n$) is true for 'arithmetic' in place of 'underlies a polarizable VHS'? $\endgroup$
    – SashaP
    Commented Jan 16, 2022 at 16:07

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