Let $X$ be a quasi-projective variety over $\mathbb{C}$, and let $\mathcal{L}$ be a rank one $\mathbb{C}$ local system on $X$. Does $H^*(X,\mathcal{L})$ come with some mixed hodge structure in general?

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    $\begingroup$ If you only have a $\mathbb{C}$-local system, you cannot expect to have a mixed Hodge structure since the cohomology has no rational structure. (It seems likely though that there should be a natural $\mathbb{C}$-MHS, but I do not know any reference for this.) $\endgroup$ – ulrich Nov 25 '18 at 7:31
  • $\begingroup$ @ulrich I get your point about the rational structure. Can you explain what you mean by a $\mathbb{C}$-MHS? $\endgroup$ – user2520938 Nov 25 '18 at 10:43
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    $\begingroup$ Take a look at my old paper "Geometry of cohomology support loci I" for an explanation of what a $\mathbb{C}$-MHS is, and how you get one on the cohomology of $\mathcal{L}$ (which needs to be unitary). The arXiv version is arxiv.org/pdf/alg-geom/9612006.pdf $\endgroup$ – Donu Arapura Nov 25 '18 at 13:30

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