# Does local system cohomology come equiped with a mixed hodge structure?

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?

• 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.) – ulrich Nov 25 '18 at 7:31
• @ulrich I get your point about the rational structure. Can you explain what you mean by a $\mathbb{C}$-MHS? – user2520938 Nov 25 '18 at 10:43
• 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 – Donu Arapura Nov 25 '18 at 13:30