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Let $K$ be a function field over complex numbers i.e. the fraction field of a complex variety. Then one can define the Deligne-Beilinson cohomology and mixed Hodge modules for $K$ as the direct limit of those for the 'models' of $K$. My questions is: did anybody study these matters; are there any related vanishing results for $K$ that are not valid for (general) smooth complex varieties? Probably, calculating 'stalks' of Hodge modules is connecting with certain vanishing cycles functor; yet I don't know much about these matters.

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