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 related to certain vanishing cycles functors; yet I don't know much about these matters.
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asked Jul 27, 2014 at 7:39
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$\begingroup$ What if you replace Deligne-Beilinson cohomology by de Rham cohomology? Apparently, this looks a bit like Beilinson's isomorphism $B_{\operatorname{dR}}^+\simeq\widehat{\operatorname{dR}}_{\overline{\mathbb Q_p}/\mathbb Q_p}$. $\endgroup$– Z. MCommented May 10, 2021 at 14:19
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$\begingroup$ Why?:) Do you have any references or explanations for this association? $\endgroup$– Mikhail BondarkoCommented May 11, 2021 at 19:06
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$\begingroup$ I did not claim that I see the precise link. I would like to replace the Deligne-Beilinson by de Rham because it is simpler (which could be understood as an "enriched" version of de Rham). What it seems to me is that, by taking filtered colimit of "models" (I don't know whatever it is), the result for de Rham cohomology might be very similar to Hodge-completed derived de Rham cohomology, and in that sense, Beilinson's isomorphism seems related. $\endgroup$– Z. MCommented May 12, 2021 at 11:05
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