7 Nov 2008 (never got a reply)
Dear Professor Levine,
You might or might not remember me from brief encounters in Toronto and Oslo last year. I’m a graduate student in Cambridge under Tony Scholl, and would like to ask you some questions, if you have time.
First, I would like to enquire about the possibilities for participating in the Oberwolfach workshop on algebraic K-theory and motivic cohomology next summer, for which you are one of the organizers. I know the number of participants in these workshops is limited, and that they do not always admit graduate students, but if there is a place open, or if there are any late cancellations, I would love to take part.
I would also like to ask you something mathematical. Any answer would be of great help for me. Is it possible to represent Deligne cohomology/Deligne-Beilinson cohomology/absolute Hodge cohomology in the (stable or unstable) A1-homotopy category? If yes, could one then study the Beilinson regulator in this category? I guess (correct me if I’m wrong) that the answer to the question boils down to verifying Nisnevich excision for these theories, but it is not clear to me how to do this. Is this easy or hard, known or unknown? A related question is the following: If we consider (complexes of) sheaves of differential forms, for example some complex defining Deligne cohomology, does it admit transfers?
Part of the reason for asking about this is that I am starting to look at ways to construct higher arithmetic/Arakelov Chow groups for arithmetic schemes which are more general than the ones considered by Feliu in her thesis (http://atlas.mat.ub.es/personals/efeliu/research.html). I recently started corresponding with Burgos Gil about this, but otherwise I have not discussed the problem with anyone else except my supervisor.
If you have any hints or references which would give me a better understanding of these questions I would be incredibly grateful.
Best regards,
Andreas Holmstrom
nLab page on Levine