Beilinson's regulator map

[LorenzoMath]

X is a smooth quasi-projective variety over Q.

Beilinson's regulator map is a map from the motivic cohomology H to the Deligne cohomology H_D. Originally the motivic cohomology was defined by Beilinson as an eigenspace of an Adams operation on an algebraic K-group. Bloch (or Levine or someone else) showed the lambda piece of K coincides with Bloch's higher Chow group. Then Voevodsky claims higher Chow group is the same as his motivic cohomology defined by his motivic complex.

My question is is it possible to give explicit formula of Beilinson's regulator map as a morphism between Voevodsky's motivic complex (over a Zariski, etale site) and Deligne complex (over an analytic manifold)? Even for a certain limited case...

 

[DeadWolfe]

You're clearly making all those terms up.

 

[mathwonk]

you are a little over most of our heads here. you could teach us the definitions, and maybe get up a discussion, or if you just want an answer, you might just ask mark levine, or spencer bloch, or steve gersten. ( i know them and they are nice guys, although i don't know the others).

or maybe some experts will chime in here after a while. but if know all these definitions, your question sounds like a natural research problem. are you asking if this formulation of beilinson's map is known?welcome to the forum, and i hope you stick around. even if we don't get a lot of questions at this level, you will greatly enrich our knowledge base.

 

[LorenzoMath]

my second thought...

Yes, I was wondering how much is already known about explicit description of the regulator map... I looked into the case of "compact" Riemann surfaces as a first step. Then I sensed an explicit description for quasi-projective case would be a complete mess... As for proj varieties of dim greater than 1 would involve spectral sequences of motivic cohomology... hmmm...

I have just got to be interested in this part of the world, i.e. Beilinson's conjectures. So, maybe i should immerse myself into the existing articles first before attempting some possibly pointless thing.