Subject: Re: Response to Question about vector bundles over varieties
Date: Fri, 31 Aug 2001 17:44:16 +0800
From: LAM SIU POR <siuplam@netvigator.com>
To: dmd1@lehigh.edu
CC: pjz <pjz@mail.math.ac.cn>

Original question:

Subject: K-theory
Date: Thu, 16 Aug 2001 20:57:40 +0800
From: "pjz" <pjz@mail.math.ac.cn>
To: <dmd1@lehigh.edu>

Here is a question for topology list.

Given a projective variety M, is the Grothendieck group of
"vector bundles" over M the same as the topological K-theory
of M as a manifold?

Note that the notion of vector bundle over a variety is different
from that over a manifold.

Best

Jianzhong Pan
_____________________________________________________________________
Hi,

                Re: Question about vector bundles over varieties

The question is about whether the forgetful map

                K_alg(X) --> K_top(X)

is iso when X is a (smooth) projective variety.

(1) The map is an iso when X = G/P where G is an algebraic group and P a
parabolic subgroup (see the remark after Definition 5 of the paper
"Holomorphic K-theory, algebraic cocycles and loop groups by Ralph Cohen
and Paulo Lima-Filho at
http://hopf.math.purdue.edu/pub/CohenR-Lima-Filho/holo-k-th.pdf).

(2) In general, they call X flag-like if amongst other things, this
forgetful map is iso. So, while I don't know if there is any example
where this map isn't iso, I guess at least it is not proved to be an iso
in general.

(3) The ppreprint "Semi-topological K-theory using function complexes"
by Friedlander and Walker (see K-theory archive 1999) also contains
related information.

Best,

Siu-Por (LAM)