### Higher K-theory of toric varieties, by Joseph Gubeladze

A natural higher K-theoretic analogue of the triviality of vector bundles on
affine toric varieties is the conjecture on nilpotence of the multiplicative
action of the natural numbers on the K-theory of these varieties. This
includes both Quillen's fundamental result on K-homotopy invariance of
regular rings and the stable version of the triviality of vector bundles on
affine toric varieties. Moreover, it yields a similar behavior of not
necessarily affine toric varieties and, further, of their equivariant closed
subsets. The conjecture is equivalent to the claim that the relevant
admissible morphisms of the category of vector bundles on an affine toric
variety can be supported by monomials not in a non-degenerate corner subcone
of the underlying polyhedral cone. We prove that one can in fact eliminate
all lattice points in such a subcone, except maybe one point. The elimination
of the last point is also possible in 0 characteristic if the action of the
big Witt vectors satisfies a very natural condition. A partial result on this
in the arithmetic case provides first non-simplicial examples -- actually, an
explicit infinite series of combinatorially different affine toric varieties,
verifying the conjecture for all higher groups simultaneously.

The paper is here:
http://xxx.uni-augsburg.de/abs/math.KT/0104166.

Joseph Gubeladze <Joseph.Gubeladze@mathematik.uni-osnabrueck.de>