  CORRECTION. One of the main results in this paper contains a fatal error. We
cannot conclude the existence of nontrivial vector bundles on X from the
nontriviality of its K-group. The K-group that is computed here is the
Grothendieck group of perfect complexes and not vector bundles. Since the
varieties are not quasi-projective, existence of nontrivial perfect complexes
says nothing about the existence of nontrivial vector bundles. We thank Sam
Payne for drawing our attention to the error and Christian Haesemeyer for
explanations about the K-theory.
  Abstract: Following Sam Payne's work, we study the existence problem of
nontrivial vector bundles on toric varieties. The first result we prove is that
every complete fan admits a nontrivial conewise linear multivalued function.
Such functions could potentially be the Chern classes of toric vector bundles.
Then we use the results of Corti\~nas, Haesemeyer, Walker and Weibel to show
that the (non-equivariant) Grothendieck group of the toric 3-fold studied by
Payne is large, so the variety has a nontrivial vector bundle. Using the same
computation, we show that every toric 3-fold X either has a nontrivial line
bundle, or there is a finite surjective toric morphism from Y to X, such that Y
has a large Grothendieck group.
