This is an enhanced version of the author's preprint
http://wwww.math.uiuc.edu/K-theory/0296 presenting
a greatly expanded version of Bogomolov's conjecture.
A somewhat weaker, but still very strong version of
the conjecture is shown to follow from certain
hypotheses about the Milnor K-theory ring.
Update: Ido Efrat informed me that the proofs of main
results of his and Koenigsmann's papers that I quoted
in the last subsection of the Appendix appear to be
erroneous. Accordingly, the subsection is removed.