  In the first section of his seminal paper on height pairings, Beilinson
constructed an $\ell$-adic height pairing for rational Chow groups of
homologically trivial cycles of complementary codimension on smooth projective
varieties over the function field of a curve over an algebraically closed
field, and asked about an generalization to higher dimensional bases. In this
paper we answer Beilinson's question by constructing a pairing for varieties
defined over the function field of a smooth variety $B$ over an algebraically
closed field, with values in the second $\ell$-adic cohomology group of $B$.
Over $\Bbb C$ our pairing is in fact $\Bbb Q$-valued, and in general we
speculate about its geometric origin.
