A generalization of Beilinson's geometric height pairing

In the first section of his seminal paper on height pairings, Beilinson constructed an ℓ-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth proper varieties over the function field of a curve over an algebraically closed field, and asked about a 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 ℓ-adic cohomology group of B. Over C our pairing is in fact Q-valued, and in general we speculate about its geometric origin.