Generalised height pairings over function fields

In the recent preprint https://arxiv.org/abs/2009.01191, T. Szamuely and D. Rössler gave a construction of a pairing on homologically trivial cycles, which generalises and refines the classical height paring over function fields of transcendence degree one. In more detail, one starts with a smooth variety B over a perfect field and a with a geometrically smooth and proper variety X over the function K of the latter. The construction then provides a pairing on homologically trivial cycles on X of appropriate codimensions, with values in the Picard group of B (tensored with the rational numbers). The construction uses l-adic cohomology and the theory of perverse sheaves and follows a suggestion made by the Russian mathematician A. Beilinson in the late eighties.

Aims and objectives

There are several natural and interesting questions that one can ask about this pairing:

(1) Is it base change invariant? One expects it to be.

(2) Can it be constructed in a completely geometric fashion (in particular, without l-adic cohomology), provided one makes a natural conjecture on models of homologically trivial cycles inside a regular model of X over B?

(3) What is the link between (2) and the standard conjectures on algebraic cycles (more concretely, do these conjectures provide an unconditional construction of the type envisaged in (2)?)

(4) In the context of (3), is it possible to adapt the methods of the recent article https://arxiv.org/abs/2009.07089 by S. Zhang?

The research will aim to answer these questions.. A secondary aim of the project is to relate the construction of the height pairing to a different conjectural construction described in https://arxiv.org/abs/2009.00533 (an article by B. Kahn). This construction is geometrical and it is important to understand how it relates to questions (2) and (3).

Height pairings have classically been studied using geometrical methods only. A. Beilinson was the first one (see above) to see the role of perverse sheaves in this context, but it is only in the article https://arxiv.org/abs/2009.01191 that this idea was picked up again. So the research methodology of this project is definitely new.


This project falls within the EPSRC research area "Algebra".