Generalised height pairings over function fields

在最近的预印本https://arxiv.org/abs/2009.01191中，T. Szamuely和D. Rössler给出了在同调平凡环上的配对的构造，它推广和改进了超越度为1的函数场上的经典高度对偶。更详细地说，我们从一个完美域上的光滑变量B开始，从一个几何上光滑的固有变量X开始，在后者的函数K上。然后，该构造提供了X上具有适当余维的同调平凡环的配对，其值在B的Picard群中（由有理数组成）。该构造使用了l进上同论和反常束理论，并遵循了俄罗斯数学家a . Beilinson在80年代末提出的建议。关于这种配对，人们可以问几个自然而有趣的问题：(1)它是碱基变化不变的吗？这是预料之中的。(2)如果对X / B的正则模型内的同调平凡环的模型作出自然的猜想，它能否以完全几何的方式（特别是不含l进上同调）构造？(3)(2)和代数循环的标准猜想之间的联系是什么（更具体地说，这些猜想是否提供了(2)中设想的类型的无条件构造？）(4)在(3)的背景下，是否有可能采用S. Zhang最近的文章https://arxiv.org/abs/2009.07089的方法？这项研究旨在回答这些问题。该项目的第二个目的是将高度配对的构造与https://arxiv.org/abs/2009.00533 （B. Kahn的一篇文章）中描述的不同推测构造联系起来。这个结构是几何的，理解它与问题(2)和(3)的关系是很重要的。高度配对传统上只使用几何方法进行研究。A. Beilinson是第一个（见上文）在这种情况下看到反常的麦穗的作用的人，但只有在文章https://arxiv.org/abs/2009.01191中，这个想法才被重新拾起。所以这个项目的研究方法绝对是新的。该项目属于EPSRC研究领域“代数”。

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 objectivesThere 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".