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在同调平凡圈上构造了一个对，它推广和改进了超越度为1的函数域上的经典高度对。更详细地说，我们从一个完美域上的光滑簇B开始，并从后者的函数K上的几何光滑和真簇X开始。然后，该构造提供了X上具有适当余维的同调平凡圈的配对，其值在皮卡德群B中（用有理数张量）。这种构造使用了l-adic上同调和反常层理论，并遵循了俄国数学家A。80年代末的贝林森。目的和目标关于这种配对，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？人们期望它是。(2)如果人们对B上X的正则模型中的同调平凡圈模型进行自然猜想，那么它能以完全几何的方式构造出来吗（特别是，不需要l-adic上同调）？(3)（2）和代数圈上的标准代数图之间的联系是什么（更具体地说，这些代数图是否提供了（2）中设想的那种类型的无条件构造？）(4)在（3）的上下文中，是否可以采用S. https://arxiv.org/abs/2009.07089张？这项研究将致力于回答这些问题。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的一篇文章）中描述的不同结构构造联系起来。Kahn）。这种构造是几何的，理解它与问题（2）和（3）的关系是很重要的。A. Beilinson是第一个看到反常层在这种情况下的作用的人（见上文），但只有在https://arxiv.org/abs/2009.01191文章中，这个想法才被再次提起。所以这个项目的研究方法肯定是新的。这个项目福尔斯属于EPSRC的研究领域“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".