Canonical isomorphisms of determinant line bundles

这个项目属于EPSRC几何和拓扑学的研究领域。60年代初，Grothendieck将经典的Hirzebruch-Riemann-Roch定理推广到相对情形。在八十年代，公式(2)在Arakelov理论中的一个类比被Faltings(见[4])和Bismut-Gillet-Soulé(见[5])证明。大约在同一时间，Deligne(见[3])注意到，在曲线族的情况下，Faltings-Bismut-Gillet-Soulé公式可以理解为纯几何公式的解析“实现”的特例。他的公式可以看作是公式(2)的改进，其中两边现在(通过适当的构造)被标识为正则线丛(相对于线丛的同构类)，并且这些线丛之间的正则同构被构造出来。上面提到的解析“实现”是使用分析工具计算这种正则同构的范数，当双方都被赋予一定的厄米特度量时。Deligne定理在Franke(未发表)和Eriksson(也未发表！)的工作中被推广到更高的相对维。但他们的方法依赖于许多选择(特别是选择将X嵌入到相对射影空间中)，这使得计算相应的解析实现变得困难。在这种情况下，E.Gomezllata M.将致力于的项目如下：使用一种新的证明方法(在公式(1)的原始上下文中也是新的)，给出Franke和Eriksson公式的一个变体的新证明，该方法非常直接，不需要任何辅助选择。这种方法是基于Nori的见解，他指出Grothendieck-Riemann-Roch公式可以被视为Lefschetz相对固定公式的特例(见[7])。使用Bismut开发的分析工具来计算得到的线丛的正则同构的解析“实现”。阿拉克洛夫理论预测了这种实现应该是什么，因此应该可以避免计算中的任何错误。这个项目的先决条件是哈特肖恩水平的图式理论(见[6])和熟悉代数K-理论(如[8])。在分析层面上，它需要黎曼几何的实用知识和[1]层面上的局部指数定理(但我不认为在解析层面上需要做很多工作--铋的结果应该足以进行计算)。

This project falls within the EPSRC Geometry and Topology research area.In the early sixties, Grothendieck generalized the classical Hirzebruch-Riemann-Roch theorem to a relative situation. In the eighties, an analogue in Arakelov theory of the formula (2) was proven by Faltings (see [4]) and Bismut-Gillet-Soulé (see [5]). Around the same time Deligne (see [3]) noticed that in the situation of a family of curves, the formula of Faltings-Bismut-Gillet-Soulé could be understood as a special case of the analytic "realisation" of a purely geometric formula. His formula can be seen as a refinement of the formula (2), where both sides are now identified (via a suitable construction) as canonical line bundles (as opposed to isomorphism classes of line bundles) and a canonical isomorphism between these bundles is constructed. The analytic "realisation" alluded to above is then the computation, using analytic tools, of the norm of this canonical isomorphism, when both sides are endowed with certain hermitian metrics.The theorem of Deligne was generalised to higher relative dimensions in the work of Franke (unpublished) and the work of Eriksson (also unpublished!) but their approach depends on many choices (in particular the choice of an embedding of X into a relative projective space), which makes it difficult to compute the corresponding analytic realisation.In this context, the project that E. Gomezllata M. will work on is the following Give a new proof of a variant of the formula of Franke and Eriksson, using a new method of proof (which is also new in the original context of the formula (1)), which is very direct and does not require any auxiliary choices. This method is based on an insight of Nori, who showed that the Grothendieck-Riemann-Roch formula can be seen as a special case of a relative fixed formula à la Lefschetz (see [7]). Use the analytic tools developed by Bismut to compute the analytic "realisation" of the resulting canonical ismorphism of line bundles. Arakelov theory predicts what this realisation should be so it should be possible to avoid any mistakes in the computation. The prerequisites for this project are scheme theory at the level of Hartshorne (see [6]) and familiarity with algebraic K-theory (as in [8]). At the analytic level, it requires a working knowledge of Riemannian geometry and the local index theorem at the level of [1] (but I don't expect that a lot of work will be required on the analytic side - the results of Bismut should suffice to carry through the computation).