The behaviour of local Riemann-Roch isomorphisms near singular fibres

在证明一阶的亚当斯-黎曼-罗克定理（即在皮卡德群的水平上）可以提升到与基变化相容的线丛同构时，达米安·罗斯勒定义了与光滑和投射纤维化相关的某些行列式线丛的规范同构（“A local refinement of the Adams-Riemann-Roch theorem in degree one.”，算术L-函数和微分几何方法，213-246，Progr。数学，338）。这些同构可以看作是Grothendieck-Riemann-Roch定理的一阶改进。格罗滕迪克-黎曼-罗克定理描述了证明了Chern特征标在沿着真映射推进下的行为的自然性，是经典Riemann-Roch定理的推广，该定理将Riemann曲面的复分析与其拓扑亏格联系起来。当纤维化具有奇异纤维时（例如半稳定纤维），这些同构提供了在奇异纤维下面的点周围具有极点或零点的有理截面。本文的目的是计算极点的阶数作为几何数据的函数。这不仅适用于经典的高度理论，也适用于镜像对称。我也可以提供一个代数证明某些公式由于Ken-ichi Yoshikawa（“论奎伦度量的奇异性。“Math.Ann.337，61-89（2007））描述了在相同情况下Quillen度量的奇异性。研究方法是新的，因为它依赖于一种新的（更明确的）方法来构造局部Riemann-Roch同构，该方法基于通过对合的研究对合的不动点公式的直接证明。Jens Franke的早期工作（“函式形式的Riemann-Roch。”预印本，国际会计准则，九十年代初）和丹尼斯埃里克森（“一个同构德德利涅黎曼洛克。“Thesis，Universite巴黎6，2008）也解决了这个问题，但他们的方法不那么基本，需要更重的机器;他们的方法需要一个巨大的分类装置，并使用更高的K理论，分别是同伦理论的计划。本论文可能会在最后结合联合收割机基本和不太基本的方法，这取决于什么是更有效的。这个项目福尔斯属于EPSRC几何研究领域，除了我的主管达米安罗斯勒，有没有其他合作者或公司参与在这一刻。

In proving that the Adams-Riemann-Roch theorem in degree one (i.e. at the level of the Picard group) can be lifted to an isomorphism of line bundles compatibly with base changes, Damian Rössler defined canonical isomorphisms of certain determinant line bundles associated with smooth and projective fibrations ("A local refinement of the Adams-Riemann-Roch theorem in degree one.", Arithmetic L-functions and differential geometric methods, 213-246, Progr. Math., 338). These isomorphisms can be seen as refinements of the Grothendieck-Riemann-Roch theorem in degree one. The Grothendieck-Riemann-Roch theorem describes (the failure of) the naturality of the behaviour of a Chern character under the push-forward along proper maps, and is a generalisation of the classical Riemann-Roch theorem which relates the complex analysis of Riemann surfaces with their topological genus.When the fibrations have singular fibres (e.g. semistable fibres), these isomorphisms provide rational sections with poles or zeroes around the point below the singular fibre. The aim of the dissertation is to compute the order of the pole as a function of geometric data. This would have application to the classical theory of heights but also to mirror symmetry. I could also provide an algebraic proof for certain formulae due to Ken-ichi Yoshikawa ("On the singularity of Quillen metrics." Math. Ann. 337, 61-89 (2007)) describing the singularities of the Quillen metric in the same situation.The research methodology is new because it relies on a new (much more explicit) approach to the construction of local Riemann-Roch isomorphisms, based on a direct proof of a fixed point formula for an involution via a study of quotients. Earlier work by Jens Franke ("Riemann-Roch in functorial form." Preprint, IAS, early nineties) and Dennis Eriksson ("Un isomorphisme de Deligne-Riemann-Roch." Thesis, Universite Paris 6, 2008) also addressed this problem but their methods were a lot less elementary and required much heavier machinery; their approaches required a vast categorical apparatus and used higher K-theory, respectively the homotopy theory of schemes. The dissertation might in the end combine elementary and less elementary methods, depending on what is more efficient.This project falls within the EPSRC Geometry research area, and with the exception of my supervisor Damian Rössler, there are no other collaborators or companies involved at this moment.