The behaviour of local Riemann-Roch isomorphisms near singular fibres

在证明一级Adams-Riemann-Roch定理（即在Picard群的水平上）可以提升为与基变化相容的线束同构时，Damian Rössler定义了与光滑和射影纤维相关的某些行列式线束的正则同构(“一级Adams-Riemann-Roch定理的局部改进”，算术l函数和微分几何方法，2013 -246,Progr。数学。, 338)。这些同构可以看作是格罗滕迪克-黎曼-洛克定理在一级上的改进。Grothendieck-Riemann-Roch定理描述了（失败的）在沿着固有映射的推进下Chern特征的行为的自然性，并且是经典Riemann- roch定理的推广，该定理将Riemann曲面的复分析与其拓扑格联系起来。当纤维具有奇异纤维（例如半稳定纤维）时，这些同构提供在奇异纤维以下的点周围具有极点或零的有理部分。本文的目的是计算作为几何数据函数的极点的阶数。这既适用于经典的高度理论，也适用于镜像对称。我还可以为吉川健一（kenichi Yoshikawa）的某些公式提供代数证明（“关于Quillen度量的奇异性”）。数学。Ann. 337, 61-89(2007))描述了在相同情况下Quillen度规的奇异性。研究方法是新的，因为它依赖于一种新的（更明确的）方法来构造局部黎曼-洛克同构，基于通过商数研究对对合的不动点公式的直接证明。延斯·弗兰克（Jens Franke）早期的作品（函数形式的黎曼-罗氏）。预印本，IAS，九十年代早期)和Dennis Eriksson（“unisomorphisme de Deligne-Riemann-Roch”）。论文，Universite Paris 6, 2008)也解决了这个问题，但他们的方法不太基本，需要更重的机械；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，分别是方案的同伦理论。本文最终可能会结合初等和非初等方法，这取决于哪种方法更有效。这个项目属于EPSRC Geometry研究领域，除了我的导师Damian Rössler之外，目前没有其他合作者或公司参与。

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.