The behaviour of local Riemann-Roch isomorphisms near singular fibres

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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.