# Canonical isomorphisms of determinant line bundles

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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The theorem of Deligne was generalised to higher relative dimensions in the work of Franke (unpublished) and the work of Eriksson (also unpublished!)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).

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).

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513295/1 | 01/10/2018 | 30/09/2023 | |||

2099846 | Studentship | EP/R513295/1 | 01/10/2018 | 31/03/2022 | Esteban Gomezllata Marmolejo |