On the deformation to the normal cone in Arakelov geometry

TL;DR

This paper develops an Arakelov geometric framework for deformation to the normal cone, introducing arithmetic invariants and establishing their conservation, which paves the way for proving an arithmetic Hilbert-Samuel theorem.

Contribution

It introduces an Arakelov theoretic deformation to the normal cone with Hermitian line bundle deformations and defines new arithmetic invariants conserved during the deformation.

Findings

Arithmetic Hilbert invariants are conserved along the deformation

Framework sets the stage for proving the arithmetic Hilbert-Samuel theorem

Enriches geometric data with Hermitian line bundle deformations

Abstract

We present an Arakelov theoretic version of the deformation to the normal cone. In particular, the geometric data is enriched with a deformation of a Hermitian line bundle. We introduce numerical invariants called arithmetic Hilbert invariants and prove the conservation of these invariants along the deformation. In a following article, this conservation of number theorem will allow a demonstration of the arithmetic Hilbert-Samuel theorem.