On the arithmetic Hilbert-Samuel theorem : a proof by deformation

TL;DR

This paper presents a novel proof of the arithmetic Hilbert-Samuel theorem utilizing deformation theory and Arakelov geometry, emphasizing the invariance of arithmetic Hilbert invariants during deformation processes.

Contribution

It introduces a new proof method combining classical sheaf reductions, deformation to the projective space, and invariance of arithmetic Hilbert invariants, bridging deformation theory and Arakelov geometry.

Findings

New proof of the arithmetic Hilbert-Samuel theorem

Demonstrates invariance of arithmetic Hilbert invariants during deformation

Connects deformation theory with Arakelov geometry

Abstract

We give a new proof the arithmetic Hilbert-Samuel theorem by using classical reductions in the theory of coherent sheaves, a direct proof in the case of the projective space and the conservation of some numerical invariants, called arithmetic Hilbert invariants, through the deformation to the projective completion of the cone. This construction lies at the intersection of deformation theory and Arakelov geometry. It provides a deformation of a Hermitian line bundle over the deformation to the normal cone.