  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.
