Arakelov-Green's functions for dynamical systems on projective varieties

We introduce functions associated to polarized dynamical systems that generalize averages of the dynamical Arakelov-Green's functions for rational functions due to Baker and Rumely. For a polarized dynamical system X→ X over a product formula field, we prove an Elkies-style lower bound for these functions evaluated on the adelic points of X. As an application, we prove a Lehmer-type lower bound on the canonical height of a non-torsion point P on an abelian variety $A/K$, where K is a product formula field having perfect residue fields at its completions (for instance, K may be a number field or the function field of a curve over C or Fₚ). For A of dimension g, the lower bound has the form \[{h}(P)≥{C}{D²ᵍ⁺³(log D)²ᵍ},\] where C=C(A,K,h)>0, D=[K(P):K]≥2, and P∈ A(K̄) A(K̄)ₜₒᵣₛ is not contained in a torsion translate of an abelian subvariety of A having everywhere potential good reduction.