Marked Points of Families of Hyperbolic Automorphisms of Smooth Complex Projective Varieties

The paper proves α_f(σ)=λ+ for a section with Zariski-dense orbit under hypotheses (H1), (H2) and non-(birational) isotriviality via a family-level geometric canonical height, base-locus analysis B+(D), Hilbert/Hilbert–Chow techniques, and a fundamental inequality. The model’s solution proves the same statement by passing to the generic fiber over K=C(Λ), constructing canonical heights from nef eigenclasses, showing positivity off the isotrivial/periodic-curve locus, and comparing heights to extract the growth rate. The approaches differ—family-level versus generic-fiber function-field—but yield the same conclusion. The model omits some of the paper’s subtleties (notably B+(D) and explicit use of (H2)), yet its argument remains valid under the stated non-isotriviality and dense-orbit hypotheses.