Positivity of Relative Pluricanonical Direct Images via Fixed-Point Metrics

We develop a direct analytic approach to positivity of relative pluricanonical direct images through a fixed-point Hermitian metric by direct differential curvature calculations. The curvature resolvent formula from fixed-point equation leads to two-sided estimates of entire Nakano curvature by determinant curvature. In this way, we identify generic rank of determinant curvature with birational variation dimension for a projective algebraic fibration between smooth connected projective varieties with general fiber of general type and maximal birational variation, assuming mKF is globally generated or the m-canonical map is birational for general fiber F. In the global generation case, Green--Griffiths Conjecture asks for the tautological bigness of the ordinary symmetric powers Sym^f_*ωX/Y⊗ m, without passing to reflexive hulls, for ≥ Pₘ(F)=h⁰(F,mKF) whenever m≥ 1. We prove it for all ≥1 at levels m≥ 2, also in the case of birational m-canonical map, but construct a counterexample at level $m=1$. Over curves, curvature and boundary estimates give a direct analytic proof of Viehweg's ampleness theorem requested by Fujino. For smooth projective families, we also establish numerical dimension and vanishing results related to the birational variation dimension.