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 $mK_F$ 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 $\operatorname{Sym}^{\ell}f_*ω_{X/Y}^{\otimes m}$, without passing to reflexive hulls, for $\ell\geq P_m(F)=h^0(F,mK_F)$ whenever $m\geq 1$. We prove it for all $\ell\geq1$ at levels $m\geq 2$, 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.