Positivity of Relative Pluricanonical Direct Images via Fixed-Point Metrics
Abstract
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 Symℓf*ωX/Y⊗ m, without passing to reflexive hulls, for ℓ≥ Pm(F)=h0(F,mKF) whenever m≥ 1. We prove it for all ℓ≥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.
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BibTeX
@misc{fu2026positivity,
title = {{Positivity of Relative Pluricanonical Direct Images via Fixed-Point Metrics}},
author = {Jixiang Fu and Yunyang Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35590},
eprint = {2610.35590},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35590},
}
RIS
TY - UNPB
TI - Positivity of Relative Pluricanonical Direct Images via Fixed-Point Metrics
AU - Jixiang Fu
AU - Yunyang Xiao
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35590
UR - https://arxiv.org/abs/2610.35590
AB - 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 Sym^ℓf_*ω_X/Y^⊗ m, without passing to reflexive hulls, for ℓ≥ P_m(F)=h^0(F,mK_F) whenever m≥ 1. We prove it for all ℓ≥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.
ER -