One complex metric and geometric adelic nefness
We prove that an arithmetic adelic line bundle on a smooth quasi-projective variety over a number field has nef geometric part if its prescribed continuous metric at one complex embedding has semipositive first Chern current. The proof constructs semiample approximations to every positive rational ample perturbation by boundary extension, multiplier ideals, and Nadel vanishing, followed by specialization and Galois descent. This gives geometric nefness of Gao--Zhang's canonical Gross--Schoen adelic line bundle.
math.AG↗