Deformation of nef adjoint canonical line bundles

Abstract

Much inspired by J. A. Wi\'sniewski's nef-value function method, we prove that in a smooth projective family over the unit disk, if the adjoint bundle of the canonical line bundle with a relatively semiample line bundle is nef on one fiber, then it remains nef on all fibers. We further extend this result to the semiampleness of the adjoint canonical line bundles. Using these, we prove the deformation invariance of any generalized plurigenera by assuming that only one fiber admits the semiample canonical line bundle and improve the first author--Xiao-Lei Liu's recent deformation rigidity of projective manifolds with semiample canonical line bundles. In particular, also by E. Viehweg--K. Zuo's result on the minimal number of singular fibers in a family and the first author--X. Liu's isotriviality result, if a projective family over $\mathbb{P}^1$ or an elliptic curve has one fiber with the big and nef (or more generally semiample) canonical line bundle, then all fibers are isomorphic to this fiber. Next, much inspired by M. Andreatta--T. Peternell's deformation theoretical approach, we prove that, in a smooth K\"ahler family of threefolds, if the canonical line bundle of one fiber is not nef, then none of its small deformations admits a nef canonical line bundle either. This partially confirms a problem posed by F. Campana--T. Peternell and the global stability of semiampleness of canonical line bundles of threefolds under a K\"ahler smooth deformation.