Deformation of nef adjoint canonical line bundles
Mu-Lin Li, Sheng Rao, Kai Wang
TL;DR
This work analyzes how nefness and semiampleness of adjoint canonical bundles behave under smooth deformations in both projective and Kähler settings. It develops a nef-value function approach within the analytic MMP framework to show that the nefness of \\ K_{X_t}+L_t \\ is deformation-stable, with the nef locus either empty or the entire base; it extends these results to semiampleness and generalized plurigenera. The authors then derive global stability results for adjoint canonical bundles on threefolds, prove deformation invariance of generalized plurigenera, and establish rigidity phenomena for projective manifolds with semiample canonical bundles, including isotriviality in families over \\mathbb{P}^1 or elliptic curves. Collectively, these results advance understanding of how canonical positivity properties persist under smooth deformations and have implications for moduli, minimal models, and global geometry in both projective and Kähler contexts.
Abstract
Much inspired by J. A. Wiśniewski'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 Kähler 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 Kähler smooth deformation.