Deformation of nef adjoint canonical line bundles

TL;DR

This paper proves that nefness and semiampleness of adjoint canonical line bundles are preserved under deformation in smooth projective families, leading to invariance of generalized plurigenera and insights into the stability of canonical bundles in Kähler families.

Contribution

It extends deformation invariance results for nef and semiample adjoint canonical line bundles and explores their stability in Kähler families of threefolds, partially confirming existing conjectures.

Findings

Nefness of adjoint bundles is preserved in smooth projective families.

Semiampleness of adjoint canonical line bundles remains invariant under deformation.

Nef canonical line bundles do not appear in small deformations if initially absent.

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.