Invariance of logarithmic plurigenera and adjoint volumes in Kähler families
Abstract
We consider smooth proper families of compact complex manifolds over a disk whose total space is Kähler, together with a boundary divisor with relative simple normal crossings. We prove that, if the boundary is a rational divisor with coefficients in [0,1), every logarithmic pluricanonical form on a fiber extends to a neighborhood of that fiber, and hence that the logarithmic plurigenera are constant. No positivity of the log canonical class is assumed. We prove further that, for real boundary coefficients in [0,1], a log-adjoint class twisted by a smooth semipositive form has the same volume on every fiber, and that it is pseudoeffective either on every fiber or on none. When the log-adjoint class is nef on every fiber, we give a second proof of the extension theorem, which does not use the minimal model program.
Related
- Psh variation of singular Kähler-Einstein metrics and Invariance of plurigenera for smooth Kähler families
- Invariance of plurigenera for Kähler families with nef canonical bundles
- Aspects of deformation invariance of plurigenera of Kähler varieties
- Non-critical p-adic twisted adjoint L-function for ordinary families on definite unitary groups
- On adjoint double shuffle relations and adjoint reduction relations
BibTeX
@misc{liu2026invariance,
title = {{Invariance of logarithmic plurigenera and adjoint volumes in Kähler families}},
author = {Kefeng Liu and Xueyuan Wan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26519},
eprint = {2610.26519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26519},
}
RIS
TY - UNPB
TI - Invariance of logarithmic plurigenera and adjoint volumes in Kähler families
AU - Kefeng Liu
AU - Xueyuan Wan
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26519
UR - https://arxiv.org/abs/2610.26519
AB - We consider smooth proper families of compact complex manifolds over a disk whose total space is Kähler, together with a boundary divisor with relative simple normal crossings. We prove that, if the boundary is a rational divisor with coefficients in [0,1), every logarithmic pluricanonical form on a fiber extends to a neighborhood of that fiber, and hence that the logarithmic plurigenera are constant. No positivity of the log canonical class is assumed. We prove further that, for real boundary coefficients in [0,1], a log-adjoint class twisted by a smooth semipositive form has the same volume on every fiber, and that it is pseudoeffective either on every fiber or on none. When the log-adjoint class is nef on every fiber, we give a second proof of the extension theorem, which does not use the minimal model program.
ER -