Matsumura's extension problem for pluricanonical forms in Kähler families I: the smooth and essentially Moishezon cases
Abstract
In this paper, we study a problem posed by Matsumura on the extension of pluricanonical forms in Kähler families with a relatively nef canonical bundle. We give an affirmative answer in the smooth case, and for one-parameter degenerations under the additional assumption that the relative canonical bundle restricts to a big line bundle on an irreducible component of the central fiber. In particular, we confirm Siu's conjecture on the invariance of plurigenera for a smooth Kähler family with one fiber admitting the nef canonical bundle. The key step in the smooth case is to construct a semipositively curved singular metric on the relative canonical bundle with the integrability required for Cao's L2 extension theorem, using Păun's twisted form of Schumacher's curvature formula and a uniform Monge--Ampère estimate obtained by adapting the capacity method of Boucksom--Eyssidieux--Guedj--Zeriahi. For one-parameter degenerations, we propagate bigness from an irreducible component of the central fiber to nearby smooth fibers and then apply Matsumura's relative Kawamata--Viehweg vanishing theorem on a projective modification to obtain the desired extension.
Related
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BibTeX
@misc{chen2026matsumuras,
title = {{Matsumura's extension problem for pluricanonical forms in Kähler families I: the smooth and essentially Moishezon cases}},
author = {Jian Chen and Sheng Rao and Kai Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40040},
eprint = {2609.40040},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40040},
}
RIS
TY - UNPB
TI - Matsumura's extension problem for pluricanonical forms in Kähler families I: the smooth and essentially Moishezon cases
AU - Jian Chen
AU - Sheng Rao
AU - Kai Wang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40040
UR - https://arxiv.org/abs/2609.40040
AB - In this paper, we study a problem posed by Matsumura on the extension of pluricanonical forms in Kähler families with a relatively nef canonical bundle. We give an affirmative answer in the smooth case, and for one-parameter degenerations under the additional assumption that the relative canonical bundle restricts to a big line bundle on an irreducible component of the central fiber. In particular, we confirm Siu's conjecture on the invariance of plurigenera for a smooth Kähler family with one fiber admitting the nef canonical bundle. The key step in the smooth case is to construct a semipositively curved singular metric on the relative canonical bundle with the integrability required for Cao's L^2 extension theorem, using Păun's twisted form of Schumacher's curvature formula and a uniform Monge--Ampère estimate obtained by adapting the capacity method of Boucksom--Eyssidieux--Guedj--Zeriahi. For one-parameter degenerations, we propagate bigness from an irreducible component of the central fiber to nearby smooth fibers and then apply Matsumura's relative Kawamata--Viehweg vanishing theorem on a projective modification to obtain the desired extension.
ER -