发表机构
Central China Normal University; Wuhan University（华中师范大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文肯定解决了 Kähler 族中多重典范形式延拓的 Matsumura 问题，包括光滑情形和单参数退化情形，并确认了 Siu 多重亏格不变性猜想。
AI 中文摘要
本文研究 Matsumura 提出的关于具有相对 nef 典范丛的 Kähler 族中多重典范形式延拓的问题。我们在光滑情形下给出了肯定回答，并在相对典范丛限制到中心纤维的不可约分量上为大线丛的附加假设下，对单参数退化情形也给出了肯定回答。特别地，我们确认了 Siu 关于多重亏格不变性的猜想在具有一个纤维承认 nef 典范丛的光滑 Kähler 族中成立。光滑情形中的关键步骤是利用 Păun 对 Schumacher 曲率公式的扭曲形式以及通过改编 Boucksom–Eyssidieux–Guedj–Zeriahi 的容量方法获得的均匀 Monge–Ampère 估计，在相对典范丛上构造满足 Cao 的 $L^2$ 延拓定理所需可积性的半正曲率奇异度量。对于单参数退化，我们将大性从中心纤维的不可约分量传播到邻近光滑纤维，然后在射影修改上应用 Matsumura 的相对 Kawamata–Viehweg 消没定理以获得所需的延拓。
英文摘要
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.
Comments27 pages
