Fujiki C类中复辛流形的Beauville--Bogomolov--Fujiki正性与Moishezon性
Beauville--Bogomolov--Fujiki positivity and Moishezonness of complex symplectic manifolds in Fujiki class $\mathscr C$
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中文总结 AI 辅助
本文研究Fujiki C类中超Fujiki流形的BBF正性与Moishezon性的关系，证明Moishezon超Fujiki流形的相关性质，给出超Fujiki 4-fold等的对应判据并应用于光滑族的Moishezon轨迹描述。
中文摘要 AI 辅助
受Beauville--Bogomolov--Fujiki（BBF）型的正性控制超Kähler流形几何这一原理的驱动，我们从双有理几何视角研究BBF正性，重点关注其与超Fujiki流形（超Kähler流形的自然双有理类比）的Moishezon性的关系。我们证明：任何Moishezon超Fujiki流形都带有一个具有正BBF平方的丰富线丛；对于超Fujiki 4-fold，以及更一般地，在弱Kähler极小模型条件下的超Fujiki 2n-fold，存在BBF正的整(1,1)-类蕴含其Moishezon性。证明主要用到B. Bakker与C. Lehn发展的本原辛簇理论、I. Biswas等人对某些K-平凡流形的小双有理模型的构造，以及S. Kebekus与C. Schnell建立的自反微分形式拉回理论。作为应用，我们结合该判据与B. Anthes等人发展的周期理论，对某些光滑族中的Moishezon轨迹给出Hodge理论描述。
英文摘要
Motivated by the principle that positivity of the Beauville--Bogomolov--Fujiki (BBF) form controls the geometry of hyperkähler manifolds, we study BBF positivity from the perspective of bimeromorphic geometry, focusing on its relation to the Moishezonness of hyperfujiki manifolds (natural bimeromorphic analogues of hyperkähler manifolds). We prove that any Moishezon hyperfujiki manifold carries a big line bundle with positive BBF square. We also prove that the existence of a BBF-positive integral $(1,1)$-class implies Moishezonness for hyperfujiki $4$-folds and, more generally, for hyperfujiki $2n$-folds under a weak Kähler minimal-model condition. The proof mainly uses the theory of primitive symplectic varieties developed by B. Bakker--C. Lehn, a construction of small bimeromorphic models for certain $K$-trivial manifolds by I. Biswas--J. Cao--S. Dumitrescu--H. Guenancia, and the theory of pull-backs of reflexive differential forms established by S. Kebekus--C. Schnell. As an application, we give a Hodge-theoretic description of the Moishezon locus in certain smooth families by combining this criterion with the period theory developed by B. Anthes--A. Cattaneo--S. Rollenske--A. Tomassini.
