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Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds

Junliang Shen Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Qizheng Yin Beijing International Center for Mathematical Research, Peking University, Beijing, China

Algebraic Geometry mathscidoc:2203.45004

Duke Math. J., 171, (1), 209-241, 2022.1
We establish a compact analogue of the P=W conjecture. For a projective irreducible holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result: one on the cohomology of the base and fibers of a Lagrangian fibration, and the other on the refined Gopakumar–Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.
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@inproceedings{junliang2022topology,
  title={Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds},
  author={Junliang Shen, and Qizheng Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316104143688548973},
  booktitle={Duke Math. J.},
  volume={171},
  number={1},
  pages={209-241},
  year={2022},
}
Junliang Shen, and Qizheng Yin. Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds. 2022. Vol. 171. In Duke Math. J.. pp.209-241. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316104143688548973.
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