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Mathematics > Algebraic Geometry

arXiv:1812.10673 (math)
[Submitted on 27 Dec 2018 (v1), last revised 23 Jan 2021 (this version, v5)]

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

Authors:Junliang Shen, Qizheng Yin
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Abstract:We establish a compact analog of the P = W conjecture. For a 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 topology of the base and fibers of a Lagrangian fibration, 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.
Comments: 29 pages. Added Appendix B by Claire Voisin. Final version accepted at Duke Mathematical Journal
Subjects: Algebraic Geometry (math.AG); Differential Geometry (math.DG)
Cite as: arXiv:1812.10673 [math.AG]
  (or arXiv:1812.10673v5 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1812.10673

Submission history

From: Junliang Shen [view email]
[v1] Thu, 27 Dec 2018 09:32:30 UTC (19 KB)
[v2] Mon, 7 Jan 2019 12:01:45 UTC (21 KB)
[v3] Wed, 27 Feb 2019 22:48:10 UTC (21 KB)
[v4] Thu, 23 May 2019 16:45:53 UTC (23 KB)
[v5] Sat, 23 Jan 2021 05:13:25 UTC (27 KB)
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