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15 January 2022 Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds
Junliang Shen, Qizheng Yin
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Duke Math. J. 171(1): 209-241 (15 January 2022). DOI: 10.1215/00127094-2021-0010

Abstract

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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Junliang Shen. Qizheng Yin. "Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds." Duke Math. J. 171 (1) 209 - 241, 15 January 2022. https://doi.org/10.1215/00127094-2021-0010

Information

Received: 20 June 2019; Revised: 2 December 2020; Published: 15 January 2022
First available in Project Euclid: 17 January 2022

Digital Object Identifier: 10.1215/00127094-2021-0010

Subjects:
Primary: 14D06
Secondary: 14F45 , 14N35

Keywords: curve counting , Hodge theory , hyper-Kähler manifolds , Lagrangian fibrations , perverse sheaves

Rights: Copyright © 2022 Duke University Press

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Vol.171 • No. 1 • 15 January 2022
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