Irreducible symplectic varieties with a large second Betti number
-
Yuchen Liu
,
Zhiyu Liu
and
Chenyang Xu
Abstract
We prove a general result on the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations. As an application, we show that the relative Jacobian fibration of cubic fivefolds containing a fixed cubic fourfold can be compactified by a ℚ-factorial terminal irreducible symplectic variety with the second Betti number at least 24, and admits a Lagrangian fibration whose base is a weighted projective space. In particular, it belongs to a new deformation type of irreducible symplectic varieties.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2237139
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 123B2002
Funding statement: Yuchen Liu was partially supported by NSF CAREER Grant DMS-2237139 and an AT&T Research Fellowship from Northwestern University. Zhiyu Liu is partially supported by NSFC Grant 123B2002. Chenyang Xu was partially supported by a Simons Investigator grant and a Simons collaboration grant.
Acknowledgements
Zhiyu Liu would like to thank his supervisor Yongbin Ruan for encouragement and support. Chenyang Xu would like to thank Daniel Huybrechts, Giulia Saccà, and Richard Thomas who are co-organizers of the Oberwolfach workshop “Algebraic Geometry: Wall Crossing and Moduli Spaces, Varieties and Derived Categories”, where his interest in this topic was raised. The collaboration was started at the National Algebraic Geometry Conference organized by Shandong University, to which we would like to thank the organizers. We would like to thank Emanuele Macrì, Kieran O’Grady, and Giulia Saccà for discussions and pointing out inaccuracies in our early draft, and Giulia Saccà for informing us about her work [51]. We would like to thank Luca Tasin for pointing out a mistake in the previous version. We would also like to thank Lie Fu, Hanfei Guo, Xiaolong Liu, Songtao Ma, Mirko Mauri, Alexander Perry, Yongbin Ruan, Junliang Shen, Rui Xiong, Qizheng Yin, and Xiaolei Zhao for many useful discussions on related topics. We thank the anonymous referee for a careful reading as well as a list of useful suggestions that improved the exposition of the paper, especially the simplification of the proof of Theorem 4.17.
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Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the 𝑆-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- Colding–Minicozzi entropies in Cartan–Hadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical Lang–Siegel problem
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the 𝑆-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- Colding–Minicozzi entropies in Cartan–Hadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical Lang–Siegel problem
