## Symplectic topology of $K3$ surfaces via mirror symmetry

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Nick Sheridan and Ivan Smith
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## Abstract:

We study the symplectic topology of certain $K3$ surfaces (including the âmirror quarticâ and âmirror double planeâ), equipped with certain KĂ€hler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic $K3$ surface of Picard rank one.## References

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## Additional Information

**Nick Sheridan**- Affiliation: School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
- MR Author ID: 962118
- ORCID: 0000-0002-6299-0682
**Ivan Smith**- Affiliation: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- MR Author ID: 650668
- Received by editor(s): December 3, 2017
- Received by editor(s) in revised form: July 11, 2019, and November 19, 2019
- Published electronically: June 9, 2020
- Additional Notes: The first author was supported in part by a Sloan Research Fellowship, a Royal Society University Research Fellowship, and by the National Science Foundation through Grant number DMS-1310604 and under agreement number DMS-1128155. The first author also acknowledges support from Princeton University and the Institute for Advanced Study.

The second author was supported in part by a Fellowship from EPSRC. - © Copyright 2020 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**33**(2020), 875-915 - MSC (2010): Primary 53D37; Secondary 53D12, 14F05, 14J28
- DOI: https://doi.org/10.1090/jams/946
- MathSciNet review: 4127905