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Symplectic topology of $K3$ surfaces via mirror symmetry


Authors: Nick Sheridan and Ivan Smith
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
Published electronically: June 9, 2020
MathSciNet review: 4127905
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Abstract: We study the symplectic topology of certain $K3$ surfaces (including the “mirror quartic” and “mirror double plane”), equipped with certain Kähler 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.


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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.
Article copyright: © Copyright 2020 American Mathematical Society