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Selecta Mathematica

, Volume 24, Issue 2, pp 1411–1452 | Cite as

Homological mirror symmetry for hypersurface cusp singularities

  • Ailsa Keating
Open Access
Article

Abstract

We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface \(Y_{p,q,r}\). By using localization techniques on both sides, we get an isomorphism between the derived wrapped Fukaya category of the Milnor fibre and the derived category of coherent sheaves on a quasi-projective surface given by deleting an anti-canonical divisor D from \(Y_{p,q,r}\). In the cusp case, the pair \((Y_{p,q,r}, D)\) is naturally associated to the dual cusp singularity, tying into Gross, Hacking and Keel’s proof of Looijenga’s conjecture.

Mathematics Subject Classification

53D37 (primary) 14J33 (secondary) 

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1. CambridgeUK

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