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Selecta Mathematica

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

Homological mirror symmetry for hypersurface cusp singularities

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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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Authors and Affiliations

  1. 1. CambridgeUK

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