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.
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Partially supported by NSF grant DMS-1505798 and by a Junior Fellow award from the Simons Foundation.
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Keating, A. Homological mirror symmetry for hypersurface cusp singularities. Sel. Math. New Ser. 24, 1411–1452 (2018). https://doi.org/10.1007/s00029-017-0334-6
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DOI: https://doi.org/10.1007/s00029-017-0334-6