Homological mirror symmetry for hypersurface cusp singularities

• Ailsa Keating ^[1]
  1. [1] University of Cambridge

     University of Cambridge

     Cambridge District, Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 2, 2018, págs. 1411-1452
• Idioma: inglés
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• Resumen

  • 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 Yp,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 Yp,q,r . In the cusp case, the pair (Yp,q,r,D) is naturally associated to the dual cusp singularity, tying into Gross, Hacking and Keel’s proof of Looijenga’s conjecture.