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.
Highlights
A landmark application of the field of mirror symmetry is the recent proof by Gross, Hacking and Keel of Looijenga’s conjecture, about pairs of cusp singularities [17]
Cusp singularities come in naturally dual pairs; in [17], this duality gets strengthened to a mirror symmetry statement, of the flavour developed by Gross–Siebert (e.g. [20,21,22]) and Kontsevich–Soibelman (e.g. [27,28])
We prove versions of Kontsevich’s Homological Mirror Symmetry Conjecture [26] for spaces appearing in Gross, Hacking and Keel’s work
Summary
A landmark application of the field of mirror symmetry is the recent proof by Gross, Hacking and Keel of Looijenga’s conjecture, about pairs of cusp singularities [17]. Cusp singularities come in naturally dual pairs; in [17], this duality gets strengthened to a mirror symmetry statement, of the flavour developed by Gross–Siebert Let Tp,q,r denote the Milnor fibre of Tp,q,r This is a Liouville domain, which, as shown in [24, Section 2], is independent of choices, including the choice of representative for a germ and the constant a. In the classification of isolated hypersurface singularities by Arnol’d and collaborators, these are all but finitely many of the modality one singularities [9, Section I.2.3 and II.2.5]; missing are the fourteen so-called ‘exceptional’ singularities (known as the object of strange duality). Homological mirror symmetry for these is comparatively well understood—see for instance [5,13,15]
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