Homological mirror symmetry for hypersurface cusp singularities

TL;DR

This paper establishes a deep connection between the symplectic geometry of Milnor fibres of certain singularities and algebraic geometry of rational surfaces, confirming instances of homological mirror symmetry.

Contribution

It demonstrates the equivalence of derived categories for specific hypersurface singularities and associated rational surfaces, extending homological mirror symmetry to new classes of singularities.

Findings

Derived directed Fukaya categories are equivalent to derived categories of coherent sheaves.

Isomorphism between wrapped Fukaya categories and coherent sheaves on quasi-projective surfaces.

Connections to dual cusp singularities and Looijenga's conjecture.

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.