Lagrangian tori in four-dimensional Milnor fibres

TL;DR

This paper constructs exact Lagrangian tori in four-dimensional Milnor fibres of non-simple singularities, revealing new structures in their Fukaya categories and advancing mirror symmetry understanding for these complex singularities.

Contribution

It introduces the first examples of Lagrangian tori in Milnor fibres of non-simple singularities, expanding the known types of Lagrangians beyond spheres.

Findings

Existence of Lagrangian tori in all non-simple four-dimensional Milnor fibres.

Fukaya categories of these fibres are not generated solely by vanishing cycles.

Progress towards mirror symmetry for unimodal singularities.

Abstract

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.