Lagrangian tori in four-dimensional Milnor fibres
This paper constructs exact Lagrangian tori in the Milnor fibres of all non-simple hypersurface singularities of dimension two. It suffices to do this for the \(T_{p,q,r}\)-singularities, as the Milnor fibre of an adjacent singularity can be symplectically embedded in the original Milnor fibre. The basis of the construction is A'Campo's explicit model for the Milnor fibre of a plane curve singularity, starting from a real divide. By suspension the result for \(T_{p,q,2}\) follows directly, while for \(r\geq 3\) a deformation of suitable suspension is used. The explicit model allows to compute Floer cohomologies, and this is used to show that the Fukaya category of the Milnor fibre of \(T_{p,q,r}\) is not split-generated by any collection of vanishing cycles. Also the bounded derived directed Fukaya category is computed, and it is shown that it is equivalent to the bounded derived category of coherent sheaves on an orbifold \(\mathbb P^1\) with \((1/p,1/q,1/r)\)-isotropy.
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