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The variation operator in singularity theory maps relative homology cycles tocompact cycles in the Milnor fiber using the monodromy. We construct itssymplectic analogue for an isolated singularity. We define a new Floercohomology, called monodromy Lagrangian Floer cohomology, which providescategorifications of the standard theorems on the variation operator and theSeifert form. The key ingredients are a special class $\Gamma$ in thesymplectic cohomology of the inverse of the monodromy and its closed-openimages. For isolated plane curve singularities whose A'Campo divide has depthzero, we find an exceptional collection consisting of non-compact Lagrangiansin the Milnor fiber corresponding to a distinguished collection of vanishingcycles under the variation operator.

Floer theory for the variation operator of an isolated singularity