Floer Cohomology and Arc Spaces

Let f be a polynomial over the complex numbers with an isolated singular point at the origin and let d be a positive integer. To such a polynomial we can assign a variety called the dth contact locus of f. Morally, this corresponds to the space of d-jets of holomorphic disks in complex affine space whose boundary `wraps' around the singularity d times. We show that Floer cohomology of the dth power of the Milnor monodromy map is isomorphic to compactly supported cohomology of the dth contact locus. This answers a question of Paul Seidel and it also proves a conjecture of Nero Budur, Javier Fernández de Bobadilla, Quy Thuong Lê and Hong Duc Nguyen. The key idea of the proof is to use a jet space version of the PSS map together with a filtration argument.