Cohomology of contact loci

Abstract

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of the their tangent cones are homotopy equivalent.

Correction: Prop. 1.2 is false: if the ample divisor W is replaced by eW with e>d, the new spectral sequence has zero differentials for the first e pages. The last sentence of the proof of Prop. 1.2 is not true. Prop 1.2 does not affect the rest of the paper