Cohomology of contact loci

TL;DR

This paper constructs a spectral sequence linking the cohomology of contact loci of complex polynomials with known Floer cohomology sequences, proposing a topological conjecture relating Milnor fibers and tangent cones.

Contribution

It introduces a spectral sequence for contact loci cohomology and connects it to Floer cohomology, proposing a new topological conjecture for holomorphic function germs.

Findings

Spectral sequence explicitly described via log resolution.

Connection established with McLean's spectral sequence.

Conjecture on homotopy equivalence of Milnor fibers based on topological equivalence.

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.