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Floer cohomology, multiplicity and the log canonical threshold

Mark McLean

Geometry & Topology 23 (2019) 957–1056
Abstract

Let f be a polynomial over the complex numbers with an isolated singularity at 0. We show that the multiplicity and the log canonical threshold of f at 0 are invariants of the link of f viewed as a contact submanifold of the sphere.

This is done by first constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose E1 page is explicitly described in terms of a log resolution of f. This spectral sequence is a generalization of a formula by A’Campo. By looking at this spectral sequence, we get a purely Floer-theoretic description of the multiplicity and log canonical threshold of f.

Keywords
log canonical threshold, Floer cohomology, singularity, Zariski conjecture, multiplicity, symplectic geometry, contact geometry
Mathematical Subject Classification 2010
Primary: 14J17, 32S55, 53D10, 53D40
References
Publication
Received: 7 March 2018
Revised: 11 August 2018
Accepted: 11 October 2018
Published: 8 April 2019
Proposed: Leonid Polterovich
Seconded: Dan Abramovich, Yasha Eliashberg
Authors
Mark McLean
Department of Mathematics
SUNY Stony Brook
Stony Brook, NY
United States