Floer cohomology, multiplicity and the log canonical threshold

McLean, Mark

doi:10.2140/gt.2019.23.957

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Floer cohomology, multiplicity and the log canonical threshold

Mark McLean

Geometry & Topology 23 (2019) 957–1056

DOI: 10.2140/gt.2019.23.957

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 $E^{1}$ 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

Volume 23, issue 2 (2019)