Geometry & Topology

Floer cohomology, multiplicity and the log canonical threshold

Mark McLean

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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 .

Article information

Geom. Topol., Volume 23, Number 2 (2019), 957-1056.

Received: 7 March 2018
Revised: 11 August 2018
Accepted: 11 October 2018
First available in Project Euclid: 17 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J17: Singularities [See also 14B05, 14E15] 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45] 53D10: Contact manifolds, general 53D40: Floer homology and cohomology, symplectic aspects

log canonical threshold Floer cohomology singularity Zariski conjecture multiplicity symplectic geometry contact geometry


McLean, Mark. Floer cohomology, multiplicity and the log canonical threshold. Geom. Topol. 23 (2019), no. 2, 957--1056. doi:10.2140/gt.2019.23.957.

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