Geometry & Topology

Floer cohomology, multiplicity and the log canonical threshold

Mark McLean

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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 .

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1555466434

Digital Object Identifier
doi:10.2140/gt.2019.23.957

Mathematical Reviews number (MathSciNet)
MR3939056

Zentralblatt MATH identifier
07056057

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.gt/1555466434


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