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Inventiones mathematicae

, Volume 204, Issue 2, pp 505–594 | Cite as

Reeb orbits and the minimal discrepancy of an isolated singularity

  • Mark McLeanEmail author
Article

Abstract

Let A be an affine variety inside a complex N dimensional vector space which has an isolated singularity at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We define an invariant of the link up to contactomorphism using Conley–Zehnder indices of Reeb orbits and then we relate this invariant with the minimal discrepancy. As a result we show that the standard contact five dimensional sphere has a unique Milnor filling up to normalization proving a conjecture by Seidel.

Notes

Acknowledgments

I would like to thank Chris Wendl, Ivan Smith, Cheuk Yu Mak and Paul Seidel (who enabled me to generalize the result from numerically Gorenstein to numerically \({\mathbb {Q}}\)-Gorenstein singularities). I would also like to thank the referees for numerous helpful comments which have improved this paper. The main work for this paper was done while I was at the University of Aberdeen and therefore I would like to thank the people at the mathematics department for providing a great working environment.

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Stony Brook UniversityStony BrookUSA

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