Symplectic properties of Milnor fibres

We present two results relating to the symplectic geometry of the Milnor fibres of isolated affine hypersurface singularities. First, given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres. In the second half of this thesis, we study exact Lagrangian tori in Milnor fibres. The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2014

Author dc:creator

• Keating, Ailsa Macgregor

Advisor dc:contributor.advisor

• Paul A. Seidel.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/90186

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/90186