Lagrangian topology and enumerative geometry

Biran, Paul¹ ; Cornea, Octav²

¹ Department of Mathematics, ETH-Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
² Department of Mathematics and Statistics, University of Montreal, CP 6128, succ. Centre-ville Montréal QC H3C 3J7, Canada

Geometry & topology, Tome 16 (2012) no. 2, pp. 963-1052.

Résumé

We analyze the properties of Lagrangian quantum homology (in the form constructed in our previous work, based on the pearl complex) to associate certain enumerative invariants to monotone Lagrangian submanifolds. The most interesting such invariant is given as the discriminant of a certain quadratic form. For $2$–dimensional Lagrangians it corresponds geometrically to counting certain types of configurations involving pseudoholomorphic disks that are associated to triangles on the respective surface. We analyze various properties of these invariants and compute them and the related structures for a wide class of toric fibers. An appendix contains an explicit description of the orientation conventions and verifications required to establish quantum homology and the related structures over the integers.

Détail

DOI : 10.2140/gt.2012.16.963

Classification : 53D12, 53D40
Keywords: Lagrangian submanifold, quantum homology, Floer homology, quadratic form, toric manifold

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