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Lagrangian topology and enumerative geometry

Paul Biran and Octav Cornea

Geometry & Topology 16 (2012) 963–1052

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.

Lagrangian submanifold, quantum homology, Floer homology, quadratic form, toric manifold
Mathematical Subject Classification 2010
Primary: 53D12, 53D40
Received: 14 June 2011
Revised: 3 February 2012
Accepted: 4 March 2012
Published: 22 May 2012
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Simon Donaldson
Paul Biran
Department of Mathematics
Rämistrasse 101
CH-8092 Zürich
Octav Cornea
Department of Mathematics and Statistics
University of Montreal
CP 6128
succ. Centre-ville Montréal QC H3C 3J7