Volume 16, issue 2 (2012)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Lagrangian topology and enumerative geometry

Paul Biran and Octav Cornea

Geometry & Topology 16 (2012) 963–1052
Abstract

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.

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