Three conjectures on lagrangian tori in the projective plane

Details Three conjectures on lagrangian tori in the projective plane Three conjectures on lagrangian tori in the projective plane

2007-12-10

arxiv.org/abs/0712.1495v1

Mathematics

Symplectic Geometry

17 pages, no figas

Scientific paper

In this paper we extend the discussion on Homological Mirror Symmetry for Fano toric varieties presented by Hori and Vafa to more general case of monotone symplectic manifolds with real polarizations. We claim that the Hori -- Vafa prediction, proven by Cho and Oh for toric Fano varieties, can be checked in much more wider context. Then the notion of Bohr - Sommerfeld with respect to the canonical class lagrangian submanifold appears and plays an important role. The discussion presents a bridge between Geometric Quantization and Homological Mirror Symmetry programmes both applied to the projective plane in terms of its lagrangian geometry. Due to this relation one could exploit some standard facts known in GQ to produce results in HMS.

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