Large Complex Structure Limits, Quantization and Compact Tropical Amoebas on Toric Varieties

We consider toric deformations of complex structures, described by the symplectic potentials of Abreu and Guillemin, with degenerate limits of the holomorphic polarization corresponding to the toric Lagrangian fibration, in the sense of geometric quantization. This allows us to interpolate continuously between quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. Under these deformations, the original toric metric is deformed to nonequivalent toric metrics. We use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. For a particularly relevant one-parameter family of deformations, the limiting tropical amoebas are compact and live naturally in the polytope, image of the moment map. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry. We also comment briefly on the relation of our results to some recent approaches to homological mirror symmetry.