Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

Abstract

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kähler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric 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. In the second part, 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. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.

v1: 32 pages, 5 figures; v2: 1 figure added; v3: 1 reference added; v4: some reorganization, 1 theorem (now 1.1) added; v5: final version, to appear in JDG

References in corpus (5)

• Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type
• Mirror Symmetry via Logarithmic Degeneration Data I
• Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers
• The spectral density function of a toric variety
• Holomorphic curves in Exploded Torus Fibrations: Compactness

Cited by in corpus (7)

• Adiabatic Limit, Theta Function, and Geometric Quantization
• Geometric quantization via SYZ transforms
• The Riemann-Roch strategy, Complex lift of the Scaling Site
• Spectral convergence in geometric quantization -- the case of non-singular Langrangian fibrations
• The geometric quantizations and the measured Gromov-Hausdorff convergences
• Spectral convergence in geometric quantization on $K3$ surfaces
• Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics