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

Abstract

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.

Complex time evolution in geometric quantization and generalized coherent state transforms

“…However, the pairing itself is in general not equal to the parallel transport of the connection it induces and is (in general) not unitary. Examples of nonunitary BKS pairing maps are given by torus-invariant Kähler structures on compact symplectic toric manifolds [BFMN11,KMN10]). …”

Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings

“…This is partly due to the fact that the observables preserving mixed polarizations are likely to be physically more interesting than those preserving a Kähler polarization. In the present paper we continue along the lines proposed in [BFMN,KW2,KMN1,Ki], which motivate the definition of the quantization for real or mixed polarizations via degeneration of quantizations on suitable families of Kähler polarizations.…”

“…This problem was studied in the toric case in [BFMN,KMN1] and in general in [MN2] and we will review now some of the results. Let us assume that the level sets L c are compact for noncritical values c ∈ R n and that a function G : R n → R exists such that h 2 = G • µ is strongly convex as a function of all action variables on equivariant neighborhoods of all regular fibers.…”

“…in an appropriate (weak) sense [BFMN,MN2]. We will also assume that there are local J 0 -holomorphic coordinates, {u j } j=1,...,n , such that pointwize, on a neighborhood of every point, one has lim t→∞ β(t) du …”

“…Building up on previous works [BFMN,KW2,KMN1,MN1], in [MN2] a general strategy is proposed to find families of well behaved polarizations degenerating to a wide class of singular real polarizations corresponding to the level sets of completely integrable systems.…”