从无穷远处、量子化与紧致热带阿米巴视角看 Toric Kähler 度量
Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas
AI总结：
本文研究了紧致 toric 流形上 toric Kähler 度量空间在无穷远处的极限，揭示了 Kähler 极化在 Lagrangian 纤维上的退化，实现了全纯与实极化几何量子化的连续插值，并利用度量退化证明了紧致超曲面阿米巴极限可由热带阿米巴描述。
摘要：
我们考虑紧致 toric 流形上所有 toric Kähler 度量构成的度量空间；当“从无穷远处观察它”（遵循 Gromov 的方法）时，我们得到了无穷远处的切锥，它由完备测地线的等价类参数化。在本文中，我们研究了 toric 簇上度量族的相关极限、其量子化以及一般除子的退化。相应 Kähler 极化的极限在由矩映射定义的 Lagrangian 纤维上变得退化。这使我们能够在全纯极化和实极化的几何量子化之间进行连续插值，并表明预量子丛的单项全纯截面收敛到支撑在 Bohr-Sommerfeld 纤维上的 Dirac delta 分布。在第二部分中，我们利用这些 toric 度量退化族来研究紧致超曲面阿米巴的极限，并表明在 Legendre 变换变量中，它们由热带阿米巴描述。我们相信，我们的方法为复代数几何与热带几何之间的关系提供了一个不同且互补的视角。
英文摘要：
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.
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