Toric K\"ahler metrics seen from infinity, quantization and compact tropical amoebas
Toric K\"ahler metrics seen from infinity, quantization and compact tropical amoebas
复制标题
从无穷大、量化和紧凑的热带变形虫中看到的 Toric K"ahler 度量
DOI：
10.4310/jdg/1335207374
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
J. P. Nunes
中科院分区：
文献类型：
--
作者：
Thomas Baier;C. Florentino;J. Mourão;J. P. Nunes
我们考虑紧致复曲面流形上的所有复曲面K\“ahler度量的度量空间;当“从无穷远看它”时（遵循Gromov），我们得到无穷远处的切锥，它由完全测地线的等价类参数化。本文研究了复曲面簇上度量族的相关极限、它的量子化和通有因子的退化。
相应的K\“ahler偏振的极限沿着由矩映射定义的拉格朗日纤维化退化。这使我们能够在全纯和真实的偏振的几何量子化之间连续插值，并表明前量子束的单项式全纯部分收敛到支持玻尔-索末菲光纤的狄拉克δ分布。
在第二部分中，我们使用这些家庭的复曲面度量退化研究的限制紧超曲面变形虫，并表明，在勒让德变换变量，他们描述的热带变形虫。我们相信，我们的方法提供了一个不同的，互补的，复杂的代数几何和热带几何之间的关系的角度。
We consider the metric space of all toric K\"ahler 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 K\"ahler 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.