TY - JOUR

T1 - The geometric quantizations and the measured Gromov-Hausdorff convergences

AU - Hattori, Kota

N1 - Funding Information:
Partially supported by Grant-in-Aid for Young Scientists (B) Grant Number 16K17598 and by Grant-in-Aid for Scientific Research (C) Grant Number 19K03474.

PY - 2020

Y1 - 2020

N2 - On a compact symplectic manifold (X,ω) with a prequantum line bundle (L,∇,h), we consider the one-parameter family of ω-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are sev-eral researches which show that the holomorphic sections of the line bundle localize at Bohr-Sommerfeld fibers. In this article we consider the one-parameter family of the Riemannian metrics on the frame bundle of L determined by the complex structures and ∇,h, and we can see that their diameters diverge. If we fix a base point in some fibers of the Lagrangian fibration we can show that they measured Gromov-Hausdorff converge to some pointed metric measure spaces with the isometric S1-actions, which may depend on the choice of the base point. We observe that the properties of the S1-actions on the limit spaces actually depend on whether the base point is in the Bohr-Sommerfeld fibers or not.

AB - On a compact symplectic manifold (X,ω) with a prequantum line bundle (L,∇,h), we consider the one-parameter family of ω-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are sev-eral researches which show that the holomorphic sections of the line bundle localize at Bohr-Sommerfeld fibers. In this article we consider the one-parameter family of the Riemannian metrics on the frame bundle of L determined by the complex structures and ∇,h, and we can see that their diameters diverge. If we fix a base point in some fibers of the Lagrangian fibration we can show that they measured Gromov-Hausdorff converge to some pointed metric measure spaces with the isometric S1-actions, which may depend on the choice of the base point. We observe that the properties of the S1-actions on the limit spaces actually depend on whether the base point is in the Bohr-Sommerfeld fibers or not.

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U2 - 10.4310/JSG.2020.v18.n6.a3

DO - 10.4310/JSG.2020.v18.n6.a3

M3 - Article

AN - SCOPUS:85100714954

SN - 1527-5256

VL - 18

SP - 1575

EP - 1628

JO - Journal of Symplectic Geometry

JF - Journal of Symplectic Geometry

IS - 6

ER -