論文

This paper is a sequel to [11]. We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of ∂-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost Kähler quantization for compatible almost complex structures, and show the analogous convergence results.

We construct smooth families of compact special Lagrangian submanifolds embedded in some toric hyper-Kähler manifolds, which never become holomorphic Lagrangian submanifolds via any hyper-Kähler rotations. These families converge to special Lagrangian immersions with self-intersection points in the sense of currents. To construct them, we apply the desingularization method developed by Joyce.

We study the geometric quantization on K3 surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the K3 surfaces and a family of hyper-Kähler structures tending to large complex structure limit, and show a spectral convergence of the ̄∂-Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr-Sommerfeld fibers.

We generalize the notion of calibrated submanifolds to smooth maps and show that several kinds of smooth maps appearing in the differential geometry are applicable to our situation. Moreover, we apply this notion to give the lower bound to some energy functionals of smooth maps in the given homotopy class between Riemannian manifolds and consider the energy functional which is minimized by the identity maps on the Riemannian manifolds with special holonomy groups.

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.

We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-Kähler metrics on the K3 surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the K3 surface with a hyper-Kähler metric to a flat Riemannian orbifold of dimension 3, then show that it gives a lower bound of the energy. Moreover, we show that the ratio of the energy to the invariant converges to 1 for Foscolo’s collapsing families.