Adiabatic limit, theta function, and geometric quantization (Q6604012)

The paper establishes the following facts related to an adiabatic limit for Lagrangian torus fibrations over complete bases.\N\NLet \((M,\omega)\) be a symplectic manifold, \(\pi: (M,\omega)\rightarrow B\) be a Lagrangian torus fibration over a complete base \(B\) with prequantum line bundle \((L,\nabla^L)\rightarrow(M,\omega), \) J\( \) be a compatible almost complex structure of \((M,\omega)\) invariant along the fiber of \(\pi\). The author presents a one-parameter family \(\{J^t\}_{t>0}\) of compatible almost complex structures of \((M,\omega)\) with \(J^1=J\) such that the fiber shrinks with respect to the associated Riemannian metrics as \(t\) goes to \(\infty\). For \(t>0\) and a positive integer \(N\), let \(D^t\) be the \(\text{Spin}^c\) Dirac operator with coefficients in \(L^{\otimes N}\) associated with \(J^t\).\N\NTheorem 1. Under the above setting, assume that \(J\) is integrable and satisfies a certain technical condition. Then, for each \(t>0\), there exists a complete orthogonal system \(\{v^t_b\}_{b\in B_{BS}}\) of holomorphic \(L^2\)-sections of \(L^{\otimes N}\rightarrow(M,N\omega,J^t)\) indexed by the Bohr-Sommerfeld points \(B_{BS}\) such that each \(v^t_b\) converges as a delta-function section supported on \(\pi^{-1}(b)\) as \(t\to\infty\) in the following sense, for any \(L^2\)-section \(s\) on \(L^{\otimes N}\), we have\N\[\N\lim\limits_{t\to\infty}\int\limits_M\left\langle s,\frac{v^t_b}{||v^t_b||_{L^1}}\right\rangle_{l^{\otimes N}}(-1)^{\frac{n(n-1)}{2}}\frac{\omega^n}{n!}=\int\limits_{\pi^{-1}(b)}\langle s,\delta_b\rangle_ {L^{\otimes N}}|dy|,\N\]\Nwhere \(\langle , \rangle\) is the Hermitian metric of \(L^{\otimes N}\), \(\delta_b\) is the covariant constant section of \((L,\nabla^L)^{\otimes N}\), and \(|dy|\) is the natural one-density on \(\pi^{-1}(b)\).\N\NTheorem 2. Under the above setting, for each \(t>0\), there exists an orthogonal family \(\{\widetilde v^t_b\}_{b\in B_{BS}}\) of \(L^2\)-sections on \(L^{\otimes N}\) indexed by \(B_{BS}\) such that\N\N\((1)\) each \(\widetilde v^t_b\) converges as a delta-function section supported on \(\pi^{-1}(b)\) as \(t\to\infty\) in the above sense, and\N\N\((2)\) \(\lim\limits_{t\to\infty}||D^t\widetilde v^t_b||_{L^2}=0\).\N\NThe basic constructions have explicit descriptions.