Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold
Yusaku Tiba
TL;DR
The paper studies the semiclassical behavior of holomorphic sections of a prequantum line bundle over a Kähler manifold near a Bohr-Sommerfeld Lagrangian submanifold X. It develops micro-local lower bounds for L^2-norms of sections in tubular neighborhoods via Bergman kernel techniques and Grauert-tube analysis, and establishes a convergence of pulled-back sections along the tangent bundle TX to fiberwise-constant data under Sobolev-type hypotheses, linking to a real polarization. Two parallel approximation frameworks are developed: (A) almost holomorphic extensions with Hörmander estimates, and (B) Bergman-kernel projections yielding s_{f,k}, with sharp lower bounds and a reproducing property guiding optimality. Finally, the work provides an asymptotic expansion for s_{f,k} in powers of k^{-1/2} using the Bergman kernel expansion, including off-diagonal decay, thereby connecting micro-local estimates, polarization limits, and semiclassical quantization in a coherent geometric-analytic framework.
Abstract
Let $(M, ω)$ be a Kähler manifold and let $(L, \nabla)$ be a prequantum line bundle over $M$. Let $X \subset M$ be a Bohr-Sommerfeld Lagrangian submanifold of $(L, \nabla)$. In this paper, we study an asymptotic behaviour of holomorphic sections of $L^k$ as $k \to \infty$. Our first result shows that the $L^2$-norm of sections of $L^k$ are bounded below around $X$ if these sections converge on $X$ under a suitable trivialization of $L^k$. Since $X$ is a Lagrangian submanifold, we consider that a neighborhood of $X$ is embedded in the tangent bundle $TX$. Let $Ψ_{k}: TX \to TX$ be a multiplication by $\frac{1}{\sqrt{k}}$ in the fibers. The pullback of the Kähler polarization by $Ψ_k$ converges to the real polarization, whose leaves are fibers of $TX$, as $k \to \infty$. Let $(f_k)_{k \in \mathbb{N}}$ be holomorphic sections of $L^k$ near $X$. By trivializing $L^k$, we consider $f_k$ as a function. In our second result, we show that $Ψ^* f_k$ converges to a fiberwise constant function on $TX$ as $k \to \infty$ under some condition on Sobolev norms of $f_k$.