Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold

TL;DR

This paper investigates the asymptotic behavior of holomorphic sections of a prequantum line bundle over a Kähler manifold near a Bohr-Sommerfeld Lagrangian submanifold, revealing convergence properties related to tangent bundle structures.

Contribution

It introduces new results on the boundedness and convergence of holomorphic sections in the large tensor power limit, connecting complex and real polarizations.

Findings

Boundedness of $L^2$-norms near $X$ for converging sections

Convergence of scaled sections to fiberwise constant functions

Link between polarization convergence and section behavior

Abstract

Let $(M, \omega)$ be a K\"ahler 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 $\Psi_{k}: TX \to TX$ be a multiplication by $\frac{1}{\sqrt{k}}$ in the fibers. The pullback of the K\"ahler polarization by $\Psi_k$ 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 $\Psi^* f_k$ converges to a fiberwise constant function on $TX$ as $k \to \infty$ under some condition on Sobolev norms of $f_k$.