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

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 $f_k$.