Legendrian Distributions with Applications to Poincar\'e Series

TL;DR

This paper develops a quantization scheme for Bohr-Sommerfeld Lagrangian submanifolds on compact Kähler manifolds, providing asymptotic estimates for associated Poincaré series and their inner products.

Contribution

It introduces a novel method to associate holomorphic sections to Bohr-Sommerfeld Lagrangians, with asymptotic analysis of their norms and inner products, including applications to Poincaré series.

Findings

Asymptotic estimates for norms of sections concentrating on Lagrangians.

Asymptotic expansion of inner products for intersecting Lagrangians.

Application of the scheme to Poincaré series on hyperbolic surfaces.

Abstract

Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |\Lambda, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|\Lambda, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $\Lambda$, and a sequence itself has a symbol which is a half-form, $\sigma$, on $\Lambda$. We prove estimates, as $k\to\infty$, of the norm squares $\langle \Lambda, k|\Lambda, k\rangle$ in terms of $\int_\Lambda \sigma\overline{\sigma}$. More generally, we show that if $\Lambda_1$ and $\Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langle\Lambda_1, k|\Lambda_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $\Lambda_1\cap\Lambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincar\'e series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.