Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds

TL;DR

This paper provides asymptotic estimates for the norms of holomorphic sections over Bohr-Sommerfeld Lagrangian submanifolds in complex manifolds, advancing understanding in geometric quantization and asymptotic analysis.

Contribution

It introduces new asymptotic estimates for holomorphic sections restricted to Bohr-Sommerfeld Lagrangian submanifolds, extending previous results in geometric quantization.

Findings

Derived explicit asymptotic formulas for section norms

Established conditions under which estimates hold

Enhanced understanding of geometric quantization on Lagrangian submanifolds

Abstract

Let $M$ be a complex manifold and $L$ be a line bundle over $M$ with a Hermitian metric $h$ whose Chern form is a K\"ahler form $\omega$. Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$. When $X$ satisfies the Bohr-Sommerfeld condition, we give an asymptotic estimate of the norm $|f|_{h^k}$ on $X$ for $f \in H^0(M, L^k)$.