Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds

TL;DR

The paper studies the semiclassical behavior of holomorphic sections of $L^k$ restricted to a Bohr-Sommerfeld Lagrangian submanifold $X$ inside a Kähler manifold $(M,\omega)$. It proves a general limsup upper bound for the normalized infimum of section norms on $X$ and then shows that this bound is sharp under several natural hypotheses (projective, Stein with Ricci bounds, or pseudoconvex domains) by constructing localized sections and correcting them with Hörmander $L^2$-estimates. A key methodological contribution is reducing the problem to a real-analytic setting via a Monge-Ampère potential and applying Demailly's Jensen-Lelong formula to obtain precise upper bounds; a complementary lower-bound argument uses localization near $X$ and holomorphic extension to achieve matching asymptotics. Together, these results provide quantitative semiclassical estimates for the quantization of Bohr-Sommerfeld Lagrangian submanifolds, connecting geometric quantization with complex-analytic methods and extending prior results in the literature.

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 Kähler form $ω$. Let $X \subset M$ be a Lagrangian submanifold of $(M, ω)$. When $X$ satisfies the Bohr-Sommerfeld condition, we give an asymptotic estimate of the norm $|f|_{h^k}$ on $X$ for $f \in H^0(M, L^k)$.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• proof
• Remark 1
• Lemma 1
• proof
• Theorem 4
• Lemma 2