Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold
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2025
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| _version_ | 1866912194468249600 |
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| author | Tiba, Yusaku |
| author_facet | Tiba, Yusaku |
| contents | Let $(M, ω)$ be a Kähler 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 $Ψ_{k}: TX \to TX$ be a multiplication by $\frac{1}{\sqrt{k}}$ in the fibers. The pullback of the Kähler polarization by $Ψ_k$ 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 $Ψ^* f_k$ converges to a fiberwise constant function on $TX$ as $k \to \infty$ under some condition on Sobolev norms of $f_k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold Tiba, Yusaku Complex Variables Symplectic Geometry 32A25, 53D12 Let $(M, ω)$ be a Kähler 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 $Ψ_{k}: TX \to TX$ be a multiplication by $\frac{1}{\sqrt{k}}$ in the fibers. The pullback of the Kähler polarization by $Ψ_k$ 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 $Ψ^* f_k$ converges to a fiberwise constant function on $TX$ as $k \to \infty$ under some condition on Sobolev norms of $f_k$. |
| title | Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold |
| topic | Complex Variables Symplectic Geometry 32A25, 53D12 |
| url | https://arxiv.org/abs/2501.11254 |