Quantization of positive Lagrangians
Abstract
A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. It is called ε-positive if the ratio of the restrictions of the imaginary and real parts of the holomorphic volume form is bounded by 1/ε. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We prove that the Riemannian metric restricted to the subset of ε-positive Lagrangians induces a path metric when the relative Kähler class of the ambient Calabi-Yau manifold lies on a line of rational slope. The proof relies on the construction of a sequence of quantization maps from a circle bundle over the space of positive Lagrangians to the dual of the space of holomorphic sections of powers of a prequantum line bundle. The rationality condition for the relative Kähler class allows us to reduce to the case of Bohr-Sommerfeld Lagrangians. Building on ideas of Tian and Nirenberg-Wells, Hörmander estimates are used to show that functions on a Bohr-Sommerfeld Lagrangian can be approximated by holomorphic sections of a sufficiently high power of the prequantum line bundle. Consequently, the quantization maps eventually separate fibers of the circle bundle.
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BibTeX
@misc{solomon2026quantization,
title = {{Quantization of positive Lagrangians}},
author = {Jake P. Solomon and Bentzion Yavetz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20177},
eprint = {2610.20177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20177},
}
RIS
TY - UNPB
TI - Quantization of positive Lagrangians
AU - Jake P. Solomon
AU - Bentzion Yavetz
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20177
UR - https://arxiv.org/abs/2610.20177
AB - A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. It is called ε-positive if the ratio of the restrictions of the imaginary and real parts of the holomorphic volume form is bounded by 1/ε. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We prove that the Riemannian metric restricted to the subset of ε-positive Lagrangians induces a path metric when the relative Kähler class of the ambient Calabi-Yau manifold lies on a line of rational slope. The proof relies on the construction of a sequence of quantization maps from a circle bundle over the space of positive Lagrangians to the dual of the space of holomorphic sections of powers of a prequantum line bundle. The rationality condition for the relative Kähler class allows us to reduce to the case of Bohr-Sommerfeld Lagrangians. Building on ideas of Tian and Nirenberg-Wells, Hörmander estimates are used to show that functions on a Bohr-Sommerfeld Lagrangian can be approximated by holomorphic sections of a sufficiently high power of the prequantum line bundle. Consequently, the quantization maps eventually separate fibers of the circle bundle.
ER -