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Geometric quantization of integrable systems with hyperbolic singularities. (English) Zbl 1191.53058
The main result of this very interesting paper is Theorem 6.1: Let \((M,\omega,F)\) be a two dimensional, compact, completely integrable system whose moment map has only nondegenerate singularities. Suppose \(M\) has a prequantum line bundle \(\mathbb L\) and let \({\mathcal J}\) be the sheaf of sections of \(\mathbb L\) flat along the leaves. The cohomology \(H^1(M, {\mathcal J})\) has two contributions of the form \(\mathbb C^{\mathbb N} \) for each hyperbolic singularity, each one corresponding to a space of Taylor series in one complex variable. It also has one \(\mathbb C\) term for each non-singular Bohr-Sommerfeld leaf. The cohomology in other degrees is zero.

MSC:
53D50 Geometric quantization
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