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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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##### References:
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