Locally toric manifolds and singular Bohr-Sommerfeld leaves by Mark D. Hamilton

By Mark D. Hamilton

A true polarization modelled on fibres of the instant map. the writer computes the implications without delay and obtains a theorem just like Sniatycki's, which supplies the quantization by way of counting Bohr-Sommerfeld leaves. notwithstanding, the count number doesn't contain the Bohr-Sommerfeld leaves that are singular. hence the quantization received isn't like the quantization got utilizing a Kohler polarization

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3. The bundle LN is trivializable over V ⊂ N . Proof. The hypotheses on our spaces guarantee that we can choose V to be of the form I m × T m × (D2 )k , with the leaf N being identified with the central torus N ∼ = {tm } × T m × {0}. A transverse neighbourhood is just I m × D2k , which is contractible; therefore LN is trivializable over it. There is a free T m action on V (just act on the T m coordinate), which “sweeps out” the transverse disc over the neighbourhood V . This action gives us a trivialization over the whole neighbourhood.

All 1-cochains are cocycles. 2. In order for the cochain {aEF , aF G , aGE } to be a coboundary, we need a 1-cochain {bE , bF , bG } which is a primitive. 10c) aEF eitn θE ei t,θ itn θF i t,θ aF G e e itn θG i t,θ aGE e e = bF eitn θF ei t,θ − bE eitn θE ei t,θ on E ∩ F itn θF i t,θ e on F ∩ G itn θG i t,θ on G ∩ E itn θG i t,θ − bF e itn θE i t,θ − bG e = bG e = bE e e e e Here each of the a’s and b’s are germs in t1 , . . tn . 2 goes through unchanged, working with germs rather than functions.

Since V does not contain 0, this means there is a disc around zero outside V ; and since ρV is zero outside V , f must therefore be zero on this disc, and therefore on any disc in B ∩ U . 2. Let U ⊂ C be either an annulus or a disc, centred at the origin. Then H 1 (U, J ) ∼ = Cm where m is the number of Bohr-Sommerfeld leaves, excluding the origin, contained in U . CHAPTER 5 Example: S 2 The simplest example of a toric manifold, which is ubiquitous1 in textbooks on symplectic geometry, is S 2 with an action of S 1 by rotations about the z-axis.

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