Torus Fibrations and Localization of Index I -- Polarization and Acyclic Fibrations

J. Math. Sci. Univ. Tokyo
Vol. 17 (2010), No. 1, Page 1--26.

Fujita, Hajime; Furuta, Mikio; Yoshida, Takahiko
Torus Fibrations and Localization of Index I -- Polarization and Acyclic Fibrations
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Abstract:
We define a local Riemann-Roch number for an open symplectic manifold when a completely integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle.

Keywords: Geometric quantization, index theory, localization.

Mathematics Subject Classification (2000): Primary 53D50, Secondary 58J20
Received: 2008-04-18