A localization technique for indices of Dirac-type operators and singular Lagrangian fibrations

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We studied applications of the theory of local indices for Dirac-type operators on open Riemannian manifolds developed by the joint work with Furuta and Fujita to the geometric quantization of Lagrangian fibrations. We computed the local index for a certain type of singular fibers of Lagrangian fibrations. We refined the theory of local indices in the case of torus actions, and, as applications, we reproved Danilov's theorem for the equivariant Riemann-Roch indices of projective toric varieties and the Guillemin-Sternberg conjecture, concerning the commutativity of the quantization and the symplectic reduction, in the case of Hamiltonian torus actions.