University of Toronto's Symplectic Geometry Seminar

January 23, 2008, 3:10pm

Bahen 3012

Takahiko Yoshida

University of Tokyo

On counting lattice points and Riemann-Roch numbers in Lagrangian fibrations.


For a compact symplectic manifold , a Riemann- Roch number is defined by


When is a prequantizable symplectic toric manifold, it is wellknown that can be computed by counting lattice points in the image of the moment map. In this talk, we observe this relationship for certain Lagrangian fibrations. Let be a 2n-dimensional closed, connected symplectic manifold and an n-dimensional manifold with corners. A map is called a locally toric Lagragian fibration if is locally identified with the map defined by


For a locally toric Lagrangian fibration , in general, no longer admits a global torus action, but locally it does. In this talk, we investigate in terms of the local torus actions in the case where is prequantizable. Under an appropriate condition, we can generalize a moment map, and we see that the above relationship still holds for some example.