ÿþ<html><head><title>Symplectic geometry seminar</title></head> <br> <br> <br> <br> <center> <h1> University of Toronto's Symplectic Geometry Seminar </h1> <hr width="100%"> <br> January 23, 2008, 3:10pm <br> <br> <br> <br> <center> <h2>Takahiko Yoshida </h2></center> <center> <h3> University of Tokyo </h3> </center> <p></p> <p> </p><h2> On counting lattice points and Riemann-Roch numbers in Lagrangian fibrations. </h2> <p></p> </center> <br> <br> <br> <blockquote> <b> Abstract: </b> <p> For a compact symplectic manifold <img src="xom.gif" align="middle" alt="(X, É)">, a Riemann- Roch number <img src="RR.gif" align="middle" alt="RR(X, É)"> is defined by</p> <center><p><img src="form.gif" align="middle" alt="RR(X, É) =+"XeÉTd(X)">.</p></center> When <img src="xom.gif" align="middle" alt="(X, É)"> is a prequantizable symplectic toric manifold, it is wellknown that <img src="RR.gif" align="middle" alt="RR(X, É)"> 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 <img src="xom.gif" align="middle" alt="(X, É)"> be a 2n-dimensional closed, connected symplectic manifold and <img src="Bn.gif" align="middle" alt="Bn"> an n-dimensional manifold with corners. A map <img src="map.gif" align="middle" alt="¼: (X, É) ’!B"> is called a locally toric Lagragian fibration if <img src="mu.gif" align="middle" alt="¼"> is locally identified with the map <img src="maptwee.gif" align="middle" alt="¼Cn : (Cn, ÉCn) ’! Rn +"> defined by</p> <center><p><img src="defmaptwee.gif" align="middle" alt="¼Cn(z) = (|z1|2, . . . , |zn|2).">.</p></center> <p> For a locally toric Lagrangian fibration <img src="map.gif" align="middle" alt="¼: (X, É) ’!B">, in general, <img src="xom.gif" align="middle" alt="(X, É)"> no longer admits a global torus action, but locally it does. In this talk, we investigate <img src="RR.gif" align="middle" alt="RR(X, É)"> in terms of the local torus actions in the case where <img src="xom.gif" align="middle" alt="(X, É)"> is prequantizable. Under an appropriate condition, we can generalize a moment map, and we see that the above relationship still holds for some example.</p> </blockquote> <br> <br> <br> <br> <br> <br> <hr width="100%"> <br> </body></html>