Communications in Number Theory and Physics

Volume 5 (2011)

Number 2

Algebraic $K$-theory of toric hypersurfaces

Pages: 397 – 600



Charles F. Doran (Department of Mathematical and Statistical Sciences, University of Alberta, Canada)

Matt Kerr (Department of Mathematics, Washington University in St. Louis, Missouri)


We construct classes in the motivic cohomology of certain1-para\-meter families of Calabi–Yau hypersurfaces in toricFano $n$-folds, with applications to local mirror symmetry(growth of genus 0 instanton numbers) and inhomogeneousPicard–Fuchs equations. In the case where the family isclassically modular the classes are related to Beilinson'sEisenstein symbol; the Abel–Jacobi map (or rationalregulator) is computed in this paper for both kinds ofcycles. For the “modular toric” families where the cyclesessentially coincide, we obtain a motivic (andcomputationally effective) explanation of a phenomenonobserved by Villegas, Stienstra, and Bertin.

Published 26 September 2011