Communications in Number Theory and Physics
Volume 5 (2011)
Algebraic $K$-theory of toric hypersurfaces
Pages: 397 – 600
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.