Toric Degenerations of Fano Varieties and Constructing Mirror Manifolds

Abstract

For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.

13 pages, AMS-Latex. This is an extended version of the author's talk given during the Summer Symposium on Algebra at University of Niigata, July 22-25, 1997