Toric Degenerations of Fano Varieties and Constructing Mirror Manifolds

TL;DR

This paper introduces small toric degenerations of Fano varieties and presents a new method for constructing mirror manifolds of Calabi-Yau complete intersections within them, facilitating Gromov-Witten invariant computations.

Contribution

It develops a general framework for mirror construction using small toric degenerations and extends the monomial-divisor mirror correspondence to compute Gromov-Witten invariants.

Findings

Defined small toric degenerations for Fano varieties

Proposed a new mirror construction method for Calabi-Yau intersections

Enabled computation of Gromov-Witten invariants via hypergeometric series

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.