Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in Grassmannians
Victor V. Batyrev, Ionunt Ciocan-Fontanine, Bumsig Kim, Duco van Straten
TL;DR
The paper develops a framework to construct mirrors and compute instanton numbers for Calabi–Yau complete intersections in Grassmannians via conifold transitions. It leverages a toric degeneration $G(k,n) o P(k,n)$ with conifold singularities, then uses a small crepant resolution to relate $X subseteq G(k,n)$ to a toric CY $Y subseteq \,widehat{P}(k,n)$ and its mirror $Y^*$. A one-parameter specialization of toric mirrors, guided by the monomial-divisor correspondence, yields mirrors $X^*$ of $X$, and the resulting period data feeds into Picard–Fuchs equations to extract instanton numbers. The construction aligns with the $Lax$ operator conjecture for Grassmannians and aims to generalize to arbitrary dimensions within the conifold web of CY complete intersections. This provides a concrete route to new mirrors and enumerative data beyond hypersurfaces in toric settings.
Abstract
In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians $G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.