Mirror symmetry for certain blowups of Grassmannians

TL;DR

The authors classify when the blowup of a Grassmannian along a smooth Schubert center is Fano, reveal the Mori cone structure, and compute key two-point genus-zero Gromov-Witten invariants for the blowups along G(k,n-1). They construct explicit bases for cohomology, prove vanishing results for higher-degree invariants, and provide precise quantum product presentations for X_{2,n}. On the mirror side, they establish a toric Landau-Ginzburg model with a superpotential f_tor whose Jacobi ring is isomorphic to the small quantum cohomology QH^*(X_{2,n}), and identify c_1(X_{2,n}) with the class of f_tor, thereby realizing A– and B–model equivalence for this family. Together, these results illuminate the interplay between birational geometry of Grassmannian blowups, their quantum invariants, and explicit mirror constructions, offering tools for broader generalizations and future quantum Schubert calculus developments.

Abstract

We classify when the blowup of a complex Grassmannian $G(k, n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k, n-1)$. We further prove a mirror symmetry statement for the blowup $X_{2, n}$ of $G(2, n)$ along $G(2, n-1)$, by introducing a toric superpotential $f_{\rm tor}$ and showing the isomorphism between the Jacobi ring of $f_{\rm tor}$ and the small quantum cohomology ring $QH^*(X_{2, n})$.

Figures (1)

• Figure 1: Ladder diagram for $X_{2, n}$ and its dual graph

Theorems & Definitions (105)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1
• Corollary 2.2
• Proposition 2.3: LaWe