A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology
Pieter Belmans, Sergey Galkin, Naichung Conan Leung, Changzheng Li, Markus Reineke, Rui Xiong
TL;DR
The paper develops general, symmetry-aware obstructions to semisimplicity of small quantum cohomology and applies them to hyperplane sections of Grassmannians and (co)adjoint Grassmannians. It establishes a sharp index–Betti obstruction and a grading-based criterion, then proves a complete Hodge–Tate classification for smooth hyperplane sections of Gr(k,n) and derives quantum Pieri rules and monodromy arguments to determine semisimplicity in key cases. The work confirms conjectures in several instances (notably for (co)adjoint Grassmannians) and identifies systematic methods—quantum Lefschetz-type tools, monodromy, and representation-theoretic techniques—for studying semisimplicity across homogeneous varieties. These results illuminate when the small quantum cohomology is generically semisimple and offer practical criteria for predicting non-semisimplicity from topological data. The findings have implications for understanding genus-zero invariants and the broader structure of quantum cohomology rings in homogeneous settings, with potential extensions to related moduli spaces and quiver varieties.
Abstract
It is known that the semisimplicity of quantum cohomology implies the vanishing of off-diagonal Hodge numbers (Hodge--Tateness). We investigate which hyperplane sections of homogeneous varieties possess either of the two properties. We provide a new efficient criterion for non-semisimplicity of the small quantum cohomology ring of Fano manifolds that depends only on the Fano index and Betti numbers. We construct a bijection between Dynkin diagrams of types A, D or E, and complex Grassmannians with Hodge-Tate smooth hyperplane sections. By applying our criteria and using monodromy action, we completely characterize the semisimplicity of the small quantum cohomology of smooth hyperplane sections in the case of complex Grassmannians, and verify a conjecture of Benedetti and Perrin in the case of (co)adjoint Grassmannians.