Semisimple quantum cohomology of some Fano varieties

Abstract

We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also give examples of Fano varieties having a non semisimple small quantum cohomology but a semisimple big quantum cohomology.

arXiv comment27 pages

References in corpus (7)

• Geometry and analytic theory of Frobenius manifolds
• Exceptional collections for Grassmannians of isotropic lines
• Quantum Pieri rules for isotropic Grassmannians
• Quantum cohomology of complete intersections
• Quantum cohomology of minuscule homogeneous spaces II : Hidden symmetries
• Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group $\mathbb{Z}$
• Small codimension subvarieties in homogeneous spaces

Cited by in corpus (10)

• Geometry of horospherical varieties of Picard rank one
• Helix Structures in Quantum Cohomology of Fano Varieties
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• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• Dubrovin's conjecture for $IG(2,6)$
• Big quantum cohomology of Fano complete intersections
• On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections
• Update on quantum cohomology of IG(2,2n)
• Derived categories of the Cayley plane and the coadjoint Grassmannian of type F
• On the big quantum cohomology of coadjoint varieties