(Semi)simple exercises in quantum cohomology

Abstract

The paper is dedicated to the study of algebraic manifolds whose quantum cohomology or a part of it is a semisimple Frobenius manifold. Theorem 1.8.1 says, roughly speaking, that the sum of $(p,p)$--cohomology spaces is a maximal Frobenius submanifold that has chances to be semisimple. Theorem 1.8.3 provides a version of the Reconstruction theorem, assuming semisimplicity but not $H^2$--generation. Theorem 3.6.1 establishes the semisimplicity for all del Pezzo surfaces, providing an evidence for the conjecture that semisimplicity is related to the existence of a full system of exceptional sheaves of the appropriate length. Finally, in §2 we calculate special coordinates for three families of Fano threefolds with minimal cohomology.

30 pp., amstex file, no figures

References in corpus (4)

• Geometry and analytic theory of Frobenius manifolds
• Hypergeometric periods for a tame polynomial
• Multiplication on the tangent bundle
• On the WDVV-equation in quantum K-theory

Cited by in corpus (13)

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Local Moduli of Semisimple Frobenius Coalescent Structures
• Quantum Witten localization and abelianization for qde solutions
• Helix Structures in Quantum Cohomology of Fano Varieties
• Gromov-Witten invariants of Fano threefolds of genera 6 and 8
• An analogue of Dubrovin's conjecture
• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• Big quantum cohomology of Fano complete intersections
• Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
• Floer cohomology of the Chiang Lagrangian
• Quantum K-Theory I: Foundations
• Computing Gromov-Witten invariants of some Fano varieties
• Elliptic Gromov-Witten Invariants of Del-Pezzo Surfaces