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Perverse schobers and GKZ systems

Špela Špenko, Michel Van den Bergh

Abstract

Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation of a reductive group. When the group is a torus the SKMS corresponds to the complement of the GKZ discriminant locus (which is a hyperplane arrangement in the quasi-symmetric case shown by Kite). We show here that a suitable variation of the perverse schober we constructed provides a categorification of the associated GKZ hypergeometric system in the case of non-resonant parameters. As an intermediate result we give a description of the monodromy of such "quasi-symmetric" GKZ hypergeometric systems.

Perverse schobers and GKZ systems

Abstract

Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation of a reductive group. When the group is a torus the SKMS corresponds to the complement of the GKZ discriminant locus (which is a hyperplane arrangement in the quasi-symmetric case shown by Kite). We show here that a suitable variation of the perverse schober we constructed provides a categorification of the associated GKZ hypergeometric system in the case of non-resonant parameters. As an intermediate result we give a description of the monodromy of such "quasi-symmetric" GKZ hypergeometric systems.

Paper Structure

This paper contains 80 sections, 71 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Perverse schobers
  3. A perverse schober using Geometric Invariant Theory
  4. The mirror picture
  5. The GKZ system
  6. Kite's result
  7. Creating a family
  8. Main result
  9. Discussion
  10. Outline of proof
  11. Acknowledgement
  12. Notation and conventions
  13. Identifications for tori
  14. Setting
  15. Quasi-symmetry
  16. ...and 65 more sections

Key Result

Proposition 1.2

GKZEuler Assume $\alpha \in Y(H)_{\mathbb C}$ is non-resonant. Then $\bar{P}(\alpha)$ is a simple perverse sheaf, in particular it is the intermediate extension of its corresponding local system. If $\alpha,\alpha'\in Y(H)_{\mathbb C}$ are non-resonant and $\alpha-\alpha'\in Y(H)$ then $\bar{P}(\alp

Theorems & Definitions (142)

  • Example 1.1
  • Proposition 1.2
  • Example 1.3
  • Theorem 1.4: Kite, §\ref{['subsec:GKZdislocus']}
  • Theorem 1.6: Theorem \ref{['thm:connect']}
  • Remark 1.7
  • Remark 3.1
  • Lemma 3.2
  • Remark 4.2
  • Theorem 4.3
  • ...and 132 more