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Geometric realisation of hypergeometric local systems

Asem Abdelraouf, Giulia Gugiatti

Abstract

We prove a realisation theorem for irreducible hypergeometric local systems defined over the rational numbers in terms of families of affine varieties in algebraic tori. The families we consider have been studied extensively in the literature and appear in mirror symmetry. Our result holds unconditionally for families with one-dimensional or even-dimensional fibres. It holds under a monodromy assumption for families with fibres of odd dimension greater than one.

Geometric realisation of hypergeometric local systems

Abstract

We prove a realisation theorem for irreducible hypergeometric local systems defined over the rational numbers in terms of families of affine varieties in algebraic tori. The families we consider have been studied extensively in the literature and appear in mirror symmetry. Our result holds unconditionally for families with one-dimensional or even-dimensional fibres. It holds under a monodromy assumption for families with fibres of odd dimension greater than one.
Paper Structure (24 sections, 29 theorems, 171 equations, 3 figures, 1 table)

This paper contains 24 sections, 29 theorems, 171 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and relation to other works
  3. Sketch of the proofs
  4. Families of affine hypersurfaces in algebraic tori
  5. Affine hypersurfaces in algebraic tori
  6. Quasi--$\Delta$-regularity
  7. Cohomology groups
  8. Families of hypersurfaces
  9. Fibrewise compactifications
  10. Relative cohomology
  11. Local systems
  12. Set up
  13. Hypergeometric local systems
  14. The pair associated to a gamma vector
  15. Toric models
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose that the local system $R^\kappa \pi_{U !} \mathbb{Q}$ has non-trivial local monodromy at $t=1$. Then the isomorphism eq:H-(Z, pi) holds.

Figures (3)

  • Figure 1: The Newton polytope of the polynomial $f_t$ in Example \ref{['exa:curve-ft']}.
  • Figure 2: The Newton polytope of the polynomial $f$ in Example \ref{['exa:nonexa']}.
  • Figure 3: The Newton polytope of the polynomial $f_t$ in $f_t$ in Example \ref{['exa:chebyshev-ft']}.

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Example 2.8
  • ...and 90 more