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Algebraic aspects of hypergeometric differential equations

Thomas Reichelt, Mathias Schulze, Christian Sevenheck, Uli Walther

Abstract

We review some classical and modern aspects of hypergeometric differential equations, including $A$-hypergeometric systems of Gel'fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler-Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

Algebraic aspects of hypergeometric differential equations

Abstract

We review some classical and modern aspects of hypergeometric differential equations, including -hypergeometric systems of Gel'fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler-Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

Paper Structure

This paper contains 24 sections, 17 theorems, 205 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Hypergeometric functions
  3. From univariate to GKZ and back
  4. Solutions
  5. Torus action and Euler-Koszul complex
  6. Torus action and $A$-grading
  7. Toric category and Euler--Koszul technology
  8. Fourier--Laplace transformed GKZ-systems
  9. Holonomicity, Rank, and Singular Locus
  10. Better behaved systems and contiguity
  11. Irregularity
  12. The Fuchs criterion and regularity
  13. Initial ideals and triangulations
  14. Slopes and the $(A,L)$-umbrella
  15. $L$-characteristic varieties
  16. ...and 9 more sections

Key Result

Theorem 2.10

Let $A\in \mathbb{Z}^{d \times n}$ be as above, then the following statements are equivalent

Figures (8)

  • Figure 1: Cone, true, and strongly resonant degrees.
  • Figure 2: The Čech complex to the 0134-curve
  • Figure 3: Two Fuchs polygons
  • Figure 4: The initial simplicial complexes $\Sigma^L_A$ for $I_A=(\partial_1\partial_2\partial_3-\partial_4\partial_5^2)$.
  • Figure 5: A non-regular triangulation of a triangle.
  • ...and 3 more figures

Theorems & Definitions (64)

  • Example 1.1: The error function, part I
  • Example 1.2: The error function, part II
  • Example 1.3: Hypergeometry and Hodge filtrations
  • Definition 1.6: $A$-hypergeometric system, GGZ87
  • Definition 1.7
  • Definition 1.8
  • Example 1.9: The GKZ-system to the Kummer confluent function
  • Example 1.10: GKZ-system to ${}_2F_1$
  • Remark 2.1
  • Definition 2.2: Degrees and Euler--Koszul complex
  • ...and 54 more