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On Hypergeometric Duality Conjecture

Lev Borisov, Zengrui Han

Abstract

We give an explicit formula for the duality, previously conjectured by Horja and Borisov, of two systems of GKZ hypergeometric PDEs. We prove that in the appropriate limit this duality can be identified with the inverse of the Euler characteristics pairing on cohomology of certain toric Deligne-Mumford stacks, by way of $Γ$-series cohomology valued solutions to the equations.

On Hypergeometric Duality Conjecture

Abstract

We give an explicit formula for the duality, previously conjectured by Horja and Borisov, of two systems of GKZ hypergeometric PDEs. We prove that in the appropriate limit this duality can be identified with the inverse of the Euler characteristics pairing on cohomology of certain toric Deligne-Mumford stacks, by way of -series cohomology valued solutions to the equations.
Paper Structure (5 sections, 17 theorems, 89 equations)

This paper contains 5 sections, 17 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Pairing of solutions
  3. Pairing of the Gamma series
  4. Euler characteristic pairing
  5. Extensions and open questions

Key Result

Lemma 2.1

Let $I\subset \{1,\ldots, n\}$ be such that $\{v_i, i\in I\}$ form a basis of $N_{\mathbb R}$. Suppose that $I$ contains $1$ and consider $j\not\in I$. Consider the spanning set $J=I\cup \{j\}$. Let $\mu$ denote the unique linear function that takes value ${\rm Vol}_I$ on $v_1$ and $0$ on $v_i,i\in

Theorems & Definitions (40)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 3.1
  • ...and 30 more