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Irregular Hodge filtration of hypergeometric differential equations

Yichen Qin, Daxin Xu

Abstract

Fedorov and Sabbah--Yu calculated the (irregular) Hodge numbers of hypergeometric connections. In this paper, we study the irregular Hodge filtrations on hypergeometric connections defined by rational parameters, and provide a new proof of the aforementioned results. Our approach is based on a geometric interpretation of hypergeometric connections, which enables us to show that certain hypergeometric sums are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ (i.e. "Frobenius Newton polygon equals to irregular Hodge polygon").

Irregular Hodge filtration of hypergeometric differential equations

Abstract

Fedorov and Sabbah--Yu calculated the (irregular) Hodge numbers of hypergeometric connections. In this paper, we study the irregular Hodge filtrations on hypergeometric connections defined by rational parameters, and provide a new proof of the aforementioned results. Our approach is based on a geometric interpretation of hypergeometric connections, which enables us to show that certain hypergeometric sums are everywhere ordinary on (i.e. "Frobenius Newton polygon equals to irregular Hodge polygon").
Paper Structure (21 sections, 24 theorems, 110 equations)

This paper contains 21 sections, 24 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Application to Frobenius slopes of hypergeometric sums
  3. Strategy of proof
  4. Organization of this article
  5. Hypergeometric connections
  6. Review of hypergeometric connections
  7. The Newton polytope of a Laurent polynomial
  8. Geometric interpretations
  9. Explicit cyclic vectors for hypergeometric connections
  10. Irregular Hodge filtration of hypergeometric connections
  11. Exponential mixed Hodge structures
  12. Objects of EMHS attached to regular functions
  13. The irregular Hodge filtration on twisted de Rham cohomology
  14. The EMHS associated with hypergeometric connections
  15. A basis in relative twisted de Rham cohomology
  16. ...and 6 more sections

Key Result

Theorem 1.0.1

Suppose $(\alpha,\beta)$ is non-resonant. We define a map $\theta\colon \{1,\ldots,n\}\to \mathbb{R}$ by Then, up to an $\mathbb{R}$-shiftOur Hodge numbers $\theta(k)$'s are normalized according to the geometric interpretation in Proposition prop::geometric-realization, and is different from those of Fedorov and Sabbah--Yu by a shift., the jumps of the irregular Hodge filtration on $\mathcal{H}yp

Theorems & Definitions (54)

  • Theorem 1.0.1: \ref{['thm:Hodge-fil']}
  • Theorem 1.1.1: \ref{['t:Frobslope']}
  • Remark 1.1.2
  • Theorem 1.2.1: \ref{['thm::hodge-number']}
  • Definition 2.1.3
  • Definition 2.2.1
  • Lemma 2.2.2
  • proof
  • Lemma 2.2.3
  • proof
  • ...and 44 more