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Shifting local exponents of Picard-Fuchs operators

Tymoteusz Chmiel

Abstract

We investigate the operation of shifting local exponents and study its effects on the monodromy representation of a one-parameter family of Calabi-Yau threefolds. The main result is a characterization of shifts of geometric operators which are also geometric. We use this description to construct some Picard-Fuchs operators with interesting properties.

Shifting local exponents of Picard-Fuchs operators

Abstract

We investigate the operation of shifting local exponents and study its effects on the monodromy representation of a one-parameter family of Calabi-Yau threefolds. The main result is a characterization of shifts of geometric operators which are also geometric. We use this description to construct some Picard-Fuchs operators with interesting properties.
Paper Structure (10 sections, 10 theorems, 41 equations)

This paper contains 10 sections, 10 theorems, 41 equations.

Table of Contents

  1. Preliminaries: Picard-Fuchs operators
  2. One-parameter families of Calabi-Yau threefolds
  3. Local exponents of geometric operators
  4. Symplectic structure of geometric operators
  5. Shift of local exponents
  6. Shifts of symplectic operators
  7. Applications: double octic operators
  8. Double octic Calabi-Yau threefolds
  9. Locally geometric operators
  10. Quadratic twists and thin monodromy

Key Result

Lemma 1

simion Let $\alpha_1\leq\alpha_2\leq\alpha_3\leq\alpha_4$ be local exponents of a geometric operator $\mathcal{P}$ at a point $s\in\mathbb{P}^1$. Put $k:=\#\{\alpha_1,\cdots,\alpha_4\}$. If $k=4$, let $N\in\mathbb{N}_{>0}$ be the order of local monodromy at $s$. If every point satisfies Assumption A, then $Mon(\mathcal{P})\subset\mathrm{Sp}(4,\mathbb{Z})$ is of infinite index. If every point sati

Theorems & Definitions (16)

  • Definition
  • Lemma 1
  • Definition
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • ...and 6 more