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Fibering out Calabi-Yau motives

Kilian Bönisch, Vasily Golyshev, Albrecht Klemm

TL;DR

The paper develops a fibering-out method to prove the modularity of mixed periods associated with the conifold fiber of a Calabi–Yau threefold family. By expressing periods as integrals of K3-surface period functions and leveraging the modularity of the K3 family, it demonstrates that the mixed period matrix $T$ (in the limit $\psi \to 1$) is governed by modular data. A key achievement is identifying a modular structure in the mixed periods through a pullback to modular forms on $\Gamma_0^*(50)$, with explicit forms $f_{50}$, $F_{50}$, and $g_{50}$ encoding the periods via CM-point integrals. The approach yields a unified framework for proving modularity of mixed period matrices for several Calabi–Yau families and points to rich connections with meromorphic modular forms, algebraic correspondences, and $L$-function phenomena, with several open questions and future directions outlined.

Abstract

We prove the modularity of mixed periods associated with singular fibers of specific families of Calabi-Yau threefolds. This is done by "fibering out", i.e. by expressing these periods as integrals of periods of families of K3 surfaces and by using modularity properties of the latter. Besides classical periods of holomorphic modular forms and meromorphic modular forms with vanishing residues, the computations lead to new interesting periods associated with meromorphic modular forms with non-vanishing residues as well as contours between CM points.

Fibering out Calabi-Yau motives

TL;DR

The paper develops a fibering-out method to prove the modularity of mixed periods associated with the conifold fiber of a Calabi–Yau threefold family. By expressing periods as integrals of K3-surface period functions and leveraging the modularity of the K3 family, it demonstrates that the mixed period matrix (in the limit ) is governed by modular data. A key achievement is identifying a modular structure in the mixed periods through a pullback to modular forms on , with explicit forms , , and encoding the periods via CM-point integrals. The approach yields a unified framework for proving modularity of mixed period matrices for several Calabi–Yau families and points to rich connections with meromorphic modular forms, algebraic correspondences, and -function phenomena, with several open questions and future directions outlined.

Abstract

We prove the modularity of mixed periods associated with singular fibers of specific families of Calabi-Yau threefolds. This is done by "fibering out", i.e. by expressing these periods as integrals of periods of families of K3 surfaces and by using modularity properties of the latter. Besides classical periods of holomorphic modular forms and meromorphic modular forms with vanishing residues, the computations lead to new interesting periods associated with meromorphic modular forms with non-vanishing residues as well as contours between CM points.

Paper Structure

This paper contains 6 sections, 2 theorems, 46 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mixed periods associated with the conifold fiber
  3. Fibering out
  4. Modularity of the mixed periods
  5. Outlook
  6. Results for other families of Calabi-Yau threefolds

Key Result

Theorem 1

The numbers $w_\pm$, $e_\pm$ and $a_\pm$ are periods of the modular forms $f_{50}$, $F_{50}$ and $g_{50}$. More precisely, there are cocycles $r_\pm : \Gamma_0^*(50) \rightarrow \mathbb{Z}^3$ such that for every $\gamma = \left(\substack{ a\, b\\ c\, d}\right) \in \Gamma_0^*(50)$ and every choice of in terms of the re-scaled periods from Equation (eq:FullNonModularPeriodIdentity).

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof