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Holomorphic linking numbers, ABC Massey products, and Calabi-Yau 3-folds

Lucía Martín-Merchán, Jonas Stelzig

TL;DR

This work demonstrates that Calabi–Yau 3-folds can host nontrivial ABC Massey products, refuting strong formality in the Calabi–Yau setting. It builds a family of simply connected projective 3-folds with trivial canonical bundle by resolving a specific (Z/2)^2-quotient of a torus, and it provides a precise formula linking ABC Massey products to holomorphic linking numbers. The central technical achievement is the intimate connection between higher-order holomorphic invariants and height-type pairings via Green currents and wave-front control, enabling explicit computations that relate ABC Massey products to the modular lambda function. The results offer a computable bridge between complex-analytic cycles, linking theory, and refined formality properties of Calabi–Yau manifolds, with potential implications for Arakelov theory and Deligne cohomology.

Abstract

On compact Kähler manifolds, we relate ABC Massey products arising from complex analytic cycles to holomorphic linking numbers. This enables us to construct a family of simply connected projective 3-folds with trivial canonical bundle, equipped with a non-vanishing ABC Massey product.

Holomorphic linking numbers, ABC Massey products, and Calabi-Yau 3-folds

TL;DR

This work demonstrates that Calabi–Yau 3-folds can host nontrivial ABC Massey products, refuting strong formality in the Calabi–Yau setting. It builds a family of simply connected projective 3-folds with trivial canonical bundle by resolving a specific (Z/2)^2-quotient of a torus, and it provides a precise formula linking ABC Massey products to holomorphic linking numbers. The central technical achievement is the intimate connection between higher-order holomorphic invariants and height-type pairings via Green currents and wave-front control, enabling explicit computations that relate ABC Massey products to the modular lambda function. The results offer a computable bridge between complex-analytic cycles, linking theory, and refined formality properties of Calabi–Yau manifolds, with potential implications for Arakelov theory and Deligne cohomology.

Abstract

On compact Kähler manifolds, we relate ABC Massey products arising from complex analytic cycles to holomorphic linking numbers. This enables us to construct a family of simply connected projective 3-folds with trivial canonical bundle, equipped with a non-vanishing ABC Massey product.

Paper Structure

This paper contains 16 sections, 10 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Holomorphic linking numbers and ABC Massey products
  3. Preliminaries
  4. Forms and currents
  5. Analytic cycles
  6. Cycles and currents
  7. Wave front sets
  8. Holomorphic linking numbers
  9. Linking numbers and Massey products
  10. Construction of the manifold
  11. Overview
  12. The orbifold
  13. The resolution
  14. The cohomology of $X$
  15. Fundamental group
  16. ...and 1 more sections

Key Result

Theorem A

There is a family $\{X_{\tau}\}_{\tau\in \mathbb{H}}$ over the upper half-plane of simply connected, projective $3$-folds with trivial canonical bundle and on each $X_\tau$ there exist divisors $D_1,...,D_4$ on $X_\tau$ such that where $\lambda$ is the modular $\lambda$-function. In particular, for general $\tau$, one has $\langle D_2, D_3, D_4\rangle_{ABC}\neq 0$ and the manifold $X_\tau$ is not

Figures (1)

  • Figure 1: The mapping torus $M$ and the four curves $N_0,\dots,N_3$.

Theorems & Definitions (28)

  • Theorem A
  • Theorem B
  • Definition 2.1: E.g. Beilinson_HeightPairing, HainBiext
  • Example 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 18 more