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Symplectic mapping class relations from pencil pairs

Russell Avdek

TL;DR

The paper develops a general framework to produce symplectic mapping class relations on affine Weinstein manifolds by translating between Lefschetz pencils on higher-dimensional projective varieties and products of Dehn twists along Lagrangian spheres. Central to the approach is the notion of pencil pairs, which, via boundary monodromies and the boundary relation, yield positive factorizations that are non-braid-like when the Euler characteristics of the projective models differ. By exploiting classifications of Fano $3$-folds and polarized $K3$ surfaces, the authors construct numerous explicit relations in dimension four and show that a single pencil pair propagates to infinitely many via cabling. This work builds bridges between algebraic geometry (Lefschetz pencils, projective models) and high-dimensional symplectic topology, with implications for constructing distinct Liouville fillings and expanding the known symplectic mapping class relations in higher dimensions.

Abstract

We describe symplectic mapping class relations between products of positive Dehn twists along Lagrangian spheres in Weinstein $4$-manifolds, all of which are affine $\mathbb{C}$ varieties. The relations are obtained by applying classification results for Fano $3$-folds and polarized $K3$ surfaces of small genus to a general methodology -- finding pencil pairs.

Symplectic mapping class relations from pencil pairs

TL;DR

The paper develops a general framework to produce symplectic mapping class relations on affine Weinstein manifolds by translating between Lefschetz pencils on higher-dimensional projective varieties and products of Dehn twists along Lagrangian spheres. Central to the approach is the notion of pencil pairs, which, via boundary monodromies and the boundary relation, yield positive factorizations that are non-braid-like when the Euler characteristics of the projective models differ. By exploiting classifications of Fano -folds and polarized surfaces, the authors construct numerous explicit relations in dimension four and show that a single pencil pair propagates to infinitely many via cabling. This work builds bridges between algebraic geometry (Lefschetz pencils, projective models) and high-dimensional symplectic topology, with implications for constructing distinct Liouville fillings and expanding the known symplectic mapping class relations in higher dimensions.

Abstract

We describe symplectic mapping class relations between products of positive Dehn twists along Lagrangian spheres in Weinstein -manifolds, all of which are affine varieties. The relations are obtained by applying classification results for Fano -folds and polarized surfaces of small genus to a general methodology -- finding pencil pairs.
Paper Structure (25 sections, 16 theorems, 66 equations, 2 figures)

This paper contains 25 sections, 16 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Acknowledgments
  4. Background
  5. Symplectic mapping class relations
  6. Known relations and counting twists
  7. Some contact-topological motivations
  8. Review of algebraic Lefschetz pencils
  9. Symplectic mapping class relations from pencil pairs
  10. Pencil pairs
  11. The case $\dim_{\mathbb{C}}X^{\pm} =2$
  12. Generalized lantern relations
  13. Pencil pairs from projective models and high-degree hypersurfaces
  14. Projective models
  15. Cabling pencil pairs
  16. ...and 10 more sections

Key Result

Theorem 1.0.1

Let $Z \subset \mathbb{CP}^{1} \times \mathbb{CP}^{1} \times \mathbb{CP}^{2}$ be the $K3$ surface cut out by a smooth complete intersection of $\deg=(1,1,2)$ and $\deg=(1,1,1)$ divisors. Let $(F, \beta_{F})$ be the Weinstein manifold given by the complement of a $\deg=(1,1,1)$ divisor in $Z$, with t

Figures (2)

  • Figure 1: The lantern relation states that the product $\tau_{c}\tau_{b}\tau_{a}$ of Dehn twists along circles on the left is boundary-relative isotopic to a product of Dehn twists along the circles on the right. Throughout, products represent compositions of maps, so $\tau_{c}\tau_{b}\tau_{a}$ means that we apply $\tau_{a}$, then $\tau_{b}$, and finally $\tau_{c}$. The surfaces depicted are oriented by a normal vector which sticks out of the page along their front halves.
  • Figure :

Theorems & Definitions (30)

  • Theorem 1.0.1
  • Definition 2.1.1
  • Lemma 2.2.1
  • proof
  • Theorem 2.3.1
  • Example 2.3.2
  • Lemma 2.3.3
  • proof
  • proof : Proof of Theorem \ref{['Thm:TwistLength']}
  • Lemma 2.4.1
  • ...and 20 more