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Fukaya A_\infty-structures associated to Lefschetz fibrations. VI

Paul Seidel

TL;DR

The work develops a comprehensive noncommutative pencil framework for Fukaya categories arising from monotone symplectic Lefschetz pencils, unifying Floer-theoretic data into a family of $A_\infty$-categories parameterised by a projective line. It introduces popsicle-based moduli, noncommutative linear systems, and localisation techniques to define fibres at infinity and zero, relating them to the Fukaya category of the fibre and of divisors, respectively. The theory connects relative and wrapped Fukaya categories, Serre functors, and noncommutative divisors, and it is illustrated through concrete examples including graded Kronecker quivers and Lefschetz pencils on quadrics. The construction yields a robust algebraic framework with potential applications to mirror symmetry and symplectic topology, providing new invariants and connections between fibre geometry, divisors, and pencils.

Abstract

To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry. One fibre of this "noncommutative pencil" is related to the Fukaya category of the open (meaning, with the base locus removed, and hence exact symplectic) fibre of the original Lefschetz pencil; the other fibres are newly constructed kinds of Fukaya categories.

Fukaya A_\infty-structures associated to Lefschetz fibrations. VI

TL;DR

The work develops a comprehensive noncommutative pencil framework for Fukaya categories arising from monotone symplectic Lefschetz pencils, unifying Floer-theoretic data into a family of -categories parameterised by a projective line. It introduces popsicle-based moduli, noncommutative linear systems, and localisation techniques to define fibres at infinity and zero, relating them to the Fukaya category of the fibre and of divisors, respectively. The theory connects relative and wrapped Fukaya categories, Serre functors, and noncommutative divisors, and it is illustrated through concrete examples including graded Kronecker quivers and Lefschetz pencils on quadrics. The construction yields a robust algebraic framework with potential applications to mirror symmetry and symplectic topology, providing new invariants and connections between fibre geometry, divisors, and pencils.

Abstract

To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry. One fibre of this "noncommutative pencil" is related to the Fukaya category of the open (meaning, with the base locus removed, and hence exact symplectic) fibre of the original Lefschetz pencil; the other fibres are newly constructed kinds of Fukaya categories.

Paper Structure

This paper contains 61 sections, 43 theorems, 336 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Floer theory relative to an anticanonical divisor
  3. Floer theory for Lefschetz fibrations
  4. Floer theory for Lefschetz pencils
  5. Content of the paper
  6. Basic notions: algebra
  7. $A_\infty$-categories
  8. $A_\infty$-bimodules
  9. Noncommutative linear systems
  10. Localisation
  11. Basic notions: geometry
  12. Riemann surfaces
  13. One-forms
  14. Discs
  15. Cauchy-Riemann equations
  16. ...and 46 more sections

Key Result

Lemma 4.1

Take a codimension one boundary face of the compactified moduli space of popsicles, determined by data as in eq:two-vertex-tree, and which is therefore of the form With our conventions, the product orientation of that space differs from the induced boundary orientation by a sign $(-1)^\ddag$, where

Figures (9)

  • Figure 1.1: The vanishing cycles from Example \ref{['th:simple-example']}.
  • Figure 4.1: The moduli space $\bar{\EuScript R}^{d,p}$ for $d = 2$ and $p: \{1,2\} \rightarrow \{1,2\}$ the identity map. Note the ${\mathbb Z}/2$-symmetry on the bottom boundary face (exchanging the two points), which is an instance of the phenomenon described in \ref{['eq:extra-symmetry']}.
  • Figure 4.2: A "shared boundary point" of two moduli spaces $\bar{\EuScript R}^{d+1,p}$, see \ref{['eq:shared']}. On the left, $p: \{1\} \rightarrow \{1,2\}$ has image $1$, and on the right it has image $2$. The broken popsicle appearing as the limit has the same structure in both cases.
  • Figure 4.3: A compactified surface $|S$, with examples of the stable curves associated to points of $|S$. Here, the round dots are the $\zeta_k$, and the square dots the extra marked points $z_{\pm}$. On the bottom right, note how the endpoints of an interval in $(|S) \setminus S$ have isomorphic associated stable curves.
  • Figure 4.4: The image of the surface $|S$ from Figure \ref{['fig:complexify']} under \ref{['eq:add-plusminus']}. In this simple case, the target space of \ref{['eq:add-plusminus']} is a fixed locus of a suitable real involution of Deligne-Mumford space $\bar{\EuScript M}_{0,5}$, and concretely is a non-orientable surface with Euler characteristic $-1$. We give two representations: as a pair-of-pants with each boundary component closed up to a Moebius band; and as a torus (represented as a hexagon with opposite sides identified by translation), again with a hole closed up to a Moebius band. The image of $|S$ is the shaded region (on the left, the front half of the pair-of-pants); the dots are the images of the corners of $|S$; and the arrows indicate identifying opposite points on each of the boundary circles.
  • ...and 4 more figures

Theorems & Definitions (93)

  • Remark 1.1
  • Example 1.2
  • Remark 1.5
  • Remark 1.6
  • Remark 1.6
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 83 more