Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constructing the big relative Fukaya category, and its open--closed maps

Nick Sheridan

Abstract

This paper continues previous work of the author with Perutz, in which the `small' version of Seidel's relative Fukaya category of a smooth complex projective variety relative to a normal crossings divisor was constructed, under a semipositivity assumption. In the present work, we generalize this to construct the `big' relative Fukaya category in the same setting, as well as its closed--open and open--closed maps, and prove Abouzaid's split-generation criterion in this context. We also establish a general framework for constructing chain-level Floer-theoretic operations in this context, and dealing efficiently with signs and bounding cochains, which will be used in follow-up work with Ganatra to construct the cyclic open--closed map and establish its properties.

Constructing the big relative Fukaya category, and its open--closed maps

Abstract

This paper continues previous work of the author with Perutz, in which the `small' version of Seidel's relative Fukaya category of a smooth complex projective variety relative to a normal crossings divisor was constructed, under a semipositivity assumption. In the present work, we generalize this to construct the `big' relative Fukaya category in the same setting, as well as its closed--open and open--closed maps, and prove Abouzaid's split-generation criterion in this context. We also establish a general framework for constructing chain-level Floer-theoretic operations in this context, and dealing efficiently with signs and bounding cochains, which will be used in follow-up work with Ganatra to construct the cyclic open--closed map and establish its properties.

Paper Structure

This paper contains 65 sections, 45 theorems, 257 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview of constructions and relation to previous work
  3. Geometric data
  4. Small and big relative Fukaya categories
  5. Bounding cochains
  6. Closed--open maps
  7. Acknowledgments:
  8. Algebraic preliminaries
  9. Gradings
  10. Koszul sign rule
  11. Coefficient rings and modules
  12. (Filtered) $A_\infty$ categories
  13. Bounding cochains
  14. Module categories
  15. Hochschild cohomology and units
  16. ...and 50 more sections

Key Result

Theorem 1.12

There is a unital graded $S$-algebra homomorphism called the closed--open map where $HH^*$ denotes Hochschild cohomology. In the case $S = R^{\operatorname{big}}$, it coincides with the first-order deformation class of the big relative Fukaya category, in the direction of the bulk variables.

Figures (3)

  • Figure 1: Stabilizing by adding an unstable disc or sphere. Note that in case the figures are to be interpreted as adding a sphere, no choice of directions is required at the nodes, so arrows at interior nodes should be deleted from the figure; and if the marked point is to be interpreted as a stabilizing marked point, then no choice of directions is required, so arrows at the corresponding marked points should be deleted from the figure.
  • Figure 2: Illustrating a smooth Whitney-stratified chain whose pullback under a forgetful map is not Whitney stratified; the empty marked point is the one which gets forgotten.
  • Figure 3: Self-gluing a disc. The arrow at the node indicates the direction from the component containing $p_-$ to the component containing $p_+$.

Theorems & Definitions (161)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Remark 1.9
  • Example 1.10
  • ...and 151 more