Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positive microlocal holonomies are globally regular

Roger Casals, Wenyuan Li

TL;DR

This work provides a general geometric criterion for turning local microlocal holonomies into globally regular functions on the moduli space of Lagrangian fillings, using Hochschild 0-cycles and derived moduli stacks of sheaves with Legendrian microsupport. By introducing an $ heta$-decorated Kashiwara–Schapira framework and an $\,\mathbb{L}$-compressing system for Lagrangian fillings, the authors lift microlocal merodromies to Hochschild classes whose HO maps yield global regular functions on $ rak{M}(\\Lambda, t)$. The main theorem shows that for any $\oldsymbol{D}$-positive relative cycle $\,\eta$, there exists $H_{\eta}$ with HO$(H_{\eta})$ equal to the trace of the microlocal merodromy on the Lagrangian chart, ensuring a global extension; this unifies previous cluster-variable regularity results and provides a conceptual framework that avoids heavy algebraic computations. Overall, the paper advances the understanding of global regularity phenomena in the moduli of Legendrian Lagrangian fillings by marrying microlocal sheaf theory, Hochschild homology, and derived stacks in a coherent geometric setting.

Abstract

We establish a geometric criterion for local microlocal holonomies to be globally regular on the moduli space of Lagrangian fillings. This local-to-global regularity result holds for arbitrary Legendrian links and it is a key input for the study of cluster structures on such moduli spaces. Specifically, we construct regular functions on derived moduli stacks of sheaves with Legendrian microsupport by studying the Hochschild homology of the associated dg-categories via relative Lagrangian skeleta. In this construction, a key geometric result is that local microlocal merodromies along positive relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for these dg-categories.

Positive microlocal holonomies are globally regular

TL;DR

This work provides a general geometric criterion for turning local microlocal holonomies into globally regular functions on the moduli space of Lagrangian fillings, using Hochschild 0-cycles and derived moduli stacks of sheaves with Legendrian microsupport. By introducing an -decorated Kashiwara–Schapira framework and an -compressing system for Lagrangian fillings, the authors lift microlocal merodromies to Hochschild classes whose HO maps yield global regular functions on . The main theorem shows that for any -positive relative cycle , there exists with HO equal to the trace of the microlocal merodromy on the Lagrangian chart, ensuring a global extension; this unifies previous cluster-variable regularity results and provides a conceptual framework that avoids heavy algebraic computations. Overall, the paper advances the understanding of global regularity phenomena in the moduli of Legendrian Lagrangian fillings by marrying microlocal sheaf theory, Hochschild homology, and derived stacks in a coherent geometric setting.

Abstract

We establish a geometric criterion for local microlocal holonomies to be globally regular on the moduli space of Lagrangian fillings. This local-to-global regularity result holds for arbitrary Legendrian links and it is a key input for the study of cluster structures on such moduli spaces. Specifically, we construct regular functions on derived moduli stacks of sheaves with Legendrian microsupport by studying the Hochschild homology of the associated dg-categories via relative Lagrangian skeleta. In this construction, a key geometric result is that local microlocal merodromies along positive relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for these dg-categories.
Paper Structure (34 sections, 19 theorems, 93 equations, 2 figures)

This paper contains 34 sections, 19 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary
  3. Scientific context
  4. Main result
  5. Outline of the argument
  6. Dg-categories and moduli of sheaves with Legendrian singular support
  7. Dg-categories of sheaves with Legendrian singular support
  8. Derived stacks of sheaves with Legendrian singular support
  9. A $\tt$-action on the derived moduli stacks of sheaves
  10. Multiple basepoints per component
  11. Microlocal rank 1 case of the basepoint action
  12. A few examples
  13. Legendrian knots
  14. Hopf link, Part I: $\mathfrak{M}_1({\Lambda,\tt})$ and $\mathfrak{M}_1({\Lambda})$
  15. Hopf link, Part II: Lagrangian fillings
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\Lambda\subset(T^*_\infty\mathbb{R}^2,\xi_\text{st})$ be a $(\tt,T)$-pointed Legendrian link and $L\subset (T^*\mathbb{R}^2,\lambda_\text{st})$ an embedded exact Lagrangian filling of $\Lambda$ endowed with an $\mathbb{L}$-compressing system $\mathscr{D}$. Then, for any $\mathscr{D}$-positive r

Figures (2)

  • Figure 1: A Lagrangian filling $L$, in blue, of $\Lambda$, in cyan, with an $\mathbb{L}$-compressing system $\mathscr{D}=\{\gamma_1,\ldots,\gamma_6\}$. Relative cycle $\eta$ in red. In Theorem \ref{['thm:main']} we lift the microlocal merodromy from a local regular function to a Hochschild 0-chain of $\operatorname{Sh}^c_{\Lambda,\tt}(\mathbb{R}^2)$. The latter allows us to construct a global regular function $A_\eta$ in $\Gamma(\mathfrak{M}{(\Lambda,\tt)},\mathcal{O}_{\mathfrak{M}{(\Lambda,\tt)}})$.
  • Figure 2: (Left) The 1-dimensional Lagrangian skeleton $I^\sigma_{p_1,\ldots,p_5}$, where all the signs of $\sigma$ are positive. (Right) The quiver associated to the 1-dimensional Lagrangian skeleton $\mathbb{K}$ given by a base $S^1$ and five positive spikes $\nu(K)$, with $K=\{p_1,\ldots,p_5\}$.

Theorems & Definitions (59)

  • Theorem 1.1: "Main extension result"
  • Corollary 1.2: "Local merodromies along positive cycles are global"
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • ...and 49 more