Duality and kernels in microlocal geometry

TL;DR

This work develops a comprehensive framework for dualities and kernel-based functors in microlocal sheaf theory, establishing a Fourier–Mukai-type correspondence that classifies colimit-preserving functors between categories of sheaves with isotropic microsupports via integral kernels. It introduces and exploits doubling and wrapping techniques to relate microsheaves to ordinary sheaves, proving Künneth formulas and a robust product theory for both sheaves and microsheaves. A central thread is the comparison between the standard duality (via convolution with kernels) and Verdier duality, mediated by wrap-once and inverse-Serre functors, with concrete criteria ensuring Verdier duality extends to compact objects in favorable Legendrian_stop settings. The results bridge microlocal geometry with non-commutative/derived geometry, offering a categorical Fourier–Mukai picture that mirrors the coherent-constructible correspondence and has implications for wrapped Fukaya categories and toric mirror symmetry.

Abstract

We study the dualizability of sheaves on manifolds with isotropic singular supports $\operatorname{Sh}_Λ(M)$ and microsheaves with isotropic supports $\operatorname{μsh}_Λ(Λ)$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports $\operatorname{Sh}_Λ^b(M)_0$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects $\operatorname{Sh}_Λ^c(M)_0$ when the wrap-once functor is an equivalence, for instance, when $Λ$ is a full Legendrian stop or a swappable Legendrian stop.

Figures (3)

• Figure 1: Let $M = N = \mathbb{R}$, $\Lambda = \{(0, -1)\}$ and $\Sigma = \{(0, 1)\} \subseteq S^*\mathbb{R}$. The figure on the left is the relative doubling construction along $\Lambda \times \Sigma \times \mathbb{R}$. The figure on the right is the union of $\Lambda_{\pm\epsilon} \times (\Sigma \times \cup_\epsilon)$ (in blue) and $(\Lambda \times \cup_\epsilon) \times \Sigma_{\pm\epsilon}$ (in red), which is now contained in a small neighbourhood of $\widehat{\Lambda}_{\pm\epsilon} \times \widehat{\Sigma}_{\pm\epsilon}$.
• Figure 2: Let $M = N = \mathbb{R}$, $\Lambda = \{(0, -1)\}$ and $\Sigma = \{(0, 1)\} \subseteq S^*\mathbb{R}$. The figure on the left is the relative doubling construction along $\Lambda \times \widehat{\Sigma}$. The figure on the right is the construction that glues together $\Lambda_{\pm\epsilon} \times \widehat{\Sigma}$ (in blue) and $(\Lambda \times \cup_\epsilon) \times \Sigma$ (in red), which is now contained in a small neighbourhood of $\widehat{\Lambda}_{\pm\epsilon} \times \widehat{\Sigma}$.
• Figure 3: Different subcategories in $\operatorname{Sh}_{\widehat{\Lambda}_\epsilon \times \widehat{\Sigma}_{\pm\epsilon}}(M \times N)$ that come from the recollements. The red piece is the essential image of the microstalk corepresentatives on $\Lambda \times \Sigma \times \mathbb{R}$. The blue pieces are the essential images of the microstalk corepresentatives of $\Lambda_{-\epsilon} \times \widehat{\Sigma}_{\epsilon}$ and $\widehat{\Lambda}_{\epsilon} \times \Sigma_{-\epsilon}$.

Theorems & Definitions (109)

• Theorem 1.1
• Theorem 1.2: The Künneth formula
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Remark 1.9
• Definition 2.1
• Definition 2.2