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Contact isotopies in the coherent-constructible correspondence

Jishnu Bose, Harold Williams

TL;DR

This work provides a geometric realization of the mirror Picard action in the coherent-constructible correspondence for toric varieties. By quantizing the flows of smoothed support functions of toric divisors and passing to the nearby-cycles limit, the authors show that the limiting kernel acts on constructible sheaves via convolution with the twisted polytope sheaf $p_! ext{P}(D)$, while on the CCC side this matches the action of $igO(D)$ under the equivalence. The results extend Hanlon’s geometric action from Fukaya–Seidel categories to the CCC framework, and they handle complete and noncomplete fans, as well as equivariant and nonequivariant versions. The analysis leverages GKS kernels, singular-support control, and Whitney-type shard resolutions to establish a canonical, functorial bridge between toric coherent and constructible categories through nearby cycles. Overall, the paper deepens the symplectic‑sheaf correspondence for toric mirror symmetry and furnishes new tools for understanding Picard actions in CCC.

Abstract

The coherent-constructible correspondence is a realization of toric mirror symmetry in which the A-side is modeled by constructible sheaves on $T^n$. This paper provides a geometric realization of the mirror Picard group action in this correspondence, characterizing it in terms of quantized contact isotopies and providing a sheaf-theoretic counterpart to work of Hanlon in the Fukaya-Seidel setting. Given a toric Cartier divisor $D$, we consider a family of homogeneous Hamiltonians $H_\varepsilon$ on $T^* T^n$. Their flows act on sheaves via a family of kernels $K_\varepsilon$ on $T^n \times T^n$. The nearby cycles kernel $K_0$ corresponds heuristically to the Hamiltonian flow of the non-differentiable function $\lim_{\varepsilon \to 0} H_\varepsilon$, which is the pullback of the support function of $D$ along the cofiber projection. We show that the action of $K_0$ coincides with the convolution action of the associated twisted polytope sheaf, hence mirrors the action of $\mathcal{O}(D)$ on coherent sheaves.

Contact isotopies in the coherent-constructible correspondence

TL;DR

This work provides a geometric realization of the mirror Picard action in the coherent-constructible correspondence for toric varieties. By quantizing the flows of smoothed support functions of toric divisors and passing to the nearby-cycles limit, the authors show that the limiting kernel acts on constructible sheaves via convolution with the twisted polytope sheaf , while on the CCC side this matches the action of under the equivalence. The results extend Hanlon’s geometric action from Fukaya–Seidel categories to the CCC framework, and they handle complete and noncomplete fans, as well as equivariant and nonequivariant versions. The analysis leverages GKS kernels, singular-support control, and Whitney-type shard resolutions to establish a canonical, functorial bridge between toric coherent and constructible categories through nearby cycles. Overall, the paper deepens the symplectic‑sheaf correspondence for toric mirror symmetry and furnishes new tools for understanding Picard actions in CCC.

Abstract

The coherent-constructible correspondence is a realization of toric mirror symmetry in which the A-side is modeled by constructible sheaves on . This paper provides a geometric realization of the mirror Picard group action in this correspondence, characterizing it in terms of quantized contact isotopies and providing a sheaf-theoretic counterpart to work of Hanlon in the Fukaya-Seidel setting. Given a toric Cartier divisor , we consider a family of homogeneous Hamiltonians on . Their flows act on sheaves via a family of kernels on . The nearby cycles kernel corresponds heuristically to the Hamiltonian flow of the non-differentiable function , which is the pullback of the support function of along the cofiber projection. We show that the action of coincides with the convolution action of the associated twisted polytope sheaf, hence mirrors the action of on coherent sheaves.
Paper Structure (10 sections, 16 theorems, 91 equations)

This paper contains 10 sections, 16 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Sheaves and spaces
  4. Kernels and integral transforms
  5. Manifolds and constructibility
  6. Quantized contact isotopies
  7. Toric varieties
  8. The coherent-constructible correspondence
  9. GKS kernels on Lie groups
  10. Main results

Key Result

Theorem 1.1

Let $D$ be a torus-invariant Cartier divisor on $X_\Sigma$. For any $\mathcal{F} \in \mathrm{Sh}(T^n)$ we have an isomorphism In particular, if $\mathcal{F} \in \mathrm{Sh}_{\Lambda_\Sigma}^c(T^n)$ we have an isomorphism

Theorems & Definitions (33)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.5
  • ...and 23 more