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Capacities from the Chiu-Tamarkin complex

Bingyu Zhang

Abstract

In this paper, we construct a sequence $(c_k)_{k\in\mathbb{N}}$ of symplectic capacities based on the Chiu-Tamarkin complex $C_{T,\ell}$, a $\mathbb{Z}/\ell$-equivariant invariant coming from the microlocal theory of sheaves. We compute $(c_k)_{k\in\mathbb{N}}$ for convex toric domains, which are the same as the Gutt-Hutchings capacities. Our method also works for the prequantized contact manifold $T^*X\times S^1$. We define a sequence of "contact capacities" $([c]_k)_{k\in\mathbb{N}}$ on the prequantized contact manifold $T^*X\times S^1$, and we compute them for prequantized convex toric domains.

Capacities from the Chiu-Tamarkin complex

Abstract

In this paper, we construct a sequence of symplectic capacities based on the Chiu-Tamarkin complex , a -equivariant invariant coming from the microlocal theory of sheaves. We compute for convex toric domains, which are the same as the Gutt-Hutchings capacities. Our method also works for the prequantized contact manifold . We define a sequence of "contact capacities" on the prequantized contact manifold , and we compute them for prequantized convex toric domains.

Paper Structure

This paper contains 8 sections, 11 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Symplectic Embedding
  3. Contact Embedding
  4. Microlocal Theory of Sheaves and the Chiu-Tamarkin complex
  5. Related works
  6. Organization and Conventions of the Paper
  7. Reminder on Sheaves and Equivariant Sheaves
  8. Microsupport of Sheaves and Functorial Estimate

Key Result

Theorem 1

Equip ${\mathbb{R}}^{2d}$ with the linear symplectic form. Let $B_{\pi r^2}=\{(x,p)\in {\mathbb{R}}^{2d} : \|x \|^2+\|p\|^2 < r^2\}$, and $Z_{\pi R^2}=\{(x,p)\in {\mathbb{R}}^{2d} : x_1^2+p_1^2< R^2 \}$. If there is a symplectic embedding $\varphi: B_{\pi r^2}\rightarrow {\mathbb{R}}^{2d}$ such t

Theorems & Definitions (16)

  • Theorem 1: gromov1985
  • Theorem 2: \ref{['capacity property symplectic']}
  • Conjecture 1
  • Remark 1
  • Definition
  • Theorem 3
  • Theorem 4: chiu2017fraser2016
  • Theorem 5
  • Theorem 6: \ref{['structure toric domain module']}
  • Definition 1: KS90
  • ...and 6 more