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A remark on monoidal structure and homological mirror symmetry

Tatsuki Kuwagaki

Abstract

For a symplectic geometry $X$, suppose the (derived) Fukaya category $\mathrm{Fuk}(X)$ of $X$ is equipped with a monoidal structure. Then its Balmer spectrum recovers a mirror $Y$ of $X$ if there exists homological mirror symmetry $\mathrm{Fuk}(X)\cong D^b\mathrm{coh}(Y)$ and the monoidal structure is the mirror of the standard one of $D^b\mathrm{coh}(Y)$. In this short note, we fill one gap of this story in the literature: we show that the monoidal structure determines the homological mirror functor $\mathrm{Fuk}(X)\to D^b\mathrm{coh}(Y)$.

A remark on monoidal structure and homological mirror symmetry

Abstract

For a symplectic geometry , suppose the (derived) Fukaya category of is equipped with a monoidal structure. Then its Balmer spectrum recovers a mirror of if there exists homological mirror symmetry and the monoidal structure is the mirror of the standard one of . In this short note, we fill one gap of this story in the literature: we show that the monoidal structure determines the homological mirror functor .
Paper Structure (8 sections, 4 theorems, 4 equations)

This paper contains 8 sections, 4 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Monoidal structure and homological mirror symmetry
  3. Main result
  4. A Consequence on homological mirror symmetry
  5. Construction
  6. Balmer spectrum
  7. The canonical functor
  8. Relation to Balmer's reconstruction

Key Result

Theorem 1.2

Suppose $\mathrm{Spc}_\otimes(\mathcal{T})$ is hypercomplete and the higher structure sheaf is classical. Then there exists a canonical functor $m_{\mathcal{T}, \otimes} \colon \mathcal{T}\to D(\mathcal{O}_{\mathrm{Spc}_{\otimes}(\mathcal{T})})$. If $\mathcal{T}$ is the derived category of perfect c

Theorems & Definitions (13)

  • Claim 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Definition 2.1: Balmer spectrum
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.6
  • proof
  • ...and 3 more