Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Categorical Torelli theorems for Gushel-Mukai threefolds

Augustinas Jacovskis, Xun Lin, Zhiyu Liu, Shizhuo Zhang

Abstract

We show that a general ordinary Gushel-Mukai(GM) threefold $X$ is reconstructed from the Kuznetsov component $\mathcal{K}u(X)$ together with an extra data coming from tautological sub-bundle of Grassmannian $\mathrm{Gr}(2,5)$. We also prove that $\mathcal{K}u(X)$ determines birational isomorphism class of $X$, while $\mathcal{K}u(X')$ determines the isomorphism class of a general special GM threefold $X'$. As an application, we prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use $\mathcal{K}u(X)$ to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds.

Categorical Torelli theorems for Gushel-Mukai threefolds

Abstract

We show that a general ordinary Gushel-Mukai(GM) threefold is reconstructed from the Kuznetsov component together with an extra data coming from tautological sub-bundle of Grassmannian . We also prove that determines birational isomorphism class of , while determines the isomorphism class of a general special GM threefold . As an application, we prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds.

Paper Structure

This paper contains 47 sections, 82 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Kuznetsov components and categorical Torelli theorems
  3. Main Results
  4. (Refined) categorical Torelli for Gushel--Mukai threefolds
  5. Birational categorical Torelli for Gushel--Mukai threefolds
  6. The Debarre--Iliev--Manivel Conjecture
  7. Uniqueness of Serre-invariant stability conditions
  8. Methods
  9. Related Work
  10. Categorical Torelli theorems
  11. Identifying classical moduli spaces as Bridgeland moduli spaces for Kuznetsov components
  12. Serre-invariant stability conditions
  13. Notation and conventions
  14. Organization of the paper
  15. Semiorthogonal decompositions
  16. ...and 32 more sections

Key Result

Theorem 1.2

Let $X$ be a general ordinary GM threefold and $\pi \colon \mathrm{D}^b(X) \rightarrow \mathcal{K}u(X)$ be the right adjoint to the inclusion $\mathcal{K}u(X) \subset \mathrm{D}^b(X)$. Then the data of $\mathcal{K}u(X)$ along with the object $\pi(\mathcal{E})$ is enough to determine $X$ up to isomor

Theorems & Definitions (160)

  • Theorem 1.2: Theorem \ref{['refined categorical torelli theorem']}
  • Theorem 1.3: Theorem \ref{['theorem_torelli_sGM']}
  • Theorem 1.4: Theorem \ref{['birational OGM torelli theorem']}
  • Conjecture 1.5: kuznetsov2019categorical
  • Theorem 1.6: Theorem \ref{['theorem_KP_conjecture']} and Corollary \ref{['prop_general_preserves']}
  • Theorem 1.7: Theorem \ref{['categorical period map']}
  • Conjecture 1.8
  • Remark 1.9
  • Theorem 1.10: Theorem \ref{['all_in_one_orbit']}
  • Definition 2.1
  • ...and 150 more