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Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

Soheyla Feyzbakhsh, Hanfei Guo, Zhiyu Liu, Shizhuo Zhang

Abstract

Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an $8$-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.

Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

Abstract

Let be a very general Gushel-Mukai (GM) variety of dimension , and let be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of and . In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an -dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.
Paper Structure (23 sections, 51 theorems, 178 equations)

This paper contains 23 sections, 51 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-orthogonal decompositions
  4. Stability conditions
  5. Stability of push-forward and pull-back objects
  6. Collection of lemmas
  7. Induction argument
  8. Gushel--Mukai varieties
  9. Gushel--Mukai varieties
  10. Stability conditions on the Kuznetsov components
  11. Stability of pull-back and push-forward
  12. General cases
  13. First application: Lagrangian families
  14. Properties of moduli spaces
  15. Lagrangian subvarieties
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $X$ be a GM variety of dimension $n\geq 4$ and $j \colon Y\hookrightarrow X$ be a smooth hyperplane section. Let $\sigma_Y$ and $\sigma_X$ be Serre-invariant stability conditions on $\mathcal{K}u(Y)$ and $\mathcal{K}u(X)$, respectively. We additionally assume that whichever of $X$ and $Y$ has ev

Theorems & Definitions (98)

  • Theorem 1.1: Theorem \ref{['thm-very-general-GM4']}
  • Theorem 1.2: Theorem \ref{['thm-general-gm4']}
  • Theorem 1.3: Theorem \ref{['thm-lagrangian-family']}
  • Corollary 1.4: Corollary \ref{['cor-rational-embedding']}
  • Corollary 1.5: Corollary \ref{['cor-projective']}
  • Corollary 1.6: Theorem \ref{['thm-CH0']}
  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Lemma 3.2
  • ...and 88 more