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Stability Conditions and Moduli Spaces on Kuznetsov Component of Cubic Fivefolds

Peize Liu

TL;DR

The paper extends stability conditions to the Kuznetsov component $\operatorname{Ku}(Y)$ of a cubic fivefold by leveraging a quadric fibration model and embedding into $\mathsf{D}^{\mathrm{b}}(\mathbb{P}^3,\mathscr{C}_0)$; it constructs a continuous family of Serre-invariant Bridgeland stability conditions on $\operatorname{Ku}(Y)$, proves the non-emptiness of all moduli spaces of stable objects, and shows the lowest-dimensional moduli is naturally identified with the Fano surface of planes $\mathcal{F}_2(Y)$, yielding a Lagrangian immersion into a hyper-Kähler manifold associated to a cubic fourfold. The approach relies on a detailed analysis of the quadric fibration geometry, mutations of derived categories, and a restricted tilt theory for Clifford-module categories, culminating in a Gepner-type stability condition with respect to the rotation and Serre functors. The results connect derived-category stability to explicit geometric moduli, extend known Serre-invariant stability phenomena to higher fractional Calabi–Yau dimensions, and reveal hyper-Kähler geometry in the moduli of objects on the Kuznetsov component. Overall, the work broadens the scope of Bridgeland stability in higher dimensions and ties moduli spaces to classical geometric objects like $\mathcal{F}_2(Y)$ and $\mathcal{F}_1(X)$.

Abstract

We study the Kuznetsov component of cubic fivefolds via their quadric fibration model. We construct a family of Serre-invariant Bridgeland stability conditions on the Kuznetsov component and establish the non-emptiness of all moduli spaces of stable objects. As an application, we identify the lowest-dimensional moduli with the Fano surface of planes of the cubic fivefold, and recover a classical result of its Lagrangian immersion into a hyper-Kähler manifold.

Stability Conditions and Moduli Spaces on Kuznetsov Component of Cubic Fivefolds

TL;DR

The paper extends stability conditions to the Kuznetsov component of a cubic fivefold by leveraging a quadric fibration model and embedding into ; it constructs a continuous family of Serre-invariant Bridgeland stability conditions on , proves the non-emptiness of all moduli spaces of stable objects, and shows the lowest-dimensional moduli is naturally identified with the Fano surface of planes , yielding a Lagrangian immersion into a hyper-Kähler manifold associated to a cubic fourfold. The approach relies on a detailed analysis of the quadric fibration geometry, mutations of derived categories, and a restricted tilt theory for Clifford-module categories, culminating in a Gepner-type stability condition with respect to the rotation and Serre functors. The results connect derived-category stability to explicit geometric moduli, extend known Serre-invariant stability phenomena to higher fractional Calabi–Yau dimensions, and reveal hyper-Kähler geometry in the moduli of objects on the Kuznetsov component. Overall, the work broadens the scope of Bridgeland stability in higher dimensions and ties moduli spaces to classical geometric objects like and .

Abstract

We study the Kuznetsov component of cubic fivefolds via their quadric fibration model. We construct a family of Serre-invariant Bridgeland stability conditions on the Kuznetsov component and establish the non-emptiness of all moduli spaces of stable objects. As an application, we identify the lowest-dimensional moduli with the Fano surface of planes of the cubic fivefold, and recover a classical result of its Lagrangian immersion into a hyper-Kähler manifold.

Paper Structure

This paper contains 18 sections, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Derived category of quadric fibrations
  3. Geometry of quadric fibrations
  4. Mutations in the derived category
  5. Numerical Grothendieck groups
  6. $\operatorname{K_{num}}$ for $\operatorname{\mathcal{K}\mkern-1mu\mathit{u}}(Y)$
  7. $\operatorname{K_{num}}$ for $\operatorname{\mathcal{K}\mkern-1mu\mathit{u}}(\mathbb{P}^3,\mathscr{C}_0)$
  8. Stability conditions on Kuznetsov component
  9. General theory
  10. Bogomolov's inequality
  11. Restricting stability conditions
  12. Serre invariance
  13. Tensor product and tilted hearts
  14. Proof of Serre invariance
  15. Moduli spaces of stable objects
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Characters in $\operatorname{K_{num}}(\operatorname{\mathcal{K}\mkern-1mu\mathit{u}}(Y))$ under the hexagonal coordinate.
  • Figure 2: The phase $\phi_P(E)$ of an object $E$ with respect to the stability condition $\widetilde{\sigma}_{P}$.

Theorems & Definitions (23)

  • Definition 8
  • Remark 9
  • Definition 18
  • Remark 21
  • Remark 23
  • Definition 24
  • Definition 28
  • Definition 29
  • Definition 30
  • Remark 31
  • ...and 13 more