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Moduli spaces on Kuznetsov components are Irreducible Symplectic Varieties

Giulia Saccà

Abstract

This article studies moduli spaces of Bridgeland semistable objects in the Kuznetsov component of a cubic fourfold that don't admit a symplectic resolution, i.e., moduli spaces of objects with non-primitve Mukai vector v=mv_0 that is not of OG10-type and where v_0^2 >0. For a generic stability condition, it is shown that these moduli spaces are projective irreducible symplectic varieties with factorial terminal singularities and that their deformation class is uniquely determined by the integers m and v_0^2. On the one hand, this generalizes the results of arXiv:1703.10839, arXiv:1912.06935, arXiv:2007.14108, which deal with moduli spaces of objects in the Kuznetsov component of a cubic fourfold which are smooth or of OG10-type; on the other hand, this extends to the Kuznetsov component of a cubic fourfold the results of arXiv:1802.01182, arXiv:2012.10649, on Gieseker moduli spaces of sheaves on K3 surfaces with non-primitive Mukai vector.

Moduli spaces on Kuznetsov components are Irreducible Symplectic Varieties

Abstract

This article studies moduli spaces of Bridgeland semistable objects in the Kuznetsov component of a cubic fourfold that don't admit a symplectic resolution, i.e., moduli spaces of objects with non-primitve Mukai vector v=mv_0 that is not of OG10-type and where v_0^2 >0. For a generic stability condition, it is shown that these moduli spaces are projective irreducible symplectic varieties with factorial terminal singularities and that their deformation class is uniquely determined by the integers m and v_0^2. On the one hand, this generalizes the results of arXiv:1703.10839, arXiv:1912.06935, arXiv:2007.14108, which deal with moduli spaces of objects in the Kuznetsov component of a cubic fourfold which are smooth or of OG10-type; on the other hand, this extends to the Kuznetsov component of a cubic fourfold the results of arXiv:1802.01182, arXiv:2012.10649, on Gieseker moduli spaces of sheaves on K3 surfaces with non-primitive Mukai vector.
Paper Structure (15 sections, 30 theorems, 31 equations)

This paper contains 15 sections, 30 theorems, 31 equations.

Table of Contents

  1. Notation, background, and recollection of the main results used
  2. The residual component of a cubic fourfold, and its Mukai lattice
  3. Bridgeland stability conditions
  4. Moduli stacks and (good) moduli spaces
  5. Results on K3 surfaces
  6. Local structure and applications
  7. Ampleness of $\ell_\sigma$
  8. Factoriality
  9. Preliminaries on Irreducible symplectic varieties
  10. Moduli spaces as Irreducible symplectic varieties
  11. Cubic fourfold case
  12. The case of twisted K3 surfaces
  13. The case of K3 surfaces
  14. Second integral cohomology of moduli spaces
  15. The case of even dimensional Gushel-Mukai manifolds

Key Result

Theorem 1.1

Let $X$ be a cubic fourfold. Then there exists a twisted K3 surface such that $\mathcal{K}u(X) \cong D^b(S, \alpha)$ if and only if there exists a class $v \in \widetilde{H}^{1,1}(\mathcal{K}u(X),\mathbb{Z})$ with $v^2=0$.

Theorems & Definitions (70)

  • Theorem 1.1: Addington-Thomas, Huybrechts, BLMNPS
  • proof
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • proof
  • Definition 1.5
  • Theorem 1.6
  • proof
  • Theorem 1.7
  • ...and 60 more