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O'Grady's tenfolds from stable bundles on hyper-Kähler fourfolds

Alessio Bottini

Abstract

We provide a modular construction of the Laza--Saccà--Voisin compactification of the intermediate Jacobian fibration of a cubic fourfold. Additionally, we construct infinitely many $20$-dimensional families of polarized hyper-Kähler manifolds of type OG10, realized as moduli spaces of stable bundles on hyper-Kähler manifolds of type $\mathrm{K3}^{[2]}$.

O'Grady's tenfolds from stable bundles on hyper-Kähler fourfolds

Abstract

We provide a modular construction of the Laza--Saccà--Voisin compactification of the intermediate Jacobian fibration of a cubic fourfold. Additionally, we construct infinitely many -dimensional families of polarized hyper-Kähler manifolds of type OG10, realized as moduli spaces of stable bundles on hyper-Kähler manifolds of type .

Paper Structure

This paper contains 41 sections, 52 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Symplectic forms
  3. Atomic, modular, and hyperholomorphic bundles
  4. Strategy of the proof
  5. The irreducible component
  6. The derived equivalence
  7. Smoothness of the moduli space
  8. Related works
  9. Structure of the paper
  10. Notations and conventions
  11. Twisted atomic objects
  12. Review: Brauer group and twisted sheaves
  13. Twisted extended Mukai lattice
  14. Twisted Poincaré sheaf
  15. Review: Tate--Shafarevich group
  16. ...and 26 more sections

Key Result

Theorem 1.1

If the cubic $Y$ is general and $\lambda$ is any polarization on $X$, then $M^{\circ}_{\mathbf v_0}(X,\lambda)$ is a connected component of $M_{\mathbf v_0}(X,\lambda)$. Moreover, it is a smooth HK manifold of type OG10 equipped with a Lagrangian fibration and there is an isomorphism $M^{\circ}_{\mathbf v_0}(X,\lambda) \cong J_Y$ compatible with the Lagrangian fibrations.

Theorems & Definitions (117)

  • Theorem 1.1: \ref{['cor:ModularLSV']} and \ref{['rem:ChangingPolarization']}
  • Theorem 1.2: \ref{['thm:ModuliOfVB']} and \ref{['cor:FamiliesOfOG10']}
  • Theorem 1.3: \ref{['thm:SmoothnessOurCase']}
  • Theorem 1.4: \ref{['thm:TwistedPoincaré']}
  • Theorem 1.5: \ref{['thm:SmoothnessGeneral']}
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • ...and 107 more