[Paper Review] Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties

Perry Alexander, Laura Pertusi+1 more|arXiv (Cornell University)|Dec 14, 2019

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel–Mukai varieties and describes the structure of moduli spaces of semistable objects in the even-dimensional case. It constructs new unirational families of polarized hyperkähler varieties of K3 type and provides a Hodge-theoretic criterion for when the Kuznetsov component is derived equivalent to a K3 surface's derived category.

ABSTRACT

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperkähler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

Motivation & Objective

• To establish the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel–Mukai varieties.
• To describe the structure of moduli spaces of Bridgeland semistable objects in the even-dimensional case.
• To construct new unirational locally complete families of polarized hyperkähler varieties of K3 type.
• To provide a Hodge-theoretic characterization of when the Kuznetsov component is derived equivalent to the derived category of a K3 surface.
• To relate moduli spaces of stable objects in the Kuznetsov component to known hyperkähler varieties such as EPW cubes and double covers of Grassmannians.

Proposed method

• Constructs Bridgeland stability conditions on the Kuznetsov component of a Gushel–Mukai variety using a generalization of the method from cubic fourfolds.
• Employs the theory of stability conditions in families to ensure the existence of such conditions in the derived category of the Kuznetsov component.
• Uses the Hodge-theoretic properties of the Kuznetsov component and its cohomology to identify numerical invariants of moduli spaces.
• Applies Verbitsky’s Torelli theorem to establish isomorphisms between moduli spaces and known hyperkähler varieties by comparing period points.
• Identifies the Mukai vector of projections of structure sheaves of conics and points in the Kuznetsov component to relate moduli spaces to geometric objects.
• Utilizes the period map and Hodge isometries to compare the cohomology of moduli spaces with that of associated hyperkähler varieties such as $\widetilde{Y}_A$ and $\widetilde{Z}_A$.

Experimental results

Research questions

• RQ1 Do Bridgeland stability conditions exist on the Kuznetsov components of Gushel–Mukai varieties?
• RQ2 What is the structure of moduli spaces of semistable objects in the Kuznetsov component for even-dimensional Gushel–Mukai varieties?
• RQ3 Can new unirational families of polarized hyperkähler varieties of K3 type be constructed from these moduli spaces?
• RQ4 Under what Hodge-theoretic conditions is the Kuznetsov component of an even-dimensional Gushel–Mukai variety derived equivalent to the derived category of a K3 surface?
• RQ5 Are the moduli spaces of stable objects in the Kuznetsov component isomorphic to known hyperkähler varieties such as EPW cubes or double covers of Grassmannians?

Key findings

• The Kuznetsov component of a Gushel–Mukai variety admits a Bridgeland stability condition, extending the result from cubic fourfolds to this broader class.
• For very general GM fourfolds, the moduli space $M_{\sigma}(\mathcal{K}u(X),\lambda_1)$ is isomorphic to either $\widetilde{Y}_A$ or $\widetilde{Y}_{A^\perp}$, depending on the action on the discriminant group.
• The moduli space $M_{\sigma}(\mathcal{K}u(X),\lambda_1)$ has a Hodge isometry to $\mathrm{H}^2(\widetilde{Y}_A,\mathbb{Z})_0$ when $X$ and $\widetilde{Y}_A$ share the same period point.
• The moduli space $M_{\sigma}(\mathcal{K}u(X),\pm(\lambda_1\pm\lambda_2))$ has the same numerical invariants as an EPW cube, suggesting it may be isomorphic to one for generic $X$.
• The projection of a point in $X$ into $\mathcal{K}u(X)$ has Mukai vector $\lambda_1 + 2\lambda_2$, and such objects are expected to be stable, giving a rational embedding of $X$ into a 12-fold moduli space.
• The construction yields a new infinite series of unirational locally complete families of polarized hyperkähler varieties of K3 type, filling a gap in known constructions.