Gushel-Mukai Threefolds

• Gushel-Mukai threefolds are smooth complex Fano threefolds of Picard rank one, degree 10, and index 1, constructed via codimension‐2 sections or double covers of Grassmannians.
• They display deep connections to Hodge theory, EPW sextics, and derived categories, offering pivotal insights into Torelli problems and rationality.
• Their 10-dimensional irreducible moduli space underscores applications in birational geometry, stability conditions, and categorical resolutions.

A Gushel–Mukai threefold (GM threefold) is a smooth complex Fano threefold of Picard rank one, index one, and degree ten. These varieties possess rich geometric, Hodge-theoretic, and categorical structures that connect them to the geometry of Grassmannians, K3 surfaces, hyperkähler manifolds, Eisenbud–Popescu–Walter (EPW) sextics, and modern developments in derived categories and Bridgeland stability. GM threefolds serve as a central example in Fano geometry and categorical birational theory, providing fertile ground for Torelli-type theorems, moduli constructions, and the study of rationality.

1. Geometric Constructions and Classification

A GM threefold $X$ is realized as either:

• an ordinary GM threefold: a smooth codimension-2 linear section of the Grassmannian $\mathrm{Gr}(2,5)$ in its Plücker embedding, cut by a quadric,
• or a special GM threefold: a double cover of a quintic del Pezzo threefold (a 3-dimensional linear section of $\mathrm{Gr}(2,5)$) branched along a smooth ordinary GM surface (degree-10 K3 surface).

Explicitly, let $V_5$ be a 5-dimensional complex vector space and $\mathrm{Gr}(2,V_5)\subset \mathbb{P}^9$ its Plücker embedding. For the ordinary case, one selects a codimension-2 linear subspace $W\subset \wedge^2 V_5$ and a quadric $Q$ in $\mathbb{P}(W)$, obtaining: $X = \mathrm{Gr}(2,V_5) \cap \mathbb{P}(W) \cap Q$ For the special case, $M = \mathrm{Gr}(2,V_5)\cap \mathbb{P}(W)$ is a Fano threefold ($Y_5$), $B\subset M$ a quadric section (K3 surface), and $X$ is the double cover $\pi: X \to M$ branched over $B$.

Both types yield smooth Fano threefolds of degree $H^3=10$, index $i=1$, and genus $g=6$ (since $H^3=2g-2$). The Mukai classification theorem asserts that all smooth Fano threefolds with these invariants are GM threefolds (Bayer et al., 27 Jan 2025, Debarre et al., 2015).

The moduli space of such varieties is irreducible and 10-dimensional, parameterized by Lagrangian data $(V_6, V_5, A)$, where $A\subset\wedge^3 V_6$ is a Lagrangian subspace, $V_5\subset V_6$ a hyperplane, and $A$ satisfies a transversality condition with $V_5$ (Debarre et al., 2015). Special GM threefolds correspond to an additional degeneracy in these data.

2. Hodge Theory, Periods, and Rationality

The Hodge diamond of a smooth GM threefold $X$ is:

• $h^{1,1}(X)=1$
• $h^{2,1}(X)=h^{1,2}(X)=10$
• $h^{3,0}(X)=h^{0,3}(X)=0$

The third cohomology group $H^3(X,\mathbb{C})$ is pure of type $(2,1)+(1,2)$ and of dimension $20$. The intermediate Jacobian $J(X)$ is a principally polarized abelian variety of dimension $10$, and the period map sending $X$ to $J(X)$ is a central tool in studies of Torelli-type problems and rationality (Debarre, 2020, Lin et al., 9 Jul 2025).

The period map for ordinary GM threefolds is generically $2$-to-$1$ on its image, with a 2-dimensional fiber arising from extra deformations that keep the Hodge structure fixed; this corresponds to a 2-dimensional kernel in the differential of the period map (Lin et al., 9 Jul 2025). For special GM threefolds, the invariant part of the infinitesimal period map is injective by Hodge-theoretic and categorical arguments.

A pivotal result in irrationality asserts that the intermediate Jacobian of a general GM threefold cannot be isogenous to a product of Jacobians of curves, and in certain cases the action of finite groups such as $\mathrm{PSL}(2, \mathbb{F}_{11})$ on $J(X)$ prohibits rationality (Debarre et al., 2021). Unirationality is classical for all smooth GM threefolds, but rationality occurs only in particular singular or specialized situations (Debarre et al., 2015).

3. EPW Sextics and Moduli Connections

To each GM threefold (or Gushel-Mukai surface) is associated a Lagrangian subspace $A \subset \wedge^3 V_6$, giving rise to an EPW sextic $Y_A$ in $\mathbb{P}(V_6)$ defined by

$$Y_A = \{ [v] \in \mathbb{P}(V_6) \mid \dim( A \cap (v\wedge\wedge^2 V_6 )) \geq 1 \}$$

The singular locus $Y_A^{\geq 2}$ parametrizes points where this intersection has dimension at least 2. O'Grady's theory supplies a canonical double cover $\widetilde{Y}_A \rightarrow Y_A$ branched along $Y_A^{\geq 2}$, which for general $A$ is a smooth irreducible holomorphic symplectic variety of K3$^{[2]}$-type (Liu et al., 15 Dec 2025).

Period partners (GM threefolds with the same $A$ but different $V_5$) and duals ($A$ vs.\ $A^\perp$) are always birational in dimension three, as are their period loci in the moduli space (Debarre et al., 2015). The geometry of lines and conics on $X$ is controlled by the associated EPW sextic.

Moduli functors for stable objects in the derived category (see below) provide further bridges between GM varieties and the geometry of their EPW data (Liu et al., 15 Dec 2025). In the case of special GM threefolds, the double EPW surface $\widetilde{Y}_A^{\geq 2}$ is realized as a Bridgeland moduli space of semistable objects in the Kuznetsov component (Liu et al., 15 Dec 2025).

4. Derived Categories, Categorical Torelli, and Moduli of Stable Objects

For any GM threefold $X$, the bounded derived category $D^b(X)$ admits a semiorthogonal decomposition

$D^b(X) = \langle \Ku(X),\ \mathcal{O}_X,\ U_X^\vee\rangle$

where $U_X^\vee$ is the dual tautological bundle, and the Kuznetsov component $\Ku(X) = \langle \mathcal{O}_X,\ U_X^\vee\rangle^\perp$ is an admissible subcategory behaving as a "fractional Calabi–Yau" category with Serre functor $S_{\Ku(X)}^2 \cong [4]$; for special GM threefolds, $\Ku(X)$ carries an involution from the covering and is thus a noncommutative analogue of a K3 (Pertusi et al., 2021, Feyzbakhsh et al., 2023).

Moduli spaces of Bridgeland-stable objects in $\Ku(X)$, constructed via double tilting of the heart of $\Coh(X)$ and an explicitly defined central charge, realize many natural geometric loci:

• For special GM threefolds, the moduli space $M_{\sigma_X}(-\kappa_1)$ coincides with the double EPW surface $\widetilde{Y}_{A^\perp}^{\geq 2}$, while $M_{\sigma_X}(-\kappa_2)$ corresponds to $\widetilde{Y}_A^{\geq 2}$ (Liu et al., 15 Dec 2025).
• For ordinary GM threefolds, the minimal model $C_m(X)$ of the Fano surface of conics is isomorphic to a Bridgeland moduli space in $\Ku(X)$ (Zhang, 2020).

A refined categorical Torelli theorem holds: for a general ordinary GM threefold, the Kuznetsov component together with the image of the tautological bundle determines $X$ up to isomorphism (Jacovskis et al., 2021, Feyzbakhsh et al., 2023). Furthermore, for special GM threefolds (under genericity conditions on lines/conics), $\Ku(X)$ fully captures the isomorphism class (Liu et al., 15 Dec 2025).

The group of Fourier–Mukai autoequivalences of $\Ku(X)$ for general $X$ coincides with the product of $\mathrm{Aut}(X)$ (the automorphism group of $X$), the Serre functor, and the shift functor, highlighting the rigidity of this construction for generic GM threefolds (Feyzbakhsh et al., 2023).

5. Stability Conditions, Wall-Crossing, and Applications

Bridgeland stability conditions on $\Ku(X)$ have been explicitly constructed by successive tilting (first slope, then tilt) of $\Coh(X)$, yielding hearts and central charges compatible with the Serre functor; these conditions give a unique $\widetilde{\mathrm{GL}}^+_2(\mathbb{R})$-orbit of Serre-invariant stability conditions (Pertusi et al., 2021).

Wall-crossing in these stability conditions controls birational modifications of moduli spaces of stable objects, such as the flop wall for length-2 subschemes (Hilbert schemes) and divisorial contractions of the Hilbert scheme of twisted cubics on special GM threefolds (Liu et al., 15 Dec 2025, Zhang, 2020). Many of these spaces are proven to be projective, smooth (in the generic case), and holomorphic symplectic.

The action of the Serre functor on $\Ku(X)$ is tightly intertwined with stability conditions, and in the case of special GM fourfolds, the Kuznetsov component of the threefold lifts to the equivariant category, providing a construction of stability conditions on higher-dimensional analogues (Pertusi et al., 2021).

6. Birational Geometry, Automorphisms, and Categorical Resolutions

GM threefolds exhibit intricate birational geometry: any two birationally equivalent GM threefolds are either period partners or duals (sharing an EPW Lagrangian $A$ or $A^\perp$). There is no birational rigidity—every smooth GM threefold admits infinitely many non-isomorphic birational models within its period/double partner locus (Debarre et al., 2015).

For singular (specifically 1-nodal) GM threefolds, categorical resolutions of the Kuznetsov component, constructed via blowups and relating to Clifford modules over quadric fibrations, still detect the birational class of the original variety. A derived equivalence of these categorical resolutions implies the threefolds are birational (Grzelakowski et al., 15 Feb 2026).

Automorphism groups of GM threefolds are finite, with generic $X$ having only the identity or order-two involution (in the special case). An exception are highly symmetric examples, such as those with a faithful $\mathrm{PSL}(2, \mathbb{F}_{11})$-action studied in (Debarre et al., 2021). These symmetries have implications for the structure of $J(X)$ and questions of rationality.

7. Moduli, K-Stability, and Deformation Theory

The moduli space of GM threefolds is irreducible of dimension 10, with the locus of special GM threefolds forming a divisor corresponding to degenerations of the associated EPW sextic (Liu et al., 2024). The K-moduli (moduli of K-stable varieties/log pairs) of special GM threefolds coincide with the moduli of log pairs $(M, cQ)$ with $M$ a quintic del Pezzo and $Q$ an ordinary GM, with explicit walls given by values of $c$ (Liu et al., 2024).

All GM threefolds (ordinary and special) are K-stable; their moduli admit a chamber structure under varying the coefficient $c$ in the log pair, interpolating between ordinary and special loci. K-stability of special GM threefolds is reduced to the log K-stability of the pair $(M, \frac{1}{2}Q)$ (Liu et al., 2024).

Infinitesimal Torelli fails for ordinary GM threefolds (kernel of dimension 2), but holds for the invariant part of special GM threefolds via both Hodge-theoretic and categorical methods (Lin et al., 9 Jul 2025). The categorical viewpoint relates the kernel of the differential of the period map to moduli of stable objects in the Kuznetsov component, providing a geometric description of the non-Torelli directions.

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References:

• (Debarre et al., 2015): Gushel–Mukai varieties: classification and birationalities.
• (Debarre, 2020): Gushel–Mukai varieties.
• (Debarre et al., 2021): Gushel-Mukai varieties with many symmetries and an explicit irrational Gushel-Mukai threefold.
• (Pertusi et al., 2021): Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor.
• (Feyzbakhsh et al., 2023): New perspectives on categorical Torelli theorems for del Pezzo threefolds.
• (Liu et al., 2024): K-stability of special Gushel-Mukai manifolds.
• (Bayer et al., 27 Jan 2025): Mukai models of Fano varieties.
• (Lin et al., 9 Jul 2025): Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives.
• (Liu et al., 15 Dec 2025): EPW varieties as moduli spaces on ordinary GM surfaces and special GM threefolds.
• (Grzelakowski et al., 15 Feb 2026): Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties.