2.1. The Gushel–Mukai varieties

In the introduction, we have defined the strongly smooth Gushel–Mukai surfaces and the special Gushel–Mukai threefolds. Here, we will provide the general definition. Let $V_5$ be a five-dimensional vector space, and consider the Plücker embedding $ {\mathrm{Gr}}(2,V_5)\subset \mathbb {P}( {\Lambda }^2 V_5)$ . Let K be a one-dimensional vector space, and consider the cone

$$ \begin{align*}{\mathrm{CGr}}(2,V_5)\subset\mathbb{P}({\Lambda}^2V_5\oplus K)\end{align*} $$

with vertex $\nu $ . Choose a subspace $W\subset {\Lambda }^2 V_5\oplus K$ and a quadratic hypersurface $Q\subset \mathbb {P}(W)$ .

Definition 2.1. A Gushel–Mukai variety X is a smooth n-dimensional intersection

$$ \begin{align*}X={\mathrm{CGr}}(2,V_5)\cap\mathbb{P}(W)\cap Q\end{align*} $$

for $2\leq n\leq 6$ , such that the dimension of the vector space W is $n+5$ .

A Gushel–Mukai variety X is canonically polarized by restriction of the hyperplane class on $\mathbb {P}(W)$ . The ample polarization H satisfies $H^n=10$ and $K_X=-(n-2)H$ .

The intersection $M_X= {\mathrm{CGr}}(2,V_5)\cap \mathbb {P}(W)$ is called the Grassmannian hull of the Gushel–Mukai variety X. A Gushel–Mukai variety is called strongly smooth if its Grassmannian hull is smooth. Any Gushel–Mukai variety of dimension $n\geq 3$ is strongly smooth and the strongly smooth Gushel–Mukai surfaces are characterized by the next statement.

In fact, due to [Reference Beri9, Remark 2.19], a Gushel–Mukai surface is possibly strongly smooth only when it is ordinary in the following definition.

The vertex $\nu $ does not belong to a Gushel–Mukai variety X as X is smooth. So the linear projection from $\nu $ defines a morphism $\gamma \colon X\rightarrow {\mathrm{Gr}}(2,V_5)$ called the Gushel map. It is an embedding when X is ordinary. Suppose that X is special, then it induces a double cover over a smooth intersection $\mathbb {P}(W/K)\cap {\mathrm{Gr}}(2,V_5)$ , branched along its intersection with a quadric.

It means that an $(n+1)$ -dimensional special Gushel–Mukai variety X admits a branched double cover over the Grassmannian hull $M_Y$ of a strongly smooth ordinary Gushel–Mukai variety Y of dimension n such that the branch locus is $Y\subset M_Y$ . Conversely, the double cover over $M_Y$ branched on an n-dimensional strongly smooth ordinary Gushel–Mukai variety $Y\subset M_Y$ is a special Gushel–Mukai variety of dimension $n+1$ .

In this case, the special Gushel–Mukai variety X is called the opposite Gushel–Mukai variety of the ordinary Gushel–Mukai variety Y and vice versa.

2.2. The Lagrangian subspaces and EPW varieties

Any Gushel–Mukai variety

$$ \begin{align*}X={\mathrm{CGr}}(2,V_5)\cap\mathbb{P}(W)\cap Q\end{align*} $$

is canonically associated with a Lagrangian subspace $A(X)\subset {\Lambda }^3 V_6$ for $V_6=H^0(\mathbb {P}(W),\mathscr {I}_X(2))$ according to [Reference Debarre and Kuznetsov15, Theorem 3.10]. One can define a series of projective varieties associated with the Lagrangian subspace. These varieties are introduced and intensively studied by O’Grady.

In general, let $V_6$ be a six-dimensional vector space and $A\subset {\Lambda }^3V_6$ be the Lagrangian subspace for the $\det (V_6)$ -valued symplectic form $\omega $ defined by $\omega (\zeta ,\eta )=\zeta \land \eta $ , then the subset

$$ \begin{align*}\{[v]\in\mathbb{P}(V_6)\,|\,\dim(A\cap \left(v\land{\Lambda}^2V_6\right))\geq\ell\}\end{align*} $$

can be endowed with a scheme structure $Y_A^{\geq \ell }$ for any integral $\ell $ according to [Reference O’Grady43, Section 2].

The subscheme $Y_A^{\geq 1}\subset \mathbb {P}(V_6)$ can a priori be the whole space $\mathbb {P}(V_6)$ . To avoid this, one needs an additional condition.

The Lagrangian subspace associated with a strongly smooth Gushel–Mukai variety does not contain any decomposable element.

The degree six hypersurface $Y_A^{\geq 1}\subset \mathbb {P}(V_6)$ contains a unique prime ample class $\mathscr {O}_{Y_A^{\geq 1}}(1)$ according to the Lefschetz theorem.

One of the most important properties is that the double EPW sextic can be determined by its associated Lagrangian subspace.

The double cover construction has been generalized in [Reference Debarre and Kuznetsov17] to other EPW varieties $Y^{\geq \ell }_A$ and the double EPW surface defined below is particularly interesting.

Theorem 2.11 [Reference Debarre and Kuznetsov17, Theorem 5.2].

Consider a Lagrangian subspace $A\subset {\Lambda }^3V_6$ which does not contain decomposable elements and the associated EPW surface $Y^{\geq 2}_A\subset \mathbb {P}(V_6)$ , then there exists a unique double covering $f_A\colon \tilde {Y}^{\geq 2}_A\rightarrow Y_A^{\geq 2}$ branched over $Y_A^{\geq 3}$ such that

$$ \begin{align*}f_{A*}\mathscr{O}_{\tilde{Y}_A^{\geq2}}\cong \mathscr{O}_{Y_A^{\geq2}}\oplus\omega_{Y_A^{\geq2}}(-3),\end{align*} $$

where $\mathscr {O}_{Y_A^{\geq 2}}(1)$ is induced by $\mathscr {O}_{\mathbb {P}(V_6)}(1)$ . Moreover, one has

1. (a) the variety $\tilde {Y}_A^{\geq 2}$ is integral and normal;

2. (b) the singular locus of $\tilde {Y}_A^{\geq 2}$ is the finite set $f^{-1}_A(Y_A^{\geq 3})$ ;

3. (c) the variety $\tilde {Y}_A^{\geq 2}$ is a smooth surface of general type once $Y_A^{\geq 3}=\varnothing $ .

The variety $\tilde {Y}_A^{\geq 2}$ is called the double EPW surface associated with the Lagrangian subspace A.

The annihilator $A^{\perp }\subset {\Lambda }^3V_6^\vee $ of a given Lagrangian subspace $A\subset {\Lambda }^3V_6$ is also a Lagrangian subspace called the orthogonal of A. It contains no decomposable elements if and only if A contains no decomposable elements. So one can define the dual EPW sextic $Y_{A^{\perp }}^{\geq 1}$ , the double dual EPW sextic $\tilde {Y}_{A^{\perp }}^{\geq 1}$ , the dual EPW surface $Y_{A^{\perp }}^{\geq 2}$ , and the double dual EPW surface $\tilde {Y}_{A^{\perp }}^{\geq 2}$ for a Lagrangian subspace $A\subset {\Lambda }^3V_6$ as the respective EPW varieties associated with $A^{\perp }$ .

2.3. The period of Gushel–Mukai threefolds

Here, let us restrict to the case of Gushel–Mukai threefolds. A Gushel–Mukai threefold $(X,H)$ determines a Lagrangian subspace A in $ {\Lambda }^3 V_6$ for the vector space $V_6=H^0(\mathbb {P}(W),\mathscr {I}_X(2))$ . Together with the vector space $V_5$ in the definition of a Gushel–Mukai variety, one gets a triple $(V_6,V_5,A)$ which in turn determines X.

One has $\dim (A\cap {\Lambda }^3V_5)=3$ for the smooth Lagrangian data set $(V_6,V_5,A)$ associated with a special Gushel–Mukai threefold and has $\dim (A\cap {\Lambda }^3V_5)=2$ for the smooth Lagrangian data set $(V_6,V_5,A)$ associated with an ordinary Gushel–Mukai threefold. Conversely, one can see the following theorem.

Since the double EPW sextic $\tilde {Y}_A^{\geq 1}$ is determined by the data $(V_6,A)$ , one cannot distinguish the Gushel–Mukai threefolds by their associated double EPW sextics directly.

Moreover, one has the following criterion by [Reference Kuznetsov and Perry33, Theorem 3.1 and Lemma 3.8]. Here, one defines $Y_{A(X)}^{\ell }:=Y_{A(X)}^{\geq \ell }-Y_{A(X)}^{\geq \ell +1}$ for $\ell \geq 1$ .

Due to [Reference Bayer and Perry7, Theorem 1.9], information about the period partners or duals for a Gushel–Mukai threefold X can be read from the Kuznetsov component ${\textbf {Ku}}(X)$ . Once an additional datum is included (serving as $V_5$ ), one can recover X from ${\textbf {Ku}}(X)$ according to [Reference Jacovskis, Liu and Zhang30].

3.1. Stability conditions on triangulated categories

To reconstruct EPW varieties, we need to use the notion of stability conditions introduced by Bridgeland [Reference Bridgeland11]. Here, we will adopt a more recent formulation of stability conditions proposed in [Reference Bayer, Macrì and Stellari6, Appendix 1] based on the locally-finite full stability conditions in [Reference Bridgeland11], [Reference Bridgeland12].

Here, one fixes a triangulated category $\mathcal {D}$ , and a group homomorphism ${\textsf {v}}\colon K(\mathcal {D})\rightarrow \Lambda $ from the Grothendieck group of $\mathcal {D}$ to a finite rank lattice $\Lambda $ .

This article will consider only certain stability conditions on triangulated categories arising from geometry.

The Euler form $\chi $ on a smooth projective variety X restricts to a geometric triangulated category $\mathcal {D}\subset {\textbf {D}}^b(X)$ and one defines $\textsf {N}(\mathcal {D}):=K(\mathcal {D})/\ker (\chi )$ .

Henceforth, a triangulated category $\mathcal {D}$ is always geometric and a stability condition is always numerical. According to [Reference Bridgeland11], the set of (numerical) stability conditions on $\mathcal {D}$ , if non-empty, admits a complex manifold structure $ {\mathrm{Stab}}(\mathcal {D})$ with dimension equal to the rank of $\textsf {N}(\mathcal {D})$ .

Also, there are two mutually-commutative group actions on the complex manifold $ {\mathrm{Stab}}(\mathcal {D})$ due to [Reference Bridgeland11, Lemma 8.2] as follows.

The group $ {\mathrm{Aut}}(\mathcal {D})$ acts on $ {\mathrm{Stab}}(\mathcal {D})$ on the left, via the mapping

$$ \begin{align*}\Phi. (\mathcal{P}, Z):= (\mathcal{P}', Z \circ \Phi_*^{-1}),\end{align*} $$

where $\Phi _*$ is the induced automorphism on $\textsf {N}(\mathcal {D})$ and $\mathcal {P}'$ is defined by $\mathcal {P}'(\phi ):=\Phi (\mathcal {P}(\phi ))$ .

The universal cover

$$ \begin{align*}{\tilde{\mathrm{GL^+}}}(2,\mathbb{R}) = \left \{ \tilde{g}=(M, f) ~ \middle\vert ~ \begin{array}{l} M \in \mathrm{GL^+}(2,\mathbb{R}), \text{ and }f \colon \mathbb{R} \rightarrow \mathbb{R} \text{ is an increasing} \\ \text{function such that for all } \phi \in \mathbb{R} \text{ we have } f(\phi+1) \\ =f(\phi) + 1 \text{ and } M \cdot e^{i\pi \phi} \in \mathbb{R}_{>0} \cdot e^{i \pi f(\phi)} \end{array} \right \} \end{align*} $$

of the group $\mathrm {GL^+}(2,\mathbb {R})=\{M\in \mathrm {GL}(2,\mathbb {R})\,|\,\det (M)>0\}$ acts on the right on $ {\mathrm{Stab}}(\mathcal {D})$ , via

$$ \begin{align*}(\mathcal{P}=\{\mathcal{P}(\phi)\}_{\phi\in\mathbb{R}}, Z). (M, f):= (\mathcal{P}'=\{\mathcal{P}(f(\phi))\}_{\phi\in\mathbb{R}}, M^{-1} \circ Z), \end{align*} $$

where we identify $\mathbb {C}$ with $\mathbb {R}\oplus \mathbb {R}\sqrt {-1}$ to make sense of the composite $M^{-1}\circ Z\colon \textsf {N}(\mathcal {D})\rightarrow \mathbb {C}$ .

3.2. Moduli spaces of semistable objects

Let $\sigma $ be a stability condition on a triangulated category $\mathcal {D}\subset {\textbf {D}}^b(X)$ and v be an element in $\textsf {N}(\mathcal {D})$ , then one can define a moduli stack $\mathfrak {M}_{\sigma }(v)$ of $\sigma $ -semistable objects with numerical class v. A comprehensive reference is [Reference Bayer, Lahoz, Macrì, Nuer, Perry and Stellari2].

Definition 3.7. The moduli stack $\mathfrak {M}_{\sigma }(v)\colon ({\textbf {Sch}}/\mathbb {C})^{{\textbf {{op}}}}\rightarrow {\textbf {Gpds}}$ is defined by setting

$$ \begin{align*}\mathfrak{M}_{\sigma}(v)(T)= \left \{ E\in{\textbf{D}}^b(X_T) ~ \middle\vert ~ \begin{array}{l} \,E|_{X_t}\text{ is in }\mathcal{D}_t\subset{\textbf{D}}^b(X_T)\text{ and} \\ \text{is }\sigma\text{-semistable with class }v \\ \text{for any closed point }t\in T \end{array} \right \} \end{align*} $$

for any locally finitely generated complex scheme T. Similarly, one defines the moduli stack $\mathfrak {M}^{\textsf {{st}}}_{\sigma }(v)$ of stable objects, which is a substack of $\mathfrak {M}_{\sigma }(v)$ .

A stability condition is called proper once, for any v, the moduli stack $\mathfrak {M}_{\sigma }(v)$ is an algebraic stack of finite type, and there is an open immersion $\mathfrak {M}_{\sigma }^{\textsf {{st}}}(v)\subset \mathfrak {M}_{\sigma }(v)$ , and $\mathfrak {M}_{\sigma }(v)$ admits a proper good moduli space $\textsf {M}_{\sigma }(v)$ in the sense of [Reference Alper1], and the open substack $\mathfrak {M}_{\sigma }^{\textsf {{st}}}(v)$ is a $\mathbb {G}_m$ -gerbe over the counterpart $\textsf {M}^{\textsf {st}}_{\sigma }(v)\subset \textsf {M}_{\sigma }(v)$ . All the stability conditions in this article are proper.

Such a good moduli space $\textsf {M}_{\sigma }(v)$ is an algebraic space and the (closed) points in $\textsf {M}_{\sigma }(v)$ are S-equivalence classes of semistable objects with phase ranging in $(-1,1]$ .

Moreover, the moduli space $\textsf {M}_{\sigma }(v)$ varies with respect to the variation of the stability condition $\sigma $ in the complex manifold $ {\mathrm{Stab}}(\mathcal {D})$ .

In general, the classification of walls is not available but one always has the following two situations for a stability condition.

The walls and wall-crossing behaviors of the moduli space $\textsf {M}_{\sigma }(v)$ are well-understood for some stability conditions on $\mathbb {P}^2$ [Reference Li and Zhao36], K3 surfaces [Reference Bayer and Macrì4], [Reference Bayer and Macrì5], [Reference Meachan and Zhang40], and Enriques surfaces [Reference Nuer and Yoshioka41]. Since a strongly smooth Gushel–Mukai surface is in particular a K3 surface, the article uses some results regarding wall-crossing for K3 surfaces which are recollected in what follows.

3.3. Some stability conditions on K3 surfaces

A connected component of the complex manifold $ {\mathrm{Stab}}({\textbf {D}}^b(S))$ for a K3 surface S is introduced in [Reference Bridgeland12]. In this case, the Mukai vector

$$ \begin{align*}v(E)={\textsf{ch}}(E).\sqrt{{\textsf{td}}(S)}=({\mathrm{rank}}(E),c_1(E),{\mathrm{rank}}(E)+c_1(E)^2/2-c_2(E))\end{align*} $$

identifies the numerical Grothendieck group $\textsf {N}(S):=\textsf {N}({\textbf {D}}^b(S))$ with the algebraic Mukai lattice

$$ \begin{align*}\tilde{\text{H}}(S,\mathbb{Z})=H^0(S,\mathbb{Z})\oplus{\mathrm{NS}}(S)\oplus H^4(S,\mathbb{Z}),\end{align*} $$

where $ {\mathrm{NS}}(S)$ is the Néron–Severi group of the K3 surface. The lattice $\tilde {\text {H}}(S,\mathbb {Z})$ is endowed with the Mukai pairing $\langle (r_1,\Delta _1,s_1),(r_2,\Delta _2,s_2)\rangle =\Delta _1.\Delta _2-r_1.s_2-r_2.s_1$ , where $\Delta _1\cdot \Delta _2$ is the usual intersection product in $H^2(S,\mathbb {Z})$ and carries a canonical weight two Hodge structure from the one on the integer cohomology group $H^0(S,\mathbb {Z})\oplus H^2(S,\mathbb {Z}) \oplus H^4(S,\mathbb {Z})$ .

An ample class $\omega \in {\mathrm{NS}}(S)_{\mathbb {R}}$ defines a notion of $\mu _{\omega }$ -slope stability on ${\textbf {Coh}}(S)$ , by declaring the slope $\mu _{\omega }(\mathscr {F}):=c_1(\mathscr {F})\cdot \omega / {\mathrm{rank}}(\mathscr {F})$ for a torsion-free sheaf $\mathscr {F}$ and $\mu _{\omega }(\mathscr {F})=+\infty $ for a torsion $\mathscr {F}$ . Truncating the HN filtrations at the number $\beta \cdot \omega $ , a pair $(\omega ,\beta )\in {\mathrm{NS}}(S)^2_{\mathbb {R}}$ with $\omega $ ample determines a torsion pair $(\mathcal {T}_{\omega ,\beta },\mathcal {F}_{\omega ,\beta })$ on ${\textbf {Coh}}(S)$ such that

$$ \begin{align*} \mathcal{T}_{\omega,\beta}&=\{\text{torsion-free part of }\mathscr{F}\text{ has }\mu_{\omega}\text{-semistable HN factors of slope}>\beta\cdot\omega\}\\ \mathcal{F}_{\omega,\beta}&=\{\mathscr{F} \text{ is torsion free and its }\mu_{\omega}\text{-semistable HN factors have slope} \leq\beta\cdot\omega\}. \end{align*} $$

This torsion pair gives a bounded t-structure on ${\textbf {D}}^b(S)$ with heart

$$ \begin{align*}\mathcal{A}_{\omega,\beta}=\{E\in {\textbf{D}}^b(S)\,|\, \mathscr{H}^i(E)=0\text{ for }i\notin\{-1,0\},\mathscr{H}^{-1}(E)\in\mathcal{F}_{\omega,\beta},\mathscr{H}^0(E)\in\mathcal{T}_{\omega,\beta}\}\end{align*} $$

according to [Reference Happel, Reiten and Smalø24], on which one can define a stability condition $\sigma _{\omega ,\beta }$ under certain conditions.

Proposition 3.12 [Reference Bridgeland12, Lemma 6.2 and Section 7].

Consider a K3 surface S and two classes $\omega , \beta \in {\mathrm{NS}}(S)_{\mathbb {R}}$ with $\omega $ ample, then there exists a stability condition $\sigma _{\omega ,\beta }$ with central charge

$$ \begin{align*}Z_{\omega,\beta}\colon \tilde{\text{H}}(S,\mathbb{Z})\rightarrow\mathbb{C},\quad (r,\Delta,s)\mapsto \frac{1}{2}(2\beta\cdot\Delta-2s+r(\omega^2-\beta^2))+(\Delta-r\beta)\cdot\omega\sqrt{-1}\end{align*} $$

and heart $\mathcal {A}_{\omega ,\beta }$ if and only if $Z_{\omega ,\beta }(\mathscr {F})\notin \mathbb {R}_{\leq 0}$ for any spherical sheaf $\mathscr {F}$ on S.

Example 3.13. Let $(S,H)$ be a strongly smooth Gushel–Mukai surface endowed with the canonical degree $10$ polarization, then one can check that $(\omega ,\beta )=(1/5 H,-2/5 H)$ defines a stability condition $\sigma _S:=\sigma _{\omega ,\beta }$ on ${\textbf {D}}^b(S)$ with central charge

$$ \begin{align*}Z_{\sigma_S}(r,\Delta,s)=-\left(\frac{3}{5}r+\frac{2}{5}H\cdot\Delta+s\right)+\left(\frac{4}{5}r+\frac{1}{5}\Delta\cdot H\right)\sqrt{-1}\end{align*} $$

using the above criterion and Proposition 2.5. Otherwise, one has $Z_{\omega ,\beta }(w)\leq 0$ for some spherical class $w=(r,\Delta ,s)$ with $r>0$ . In particular, one has

$$ \begin{align*}r>0,\quad\Delta^2=2rs-2,\quad \Delta\cdot H=-4r,\quad 3r+2\Delta\cdot H+5s\geq0\end{align*} $$

at the same time. Set $x=-r$ and $y=\Delta ^2/2$ , then the only possible integer solutions for $(x,y)$ are $(-1,0)$ and $(-2,3)$ . It follows that one has either $H\cdot \Delta =4$ and $\Delta ^2=0$ or $H\cdot \Delta =2$ and $\Delta ^2=0$ . Neither is possible due to Proposition 2.5.

The stability conditions $\sigma _{\omega ,\beta }$ on a K3 surface S are proper and contained in a connected component $ {\mathrm{Stab}}^{\dagger }(S)$ of $ {\mathrm{Stab}}({\textbf {D}}^b(S))$ according to [Reference Bridgeland12], [Reference Toda51]. Moreover, the following statement asserts that, for a certain vector $v\in \tilde {\text {H}}(S,\mathbb {Z})$ and a given polarization H on S, the moduli space $\textsf {M}_{\sigma }(v)$ is equal to the moduli space $\textsf {M}_H(v)$ of Gieseker semistable sheaves on $(S,H)$ with Mukai vector v once the stability condition $\sigma $ belongs to a special chamber in $ {\mathrm{Stab}}^{\dagger }(S)$ .

In general, the possible walls in $ {\mathrm{Stab}}^{\dagger }(S)$ with respect to a given primitive Mukai vector are classified in [Reference Bayer and Macrì4] and the criteria to determine a wall can be found in [Reference Bayer and Macrì5]. These results will be recalled in the subsequent section.

3.4. The walls and wall-crossing on K3 surfaces

Let S be a K3 surface and $v\in \tilde {\text {H}}(S,\mathbb {Z})$ be a primitive Mukai vector with $\langle v,v\rangle \geq -2$ . Suppose that $W\subset {\mathrm{Stab}}^{\dagger }(S)$ is a wall with respect to v in the sense of Proposition 3.8 and $\sigma _0$ is a generic point on W, then for any two v-generic stability conditions $\sigma _{\pm }$ in the adjacent chambers with respect to $W,$ one has a birational map

$$ \begin{align*}\textsf{M}_{\sigma_-}(v)\dashrightarrow \textsf{M}_{\sigma_+}(v)\end{align*} $$

and two contractions $\textsf {M}_{\sigma _{\pm }}(v)\rightarrow \bar {M}_{\pm }$ due to [Reference Bayer and Macrì4, Theorem 1.4].

To determine a potential wall W, one introduces a rank two primitive sublattice $\textbf {H}_W$ of the Mukai lattice $\tilde {\text {H}}(S,\mathbb {Z})$ such that $w\in \textbf {H}_W$ if and only if $\Im (Z(w)/Z(v))=0$ for any $\sigma \in W$ .

In this case, using the existence of certain semistable objects, one can determine the type of the wall W according to specific vectors in $\textbf {H}_W$ by [Reference Bayer and Macrì5, Theorem 5.7] when $\langle v,v\rangle>0$ .

Otherwise, W is either a flopping wall with respect to v or not a wall. In this article, only vectors v with $\langle v,v\rangle =2$ will be involved, so one has the following special criterion.

In this case, according to [Reference Bayer and Macrì5, Proposition 9.4 and Section 14], the Jordan–Hölder factors for a strictly semistable object A in $\textsf {M}_{\sigma _0}(v)$ are spherical $\sigma _{\pm }$ -stable objects. Moreover, one notices that the vector $w'=v-w$ also satisfies $\langle w',w'\rangle =-2$ and $\langle v,w'\rangle =1$ in the setting of Theorem 3.18, so a general flopping wall is induced by two spherical stable objects.

To identify the spherical factors for such a flopping wall, one also needs to understand the walls for $\textsf {M}_{\sigma _0}(v)$ with respect to spherical classes.

4.1. Equivariant categories

Given an action of G on $\mathcal {D}$ , one can construct the equivariant category $\mathcal {D}_G$ with respect to the G-action on $\mathcal {D}$ whose objects and morphisms are, respectively, the G-equivariant objects and G-equivariant morphisms according to [Reference Beckmann and Oberdieck8], [Reference Elagin21].

Suppose in addition that G is abelian, then there exists a natural action of the dual group $G^\vee := {\mathrm{Hom}}(G,\mathbb {C}^\times )$ on $\mathcal {D}_G$ induced by a G-action on $\mathcal {D}$ .

There are mutually left and right adjoint functors: the forgetful functor

$$ \begin{align*}{{\boldsymbol{\mathsf{Forg}}}}\colon\mathcal{D}_G\rightarrow\mathcal{D}\end{align*} $$

defined by forgetting the G-equivariant structure, and the inflation functor

$$ \begin{align*}{{\boldsymbol{\mathsf{Inf}}}}\colon\mathcal{D}\rightarrow\mathcal{D}_G\end{align*} $$

given on objects by sending A to $\bigoplus _{g \in G} g_*A$ with the natural G-equivariant structure (one can see, e.g., [Reference Elagin21, Lemma 3.8]). Moreover, one has the following observation.

Proposition 4.2. Under the assumptions of Theorem 4.1, there is an equivalence of functors

$$ \begin{align*}{{\boldsymbol{\mathsf{Forg}}}} \cong {{\boldsymbol{\mathsf{Inf}}}} \circ {{\boldsymbol{\mathsf{X}}}} \colon (\mathcal{D}_G)_{G^\vee} \rightarrow \mathcal{D}_G, \end{align*} $$

where ${{\boldsymbol{\mathsf{Forg}}}}$ is the one for the induced $G^\vee $ -action on $\mathcal {D}_G$ .

Suppose in addition that both $\mathcal {D}$ and $\mathcal {D}_G$ are triangulated categories, then $(\mathcal {D}_G)_{G^\vee }$ comes with a natural class of exact triangles, consisting of those triangles whose images under

$$ \begin{align*}{{\boldsymbol{\mathsf{Forg}}}}\colon (\mathcal{D}_G)_{G^\vee} \rightarrow \mathcal{D}_G\end{align*} $$

are exact. Then, according to Proposition 4.2, the equivalence ${{\boldsymbol{\mathsf{X}}}}$ in Theorem 4.1 respects exact triangles and is therefore an equivalence of triangulated categories.

4.2. Equivariant stability conditions and covering maps

Let G be a finite group action on a triangulated category $\mathcal {D}$ . Then one can construct new stability conditions on $\mathcal {D}_G$ out of the stability conditions on $\mathcal {D}$ under suitable compatible conditions. To avoid unnecessary technical notions, we will only consider the case where $\mathcal {D}={\textbf {D}}^b(X)$ for a smooth projective variety X and the group G is the cyclic group $\mathfrak {S}_2$ . One consults [Reference Perry, Pertusi and Zhao49] for more general statements.

Theorem 4.3 [Reference Macrì, Mehrotra and Stellari38], [Reference Perry, Pertusi and Zhao49], [Reference Polishchuk50].

Suppose that the involution $\Pi $ on ${\mathbf {D}}^b(X)$ is a generator of the group action $\mathfrak {S}_2$ on $\mathcal {D}={\textbf {D}}^b(X)$ , and let $\sigma =(\mathcal {P},Z)$ be a stability condition on $\mathcal {D}$ such that, for each $\phi \in \mathbb {R,}$ one has $\Pi (\mathcal {P}(\phi ))=\mathcal {P}(\phi )$ and $Z\circ \Pi _*=Z,$ where $\Pi _*$ is the action of $\Pi $ on $\textsf {N}(X)$ , then

$$ \begin{align*} \mathcal{P}_{\mathfrak{S}_2}(\phi)&:=\{A\in\mathcal{D}_{\mathfrak{S}_2}\,|\,{{\boldsymbol{\mathsf{Forg}}}}(A)\in\mathcal{P}(\phi)\}\\ Z_{\mathfrak{S}_2}&=Z\circ{\textsf{forg}}\colon \textsf{{N}}(\mathcal{D}_{\mathfrak{S}_2})\rightarrow \mathbb{C} \end{align*} $$

determines a stability condition $\sigma _{\mathfrak {S}_2}$ on $\mathcal {D}_{\mathfrak {S}_2}$ .

In the construction from Theorem 4.3, both the central charge $Z_{\mathfrak {S}_2}$ and the slicing $\mathcal {P}_{\mathfrak {S}_2}$ of the stability condition $\sigma _{\mathfrak {S}_2}$ are preserved by the dual group $\mathfrak {S}_2^\vee $ action on $\mathcal {D}_{\mathfrak {S}_2}$ . So we may apply the theorem again to obtain a stability condition $(\sigma _{\mathfrak {S}_2})_{\mathfrak {S}_2^\vee }$ on $(\mathcal {D}_{\mathfrak {S}_2})_{\mathfrak {S}_2^\vee }$ . This iterated construction is compatible with the equivalence in Theorem 4.1, up to a multiplication by $2$ .

4.3. The Kuznetsov components and the involutions

Let X be a Gushel–Mukai threefold endowed with the canonical degree $10$ ample polarization $\mathscr {O}_X(1)$ , and let $\mathscr {U}_X$ be the pullback of the tautological bundle on $ {\mathrm{Gr}}(2,V_5)$ along the Gushel map $\gamma \colon X\rightarrow {\mathrm{Gr}}(2,V_5)$ , then ${\textbf {D}}^b(X)$ has a geometric triangulated subcategory

$$ \begin{align*}{\textbf{Ku}}(X):=\{E\in{\textbf{D}}^b(X)\,|\,\textsf{R}{\mathrm{Hom}}(\mathscr{O}_X,E)=\textsf{R}{\mathrm{Hom}}(\mathscr{U}^\vee_X,E)=0\},\end{align*} $$

which is called the Kuznetsov component of X.

The numerical Grothendieck group $\textsf {N}({\textbf {Ku}}(X))$ has rank 2. Denote the first Chern class of the polarization $\mathscr {O}_X(1)$ by h, then one has a basis $\kappa _1,\kappa _2$ of $\textsf {N}({\textbf {Ku}}(X))$ with

$$ \begin{align*}{\textsf{ch}}(\kappa_1)=-1+\frac{1}{5}h^2\quad\text{and}\quad{\textsf{ch}}(\kappa_2)=2-h+\frac{1}{12}h^3\end{align*} $$

such that the Euler form is given in this basis by $-I_2$ . One consults [Reference Kuznetsov and Perry32], [Reference Perry, Pertusi and Zhao49] for details.

Suppose now that X is special, then the opposite of X is a strongly smooth Gushel–Mukai surface S and one has a covering involution $f\colon X\rightarrow X$ for the double cover $X\rightarrow M_S$ branched over the divisor $S\subset M_S$ . The direct image of the covering involution restricts to an involution on the Kuznetsov component ${\textbf {Ku}}(X)$ and induces an $\mathfrak {S}_2$ -action on ${\textbf {Ku}}(X)$ .

According to [Reference Kuznetsov and Perry32], [Reference Perry, Pertusi and Zhao49], the equivariant category ${\textbf {Ku}}(X)_{\mathfrak {S}_2}$ is equivalent to ${\textbf {D}}^b(S)$ and the residual $\mathfrak {S}^\vee _2$ -group action on ${\textbf {D}}^b(S)$ is induced by an involution on ${\textbf {D}}^b(S)$ say $\Pi $ . The involution induces a Hodge isometry on $\tilde {\text {H}}(S,\mathbb {Z})$ whose invariant lattice is spanned by the orthogonal Mukai vectors $v_1=(1,0,-1)$ and $v_2=(-2,H,-2)$ .

The forgetful functor ${{\boldsymbol{\mathsf{Forg}}}}\colon {\mathbf{D}}^b(S)\rightarrow {\mathbf{Ku}}(X)$ induces a group homomorphism ${\textsf {forg}}$ from the lattice $\tilde {\text {H}}(S,\mathbb {Z})$ to the lattice $\textsf {N}({\textbf {Ku}}(X))$ and the inflation functor ${{\boldsymbol{\mathsf {Inf}}}}\colon {\textbf {Ku}}(X)\rightarrow {\textbf {D}}^b(S)$ induces a group homomorphism ${\textsf {inf}}\colon \textsf {N}({\textbf {Ku}}(X))\rightarrow \tilde {\text {H}}(S,\mathbb {Z})$ .

Proposition 4.5 [Reference Liu37, Proposition 6.6].

In the setups above, one has

$$ \begin{align*} {\mathsf{forg}}(r,\Delta,s)&=(r-s)\kappa_1+(2r+2s+\Delta\cdot H)\kappa_2\\ {\mathsf{inf}}(a\kappa_1+b\kappa_2)&=av_1+bv_2 \end{align*} $$

for any vector $(r,\Delta ,s)\in \tilde {\text {H}}(S,\mathbb {Z})$ and integers $a,b$ .

Moreover, the stability condition $\sigma _S$ in Example 3.13 is invariant under the involution $\Pi $ according to [Reference Liu37, Proposition 6.2].

5.1. Birational model of the double dual EPW sextics

To achieve the isomorphism, one needs the description of the associated double dual EPW sextic in [Reference Debarre and Kuznetsov18, Section 7.5].

One notices first that each line $L\subset S$ gives a Lagrangian submanifold $\mathbb {P}^2\cong (\mathbb {P}^1)^{[2]}$ in the hyperkähler manifold $S^{[2]}$ . Then, one has a Mukai flop

of these Lagrangian planes in $S^{[2]}$ . The variety $\bar {X}_S$ is a subvariety of $ {\mathrm{Gr}}(2,W)$ and the contraction $S^{[2]}\rightarrow \bar {X}_S$ is given by sending a point in $S^{[2]}$ to the line it spans in $ {\mathrm{Gr}}(2,W)$ .

Then one considers the Fano variety of lines $F(M_S)\cong \mathbb {P}^2$ in the Grassmannian hull $M_S$ of the strongly smooth Gushel–Mukai surface S. It canonically contains the Fano variety of lines $F(S)$ on S and embeds into the contracted space $\bar {X}_S$ . So the blow-up $Z_S$ of $F(M_S)$ along the set $F(S)$ of points embeds into $S^{[2]}$ . The strict transformation of the blow-up $Z_S$ along the Mukai flop g is isomorphic to $F(M_S)$ and will be again denoted by $F(M_S)$ .

Also, one notices that a conic (including the degenerate ones) and a line in S cannot meet exactly at a length $2$ subscheme due to Theorem 2.5. So the dual plane $\Sigma _C^\vee $ parameterizing lines in the plane $\Sigma _C$ spanned by C can be realized as a subvariety in $X_S$ .

Indeed, suppose that C is smooth, then every line in $\Sigma _C$ intersects C in a length $2$ subscheme which is not contained in the exceptional locus of g as the conic C cannot meet a line in S at a subscheme of length $2$ . So $\Sigma _C^\vee \cong \Sigma _C^{[2]}\subset S^{[2]}$ is not contained in the exceptional locus of g and hence embeds into $X_S$ . The singular case requires a separate argument. Suppose, for example, that C is a union of two lines, then $\Sigma _C^{[2]}$ is the union of the two Lagrangian planes spanned by these lines and the blow-up of $\Sigma _C^\vee $ along the two points represented by the lines. So one can see that $\Sigma _C^\vee $ is the strict transformation of $\Sigma _C^{[2]}$ via the flop g and hence embeds into $X_S$ .Footnote ¹

The singular locus of $\tilde {Y}_{A(S)^{\perp }}^{\geq 1}$ is equal to $f_{A(S)^{\perp }}^{-1}(Y^{\geq 3}_{A(S)^{\perp }})$ as a set, and for any p in $Y^{\geq 3}_{A(S)^{\perp }}$ , one uses $\tilde {p}$ to denote its preimage in $f_{A(S)^{\perp }}^{-1}(Y^{\geq 3}_{A(S)^{\perp }})$ . One always has a distinguished point

$$ \begin{align*}p_S\in Y^{\geq 3}_{A(S)^{\perp}}=\{[v]\in\mathbb{P}(V_6^\vee)\,|\,\dim(A(S)^{\perp}\cap\left(v\land{\Lambda}^2V_6^\vee\right))\geq 3\}\end{align*} $$

determined by the subspace $V_5\subset V_6$ according to [Reference Debarre and Kuznetsov13, Theorem 3.10], and $Y^{\geq 3}_{A(S)^{\perp }}-\{p_S\}$ is equal to the set of conics in S. The point $p_C\in Y^{\geq 3}_{A(S)^{\perp }}$ corresponding to a conic $C\subset S$ is given by the unique five-dimensional subspace $U_5\subset V_6$ determined by the linear span of C in $\mathbb {P}(W)$ .

Theorem 5.1 [Reference Debarre and Kuznetsov18, Theorem 7.7 and Remark 7.9].

There is a diagram

$$ \begin{align*}S^{[2]}\stackrel{g}{\dashrightarrow} X_S\stackrel{f^+}{\longrightarrow}\tilde{Y}_{A(S)^{\perp}}^{\geq 1},\end{align*} $$

where the birational map g is the Mukai flop of all Lagrangian planes in $S^{[2]}$ corresponding to lines in S and the morphism $f^+$ is a symplectic resolution of singularities. In particular, the map $f^+$ contracts $F(M_S)\subset X_S$ to the singular point $\tilde {p}_S$ and, for any other point corresponding to a conic $C\subset S$ , the dual projective plane $\Sigma ^\vee _C$ to that point.

In the following sections, we will recover the diagram by studying the wall-crossing behavior of the moduli space $\textsf {M}_{\sigma _t}(v_1)$ for the stability condition $\sigma _t:=\sigma _{\sqrt {t}/5H,-2/5H}$ with $t\geq 1$ .

5.2. Exclusion of the non-flopping walls

In this section, we will show that either $\sigma _t$ is not on a wall with respect to $v_1$ or is on a flopping wall with respect to $v_1$ .

The central charge of the stability condition $\sigma _t$ is

$$ \begin{align*}Z_{\sigma_t}(r,\Delta,s)=\left((\frac{t-4}{5})r-\frac{2}{5}H\cdot\Delta-s\right)+\left(\frac{4}{5}r+\frac{1}{5}\Delta\cdot H\right)\sqrt{-t}\end{align*} $$

and the associated lattice $\textbf {H}_W$ of a potential wall W for $v_1=(1,0,-1)$ is characterized by

$$ \begin{align*}\Im\frac{Z_{\sigma_t}(r,\Delta,s)}{Z_{\sigma_t}(1,0,-1)}=\Im\left(\frac{((t-4)r-2\Delta\cdot H-5s)+(\Delta\cdot H+4r)\sqrt{-t}}{(t+1)+4\sqrt{-t}}\right)=0\end{align*} $$

or equivalently the condition $20r+(t+9)\Delta \cdot H+20s=0$ .

Proposition 5.3. There is no class $w\in \mathbf {H}_W$ such that $\langle w,w\rangle =-2$ and $\langle v,w\rangle =0$ .

Proof. Suppose otherwise that there is such a vector $w=(r,\Delta ,s)$ , then one has $(t+9)\Delta \cdot H=-40r$ and $\Delta ^2=2r^2-2$ . It follows

$$ \begin{align*}20r^2-20=10\Delta^2\leq (\Delta\cdot H)^2=\left(\frac{40r}{t+9}\right)^2\end{align*} $$

or equivalently $((t+9)^2-80)r^2\leq (t+9)^2$ . So $r^2\leq 5$ and all the possible cases are

• • $r=0$ , $t\geq 1$ and $\Delta ^2=-2$ and $\Delta \cdot H=0$ ;

• • $r=\pm 1$ , $t=1$ , $\Delta ^2=0$ and $\Delta \cdot H=\pm 4$ ;

• • $r=\pm 1$ , $t=13/3$ , $\Delta ^2=0$ and $\Delta \cdot H=\pm 3$ ;

• • $r=\pm 1$ , $t=11$ , $\Delta ^2=0$ and $\Delta \cdot H=\pm 2$ ;

• • $r=\pm 1$ , $t=31$ , $\Delta ^2=0$ and $\Delta \cdot H=\pm 1$ ;

• • $r=\pm 2$ , $t=1$ , $\Delta ^2=6$ and $\Delta \cdot H=\pm 8$ .

The first case is absurd as $|\Delta |$ should contain an effective class and H is very ample, the third case will contradict Example 2.3, and the other cases contradict Proposition 2.5 (in the last case, one has $(H\pm \Delta )^2=0$ and $(H\pm \Delta )\cdot H=2$ ).

Proposition 5.4. There is no vector $w\in \mathbf {H}_W$ such that $\langle w,w\rangle =0$ and $\langle v,w\rangle =1$ .

Proof. Suppose otherwise that there exists such a vector $w=(r,\Delta ,s)$ , then one has $(t+9)\Delta \cdot H=-40r+20$ and $\Delta ^2=2r^2-2r$ . It follows

$$ \begin{align*}20r^2-20r=10\Delta^2\leq (\Delta\cdot H)^2=\left(\frac{40r-20}{t+9}\right)^2\end{align*} $$

or equivalently $((t+9)^2-80)r^2-((t+9)^2-80)r\leq 20$ . So $r\in \{0,1\}$ and all possible cases are

• • $r=0,t=1$ and $\Delta ^2=0$ and $\Delta \cdot H=2$ ;

• • $r=0,t=11$ and $\Delta ^2=0$ and $\Delta \cdot H=1$ ;

• • $r=1,t=1$ and $\Delta ^2=0$ and $\Delta \cdot H=-2$ ;

• • $r=1,t=11$ and $\Delta ^2=0$ and $\Delta \cdot H=-1$ ;

which always conflict with Proposition 2.5.

Proposition 5.5. There is no vector $w\in \mathbf {H}_W$ such that $\langle w,w\rangle =0$ and $\langle v,w\rangle =2$ .

Proof. Suppose otherwise that there exists such a vector $w=(r,\Delta ,s)$ , then one has $(t+9)\Delta \cdot H=-40r+40$ and $\Delta ^2=2r^2-4r$ . It follows

$$ \begin{align*}20r^2-40r=10\Delta^2\leq (\Delta\cdot H)^2=\left(\frac{40r-40}{t+9}\right)^2\end{align*} $$

or equivalently $((t+9)^2-80)r^2-2((t+9)^2-80)r\leq 80$ . So $r\in \{-1,0,1,2,3\}$ and one can check as in Propositions 5.3 and 5.4 that all possible cases are invalid.

Proposition 5.6. The potential wall cannot be a totally semistable wall.

Proof. Thanks to Proposition 5.4, it suffices to show that there is no $w=(r,\Delta ,s)$ in $\textbf {H}_W$ with $\langle w,w\rangle =-2$ satisfying $r-s<0$ and $\Re (Z_{\sigma _t}(r,\Delta ,s)/Z_{\sigma _t}(1,0,-1))>0$ . Otherwise, one has

$$ \begin{align*} r&=-\frac{t+9}{40}\Delta\cdot H-\frac{1}{2}\sqrt{\frac{(t+9)^2}{400}(\Delta\cdot H)^2-4-2\Delta^2}\\ s&=-\frac{t+9}{40}\Delta\cdot H+\frac{1}{2}\sqrt{\frac{(t+9)^2}{400}(\Delta\cdot H)^2-4-2\Delta^2} \end{align*} $$

since $2rs=\Delta ^2+2$ and $20(r+s)=-(t+9)\Delta \cdot H$ .

The condition $\Re (Z_{\sigma _t}(r,\Delta ,s)/Z_{\sigma _t}(1,0,-1))>0$ is equivalent to

$$ \begin{align*}(t^2+13t-4)r+(2t-2) \Delta\cdot H -5(t+1)s>0\end{align*} $$

and, after plugging s and r, it becomes

$$ \begin{align*}(t^2+18t+1)\left(r+\frac{1}{4}\Delta\cdot H\right)>0,\end{align*} $$

which implies $\Delta \cdot H+4r>0$ as $t\geq 1$ . Together with $40r<-(t+9)\Delta \cdot H$ and $t\geq 1$ , one can see that $\Delta \cdot H<0$ and $r>0$ . Also, since $r<s,$ one has $s\geq r+1$ so that

$$ \begin{align*}16r^2>(\Delta\cdot H)^2\geq 10\Delta^2=20rs-20\geq 20r(r+1)-20,\end{align*} $$

which is not possible for integer $r>0$ .

In summary, every actual wall crossed by the path $\sigma _t$ is a flopping wall.

5.3. The wall-crossing of spherical factors

In this section, some stable spherical objects will be identified as the Jordan–Hölder factors appearing in the next section.

Proposition 5.7. Consider a line $L\subset S$ , then $\textsf {M}_{\sigma _t}(0,L,-1)=\textsf {{M}}_H(0,L,-1)$ for any $t\geq 1$ .

Proof. It suffices to show that $\textsf {M}_{\sigma _t}(0,L,-1)$ contains only stable objects in the heart of $\sigma _t$ for any $t\geq 1$ . Otherwise, one has an exact sequence $0\rightarrow A\rightarrow \mathscr {O}_L(-2)\rightarrow B\rightarrow 0$ of $\sigma _t$ -semistable objects with the same phase. Set $v(A)=(r,\Delta ,s)$ , then $r\geq 1$ and one has

$$ \begin{align*}\Im(Z_t(A))=(\frac{1}{5}\Delta\cdot H+\frac{4}{5}r)\sqrt{t}>0\quad\text{and}\quad \Im(Z_t(B))=(\frac{1}{5}-\frac{1}{5}\Delta\cdot H-\frac{4}{5}r)\sqrt{t}>0\end{align*} $$

implying that $\Delta \cdot H+4r$ cannot be an integer, which is absurd.

Proposition 5.8. Consider a line $L\subset S$ , then $\textsf {M}_{\sigma _t}(1,-L,0)=\textsf {{M}}_H(1,-L,0)$ for any $t\geq 1$ .

Proof. It suffices to show that $\textsf {M}_{\sigma _t}(1,-L,0)$ contains only stable objects in the heart of $\sigma _t$ for any $t\geq 1$ . Otherwise, one has an exact sequence $0\rightarrow A\rightarrow \mathscr {O}_S(-L)\rightarrow B\rightarrow 0$ of $\sigma _t$ -semistable objects with the same phase such that A is stable. Set $v(A)=(r,\Delta ,s)$ , then

$$ \begin{align*}\Im(Z_t(A))=\Big(\frac{1}{5}\Delta\cdot H+\frac{4}{5}r\Big)\sqrt{t}>0\quad\text{and}\quad \Im(Z_t(B))=\Big(\frac{3}{5}-\frac{1}{5}\Delta\cdot H-\frac{4}{5}r\Big)\sqrt{t}>0\end{align*} $$

so that $k:=\Delta \cdot H+4r$ is either $1$ or $2$ . It follows $3\Re (Z_t(A))=k\Re (Z_t(\mathscr {O}_S(-L)))$ and then an equation $15s=(3t+12)r-4k-kt$ .

Since $(1,-L,0)$ is a spherical vector, the stable HN factor A of $\mathscr {O}_S(-L)$ is also spherical due to Proposition 3.19 (or directly [Reference Bridgeland12, Lemma 12.2]). So the Hodge index theorem gives

$$ \begin{align*}(4r-k)^2\geq10\Delta^2=10(2rs-2)=\frac{4}{3}((3t+12)r-4k-kt)r-20\end{align*} $$

and then the inequality $60+3k^2 \geq t(12r-4k)r+8kr\geq 12r^2+4kr$ as $3r\geq 3>k$ . It follows that $r=1$ or $r=2$ . The latter case is impossible as one can always see $s\geq 2$ from $t\geq 1$ and the equality for s. So the Hodge index theorem becomes $(8-k)^2\geq 10\Delta ^2\geq 60$ , impossible.

Hence, one must have $r=1$ , then $\mathscr {H}^{-1}(B)=0$ from the destabilizing sequence. In this case, the object B is also a spherical stable factor of $\mathscr {O}_S(-L)$ by Proposition 3.19. It follows that $s=1-\Delta \cdot L$ . One notices that $\Delta \cdot H\leq -2$ as $k=1,2$ . Then one can see from the Hodge index theorem that $s=1$ , hence $\Delta ^2=0$ . So one has $\Delta \cdot H=-2$ or $\Delta \cdot H=-3$ which will contradict either Example 2.3 or Proposition 2.5.

Proposition 5.9. Consider a conic $C\subset S$ , then $\textsf {M}_{\sigma _t}(0,C,-1)=\textsf {{M}}_H(0,C,-1)$ for $t\geq 1$ .

Proof. It suffices to show that $\textsf {M}_{\sigma _t}(0,C,-1)$ contains only stable objects in the heart of $\sigma _t$ for any $t\geq 1$ . Otherwise, one has an exact sequence $0\rightarrow A\rightarrow \mathscr {O}_C(-2)\rightarrow B\rightarrow 0$ of $\sigma _t$ -semistable objects with the same phase such that A is $\sigma _t$ -stable. Set $v(A)=(r,\Delta ,s)$ , then one has

$$ \begin{align*}\Im(Z_t(A))=\Big(\frac{1}{5}\Delta\cdot H+\frac{4}{5}r\Big)\sqrt{t}>0\quad\text{and}\quad \Im(Z_t(B))=\Big(\frac{2}{5}-\frac{1}{5}\Delta\cdot H-\frac{4}{5}r\Big)\sqrt{t}>0\end{align*} $$

so that $\Delta \cdot H+4r=1$ . Then $2\Re (Z_t(A))=\Re (Z_t(\mathscr {O}_C(-2)))$ and one gets $10s=(2t+8)r-5$ .

Since $(0,C,-1)$ is a spherical vector, the stable factor A is also spherical. So the Hodge index theorem gives $(4r-1)^2\geq 10\Delta ^2=10(2rs-2)=(4t+16)r^2-10r-20$ from which one obtains $r\in \{0,1,2\}$ as $r\geq 0$ and $t\geq 1$ . Suppose that $r=0$ , then one has $2s=-1$ , which is absurd. Suppose that $r=1$ , then $\Delta \cdot H=-3$ . However, one can see from $10s=2t+3$ and the Hodge index theorem that $\Delta ^2=0$ , which is impossible as $(S,H)$ is Brill–Noether general of degree $10$ . Suppose that $r=2$ , then $s\geq 2$ and $\Delta ^2\geq 6$ , violating the Hodge index theorem.

Proposition 5.10. Consider a conic $C\subset S$ , then $\textsf {M}_{\sigma _t}(1,-C,0)=\textsf {{M}}_H(1,-C,0)$ for $t\geq 1$ .

Proof. It suffices to show that $\textsf {M}_{\sigma _t}(1,-C,0)$ contains only stable objects in the heart of $\sigma _t$ for any $t\geq 1$ . Otherwise, one has an exact sequence $0\rightarrow A\rightarrow \mathscr {O}_S(-C)\rightarrow B\rightarrow 0$ of $\sigma _t$ -semistable objects with the same phase such that A is $\sigma _t$ -stable. Set $v(A)=(r,\Delta ,s)$ , then one has

$$ \begin{align*}\Im(Z_t(A))=\Big(\frac{1}{5}\Delta\cdot H+\frac{4}{5}r\Big)\sqrt{t}>0\quad\text{and}\quad \Im(Z_t(B))=\Big(\frac{2}{5}-\frac{1}{5}\Delta\cdot H-\frac{4}{5}r\Big)\sqrt{t}>0\end{align*} $$

so that $\Delta \cdot H+4r=1$ . Then $\Re (Z_t(A))=\Re (Z_t(B))$ and one gets $10s=(2t+8)r-t-4$ .

Since $(1,-C,0)$ is a spherical vector, the stable factor A is also spherical. So the Hodge index theorem gives $(4r-1)^2\geq 10\Delta ^2=10(2rs-2)=(4t+16)r^2-(2t+8)r-20$ from which one obtains $r=1$ or $r=2$ as $r\geq 1$ and $t\geq 1$ . Suppose that $r=1$ , then $\Delta \cdot H=-3$ . On the other hand, the Hodge index theorem and $10s=t+4>0$ ensure that $\Delta ^2=0$ . It is impossible since $(S,H)$ is a Brill–Noether general K3 surface. Suppose that $r=2$ , then one can see $s\geq 1$ and $\Delta ^2\geq 6$ but it will violate the Hodge index theorem.

Now one considers the unique Gieseker stable bundle $\mathscr {U}_S$ on S with Mukai vector $(2,-H,3)$ which is known to be the restriction of the tautological bundle $\mathscr {U}_G$ on $ {\mathrm{Gr}}(2,V_5)$ .

Then one considers the unique Gieseker stable bundle $\mathscr {V}_S$ on S with Mukai vector $(3,-H,2)$ which is actually the restriction of the tautological bundle $\mathscr {V}_G$ on $ {\mathrm{Gr}}(3,V_5)\cong {\mathrm{Gr}}(2,V_5)$ .

5.4. The flopping walls and identification

In this section, we will recover the birational map by identifying the possible flopping walls for the moduli space $\textsf {M}_{\sigma _t}(v_1)$ .

Proposition 5.13. Suppose that $t>1$ , then the stability condition $\sigma _t$ is on a flopping wall for $v_1=(1,0,-1)$ if and only if $t=11$ and S contains lines. Moreover, the induced flop map

$$ \begin{align*}S^{[2]}\dashrightarrow Y_S\end{align*} $$

coincides with the Mukai flop g in Theorem 5.1.

Proof. According to Theorem 3.18, a flopping wall for $v_1$ can only be induced by a pair of vectors w and $w'$ in $\textbf {H}_W$ such that $\langle w,w\rangle =\langle w',w'\rangle =-2,\langle v_1,w\rangle =\langle v_1,w'\rangle =1$ and $w'+w=v_1$ . One writes $w=(r,\Delta ,s)$ , then $(t+9)\Delta \cdot H=-40r+20$ and $\Delta ^2=2r^2-2r-2$ . It follows

$$ \begin{align*}20r^2-20r-20=10\Delta^2\leq (\Delta\cdot H)^2=\left(\frac{40r-20}{t+9}\right)^2\end{align*} $$

or equivalently $r^2-r-1\leq 100/((t+9)^2-80)$ . It follows $t=3,11$ as $t>0$ and $r\in \mathbb {Z}$ .

Suppose that $t=3$ , then $r\in \{-1,2\}$ . It means that $\Delta ^2=2$ and $\Delta \cdot H=\pm 5$ . However, the class $2\Delta \pm H$ will have square $-2$ and degree zero. It is impossible as H is ample.

Hence, $t=11$ and $r\in \{1,0\}$ . Then all possible vectors are $(1,\Delta ,0)$ with $\Delta ^2=-2,\Delta \cdot H=-1$ and $(0,\Delta ,-1)$ with $\Delta ^2=-2,\Delta \cdot H=1$ .

The divisor class $\Delta $ in $(0,\Delta ,-1)$ is in fact the class of a line in S (see, e.g., the discussions in [Reference Huybrechts25, Section 2.1.4]) and the class $\Delta $ in $(1,\Delta ,0)$ is $-L$ for some line L in S.

According to Theorem 3.14, one has $\textsf {M}_{\sigma _t}(1,0,-1)=\textsf {{M}}_H(1,0,-1)\cong S^{[2]}$ for $t>11$ . Then the flopping walls for $\textsf {M}_{\sigma _{11}}(1,0,-1)$ are induced by the short exact sequences

$$ \begin{align*}0\rightarrow\mathscr{O}_S(-L)\rightarrow\mathscr{I}_Z\rightarrow \mathscr{O}_L(-Z)\rightarrow0,\end{align*} $$

where Z is any length $2$ subscheme on any line $L\subset S$ . Then one has a flop

$$ \begin{align*}g\colon S^{[2]}\dashrightarrow Y_S=\textsf{M}_{\sigma_t}(1,0,-1)\end{align*} $$

for any $11>t>1$ , which coincides with the Mukai flop in Theorem 5.1 by construction.

It remains to show that $Y_S\cong X_S$ contracts to the moduli space $\textsf {M}_{\sigma _S}(1,0,-1)$ through the same symplectic resolution $f^+\colon X_S\rightarrow \tilde {Y}_{A(S)^{\perp }}^{\geq 1}$ as in Theorem 5.1.

Proposition 5.14. The moduli space $\textsf {M}_{\sigma _S}(v_1)$ is isomorphic to $\tilde {Y}^{\geq 1}_{A(S)^{\perp }}$ .

Proof. According to Theorem 3.18, a flopping wall for $\textsf {M}_{\sigma _S}(v_1)$ is induced by a pair of vectors w and $w'$ in $\textbf {H}_W$ such that $\langle w,w\rangle =\langle w',w'\rangle =-2,\langle v_1,w\rangle =\langle v_1,w'\rangle =1,$ and $w'+w=v_1$ . One writes $w=(r,\Delta ,s)$ , then $\Delta \cdot H=-4r+2$ and also $\Delta ^2=2r^2-2r-2$ . It follows

$$ \begin{align*}20r^2-20r-20=10\Delta^2\leq (\Delta\cdot H)^2=(4r-2)^2\end{align*} $$

or equivalently $r^2-r-1\leq 5$ . So one has $r\in \{-2,-1,0,1,2,3\}$ .

At first, one notices that r cannot be $-1$ or $2$ as in this case one will get a class $\Delta $ with $\Delta \cdot H=6$ and $\Delta ^2=2$ . It is impossible as otherwise $H-\Delta $ would violate Proposition 2.5.

Also, one can see that the class $\Delta $ in $w=(1,-\Delta ,0)$ and $w'=(0,\Delta ,-1)$ with $\Delta \cdot H=2$ and $\Delta ^2=-2$ is in fact the class of a conic C in S (see, e.g., [Reference Huybrechts25, Section 2.1.4]). The corresponding flopping wall is induced by the short exact sequences

$$ \begin{align*}0\rightarrow\mathscr{O}_S(-C)\rightarrow\mathscr{I}_Z\rightarrow \mathscr{O}_C(-Z)\rightarrow0\end{align*} $$

for the conic $C\subset S$ and any length two subscheme $Z\subset C\subset S$ which is not in the exceptional locus of the Mukai flop $g\colon S^{[2]}\dashrightarrow X_S$ . So the space $\Sigma _C^\vee $ is contracted to a point.

Then one considers the pair of vectors $w=(-2,\Delta ,-3)$ and $w'=(3,-\Delta ,2)$ with $\Delta \cdot H=10$ and $\Delta ^2=10$ . The Hodge index theorem ensures that $\Delta =H$ . So according to Propositions 5.11 and 5.12, the corresponding destabilizing exact sequence for $\mathscr {I}_Z$ is

$$ \begin{align*}0\rightarrow\mathscr{V}_S\rightarrow\mathscr{I}_Z\rightarrow\mathscr{U}_S[1]\rightarrow0,\end{align*} $$

where $Z\subset S$ is a length $2$ subscheme spanning a line not in S. Indeed, such a short exact sequence follows from the fact that the ideal sheaf $\mathscr {I}_{L/M_S}$ of every line $L\subset M_S$ can be written as the cokernel of a unique map $\mathscr {U}_G|_{M_S}\rightarrow \mathscr {V}_G|_{M_S}$ , and each such map has the ideal sheaf of a line as the cokernel. One just takes the restriction onto $S\subset M_S$ . The line spanned by Z is not in S because such a length two subscheme Z has already been flopped in Proposition 5.13. So one contracts the Fano variety $F(M_S)$ of lines, and therefore recovers $f^+\colon X_S\rightarrow \tilde {Y}_{A(S)^{\perp }}^{\geq 1}$ .

Moreover, the computations above show that the contracted space parameterizes all the S-equivalence classes of $\sigma _S$ -semistable objects with class $v_1$ . This proves the claim.

In conclusion, we finish the proof of Theorem 1.1.

6.1. Double EPW sextics

It is shown in [Reference Liu37, Section 6.6] that the moduli space $\textsf {M}_{\sigma _S}(v_2)$ is isomorphic to the double EPW sextic once S has Picard rank one. Here, we extend the result to the case where $Y_{A(S)}^{\geq 3}=\varnothing $ . According to [Reference Beri9, Theorem 3.3], it means that S contains no lines or quintic elliptic curves, allowing us to achieve the following two observations.

Proposition 6.1. Suppose that $(S,H)$ is a strongly smooth Gushel–Mukai surface containing no quintic elliptic curves, then $\textsf {M}_H(-v_2)$ is a hyperkähler manifold.

Proof. It suffices to show that there are no strictly semistable sheaves in $\textsf {M}_H(-v_2)$ . Suppose, by contradiction, that there exists such a strictly semistable sheaf $\mathscr {G}$ . Then one can take a Jordan–Hölder factor $\mathscr {B}$ of $\mathscr {G}$ with $v(\mathscr {B})=(1,\Delta ,s)$ . One can see

$$ \begin{align*}\Delta\cdot H=5\quad\text{and}\quad s=1\end{align*} $$

by the equality of reduced Hilbert polynomials. So $\mathscr {B}\cong \mathscr {I}_Z(\Delta )$ for some zero-dimensional subscheme $Z\subset S$ . It follows that $\Delta ^2\geq 0$ as the length of Z is non-negative.

By the Hodge index theorem, one can see that $\Delta ^2=0$ or $\Delta ^2=2$ . The case $\Delta ^2=0$ will violate our assumption, while the case $\Delta ^2=2$ will violate Example 2.3.

Proposition 6.2. Suppose that $(S,H)$ is a strongly smooth Gushel–Mukai surface containing no lines or quintic elliptic curves, then one has $\textsf {M}_{\sigma _S}(v_2)\cong \textsf {M}_H(-v_2)$ .

Proof. It suffices to show that $\sigma _t$ is $v_2$ -generic for any $t\geq 1$ . Otherwise, one chooses the largest $t\geq 1$ such that $\sigma _t$ is on a wall with respect to $v_2$ . Then a strictly $\sigma _t$ -semistable object in the heart with Mukai vector $v_2$ has the form $\mathscr {G}[1]$ for some $\mathscr {G}$ in $\textsf {M}_H(-v_2)=\textsf {{M}}_H(2,-H,2)$ .

One finds a short exact sequence $0\rightarrow A\rightarrow \mathscr {G}[1]\rightarrow B\rightarrow 0$ of $\sigma _t$ -semistable objects with the same phase such that A is stable. One can first see $r\geq -1$ for $v(A)=(r,\Delta ,s)$ from the long exact sequence of cohomology sheaves $0\rightarrow \mathscr {H}^{-1}(A)\rightarrow \mathscr {G}\rightarrow \mathscr {H}^{-1}(B)\rightarrow \mathscr {H}^0(A)\rightarrow 0$ .

Also, one can obtain

$$ \begin{align*}\Delta\cdot H=1-4r\quad\text{and}\quad s=\frac{t+4}{5}r+\frac{t-1}{5} \end{align*} $$

through $\mu (A)=\mu (B)=\mu (\mathscr {G}[1])<+\infty $ . Then one can see

$$ \begin{align*}(4r-1)^2=(\Delta\cdot H)^2\geq10\Delta^2\geq 20rs-20=4(t+4)r^2+4(t-1)r-20\end{align*} $$

using the Hodge index theorem and Mukai lemma. It follows

$$ \begin{align*}\frac{25}{4}\geq (tr+1)(r+1)\end{align*} $$

and one can see $r\leq 1$ from this inequality and $t\geq 1$ .

Suppose that $r=1$ , then $\Delta \cdot H=-3$ and $9\geq 10\Delta ^2\geq 8t-8\geq 8$ . It follows $\Delta ^2=0$ , but then the class $-\Delta $ will violate Example 2.3.

Suppose that $r=0$ , then $\Delta \cdot H=1$ and $1\geq 10\Delta ^2\geq -20$ . It follows $\Delta ^2=-2$ due to Proposition 2.5, but then the class $\Delta $ is a line which is impossible.

Suppose that $r=-1$ , then $\Delta \cdot H=5$ and $25\geq 10\Delta ^2\geq 0$ . It follows $\Delta ^2=2$ , as S also does not contain quintic elliptic curves. However, in this case, the class $D:=H-2\Delta $ is a degree zero effective class, which is impossible as H is ample.

Using the Mukai isomorphism

$$ \begin{align*}\theta_{-v_2}\colon(-v_2)^{\perp}\stackrel{\sim}{\rightarrow} H^2(\textsf{M}_H(-v_2),\mathbb{Z})\end{align*} $$

in [Reference Bayer and Macrì4], [Reference Bayer and Macrì5], one can define a square two class $\theta _S:=\theta _{-v_2}(v_1)$ . According to [Reference Bayer and Macrì5, Theorem 12.1] and the wall-crossing criterion in Section 3.4, one can verify that $\theta _S$ is ample by Proposition 6.2.

Proposition 6.3. Suppose that $(S,H)$ is a strongly smooth Gushel–Mukai surface containing no lines or quintic elliptic curves, then one has a polarized isomorphism

$$ \begin{align*}(\textsf{M}_H(-v_2),\theta_S)\stackrel{\sim}{\rightarrow} (\tilde{Y}_{A(S)}^{\geq 1},h_{A(S)})\end{align*} $$

of hyperkähler manifolds where $h_{A(S)}$ is the ample class corresponding to $\mathscr {O}_{\tilde {Y}_{A(S)}^{\geq 1}}(1)$ .

Proof. The statement holds when S has Picard rank one according to [Reference O’Grady42, Section 5.4] and the discussion in [Reference Liu37, Section 6.6]. Then we want to spread this polarized isomorphism.

To save notation, let us set the hyperkähler manifolds

$$ \begin{align*}M_S:=\textsf{M}_H(-v_2)\quad\text{and}\quad Y_S:=\tilde{Y}_{A(S)}^{\geq 1} \end{align*} $$

for a strongly smooth Gushel–Mukai surface S containing no lines or quintic elliptic curves.

Now pick any such Gushel–Mukai surface $(S,H)$ , and consider a small irreducible analytic neighborhood B of the point represented by $(S,H)$ in the moduli space of Gushel–Mukai surfaces such that any point in B corresponds to a strongly smooth Gushel–Mukai surface containing no lines or quintic elliptic curves. After shrinking B, we may assume that there is a family

$$ \begin{align*}(\mathcal{S},\mathcal{H})\rightarrow B \end{align*} $$

whose fiber over $b\in B$ is $(S_b,H_b)$ . The associated Lagrangian subspaces $A(S_b)$ vary in families according to [Reference Debarre and Kuznetsov16], [Reference Debarre and Kuznetsov17], and hence give a smooth projective family

$$ \begin{align*}\mathcal{Y}\rightarrow B \end{align*} $$

together with a relative EPW polarization $h_B$ on $\mathcal {Y}$ such that $(\mathcal {Y}_b,h_b)=(\tilde {Y}^{\geq 1}_{A(S_b)},h_{A(S_b)})$ .

Similarly, the relative moduli spaces (see [Reference Huybrechts and Lehn26]) parameterizing $H_b$ -stable sheaves on $S_b$ with Mukai vector $(2,-H_b,2)$ form a smooth projective family

$$ \begin{align*}\mathcal{M}:=\mathcal{M}_{\mathcal{H}}(2,-\mathcal{H},2)\rightarrow B, \end{align*} $$

which carries relative polarization class $\theta _B$ such that $\theta _b:=\theta _{S_b}$ for any $b\in B$ .

By construction, both families define holomorphic maps

$$ \begin{align*}B\longrightarrow \mathfrak{N} \end{align*} $$

to the same connected component $\mathfrak {N}$ of the moduli space of polarized hyperkähler manifolds of $K3^{[2]}$ -type with square two polarizations. These holomorphic maps agree on the analytically dense subset $B^{\circ }\subset B$ containing Picard rank one K3 surfaces. Since $\mathfrak {N}$ is quasi-projective (see, e.g., [Reference Debarre and Macrì19]), it is separated. Then the two holomorphic maps agree on all of B as B is irreducible. In particular, the polarized periods of $(M_S,\theta _S)$ and $(Y_S,h_{A(S)})$ agree.

In this case, the global Torelli theorem for hyperkähler manifolds [Reference Markman39, Theorem 1.3] ensures the desired isomorphism.

Using the two identifications and Remark 5.15, one finishes Proposition 1.2.

6.2. Double dual EPW surfaces

The statement in this section is essentially a corollary of [Reference Debarre and Kuznetsov18, Theorem 7.3], which describes the double dual EPW surface associated with a Gushel–Mukai threefold, and [Reference Jacovskis, Lin, Liu and Zhang29, Theorem 7.12], which describes the corresponding moduli space.

Here, we only elaborate on the case for a special Gushel–Mukai threefold X over S and the starting point is the Hilbert scheme $C(X)$ of conics on X. The scheme $C(X)$ is a normal projective surface which has two components $C_{\sigma }(X)$ and $C_0(X)$ . The component

$$ \begin{align*}C_{\sigma}(X)\cong F(M_S)\cong\mathbb{P}^2\end{align*} $$

parameterizes the preimage of lines on $M_S$ . The other component $C_0(X)$ is isomorphic to the blow-up of the double dual EPW surface $\tilde {Y}^{\geq 2}_{A(X)^{\perp }}$ associated with X at the distinguished point $\tilde {p}_S$ mentioned in Theorem 5.1 according to [Reference Debarre and Kuznetsov18, Theorem 7.3]. The two components intersect transversely along the exceptional curve on $C_0(X)$ .

By [Reference Jacovskis, Lin, Liu and Zhang29, Theorem 7.12], the moduli space $\textsf {M}_{\sigma _X}(\kappa _1)$ is a contraction from $C(X)$ such that the component $C_{\sigma }(X)$ is contracted to the distinguished singular point $\tilde {p}_S$ . It follows that the moduli space $\textsf {M}_{\sigma _X}(\kappa _1)$ is exactly the double dual EPW surface associated with X.

In conclusion, we can finish Corollary 1.3.

Proposition 6.5. Consider a strongly smooth Gushel–Mukai surface S and the special Gushel–Mukai threefold X over it, then $\mathsf {M}_{\sigma _X}(\kappa _1)$ is isomorphic to $\tilde {Y}^{\geq 2}_{A(S)^{\perp }}$ and the branched cover

$$ \begin{align*}\mathsf{M}_{\sigma_X}(\kappa_1)\rightarrow \mathsf{M}_{\sigma_S}(v_1)_{\mathfrak{S}^\vee_2}\end{align*} $$

induced by the forgetful functor coincides with $\tilde {Y}^{\geq 2}_{A(S)^{\perp }}\rightarrow Y^{\geq 2}_{A(S)^{\perp }}$ .

Proof. The isomorphism $\textsf {M}_{\sigma _X}(\kappa _1)\cong \tilde {Y}^{\geq 2}_{A(S)^{\perp }}$ follows from [Reference Debarre and Kuznetsov18], [Reference Jacovskis, Lin, Liu and Zhang29] as above. The identification of the branched double cover is similar to [Reference Liu37, Proposition 5.20]. One can do this because the involution on $C(X)$ induced by the branched cover $X\rightarrow M_S$ coincides with the one induced by the natural covering involution on $\tilde {Y}^{\geq 2}_{A(S)^{\perp }}$ according to [Reference Debarre and Kuznetsov18, Section 7.3]. In particular, the branched double cover for the double dual EPW surface $\tilde {Y}^{\geq 2}_{A(S)^{\perp }}$ can be described in terms of conics on an open subset and therefore be identified.

The double cover can be described pointwise. The dual EPW surface $Y^{\geq 2}_{A(S)^{\perp }}$ is the union of a distinguished point $p_S$ and the surface $BTC_S$ of conics bitangent to S, where we adopt the convention that $BTC_S$ also parameterizes the conics in S. Any singular point $p_C$ in $ Y^{\geq 2}_{A(S)^{\perp }}$ other than $p_S$ corresponds to a unique conic C contained in S.

The singular point in $\tilde {Y}^{\geq 2}_{A(S)^{\perp }}$ induced by the contraction from $C(X)$ is precisely $\tilde {p}_S$ , and any other singular point corresponds to a conic contained in the ramification locus of the double cover $X\rightarrow M_S$ . In this case, a smooth point in $\tilde {Y}^{\geq 2}_{A(S)^{\perp }}$ is represented by a conic bitangent to the ramification locus and corresponds to a conic in $M_S$ bitangent to S. Conversely, the preimage of a conic in $M_S$ bitangent to S is the union of two distinct conics.

7.1. The duality conjecture

Now let us explain our Conjecture 1.4.

It is in fact a refinement of the homological projective duality theorem in [Reference Kuznetsov and Perry35].

The equivalence is not canonically constructed in general. In fact, the theorem constructs a canonical equivalence using [Reference Kuznetsov and Perry34] for special period duals, called the direct period duals, and then uses Proposition 2.15 to conclude the general case.

The conjecture is true if X is very general in the sense that it is not the period dual of itself. Hence, the canonical equivalence ${\textbf {Ku}}(X)\cong {\textbf {{Ku}}}(X')$ constructed in Theorem 7.2 satisfies Conjecture 7.1 for a very general direct period dual X and $X'$ .

It is shown in [Reference Debarre and Kuznetsov13, Appendix C] and [Reference Debarre and Kuznetsov16] that the construction of Gushel–Mukai varieties can be generalized to the relative setting. In particular, one has the classical projective duality for relative direct period duals according to [Reference Debarre and Kuznetsov13, Lemma C.6].

Suppose that one has a relative version of Theorem 7.2 for a certain small base with positive dimension, then one can spread Proposition 7.4 to any direct period duals. In this case, it is not difficult to verify Conjecture 7.1. This will appear in future work.

7.2. Possible self-dual surfaces

If there are no self-dual Gushel–Mukai threefolds, then it follows from Theorem 2.10 that the equivalence ${\textbf {Ku}}(X)\simeq {\textbf {{Ku}}}(X')$ would send $\kappa _1$ to $\kappa _2'$ and $\kappa _2$ to $\kappa _1'$ , up to sign. Unfortunately, this is not true by the example constructed in [Reference Debarre and Mongardi20]. We want to discuss the possible self-dual examples in our setting.

Let S be a strongly smooth Gushel–Mukai surface, and let X be the special Gushel–Mukai threefold over it. Suppose that one has an autoequivalence on ${\textbf {Ku}}(X)$ which exchanges $\kappa _1$ and $\kappa _2$ up to signs. Then, according to [Reference Bayer and Perry7, Lemma 4.9], the autoequivalence descends to an autoequivalence $\Psi $ on ${\textbf {D}}^b(S)$ which exchanges $v_1$ and $v_2$ up to signs.

Set $\sigma ^{\prime }_S:=\Phi .\sigma _S$ , then one should have $\textsf {M}_{\sigma _S}(0,0,1)\cong \textsf {M}_{\sigma ^{\prime }_S}(\psi (0,0,1))$ . Suppose further that the stability condition $\sigma ^{\prime }_S$ is still in $ {\mathrm{Stab}}^{\dagger }(S)$ and one has a birational map

$$ \begin{align*}\textsf{M}_{\sigma^{\prime}_S}(\psi(0,0,1))\dashrightarrow\textsf{{M}}_H(\psi(0,0,1))\end{align*} $$

coming from Theorem 3.14. It will lead to an isomorphism $\textsf {{M}}_H(\psi (0,0,1))\cong S$ .

Example 7.6. The moduli space $\textsf {{M}}_H(0,E,-2)$ is the relative compactified Jacobian

$$ \begin{align*}\overline{{\mathrm{Jac}}}_{-2}(S/|E|)\rightarrow |E|=\mathbb{P}^1,\end{align*} $$

which is in general not canonically isomorphic to S. In fact, one has $S\cong \textsf {M}_H(0,E,-1)$ .

7.3. The implications of the conjecture

Now let us return to the EPW varieties. Assume that Conjecture 7.1 is true, we are able to generalize Proposition 1.2 and Corollary 1.3.

Proposition 7.8. Assume that Conjecture 7.1 holds, the moduli space $\textsf {M}_{\sigma _S}(v_2)$ is isomorphic to the double EPW sextic for any strongly smooth Gushel–Mukai surface S.

Proof. According to Proposition 6.4 and [Reference Kuznetsov and Perry33, Lemma 3.8], it remains to prove the statement when the special Gushel–Mukai threefold X over S admits a period dual $X'$ which is a special Gushel–Mukai threefold over $S'$ . Using the conjecture, one finds an equivalence

$$ \begin{align*}{\textbf{Ku}}(X)\cong{\textbf{{Ku}}}(X'),\end{align*} $$

which sends $\kappa _1$ to $\kappa ^{\prime }_2$ and $\kappa _2$ to $\kappa ^{\prime }_1$ up to signs. It descends to an $\mathfrak {S}_2^\vee $ -equivariant equivalence

$$ \begin{align*}{\textbf{D}}^b(S)\cong{\textbf{{D}}}^b(S')\end{align*} $$

by [Reference Bayer and Perry7, Lemma 4.9], such that the stability condition $\sigma _S$ is sent to $\sigma _{S'}$ possibly up to a ${\tilde {\mathrm {GL^+}}}(2,\mathbb {R})$ -action and $v_2$ is sent to $v^{\prime }_1$ up to a sign. Then one concludes by Proposition 5.14.

Now let us prove Proposition 1.6.

Proposition 7.10. Assume that Conjecture 7.1 holds, then the double cover $\textsf {M}_{\sigma _X}(\kappa _2)\rightarrow \textsf {M}_{\sigma _S}(v_2)_{\mathfrak {S}_2^\vee }$ coincides with $\tilde {Y}^{\geq 2}_{A(X)}\rightarrow Y^{\geq 2}_{A(X)}$ .

Proof. As in Proposition 7.8, the case where X has a special Gushel–Mukai threefold as its period dual can be proved using Proposition 6.5. In general, one can take an ordinary Gushel–Mukai threefold $X'$ as the period dual according to Proposition 2.15. Then the conjecture provides an equivalence

$$ \begin{align*}{\textbf{Ku}}(X)\cong{\textbf{Ku}}(X'),\end{align*} $$

which sends $\sigma _X$ to a stability condition $\sigma _{X'}$ on ${\textbf {Ku}}(X')$ and sends $\kappa _2$ to $\kappa ^{\prime }_1$ . One can see

$$ \begin{align*}\textsf{M}_{\sigma_{X'}}(\kappa^{\prime}_1)\cong\tilde{Y}^{\geq 2}_{A(X')^{\perp}}\end{align*} $$

as [Reference Debarre and Kuznetsov18, Theorem 7.3] and [Reference Jacovskis, Lin, Liu and Zhang29, Theorem 7.12] also apply to the ordinary case. Then one can conclude that the double cover $\textsf {M}_{\sigma _X}(\kappa _2)\rightarrow \textsf {M}_{\sigma _S}(v_2)_{\mathfrak {S}_2^\vee }$ is the one we want.

As a result, one can easily conclude the following corollary using Proposition 4.5 and the identifications of $\textsf {M}_{\sigma _S}(v_1)$ and $\textsf {M}_{\sigma _S}(v_2)$ . It is exactly Corollary 1.7.

7.4. Description of the special period partners

It is shown in [Reference Jacovskis, Liu and Zhang30], [Reference Liu37] that the stable object $G_X$ representing the distinguished singular point $\tilde {p}_S$ in $\textsf {M}_{\sigma _X}(\kappa _1)$ has the form

$$ \begin{align*}\mathscr{U}_X[2]\rightarrow G_X\rightarrow\mathscr{V}_X^\vee[1]\rightarrow\mathscr{U}_X[3],\end{align*} $$

where $\mathscr {U}_X$ and $\mathscr {V}_X$ are restrictions of tautological bundles on $ {\mathrm{Gr}}(2,V_5)$ and $ {\mathrm{Gr}}(3,V_5)\cong {\mathrm{Gr}}(2,V_5)$ .

Theorem 7.12 [Reference Jacovskis, Liu and Zhang30, Theorem 1.1].

Consider a special Gushel–Mukai threefold X and the special stable object $G_X$ in $\textsf {M}_{\sigma _X}(\kappa _1)$ , then the Brill–Noether locus

$$ \begin{align*}{\mathrm{BN}}_X(G_X)=\{A\in\textsf{M}_{\sigma_X}(\kappa_1+2\kappa_2)\,|\,{\mathrm{Hom}}(A,G_X[2])=\mathbb{C}^{\oplus 3}\}\subset\textsf{M}_{\sigma_X}(\kappa_1+2\kappa_2)\end{align*} $$

consists of the objects $K_x={{\boldsymbol{\mathsf{P}}}}\mathscr {O}_x$ for any $x\in X$ , where ${{\boldsymbol{\mathsf{P}}}}$ is the left adjoint functor of the inclusion ${\mathbf {Ku}}(X)\subset {\textbf {D}}^b(X)$ . Moreover, one has $ {\mathrm{BN}}_X(G_X)\cong X$ .

Proof. One notices that the Kuznetsov component used in [Reference Jacovskis, Liu and Zhang30] is different from ${\textbf {Ku}}(X)$ but equivalent. The relation between the distinguished stable object $G_X$ and the characterizing object in [Reference Jacovskis, Liu and Zhang30, Theorem 1.1] is explained in [Reference Jacovskis, Lin, Liu and Zhang29, Section 5]. The statement about the stable objects in the Brill–Noether locus follows from the argument for [Reference Jacovskis, Liu and Zhang30, Theorem 1.1].

The representatives for other singular points in the moduli space $\textsf {M}_{\sigma _X}(\kappa _1)$ are determined in [Reference Liu37, Corollary 6.11]. Each of them is given by a conic in the ramification locus of the covering involution on X and has the form $\mathscr {I}_C[1]$ . Consequently, one can see the following corollary.

Corollary 7.13. Assume that Conjecture 7.1 holds. Then a non-trivial special Gushel–Mukai threefold period partner of X is isomorphic to the Brill–Noether locus

$$ \begin{align*}{\mathrm{BN}}_X(\mathscr{I}_C[1])=\{A\in\textsf{M}_{\sigma_X}(\kappa_1+2\kappa_2)\,|\,{\mathrm{Hom}}(A,\mathscr{I}_C[3])=\mathbb{C}^{\oplus 3}\}\end{align*} $$

in the moduli space $\textsf {M}_{\sigma _X}(\kappa _1+2\kappa _2)$ , where C is a conic in the ramification locus of the branched double cover $X\rightarrow M_S$ . This period partner is denoted by $X_C$ .

Proof. Let $X'$ be such a period partner, then one can find an equivalence

$$ \begin{align*}{\textbf{Ku}}(X)\cong{\textbf{Ku}}(X')\end{align*} $$

using Conjecture 7.1, such that $\kappa _i$ is sent to $\kappa ^{\prime }_i$ for $i=1,2$ . It follows that

$$ \begin{align*}\textsf{M}_{\sigma_X}(\kappa_1)\cong\textsf{M}_{\sigma_{X'}}(\kappa^{\prime}_1)\quad\text{and}\quad \textsf{M}_{\sigma_X}(\kappa_1+2\kappa_2)\cong\textsf{M}_{\sigma_{X'}}(\kappa^{\prime}_1+2\kappa^{\prime}_2)\quad\end{align*} $$

according to [Reference Jacovskis, Lin, Liu and Zhang29, Appendix]. In particular, the singular point in $\textsf {M}_{\sigma _{X'}}(\kappa ^{\prime }_1)$ represented by $G_{X'}$ is sent to a singular point p in $\textsf {M}_{\sigma _X}(\kappa _1)$ . Suppose that p is represented by $G_X$ , then one has an isomorphism $X\cong X'$ by Theorem 7.12, violating our assumption. So p must be represented by $\mathscr {I}_C[1]$ and one concludes that $X'\cong {\mathrm{BN}}_{X'}(G_{X'})$ is isomorphic to $ {\mathrm{BN}}_X(\mathscr {I}_C[1])$ .

One can see that $ {\mathrm{BN}}_X(\mathscr {I}_C[1])$ does not intersect $ {\mathrm{BN}}_X(G_X)$ in $\textsf {M}_{\sigma _X}(\kappa _1+2\kappa _2)$ as

$$ \begin{align*}\dim{\mathrm{Hom}}({{\boldsymbol{\mathsf{P}}}}\mathscr{O}_x,\mathscr{I}_C[3])=\dim{\mathrm{Hom}}(\mathscr{O}_x,\mathscr{I}_C[3])=\dim{\mathrm{Hom}}(\mathscr{I}_C,\mathscr{O}_x)<3\end{align*} $$