Gushel--Mukai varieties: intermediate Jacobians

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds.


Introduction
This article is an addition to the series [DK18, DK19, DK20b,KP18] on the geometry of Gushel-Mukai varieties. For an introduction, we recommend the survey [Deb20].
A smooth complex Gushel-Mukai (GM for short) variety of dimension n ∈ {2, 3, 4, 5, 6} is a smooth dimensionally transverse intersection where V 5 is a 5-dimensional complex vector space, CGr(2, V 5 ) ⊂ P(C ⊕ 2 V 5 ) is the cone over the Grassmannian Gr(2, V 5 ) in its Plücker embedding, W ⊂ C ⊕ 2 V 5 is a linear subspace of dimension n + 5, and Q ⊂ P(W ) is a quadratic hypersurface. GM varieties of dimension 2 are Brill-Noether general K3 surfaces of genus 6. GM varieties of dimension 4 or 6 are Fano varieties but they share some properties with K3 surfaces. The key object we use to study a GM variety X is its associated Lagrangian data set constructed in [DK18]. It is a triple (V 6 (X), V 5 (X), A(X)) (or (V 6 , V 5 , A) for short) that consists of a 6-dimensional vector space V 6 , a hyperplane V 5 ⊂ V 6 , and a subspace A ⊂ 3 V 6 which is Lagrangian with respect to the symplectic form on 3 V 6 given by exterior product, and contains no decomposable vectors (this means no nonzero products v 1 ∧ v 2 ∧ v 3 , where v 1 , v 2 , v 3 ∈ V 6 ). A GM variety can be reconstructed from its Lagrangian data set ([DK18, Theorem 3.6]). Moreover, Lagrangian data sets can be used to describe the moduli stack of smooth GM varieties and its coarse moduli space ( [DK20b]).
It is not surprising then that many geometric properties of a GM variety can be described in terms of its Lagrangian data set, particularly in terms of the Eisenbud-Popescu-Walter (EPW for short) varieties A is a sextic hypersurface (called an EPW sextic) with singular locus Y ≥2 A , itself an integral surface with singular locus the finite set Y ≥3 A , and the dual EPW varieties ) associated with the Lagrangian subspace A ⊥ ⊂ 3 V ∨ 6 (see Section 2.2 for the definitions). Let A be a Lagrangian with no decomposable vectors (such as A(X) and A(X) ⊥ ). O'Grady constructed a canonical double covering A is empty (this holds for A general), Y ≥1 A is a hyperkähler fourfold called a double EPW sextic. When X is a GM variety of even dimension, the double EPW sextic Y ≥1 A(X) is the hyperkähler fourfold mentioned above whose primitive second cohomology is isomorphic to a Tate twist of the vanishing middle cohomology of X.
We also defined in [DK20a, Theorem 5.2(2)] a canonical double covering étale away from the finite set Y ≥3 A , where Y ≥2 A is an integral surface (called a double EPW surface) which has an ordinary double point over each point of Y ≥3 A and is smooth elsewhere; in particular, Y ≥2 A is smooth for A general. It has a 10-dimensional Albanese variety Alb( Y ≥2 A ) which, when Y ≥2 A is singular, can be defined as the Albanese variety of any desingularization.
The first main result of this article is the following.
Theorem 1.1. For any Lagrangian subspace A ⊂ 3 V 6 with no decomposable vectors, the Albanese variety Alb( Y ≥2 A ) has a canonical principal polarization such that there is an isomorphism We prove this in Theorem 4.4 for GM threefolds X with Y ≥3 A(X) = ∅ and in Theorem 5.3 for GM fivefolds X with Y ≥3 A(X) = ∅. In particular, we use the natural principal polarization of the intermediate Jacobian Jac(X) to produce a principal polarization on Alb( Y ≥2 A(X) ) and we deduce the isomorphism (1.2) from a birational isomorphism (a line transform) between two GM threefolds X and X such that A(X ) = A(X) ⊥ (such pairs are called period duals in [DK18]). The extension to arbitrary GM threefolds and fivefolds is given in Section 6.
Remark 1.2. We are not aware of a direct proof of the isomorphism (1.2). It can be thought of as a Hodge-theoretic incarnation of the equivalence between the nontrivial components of derived categories of odd-dimensional GM varieties conjectured in [KP18,Conjecture 3.7] and proved in [KP19, Corollary 6.5]. It would be interesting to extract the principal polarization on Alb( Y ≥2 A ) from the categorical data and to deduce the isomorphism (1.2) from the equivalence of categories.
One can also think of the principal polarization on Alb( Y ≥2 A ) as of an element of H 2 (Alb( Y ≥2 A ), Z) = 2 H 1 (Alb( Y ≥2 A ), Z) = 2 H 1 ( Y ≥2 A , Z). The latter group maps, by cup-product, to H 2 ( Y ≥2 A , Z) + (the invariant space for the canonical covering involution on Y ≥2 A ) with finite (but nontrivial) cokernel. The principal polarization maps to a class 3ν (1) = 2ν (thus, ν is the class of one of the components of the curve (1.5) discussed below); this follows from Welters' work on the variety of lines on quartic double solids (see the proof of Proposition 2.5 and [Wel81, (3.32), Proposition (3.60), and p. 70]).
Let X be a smooth GM variety of dimension n ∈ {3, 5} and assume Y 3 A(X) = ∅. To prove the isomorphisms (1.3) and (1.4), it is natural to construct a subscheme or a cycle of codimension n+1 2 and use the Abel-Jacobi map AJ Z : H 1 ( Y ≥2 A(X) , Z) → H n (X, Z). For this, one needs an interpretation of the double EPW surface Y ≥2 A(X) (or some other closely related surface) as a moduli space of sheaves or as a parameter space of cycles on X.
When X is a GM threefold, the most natural moduli space of sheaves to consider is the Hilbert scheme of conics on X. This scheme was thoroughly studied in [Log12] and [DIM12,Section 6]; in [DK20+], we prove that it is isomorphic to the blow up of a point of the dual double EPW surface Y ≥2 A(X) ⊥ . Similarly, for a GM fivefold X, one could use the Hilbert scheme of quadric surfaces in X; we proved in [DK20+] that it has a connected component isomorphic to a P 1 -bundle over Y ≥2 A(X) ⊥ . In the case of GM threefolds, it is claimed in [IM11,Section 5.1] and [IM07,Theorem 9] that the Clemens-Letizia degeneration method can be applied to prove that the Abel-Jacobi map given by the universal conic is an isomorphism; however, it is not clear whether this method would work in the case of GM fivefolds, so we need a different approach.
Another possible approach in the case of a GM threefold X would be to use the moduli space M X (2; 1, 5) of Gieseker semistable rank-2 torsion-free sheaves on X with c 1 = 1, c 2 = 5, and c 3 = 0. This space was shown in [DIM12, Section 8] to be birational to the Hilbert scheme of conics on X , a line transform of X (see Section 4.5), hence to the double EPW surface Y ≥2 A(X) ; the natural correspondence is provided by the second Chern class of the universal sheaf on the product X × M X (2; 1, 5). We use a small modification of this construction in which the moduli space of sheaves is kept implicit. We explain it below.
If X is a GM threefold and L 0 ⊂ X is a line, the corresponding (inverse) line transform X X takes a general conic C ⊂ X to a rational quartic curve C ⊂ X to which the line L 0 is bisecant (the corresponding rank-2 sheaf on X can then be obtained by Serre's construction applied to C ∪ L 0 ; in particular, the curve C ∪ L 0 represents the second Chern class of this sheaf). We consider the union C ∪ L 0 as a quintic curve of arithmetic genus 1 on X containing the line L 0 and construct the correspondence Z as the closure of a family of such curves parameterized by an open subscheme of Y ≥2 A(X) . To prove that the Abel-Jacobi map AJ Z associated with this family of curves is an isomorphism, we make the crucial observation that over the curve A(X) splits, and that over a general point y of one of the components of its preimage, there is a relation Z y + L y = S y ∩ X in the Chow group CH 1 (X) of 1-cycles. Here Z y is the fiber of the correspondence Z over y, L y is a line on X, and S y is a cubic surface scroll on the fourfold M X := CGr(2, V 5 ) ∩ P(W ). Moreover, the curve (1.5) is birational to the Hilbert scheme F 1 (X) of lines on X and the line L y comes from the universal family of lines over F 1 (X).
From these observations and from the vanishing of the odd cohomology of M X , it follows that for X general, there is a morphism φ : F 1 (X) → Y ≥2 A(X) such that the composition is the opposite of the Abel-Jacobi map defined by the universal family of lines. The latter map is surjective by an argument of Clemens-Tyurin (see Section 3.3), hence AJ Z is surjective as well. It is not hard to check that the source and target of AJ Z are free abelian groups of rank 20, hence AJ Z is an isomorphism. A similar argument works for GM fivefolds: rational quartic curves are replaced by rational quartic surface scrolls, reducible quintic curves by reducible quintic del Pezzo surfaces, the Hilbert scheme of lines by a component of the Hilbert scheme of planes, and a higher-dimensional analogue of the Clemens-Tyurin argument is applied.
For GM fivefolds X, the isomorphism (1.4) may be proved by a completely different topological argument. When X is general, we consider the double cover Y ≥2 A(X),V 5 (X) of the curve (1.5) induced by the double covering Y ≥2 A(X) → Y ≥2 A(X) ; in contrast with the case of GM threefolds, this is a smooth curve of genus 161. Using classical monodromy arguments, we prove that its Jacobian has three simple factors: the Jacobian of the curve Y ≥2 A(X),V 5 (X) (of dimension 81), the Albanese variety of the surface Y ≥2 A(X) (of dimension 10), and a simple factor of dimension 70. The curve Y ≥2 A(X),V 5 (X) parameterizes planes on X (see Section 2.5.1) and the corresponding Abel-Jacobi map is surjective by a generalization of the Clemens-Tyurin argument. The induced surjective morphism Jac( Y ≥2 A(X),V 5 (X) ) −→ Jac(X) therefore has connected kernel. The description of the simple factors implies that it has to be isogeneous to the product of the 81-dimensional and 70-dimensional factors. Therefore, Jac(X) is isomorphic to the remaining 10-dimensional factor Alb( Y ≥2 A(X) ). To complete the proof of Theorem 1.1 and to describe the period maps for GM varieties of dimension 3 or 5, we investigate the rational map from the coarse moduli space of EPW sextics (constructed in [O'Gr08a, Section 6]) to the coarse moduli space of principally polarized abelian varieties of dimension 10 defined by is the Albanese variety of (any desingularization of) the double EPW surface Y ≥2 A . Let now M GM n be the coarse moduli space of GM varieties of dimension n (see [DK20b] and Section 2.3). We use the above result to prove the following.
This factorization of the period map for GM threefolds was discussed in the introduction of [DIM12] (see also [DIM12,Remark 7.5]); moreover, it was conjectured there that the map℘ is generically injective (the computation in [DIM12, Theorem 5.1] shows that it has finite fibers).
The story of GM threefolds is very similar to the story of quartic double solids. The articles [Wel81,Voi88] were an inspiration to us; in particular, we took the idea of using the Clemens-Tyurin argument from [Wel81].
The article is organized as follows. In Section 2, we review the theory of GM varieties, EPW varieties, and their double covers. In particular, we describe the Hilbert scheme of lines on GM threefolds, the Hilbert scheme of σ -planes on GM fivefolds, and we identify double EPW varieties with the canonical covers of degeneracy loci for the family of quadrics containing a GM variety.
In Section 3, we recall basic facts about Abel-Jacobi maps, prove a generalization of the Clemens-Tyurin argument, and discuss the endomorphism ring of intermediate Jacobians; in particular, we check that the intermediate Jacobian of a very general GM variety of odd dimension is simple and has Picard number 1 (this had already been proved for GM threefolds by a different argument in [DIM12, Corollary 5.3]).
In Section 4, we construct, for any GM threefold X, a cycle Z ⊂ X × Y ≥2 A(X) , and prove that the Abel-Jacobi map defined by Z is an isomorphism when Y ≥3 A(X) = ∅. We also describe how the line transform of GM threefolds acts on their coarse moduli spaces. In Section 5, we prove analogous results for GM fivefolds. Finally, we describe in Section 6 the period map for GM threefolds and fivefolds and prove Theorems 1.1 and 1.3.

Geometry of Gr(2, 5)
Let V 5 be a 5-dimensional vector space. A subspace of V 5 of dimension k will usually be denoted by V k or U k . We denote by the Grassmannian of 2-dimensional vector subspaces in V 5 in its Plücker embedding.
We recall some standard facts about its geometry. It has codimension 3 and degree 5 and is the intersection, for v ∈ V 5 {0}, of the Plücker quadrics In the next lemma, we describe Hilbert schemes of linear subspaces on Gr(2, V 5 ).
If a finite morphism γ : X → Gr(2, V 5 ) is compatible with the polarizations, it induces a morphism F k (γ) : F k (X) → F k (Gr(2, V 5 )) between Hilbert schemes and we denote by ). We will need the following classical result.
Lemma 2.2. Let V 2 ⊂ V 5 be a 2-dimensional subspace. We have the equality in P( 2 V 5 ), where the right side is the cone over the cubic scroll.
Assume now dim(T ) = 2. Since Gr(2, V ∨ 5 ) is an intersection of quadrics, so is T , hence T is either an irreducible quadric surface or contains a plane.
If T contains a plane, we have, by Lemma 2.
, hence is the union of a hyperplane section of Gr(2, V 4 ) and of a linear 3-space.
If T is an irreducible quadric, we have P(W ⊥ 6 ) ⊂ P( 2 V ⊥ 1 ) and Gr(2, V 5 )∩P(W 6 ) is the union of P(V 1 ∧V 5 ) and two planes (or a double plane) corresponding to the intersection of Gr(2, V 5 /V 1 ) with the line P 1 given by the orthogonal of W ⊥ 6 ⊂ 2 V ⊥ 1 . Therefore, the only 3-dimensional component of Gr(2, V 5 ) ∩ P(W 6 ) has degree 1.
Finally, assume dim(T ) = 3. By Lemma 2.1(c), we have Therefore, the only case when Gr(2, V 5 ) ∩ P(W 6 ) has a 3-dimensional component of even degree is the case when T contains a σ -plane, and in this case, this component is a hyperplane section of Gr(2, V 4 ).
We will also need the following standard locally free resolution for the cone CGr(2, V 5 ).

Lemma 2.4. There is an exact sequence
of coherent sheaves on P(C ⊕ 2 V 5 ).

Eisenbud-Popescu-Walter varieties and their double coverings
Let V 6 be a 6-dimensional vector space. We consider subspaces A ⊂ 3 V 6 that are Lagrangian for the symplectic form given by exterior product. Those that contain no decomposable vectors (that is, such that P(A) ∩ Gr(3, V 6 ) = ∅) are parameterized by the complement of a hypersurface in the Lagrangian Grassmannian LGr( 3 V 6 ). Given a Lagrangian subspace A ⊂ 3 V 6 , one defines its EPW varieties; they form a chain of closed subschemes, where Y ≥k A is, at least set-theoretically, the set of points [v] in P(V 6 ) such that dim(A ∩ (v ∧ 2 V 6 )) ≥ k (the scheme structures were defined in [DK20a,(18) The Lagrangian subspace A ⊂ 3 V 6 defines a Lagrangian subspace A ⊥ ⊂ 3 V ∨ 6 hence, as above, dual EPW varieties Set-theoretically, the variety Y ≥k A ⊥ is the set of points [V 5 ] in Gr(5, V 6 ) = P(V ∨ 6 ) such that dim(A ∩ 3 V 5 ) ≥ k. We also use the notation Assume for the rest of this section that A contains no decomposable vectors. A combination of results of O'Grady (see [DK18,Theorem B.2]) gives the following: A is a normal sextic hypersurface (called an EPW sextic); is a normal integral surface of degree 40 (called an EPW surface); A is empty. Note that the Lagrangian subspace A ⊥ also contains no decomposable vectors and analogous statements hold for dual EPW varieties.
EPW varieties have canonical double coverings. First, there is a double covering A is an integral normal surface (called a double EPW surface), and proved an isomorphism The double coverings π A are the main characters of this article. We now prove some results about the surface Y ≥2 A that will be needed later on.

Proposition 2.5. Let A be a Lagrangian subspace with no decomposable vectors and assume Y ≥3
A are smooth connected projective surfaces. One has (2.6) where the isomorphism is canonical, and the abelian group H 1 ( Y ≥2 A , Z) is free of rank 20. Proof. From (2.5) and Serre duality, we deduce that there are isomorphisms From the table in [DK19, Corollary B.5], we see that the first summand vanishes, whereas the second summand is canonically isomorphic to A. This proves the statements (2.6) of the proposition.
To prove that H 1 ( Y ≥2 A , Z) is torsion-free, we use a degeneration argument. Let S ⊂ P 3 be a smooth quartic surface containing no lines. They are therefore diffeomorphic. The statement that H 1 ( Y ≥2 A , Z) is free of rank 20 then follows from the analogous statement for H 1 (F 1 (X), Z) proved in [Wel81, Section 6, Proposition, p. 71].

Gushel-Mukai varieties
Let n ∈ {3, 4, 5, 6}. As recalled in the introduction (see (1.1)), a Gushel-Mukai variety of dimension n is a dimensionally transverse intersection It is the intersection in P(W ) of the 6-dimensional space V 6 (X) ⊂ Sym 2 (W ∨ ) of quadrics containing X, generated by the space ) of (the restrictions to W of) Plücker quadrics and the quadric Q. In particular, one can replace Q by any other quadric in the space V 6 (X) V 5 (X). The intersection is called the Grassmannian hull of X. There are two types of GM varieties: • if M X does not contain the vertex of the cone CGr(2, V 5 ), then M X Gr(2, V 5 ) ∩ P(W ) is a linear section of Gr(2, V 5 ) and X = M X ∩ Q is a quadratic section of M X ; these GM varieties are called ordinary; • if M X contains the vertex of the cone CGr(2, V 5 ), then M X is a cone over M X = Gr(2, V 5 ) ∩ P(W ), a linear section of Gr(2, V 5 ), and X → M X is a double covering branched along a quadratic section X = M X ∩ Q ; these GM varieties are called special.
When X is a special GM variety of dimension n, the variety X is an ordinary GM variety of dimension n − 1; the varieties X and X are called opposite GM varieties. With every GM variety X, we associated in [DK18, Section 3.2] a Lagrangian data set (V 6 (X), V 5 (X), A(X)) consisting of • the 6-dimensional space V 6 (X) of quadrics containing X, • the hyperplane V 5 (X) ⊂ V 6 (X) of Plücker quadrics, • a Lagrangian subspace A(X) ⊂ 3 V 6 (X).
The Lagrangian data sets of a GM variety and of its opposite GM variety coincide.
Many properties of X are related to properties of its Lagrangian data set. For instance, when X is smooth and dim(X) ≥ 3, the space A(X) contains no decomposable vectors ([DK18, Theorem 3.16]) and dim(A(X) ∩ 3 V 5 (X)) ≤ 3.
Conversely, if (V 6 , V 5 , A) is a Lagrangian data set such that A contains no decomposable vectors and = dim(A ∩ 3 V 5 ) ≤ 3, there are exactly two smooth GM varieties X such that (V 6 (X), V 5 (X), A(X)) = (V 6 , V 5 , A) : one ordinary GM variety of dimension 5 − and one special GM variety of dimension 6 − ([DK18, Theorem 3.10]); they are opposite of one another.
In [DK20b], we upgraded the above constructions to a description of the moduli stack M GM n of smooth GM varieties of dimension n and its coarse moduli space M GM n . In particular, we showed in [DK20b, Theorem 5.15(a)] that the coarse moduli space of smooth GM varieties of dimension n ≥ 3 is the quasiprojective GIT quotient (see (2.3) for the notation). In particular, as explained in [DK20b, Section 6.1], there is a map and (2.10)  [DK20b,Proposition 6.1]) that the map p n can be thought of as the period map for GM n-folds.
We computed in [DK19] the integral cohomology groups of GM varieties of dimensions 3 or 5 and of their Grassmannian hulls. We denote by F • H n (X, C) the Hodge filtration on the cohomology H n (X, C).
Proposition 2.6. Let X be a smooth GM variety of odd dimension n ∈ {3, 5}.
Proof. The first part follows from [DK19, Propositions 3.1 and 3.4]. The second part is a standard consequence of the Lefschetz Theorem.
We will also need the following result.
Lemma 2.7. A smooth GM fivefold contains no quadric threefold whose image in Gr(2, V 5 ) is a hyperplane section of some Gr(2, V 4 ).
Proof. Let Q ⊂ CGr(2, V 5 ) be a quadric threefold contained in a smooth GM fivefold X. Then Q does not contain the vertex of the cone CGr(2, V 5 ) (because X does not), hence its projection from the vertex to Gr(2, V 5 ) is well defined. Assume it is a hyperplane section of some Gr(2, V 4 ). Then Q is a local complete intersection and its normal bundle splits as where U is the restriction of the tautological bundle of Gr(2, V 5 ). Any GM fivefold X is the intersection of a hyperplane and a quadric in CGr(2, V 5 ). If X contains Q, the differentials of the equations of X give a morphism Clearly, X is singular at any degeneracy point of that morphism. If X is smooth, this morphism is therefore surjective, hence its kernel is a vector bundle of rank 2. This is absurd, since a simple computation shows that its third Chern class is nonzero. Therefore, X cannot contain Q.

Linear spaces on quadrics containing GM varieties
If X ⊂ P(W ) is a GM variety and V 6 (X) is the space of quadrics in P(W ) containing X, we denote by (2.11) Q ⊂ P(W ) × P(V 6 (X)) the total space of this family of quadrics and, for v ∈ V 6 (X) nonzero, by Q v the corresponding quadric in P(W ).
The Lagrangian data set associated with X can be used to describe the ranks of the family of quadrics (2.11): by [DK18, Proposition 3.13(b)], we have In particular, Y k A(X) P(V 5 (X)) is the locus of non-Plücker quadrics of corank k containing X. In fact, the family of quadrics (2.11) itself can be reconstructed from the Lagrangian data set, which allows us to relate the double covering (2.4) to the coverings associated with the family of quadrics by [DK20a, Theorem 3.1]. Note that (2.11) corresponds to an embedding O P(V 6 (X)) (−1) → Sym 2 W ∨ ⊗ O P(V 6 (X)) .
Lemma 2.8. Let X be a smooth GM variety of dimension n, with associated Lagrangian data set (V 6 , V 5 , A) and let k ∈ {0, 1, 2}. Over P(V 6 ) P(V 5 ), the canonical double covering of the k-th degeneracy locus Y ≥k A P(V 5 ) of the family of quadrics (2.11) coincides with the base change Y ≥k of 3 V 6 ⊗ O over P(V 6 ). To identify the double coverings, we will show that the family of quadrics (2.11) can be related to this pair by the isotropic reduction procedure of [DK20a, Section 4.2]. We will use freely the notation from [DK20a]. Asssume first that X is ordinary. We restrict A 1 and A 2 to P(V 6 ) P(V 5 ) and consider the third Lagrangian subbundle We apply isotropic reduction with respect to the rank-2 subbundle in the sense of [DK20a, Section 4.2] and obtain three Lagrangian subbundles A 1 , A 2 , A 3 in a symplectic vector bundle V . We describe below all these bundles explicitly.
and the respective fibers at [v] of the bundles A 1 , A 2 , A 3 are A, v ∧ 2 V 5 (the second summand), and 3 V 5 (the first summand). Furthermore, the fiber of I is the subspace I := A ∩ 3 V 5 of the first summand. By [DK18, Proposition 3.13(a)], the space I can be identified with the space of linear functions on 2 V 5 vanishing on W , hence 3 V 5 /I W ∨ . Thus and the fibers at [v] of the bundles A 1 , A 2 , A 3 are A/I, W (the second summand), and W ∨ (the first summand). In particular, A 2 ∩ A 3 = 0. Moreover, A 1 ∩ A 3 = 0. Indeed, any vector in the intersection comes from (A + I) ∩ 3 V 5 , and since I ⊂ A, it belongs to A ∩ 3 V 5 = I, hence corresponds to the zero vector in A 1 ∩ A 3 . This implies A 3 W ∨ ⊗ O , and the maps [DK20a, (23)] for the triple A 1 , A 2 , A 3 and the trivial line bundle L = O are isomorphisms over P(V 6 ) P(V 5 ). Therefore, the construction [DK20a, (24)] defines a family of quadratic forms on the trivial vector bundle A ∨ 3 W ⊗ O over P(V 6 ) P(V 5 ). By [DK18, proof of Theorem 3.6 and Appendix C], this family of quadrics coincides with the restriction of Q to P(V 6 ) P(V 5 ). Applying [DK20a, Propositions 4.5 and 4.7], we see that the associated double coverings coincide with Y ≥k A × P(V 6 ) (P(V 6 ) P(V 5 )). This completes the proof of the lemma for ordinary GM varieties.
Assume now that X is a special GM variety. By [DK18, Lemma 2.33], there is a canonical direct sum decomposition W = W 0 ⊕ W 1 , with dim(W 1 ) = 1, such that for the family of quadrics q : V 6 → Sym 2 W ∨ defining X, we have , and the family of quadrics q 0 corresponds to the ordinary GM variety X 0 opposite to X. By [DK18, Proposition 3.14(c)], the varieties X 0 and X have the same Lagrangian data sets, hence they give rise to the same double coverings Y ≥2 A → Y ≥2 A . Also, the family of quadrics q 1 is nondegenerate over P(V 6 ) P(V 5 ), hence the families of quadrics q and q 0 have the same degeneration loci and isomorphic cokernel sheaves. By [DK20a, Theorem 3.1], they induce the same double covers of degeneration loci. We conclude by the argument of the first part of the proof.
We will need the following consequence of the above lemma. Set ) and the base change Q 0 := Q × P(V 6 ) S 0 of the family of quadrics (2.11).
Corollary 2.9. Let X be a smooth GM variety of odd dimension n = 2s + 1 with Lagrangian data set (V 6 , V 5 , A). Let Π 0 ⊂ X be a linear subspace of dimension s. There is an isomorphism where the left side is the subscheme of the relative Hilbert scheme of linear spaces of dimension s+3 contained in the fibers of Q 0 → S 0 and containing Π 0 .
As the proof of [DK20a, Proposition 3.10] shows, we have an isomorphism whereQ 0 → S 0 is the family of nondegenerate (2s + 2)-dimensional quadrics obtained from Q 0 by passing to the quotients by the kernel spaces of quadrics. Moreover, for any hence X ∩ Sing(Q v ) = ∅ (otherwise X would be singular). Therefore, the space Π 0 intersects none of these kernel spaces hence projects isomorphically onto a spaceΠ 0 in the fiberQ v ofQ 0 at [v], so that a (s + 3)-space in Q v contains Π 0 if and only if the corresponding (s + 1)-space inQ 0 containsΠ 0 . This proves that we also have an isomorphism of S 0 -schemes. Finally, since the family of (2s + 2)-dimensional quadricsQ 0 is everywhere nondegenerate, it follows from [KS18, Lemma 2.12] that F s+1 from the Stein factorization of the map F s+1 (Q 0 /S 0 ) → S 0 , which, because of the isomorphism (2.15) and the observation made at the beginning of the proof, is isomorphic to

Linear spaces on ordinary GM threefolds and fivefolds
In [DK19], we described the Hilbert schemes of linear spaces on a smooth GM variety X of dimension at least 3 in terms of its Lagrangian data set (V 6 , V 5 , A), its EPW varieties, and their double covers. We focus here on the Hilbert schemes of lines on an ordinary GM threefold and σ -planes on an ordinary GM fivefold (see (2.2) for the definition). The description of [DK19] was given in terms of the first quadratic fibration where U X is the pullback to X of the tautological rank-2 vector bundle U on Gr(2, V 5 ) and the map ρ 1 is the pullback along the embedding X → Gr(2, V 5 ) of the projectioñ . The Hilbert scheme F 1 (X) of lines on X was identified in [DK19, Proposition 4.1] with the relative Hilbert scheme of lines of the map ρ 1 and the Hilbert scheme F 2 σ (X) of σ -planes on X with the relative Hilbert schemes of planes of the map ρ 1 . This defines maps (2.17) σ : F 1 (X) → P(V 5 ) and σ : F 2 σ (X) → P(V 5 ). To better describe these maps, we set (see (2.13) and (2.4) for the notation) [DK18,Proposition 4.5], the fibers of the first quadratic fibration ρ 1 defined in (2.16) are

σ -planes on ordinary GM fivefolds.
Using this, we proved in [DK19, Theorem 4.3(b)] and [DK20a, Corollary 5.5] that there is an isomorphism A,V 5 is the second map σ from (2.17). This has the following simple consequence (compare with [Nag98, Lemma 2.2]). Lemma 2.10. Let A ⊂ 3 V 6 be a Lagrangian subspace with no decomposable vectors. For a general GM fivefold X such that A(X) = A, the Hilbert scheme F 2 σ (X) of σ -planes contained in X is a smooth connected curve of genus 161.
Proof. According to the description of the moduli space in (2.8), a general GM fivefold X such that A(X) = A corresponds to a general point

Lines on ordinary GM threefolds.
Let X = Gr(2, V 5 ) ∩ P(W ) ∩ Q be a smooth ordinary GM threefold, where W ⊂ 2 V 5 has codimension 2. By (2.8), we have is a pencil of skew-symmetric forms on V 5 . Since X is smooth, these forms all have one-dimensional kernels ([DK18, Remark 2.25]) and these kernels form a smooth conic Using this, we proved the following result.
A,V 5 ). If also A is general, the curve F 1 (X) is a smooth irreducible curve of genus 71, the map σ : The , which is the intersection number in Y A of the curve Σ 1 (X), viewed as above as a fiber of q, with the divisor E, is A,V 5 and F 1 (X) is its normalization.
Lemma 2.13. For any smooth GM threefold X, the curve F 1 (X) is connected.
Proof. Consider a general deformation : X → B with central fiber X, parameterized by a smooth irreducible curve B. For any line L ⊂ X, there is an exact sequence

Topological preliminaries
In [Tju72, Section 4.3], Tyurin gave a beautiful argument (which he attributed to Clemens) proving the surjectivity of the Abel-Jacobi map given by the universal line on a threefold. In this section, we recall this argument and prove the generalization on which our results about GM threefolds and fivefolds are based. In Section 3.4, we use Picard-Lefschetz theory to show that intermediate Jacobians of very general GM varieties of dimensions 3 or 5 have trivial endomorphism rings.

Abel-Jacobi maps
We start by recalling a few properties of Abel-Jacobi maps. Let X and Y be smooth proper varieties of respective dimensions d X and d Y , let Z be an algebraic cycle of dimension d Y + c on X × Y , and let k be an integer. The Abel-Jacobi map where p X and p Y are the projections from X ×Y onto the factors and the isomorphisms are given by Poincaré duality and the middle map is the cup-product with the cohomology class of the cycle Z.
We will use the following functoriality properties of the Abel-Jacobi map.
Proof. All these statements follow from base change and the projection formula.

Generalized blow up decomposition
We will need the following (co)homological result generalizing the formula for the (co)homology of a smooth blow up (see [BFM19, Proposition 46] for another proof).

Lemma 3.2. Let S be a smooth proper variety and let E be a rank-r vector bundle on S. Let us consider
and Proof. We have a commutative diagram where π is a P r−2 -fibration away from Z and a P r−1 -fibration over Z. In particular, S is a smooth hypersurface over S Z. Therefore, for the first statement, we have to check that S is smooth of codimension 1 where ds is the differential of s considered as a section of O (1). The restriction of the map ds to E x is zero, hence it factors through the dashed arrow, which can be identified with the differential of s considered as a section of E ∨ evaluated at e ∈ E x . Thus, the vertical arrow is surjective at (x, e) for any e ∈ E x {0} if and only if ds : Denote by H ∈ H 2 ( S, Z) the restriction to S of the relative hyperplane class of P S (E). Then P Z (E) has codimension r − 1 in S and Indeed, this follows from the standard exact sequence and the Whitney formula. In particular, We will now prove the direct sum decomposition of cohomology; the homological decomposition is proved analogously or follows from Poincaré duality. Consider the maps We claim that To prove that, we define maps If k ≤ l, we have Using (3.1), we obtain If we define maps it follows that the map ψ • φ is lower triangular with ±1 on the diagonal, hence invertible, so that φ is injective and ψ is surjective. The injectivity of ψ (hence the surjectivity of φ) follows from projective bundle formulas for the maps S P Z (E) → S Z and P Z (E) → Z, and excision.

The Clemens-Tyurin argument
The following result is a generalization of [Tju72, Section 4.3] (see also [Wel81, Lemma (4.6)]); the original result is the case m = 1.
be a smooth closed subscheme of the Hilbert scheme of m-dimensional linear projective spaces in Y , let F X ⊂ F Y be the closed subscheme parameterizing projective spaces contained in X, and let L Y ⊂ F Y × Y and L X ⊂ F X × X be the pullbacks of the corresponding universal families of projective spaces. Assume that Then the Abel-Jacobi map Set X := q −1 (X) andp := p| X ,q := q| X , and consider the restricted diagram Since L Y → F Y is the projectivization of a rank-(m + 1) vector bundle and X ⊂ L Y is a relative hyperplane section, the hypotheses of Lemma 3.2 are satisfied by assumptions (a) and (c), hence X is smooth and there is a direct sum decomposition , which vanishes by assumption (d). The surjectivity of AJ L X therefore will follow from the surjectivity of To prove this surjectivity, we note that by assumption (b), the mapq is dominant and generically finite hence, by [BM79,Lemma 7.15], the image ofq * contains the vanishing cycles, that is, the kernel of the map By (d), the target is zero, so this proves the surjectivity ofq * and of AJ L X .

Intermediate Jacobians and their endomorphisms
Let X be a smooth projective variety of dimension 2m − 1. We consider the middle cohomology H 2m−1 (X, Z) with its natural Hodge structure of weight 2m − 1. The complex torus In particular, the Hodge structure H 2m−1 (X, Q) van acquires a polarization from Poincaré duality on X and we denote by Jac(X) van the corresponding isogeny class of nondegenerate complex tori. We will say that the endomorphism ring of a complex torus T is trivial if any endomorphism of T is the multiplication by an integer. If T 0, this means that the endomorphism ring End(T ) is isomorphic to Z or, equivalently, that the rational endomorphism ring End(T ) ⊗ Q is isomorphic to Q; so we can extend this terminology to isogeny classes of complex tori.
If Proof. Assume first that there is a rational map f : M P 1 , which we resolve by blowing up a smooth codimension-2 subvariety to obtain a morphismf : M → M f P 1 with critical values t 1 , . . . , t r ∈ P 1 , and that the strict transform of X is the fiber over 0 ∈ P 1 {t 1 , . . . , t r }. Let : X → M be the embedding and let ρ : π 1 (P 1 {t 1 , . . . , t r }, 0) −→ Sp(H 2m−1 (X, Q)) be the monodromy representation.
Assume moreover that the only singularities of the fibers off are nodes. As explained in [Voi03, Sections 2.2.2 and 2.3.1], one can then attach to each singular point of a fiber a (noncanonically defined) vanishing cycle in H 2m−1 (X, Q) van and the vanishing cycles span the vector space H 2m−1 (X, Q) van ([Voi03, Lemma 2.26]; this reference deals with the case where each singular fiber has a single node but the proofs extend to the general case).
Assume that f : M P 1 is a Lefschetz pencil. For each i ∈ {1, . . . , r}, the singular fiber X t i has a single node and there exists an element γ i of π 1 (P 1 {t 1 , . . . , t r }, 0) that acts on H 2m−1 (X, Q), via the monodromy representation, as the transvection It follows that the monodromy is "big": the Zariski closure of its image is the full symplectic group Sp(H 2m−1 (X, C) van ) ([PS03, Lemma 4]). As in the proof of [PS03, Theorem 17], for t ∈ P 1 very general, any endomorphism of Jac(X t ) van intertwines every element of the monodromy group, hence every element of the symplectic group. It must therefore be a multiple of the identity: the endomorphism ring of Jac(X t ) van is trivial.  End(Jac(X)) Z.
In particular, the Picard number of Jac(X) is 1.
Proof. For any GM variety X of dimension 3 or 5, condition (3.2) holds by Proposition 2.6, hence the abelian variety Jac(X) is principally polarized; its dimension is 10 again by Proposition 2.6.
For the second statement, we may assume that X is ordinary; it is then a very ample divisor in its Grassmannian hull M X , which is the Grassmannian Gr(2, V 5 ) when n = 5 or the fixed smooth fourfold Gr(2, V 5 ) ∩ P 7 ⊂ P( 2 V 5 ) when n = 3. We have H n (M X , Q) = 0 in both cases by Proposition 2.6, so Corollary 3.5 implies the first claim. A standard result then implies that the Picard number of Jac(X) is 1.

Intermediate Jacobians of GM threefolds
In this section, we study the intermediate Jacobians of GM threefolds. The main result (Theorem 4.4) is stated at the end of Section 4.1 and its proof takes up the rest of Section 4.

Family of curves
Let X be an arbitrary smooth GM threefold. Its associated Lagrangian subspace A ⊂ 3 V 6 has no decomposable vectors (Section 2.3). Let Y ≥2 A ⊂ P(V 6 ) be the corresponding EPW surface and let A be the double covering from (2.4), which is connected and étale away from the finite set Y 3 A . We are going to construct a subvariety A is, away from a finite subset of Y ≥2 A , a family of quintic curves of arithmetic genus 1 containing a fixed line L 0 ⊂ X. We will then check that the associated Abel-Jacobi map gives an isomorphism between Alb( Y ≥2 A ) and Jac(X). We start by choosing a line L 0 ⊂ X. It is for the moment arbitrary, but we will impose some restrictions in Section 4.2. We consider the open surface S 0 ⊂ Y ≥2 A defined by (2.14) and the family of quadrics Q 0 → S 0 obtained by base change to S 0 of the family (2.11). We denote by F 4 (Q 0 /S 0 ) → S 0 the relative Hilbert scheme of linear 4-spaces in the fibers of Q 0 → S 0 and by F 4 L 0 (Q 0 /S 0 ) → S 0 the subscheme parameterizing those 4-spaces that contain the line L 0 . Applying Corollary 2.9, we obtain an isomorphism of schemes over S 0 . In particular, the canonical map F 4 L 0 (Q 0 /S 0 ) → S 0 can be identified with the double étale covering π : S 0 → S 0 induced by the double covering π A . Note that S 0 is a smooth surface. Let be the base change of the family of quadrics Q 0 → S 0 along π. We have a canonical map , where the first map is the product of the isomorphism (4.1) with the identity map. By construction, it is a section of the projection F 4 ( Q 0 / S 0 ) → S 0 .
Let P 4 ⊂ Q 0 ⊂ P(W ) × S 0 be the pullback of the universal family of projective 4-spaces over F 4 ( Q 0 / S 0 ) along the section S 0 → F 4 ( Q 0 / S 0 ) constructed above. Set Proof. Let y ∈ S 0 and set [v] := π(y) ∈ P(V 6 ) P(V 5 ). The fiber of Z 0 over y is The cone CGr(2, V 5 ) ⊂ P(C ⊕ 2 V 5 ) has codimension 3 and degree 5. Therefore, CGr(2, V 5 ) ∩ P 4 y has dimension at least 1 and degree at most 5 (and if the dimension is 1, the degree is 5). Furthermore, P 4 y ⊂ Q v , hence Z 0,y ⊂ M X ∩ Q v = X. Since X contains no surfaces of degrees less than 10 ([DK19, Corollary 3.5]), Z 0,y is a local complete intersection curve in X of degree 5. This also proves the inclusion Z 0 ⊂ X × S 0 .
Since the curve Z 0,y is a dimensionally transverse linear section of CGr(2, V 5 ), the resolution of Lemma 2.4 restricts on P 4 y P 4 to a resolution y is a connected curve of arithmetic genus 1; in particular, its Hilbert polynomial is h Z 0,y (t) = 5t. Since the Hilbert polynomial does not depend on y, the family of curves Z 0 is flat over S 0 . Finally, L 0 ⊂ M X and L 0 ⊂ P 4 y by construction, hence L 0 ⊂ Z 0,y .
We now extend the family of curves Z 0 → S 0 to a family defined over the entire surface Y ≥2 A . We will need the following construction.
Definition 4.2. Let Z ⊂ X × S be an S -flat family of subschemes in a projective variety X , let ϕ : S → Hilb(X ) be the induced morphism, and let S ⊂ S be a partial compactification of S . Then ϕ can be considered as a rational map S Hilb(X ). Let S ⊂ S × Hilb(X ) be the graph of ϕ and letφ : S → Hilb(X ) be the projection. Let Z ⊂ X × S be the pullback of the universal subscheme in X × Hilb(X ) and let Z ⊂ X × S be the scheme-theoretic image of Z by the morphism X × S → X × S . We will call the subscheme Z the Hilbert closure of Z with respect to the embedding S ⊂ S .
We apply this construction to the subscheme Z 0 ⊂ X × S 0 and the embedding S 0 ⊂ Y ≥2 A .

Lemma 4.3. Let
A , the scheme Z is a flat family of curves of degree 5 and arithmetic genus 1 containing the line L 0 . Moreover, we have Hilb(X) defined by the subscheme Z 0 extends regularly to all codimension-1 points of Y ≥2 A . The nonflat locus of the morphism Z → Y ≥2 A is therefore supported in codimension 2, hence is a finite subscheme. All the remaining properties of Z are clear from the construction of the Hilbert closure.
By Proposition 4.1, every irreducible component of Z 0 has dimension 3. By definition of the Hilbert closure, the same is true for Z.
The main result of this section is the following (recall that by Propositions 2.5 and 2.6, the abelian groups H 1 ( Y ≥2 A(X) , Z) and H 3 (X, Z) are both free of rank 20).

Theorem 4.4. For any Lagrangian subspace A ⊂ 3 V 6 such that A has no decomposable vectors and Y ≥3 A = ∅, the abelian variety Alb( Y ≥2
A ) has a canonical principal polarization. If moreover Y ≥3 A ⊥ = ∅, there is an isomorphism

of principally polarized abelian varieties. Furthermore, if X is any smooth GM threefold with associated Lagrangian subspace
A is the subscheme defined in Lemma 4.3, the Abel-Jacobi map (4.5) is an isomorphism of integral Hodge structures which induces an isomorphism

of principally polarized abelian varieties.
This theorem is a more precise form of Theorem 1.1 for threefolds and its proof takes up the rest of Section 4. Note that if A is very general, the principal polarization of Alb( Y ≥2 A ) is unique by Corollary 3.6.

The boundary of the family
To prove Theorem 4.4, we study, over the boundary Y ≥2 A S 0 , the family of curves Z constructed in Lemma 4.3. By (2.14), this boundary consists of the curve Y ≥2 A,V 5 and the finite set Y 3 A . As we will see in the proof of Proposition 4.15, finite sets are not important for the Abel-Jacobi map, so we will concentrate on a dense open subset (denoted by S 0,V 5 and defined in Definition 4.7) of the curve Y ≥2 A,V 5 . We will construct a diagram (4.7) where S 0+ = S 0 ∪ S 0,V 5 and all squares are cartesian. The lower vertical arrows are double coverings (étale except for the right one, which is only étale away from Y ≥3 A ). We want to emphasize that the schemes Z 0+ and Z are different over the boundary S 0+ S 0 ⊂ Y ≥2 A S 0 , and this difference will be crucial for the rest of the proof. In fact, the map Z 0,V 5 → S 0,V 5 is a flat family of surfaces in M X , while the map Z → Y ≥2 A is a family of curves in X.
To construct the diagram, we need to impose some restrictions on X and L 0 . First, we will assume from now on that X is ordinary. To explain the restriction imposed on L 0 , we will need the following definition (the map σ : A,V 5 and the conic Σ 1 (X) ⊂ P(V 5 ) were defined in (2.23) and (2.21)).

Definition 4.5. A line
We will use the following simple observation. In particular, the subspace P(W ) is transverse to P(v ∧ V 5 ) ⊂ P( 2 V 5 ).
From now on, we will assume that L 0 is a nice line and set Recall that Y ≥2 A,V 5 ∩ Σ 1 (X) is a finite scheme (Lemma 2.12).
Definition 4.7. Denote by S 0,V 5 the dense open complement in the curve Y ≥2 A,V 5 of the finite set Y ≥2 A,V 5 ∩Σ 1 (X) and of the finite subset of Y ≥2 A,V 5 corresponding to lines intersecting L 0 (including the line L 0 itself). Set

This is a smooth open subscheme of Y ≥2
A containing S 0 and with finite complement.
Note that each point of the curve S 0,V 5 corresponds to a nice line on X.
Lemma 4.8. The double covering π A : Y ≥2 A → Y ≥2 A splits over the curve Y ≥2 A,V 5 .
Proof. As we saw in the proof of Corollary 2.9, the double covering π : S 0 → S 0 induced by π A agrees with the relative Hilbert scheme F 2 L 0 (Q 0 /S 0 ) → S 0 of planes containing the projectionL 0 of the line L 0 , where the family of quadricsQ 0 → S 0 is obtained from the family (2.11) by restricting to S 0 and passing to the quotients with respect to the 2-dimensional kernel spaces of quadrics. We will prove that this identification also holds over S 0+ .
Denote by Q 0+ → S 0+ the restriction of the family of quadrics (2.11) to S 0+ . For [v] ∈ S 0,V 5 , the quadric Q v is the restriction to P(W ) of the corresponding Plücker quadric (see (2.1)), that is By Definition 4.7, the line L v corresponding to the point A,V 5 is nice hence, by Lemma 4.6, the space P(W ) intersects P(v ∧ V 5 ) transversely along the line L v , so that (4.8) Q v = Cone L v (Gr(2, V 5 /Cv)).
In particular, Q v has corank 2 and its vertex L v does not meet L 0 (by Definition 4.7). Therefore, by passing to the quotients with respect to the kernel spaces of quadrics, we obtain, as in the proof of Corollary 2.9, a familyQ 0+ → S 0+ of nondegenerate 4-dimensional quadrics over S 0+ and conclude that the Hilbert scheme F 2 L 0 (Q 0+ /S 0+ ) of planes in its fibers containingL 0 is isomorphic to F 4 L 0 (Q 0+ /S 0+ ) and gives an étale double covering of S 0+ . Over the dense open subset S 0 ⊂ S 0+ , this covering is induced by π A , hence the same is true over S 0+ , that is, Therefore, to prove that the covering π A : it is enough to check that the covering F 2 L 0 (Q 0+ /S 0+ ) → S 0+ splits over S 0,V 5 . We do that by constructing a section of this covering over S 0,V 5 as follows: for [v] ∈ S 0,V 5 , consider the plane (4.10) ∈ S 0,V 5 by Definition 4.7). The lineL 0 is contained in this plane because L 0 = P(W ) ∩ P(v 0 ∧ V 5 ) by Lemma 4.6. Therefore, we obtain a regular map (4.11) S 0,V 5 −→ F 2 L 0 (Q 0 /S 0 ) = F 4 L 0 (Q 0+ /S 0+ ) S 0+ which gives the required section.
Remark 4.9. The map (4.11) is the restriction of the map

hence it is well defined even if the curve Y ≥2
A,V 5 is not reduced.
We still denote by π the double covering S 0+ → S 0+ constructed above. Note that S 0+ is a smooth irreducible open surface in Y ≥2 A containing S 0 as an open subscheme. Let be the base change of the family Q 0+ → S 0+ along π. The isomorphism (4.9) induces a section S 0+ −→ F 4 L 0 ( Q 0+ / S 0+ ) of its relative Hilbert scheme of 4-spaces and we denote by the corresponding family of projective 4-spaces over S 0+ , which agrees by construction with the family P 4 over S 0 ⊂ S 0+ . We set (4.12) Z 0+ := P 4 + ∩ (M X × S 0+ ).
This defines the middle column of the diagram (4.7). We denote by Z 0+,y = P 4 +y ∩ M X ⊂ M X the fiber of Z 0+ over a point y ∈ S 0+ . Lemma 4.10. We have Z 0+ × S 0+ S 0 = Z 0 and, for general points y of every irreducible component of S 0+ S 0 , we have Z y ⊂ Z 0+,y , where Z y is the fiber of the scheme Z defined in (4.3).
Proof. The equality follows from the fact that the family of 4-spaces P 4 + agrees with P 4 over S 0 . Let y be a general point of an irreducible component of S 0+ S 0 . By continuity, we obtain Z y ⊂ P 4 +,y . Since Z y ⊂ X ⊂ M X , we also get Z y ⊂ Z 0+,y .
We denote by (4.13) S 0,V 5 ⊂ S 0+ the image of the map (4.11) and by the restriction of the family (4.12) to the curve S 0,V 5 .
Proposition 4.11. The map Z 0,V 5 → S 0,V 5 is a flat family of surfaces which are isomorphic to hyperplane sections of a cubic scroll P 1 × P 2 . For each point y ∈ S 0,V 5 , the fiber Z y contains the corresponding nice line L π(y) and the line L 0 .
We will give a more detailed description of the fibers of Z 0,V 5 in Lemma 4.13.
Proof. Let y ∈ S 0,V 5 and set again [v] = π(y) ∈ P(V 5 ). The proof of Lemma 4.8 shows that the quadric Q v has the form (4.8) and that the point y ∈ S 0,V 5 corresponds to the plane (4.10) inQ v . Its preimage in Q v is the 4-space where V 2 ⊂ V 5 is the subspace spanned by v 0 and v (they are linearly independent by Definition 4.7). Furthermore, by Lemma 2.2, the fiber of Z 0,V 5 at y can be written as (4.14) so it is a linear section of a cone over the 3-dimensional cubic scroll. The vertex [ 2 V 2 ] of the cone does not belong to P(W ). Indeed, since both L 0 and L v are nice lines (in the sense of Definition 4.5), we have by Lemma 4.6 (4.15) , so if it also belongs to P(W ), we get L 0 ∩ L v ∅, which contradicts Definition 4.7.
Since W has codimension 2 in 2 V 5 and is transverse to V 2 ∧ V 5 by Lemma 4.6, it follows that Z y is isomorphic to a hyperplane section of P(V 2 )×P(V 5 /V 2 ) P 1 ×P 2 . It is easy to see that its Hilbert polynomial is h Z y (t) = (t + 1)( 3 2 t + 1). Since it does not depend on y, the family of surfaces Z 0,V 5 is flat over S 0,V 5 . A combination of (4.14) and (4.15) shows that L v , L 0 ⊂ Z y . Remark 4.12. As in Remark 4.9, the family Z 0,V 5 is the restriction of the family of surfaces , these surfaces are hyperplane sections of the cubic scroll P 1 × P 2 . This proves flatness of the family Z 0,V 5 even when the curve S 0,V 5 is not reduced.
Propositions 4.1 and 4.11 show that the components of Z 0 and Z 0,V 5 all have dimension 3. They are components of the scheme Z 0+ , which has other 3-dimensional components over the curve S 0,V 5 defined by S 0,V 5 = S 0+ ( S 0 ∪ S 0,V 5 ), but we will not need this fact.
Finally, to construct the left column of (4.7), recall that the curve S 0,V 5 is by definition isomorphic to the with an open subscheme in the Hilbert scheme of lines F 1 (X) (see Proposition 2.11). Applying to Z 0,V 5 the construction of Hilbert closure from Definition 4.2, we obtain a subscheme Note that Z F may be not flat over the singular locus of the curve F 1 (X).

A relation between the subschemes
Let X be a smooth ordinary GM threefold and let L 0 be a nice line on X. In (4.3) and (4.16), we constructed subschemes Z ⊂ X × Y ≥2 A and Z F ⊂ M X × F 1 (X). The proof of Theorem 4.4 is based on a relation between the schemes Z ∩ (X × S 0,V 5 ) and where the curve S 0,V 5 defined in (4.13) is considered as a subscheme of Y ≥2 A and F 1 (X). To prove such a relation, we will assume that the Hilbert scheme of lines F 1 (X) is a smooth curve (by Lemma 2.13, it is then irreducible). This assumption implies that the open curves Y 2 A,V 5 Σ 1 (X) and S 0,V 5 are also smooth and irreducible. The next lemma sharpens the results of Proposition 4.11 under this assumption.
Lemma 4.13. Assume that F 1 (X) is smooth. For a general point y in S 0,V 5 , the fiber Z y of Z 0,V 5 → S 0,V 5 is a smooth cubic surface scroll and the lines L 0 and L π(y) are distinct fibers of the ruling of this scroll.
Proof. We saw at the end of the proof of Proposition 4.11 that Z y is a hyperplane section of P(V 2 ) × P(V 5 /V 2 ). These hyperplane sections come in two kinds: (a) smooth cubic scrolls with projection to P 1 induced by P( where pr v 0 : P(V 5 ) P(V 5 /Cv 0 ) is the projection from v 0 . Since Y ≥2 A,V 5 is an integral curve of degree 40, its image by pr v 0 is contained in the image of the conic Σ 1 (X) only if the line connecting v 0 with a general point of Σ 1 (X) intersects Y ≥2 A,V 5 in 20 points. But the surface Y ≥2 A is an intersection of hypersurfaces of degree 6 by [DK19, (33)], hence the same is true for its hyperplane section Y ≥2 A,V 5 , and the curve Y ≥2 A,V 5 would then contain the cone Cone [v 0 ] (Σ 1 (X)), which is absurd. Therefore, for y general in S 0,V 5 , we are in case (a).
By (4.15), the lines L 0 and L v are contained in fibers of the map P(V 2 ) × P(V 5 /V 2 ) → P(V 2 ). In case (a), they are therefore the fibers of the ruling of the scroll.
Since the curve S 0,V 5 is isomorphic to a dense open subscheme of the smooth curve F 1 (X), the locally closed embedding S 0,V 5 → Y ≥2 A extends to a regular map We combine all these maps into the commutative diagram (4.18) where L 1 (X) ⊂ X × F 1 (X) is the universal family of lines, i : X → M X is the embedding, and the schemes Z and Z F are defined by (4.3) and (4.16) respectively. We make the following key observation.
Proposition 4.14. Assume that F 1 (X) is smooth. There is a dense open subscheme U ⊂ F 1 (X) such that as cycles.
Proof. Since we only need an equality over a dense open subset of F 1 (X), we may base change both sides along the open embedding S 0,V 5 → F 1 (X). The left side can then be rewritten as By Proposition 4.11 and Lemma 4.13, the morphism Z 0,V 5 → S 0,V 5 is a flat family of surfaces whose general fiber is a smooth cubic surface scroll. Since X contains no surfaces of degrees less than 10 ([DK19, Corollary 3.5]), it contains no components of any fiber of Z 0,V 5 . Therefore, the morphism is a flat family of curves whose fiber over a general point y ∈ S 0,V 5 is the intersection of the smooth cubic surface scroll Z y with any non-Plücker quadric Q 0 containing X. Such an intersection is a connected curve of degree 6 and arithmetic genus 2. Since the lines L 0 and L π(y) are contained both in the scroll Z y and the quadric Q 0 , they are components of C y . To describe the remaining components, we denote by e the class of the exceptional section L e of the scroll Z y and by f the class of a fiber of its ruling. We have The hyperplane class is equal to e + 2f hence, the class of C y in Z y is 2e + 4f . As we observed above, the lines L 0 and L π(y) are fibers of the ruling, hence is an effective divisor on Z y with class 2e + 2f . If C y contains a line, the class of this line is either f (the class of a fiber of the ruling), or e (the class of the exceptional section L e of Z y ). If it is f , the residual components have class 2e + f , and since (2e + f ) · e = −1, the section L e is in both cases a component of C y , hence a line on X. The line L e is in the finite set of lines on X intersecting L 0 , and L π(y) is in the finite set of lines on X intersecting a line that intersects L 0 . It follows that for y general, the curve C y contains no lines.
By Lemma 4.10, the curve Z y is contained in the surface Z y for general y ∈ S 0,V 5 . Therefore, by Lemma 4.3, for general y ∈ S 0,V 5 , the sextic curve C y contains the quintic curve Z y , hence C y = Z y + L y , where L y is a line. Since Z y contains L 0 and C y contains no lines, the line L y must be L π(y) . Thus, C y = Z y + L π(y) .
Since this holds for y general in S 0,V 5 , it follows that the equality of cycles (4.19) holds over a dense open subscheme U ⊂ S 0,V 5 ⊂ F 1 (X).

Abel-Jacobi maps
Let X be a smooth GM threefold with associated Lagrangian A. Assume that Y 3 A = ∅, so that Y ≥2 A is a smooth surface, and that the Hilbert scheme of lines F 1 (X) is smooth. The subscheme Z ⊂ X × Y ≥2 A was constructed in Lemma 4.3. Consider the universal line L 1 (X) ⊂ X × F 1 (X), the Abel-Jacobi maps A defined in (4.17). Proposition 4.15. Let X be a smooth ordinary GM threefold with associated Lagrangian A satisfying Y 3 A = ∅. Assume that F 1 (X) is smooth and let L 0 be a nice line on X. The composition of maps is equal to the map − AJ L 1 (X) .
Proof. By Lemma 3.1(a), it is enough to check that the Abel-Jacobi map given by the image of [Z] with respect to the pullback map Equality (4.19) implies that there is a cycle Z D supported on X × D, where the Let us show that the Abel-Jacobi map defined by the right side of this equality is zero.
By Lemma 3.1(b), the Abel-Jacobi map corresponding to the cycle (i × Id F 1 (X) ) * ([Z F ]) is equal to the composition Since H 5 (M X , Z) = 0 by Proposition 2.6, this map vanishes. Similarly, the Abel-Jacobi map corresponding to the cycle (Id X ×δ) * ([Z D ]) is equal to the composition hence vanishes as well. This completes the proof of the proposition.
The above proposition relates the Abel-Jacobi maps AJ Z and AJ L 1 (X) . The next lemma uses the Clemens-Tyurin argument (Section 3.3) to show that the latter is surjective. Proof. Let Y be a general GM fourfold and let X ⊂ Y be a general hyperplane section, so that X is a general GM threefold. Let F Y = F 1 (Y ) be the Hilbert scheme of lines contained in Y . We check that the assumptions of Proposition 3.3 (with m = 1) are satisfied.
Combining the above results, we can now prove Theorem 4.4.
Proof of Theorem 4.4. Let A ⊂ 3 V 6 be a general Lagrangian subspace. As recalled in Section 2.3, any hyperplane V 5 ⊂ V 6 corresponding to a point of Y ≥2 A ⊥ ⊂ P(V ∨ 6 ) defines a smooth GM threefold X with A(X) = A. We choose a general such [V 5 ], so that X is a general GM threefold. We also choose a nice line L 0 ⊂ X. A combination of Proposition 4.15 and Lemma 4.16 proves that the map AJ Z is surjective. By Propositions 2.5 and 2.6, its source and target are free abelian groups of rank 20. Therefore, the Abel-Jacobi map (4.5) is an isomorphism.
The Abel-Jacobi map is defined by the cohomology class of an algebraic cycle, hence it preserves the Hodge structures and induces an isomorphism (4.6) between the corresponding abelian varieties: the Albanese variety of Y ≥2 A and the intermediate Jacobian of X.
A is defined in Section 4.1 for all smooth X and all lines L 0 ⊂ X, and since this definition works in families, the maps (4.5) and (4.6) are by continuity isomorphisms for any A such that Y 3 A = ∅ (so that the surface Y ≥2 A is smooth), any smooth X such that A(X) = A, and any line L 0 ⊂ X. The abelian variety Jac(X) carries a canonical principal polarization and the isomorphism (4.6) transports it to a principal polarization on the abelian variety Alb( Y ≥2 A ). Since Y ≥2 A ⊥ is connected and the isomorphism (4.6) depends continuously on X, this principal polarization is independent of the choice of [V 5 ] ∈ Y ≥2 A ⊥ . It is therefore canonical.
It remains to construct an isomorphism (4.4). Choose a point . Let X and X be the smooth ordinary GM threefolds corresponding to the Lagrangian data sets (V 6 , V 5 , A) and (V ∨ 6 , V ⊥ 1 , A ⊥ ) respectively. We will prove in Proposition 4.17 that there is a diagram where both vertical morphisms are blow ups of smooth rational curves and ψ is a flop. By [FW08, Proposition 3.1], the morphism H 3 ( X; Z) → H 3 ( X ; Z) induced by the correspondence defined by the graph of ψ is an isomorphism of polarized Hodge structures. It induces in particular an isomorphism Jac( X) ∼ → Jac( X ) of principally polarized abelian varieties between intermediate Jacobians. Therefore, there is a chain of isomorphisms

The line transform
In this section, we revisit the birational isomorphism of [DK18,Proposition 4.19] and identify it with an elementary transformation along a line, a birational transformation between GM threefolds defined in [IP99,Proposition 4.3.1] and [DIM12, Section 7.2] (this relation was mentioned without proof in [DK18, Section 4.6]).
Proposition 4.17. Let A ⊂ 3 V 6 be a Lagrangian subspace with no decomposable vectors. Consider subspaces . Let X and X be the smooth ordinary GM threefolds corresponding to the Lagrangian data sets (V 6 , V 5 , A) and (V ∨ 6 , V ⊥ 1 , A ⊥ ), and let L 0 ⊂ X and L 0 ⊂ X be the lines corresponding to the points There is a diagram of birational maps (4.21) where β and β are the respective blow ups of X and X along the lines L 0 and L 0 , the birational maps and are the respective linear projections of X from L 0 , and of X from L 0 , the morphisms ρ X and ρ X are small crepant extremal birational contractions, the varietyX is a normal, Gorenstein, cubic hypersurface in Gr(2,  Proof. Let M GM 3 be the moduli stack of smooth GM threefolds (see [DK20b]), let X → M GM 3 be the universal family of threefolds over it, and let F 1 (X /M GM 3 ) be the relative Hilbert scheme of lines. As we already mentioned, by [IP99, Section 4.1], the elementary transformation is defined for any line contained in any smooth GM threefold X. Moreover, this transformation can be performed for a family of lines and will produce a family of GM threefolds. This defines a morphism By [DK20b, Theorem 5.11] and [DK19, Theorem 4.7], the left side is irreducible and birational to To prove Proposition 4.17, we start with some preliminaries. First, subspaces V 1 and V 5 satisfying the conditions (4.20) exist: this follows from [DK18, Lemma B.5] (where Y A is defined in [DK18,(B.5

)]). More exactly, for [V 5 ] general in Y 2
A ⊥ (so that X is a general GM threefold associated with the fixed Lagrangian A), these conditions will be satisfied for a general [V 1 ] ∈ Y 2 A,V 5 , corresponding to a general line L 0 ⊂ X. As explained in the proof of [DK18,Theorem 4.20], the conditions (4.20) are equivalent to hence the lines L 0 and L 0 are nice and the assumptions of [DK18, Proposition 4.19] are satisfied. It was explained in the proof of that proposition that these lines can be written as Following [DK18,Section 4.4], we introduce the second quadratic fibration and analogously for X , and study the diagram [DK18, (4.5)] (4.23) whereρ 2 is obtained from ρ 2 by restriction to Gr(2, V 5 /V 1 ) ⊂ Gr(3, V 5 ) and analogously forρ 2 . The next lemma is a refinement of [DK18, Lemma 4.18].
Lemma 4.19. The scheme X has two irreducible components, Bl L 0 (X) and f −1 (L 0 ). They are both smooth of dimension 3 and meet transversely along the exceptional divisor of Bl L 0 (X). Moreover, the mapρ 2 : X → Gr(2, V 5 /V 1 ) is induced by the linear projection from the line L 0 .

Consider the commutative diagram
where M X was defined in (2.7) and the vertical arrow is the canonical projection, which induces the diagonal arrows (the left diagonal arrow factors through Z A,V 5 by [DK18, Proposition 4.10]). Pulling this diagram back by the inclusion Gr(2, V 5 /V 1 ) ⊂ Gr(3, V 5 ), we obtain the diagram Indeed, Gr(2, V 5 /V 1 ) ⊂ Gr(3, V 5 ) is the zero-locus of the section of the vector bundle V 5 /U 3 corresponding to V 1 and, by [Kuz16, Lemma 2.1], the zero-locus of the corresponding section on P Gr(2,V 5 ) (V 5 /U ) is the blow up of Gr(2, V 5 ) along the zero-locus of induced section of V 5 /U , which is equal to the locus P(V 1 ∧ V 5 ) of 2-dimensional subspaces in V 5 containing V 1 . Note also that the map is induced by the linear projection V 5 → V 5 /V 1 from V 1 , or equivalently by the linear projection P( 2 V 5 ) P( 2 (V 5 /V 1 )) from P(V 1 ∧ V 5 ). Furthermore, M X ⊂ Gr(2, V 5 ) is the linear section by the subspace P(W ) ⊂ P( 2 V 5 ) which is transverse to P(V 1 ∧V 5 ) by Lemma 4.6, because the line L 0 is nice; the pullback of P M X (V 5 /U M X ) is therefore Bl L 0 (M X ). Moreover, the map Bl L 0 (M X ) → Gr(2, V 5 /V 1 ) is induced by the linear projection from P(W ∩(V 1 ∧V 5 )) = L 0 .
To prove the lemma, it remains to invoke the equality which holds because the first component is the strict transform of X and the second component is the It was proved in [DK18, Lemma 4.18 and Proposition 4.19] that •ρ 2 maps f −1 (L 0 ) birationally onto the Schubert hyperplane divisor D ⊂ Gr(2, V 5 /V 1 ) parameterizing subspaces intersecting V 3 /V 1 , where V 3 was defined in (4.22); •ρ 2 maps Bl L 0 (X) birationally onto a cubic hypersurfaceX ⊂ Gr(2, V 5 /V 1 ); • the image ofρ 2 is the quartic hypersurface Z A,V 1 ,V 5 ; it is therefore equal toX ∪ D. We denote by ρ X : Bl L 0 (X) →X the (birational) restriction ofρ 2 and we define ρ X similarly.
Proof of Proposition 4.17. We have already constructed the left part of the diagram (4.21). The right part is constructed analogously. It remains to prove that ψ is a flop.
As explained in [IP99, Lemma 4.1.1 and Corollary 4.3.2], ρ X is a flopping contraction. The same is true for ρ X , so ψ is either a flop or an isomorphism. If ψ is an isomorphism, the maps β and β are the contractions of the same extremal ray, hence X X . Let us show that this is impossible.
Indeed, we can perform this construction on a fixed X (that is, with A and V 5 fixed) but with [V 1 ] varying in the curve Y 2 A,V 5 . Locally, the map ψ will remain an isomorphism and the threefolds X obtained by the construction will be all isomorphic to X. But this is impossible: by definition, X is the ordinary GM threefold corresponding to the Lagrangian data set (V ∨ 6 , V ⊥ 1 , A ⊥ ), hence its moduli point describes the curve (recall that the map p 3 was defined in (2.9) and that the group PGL(V 6 ) A is finite). It follows that ψ is a flop and the proof of the proposition is complete.

Intermediate Jacobians of GM fivefolds
In this section, we perform, for GM fivefolds, a construction analogous to what we did in Section 4 for threefolds. The curves in the construction are replaced by surfaces: lines by planes, elliptic quintic curves by quintic del Pezzo surfaces, and rational quartic curves by rational quartic surface scrolls. In Section 5.5, we give an alternative proof of the main result.

Family of surfaces
Given an arbitrary GM fivefold X with associated Lagrangian A, we begin by choosing an arbitrary σplane Π 0 ⊂ X (that is, a point of F 2 σ (X); see (2.2)). We consider the open surface S 0 ⊂ Y ≥2 A defined by (2.14) and the family of quadrics Q 0 → S 0 obtained by restricting to S 0 the universal family (2.11) of 8-dimensional quadrics containing X. We denote by F 5 (Q 0 /S 0 ) → S 0 the relative Hilbert scheme of linear 5-spaces in the fibers of Q 0 → S 0 and by F 5 Π 0 (Q 0 /S 0 ) → S 0 the subscheme parameterizing those 5-spaces which contain the plane Π 0 . Applying Corollary 2.9, we obtain an isomorphism of schemes over S 0 . In particular, the canonical map F 5 Note that S 0 is a smooth surface. Let Q 0 := Q 0 × S 0 S 0 be the base change of the family of quadrics Q 0 → S 0 along π. We have a canonical map , where the first map is the product of the isomorphism (5.1) with the identity map. By construction, it is a section of the projection F 5 ( Q 0 / S 0 ) → S 0 .
Let P 5 ⊂ Q 0 ⊂ P(W ) × S 0 be the pullback of the universal family of linear 5-spaces over F 5 ( Q 0 / S 0 ) along this section. Set where the Grassmannian hull M X ⊂ P(W ) was defined in (2.7).
Proof. Let y ∈ S 0 and set [v] := π(y) ∈ P(V 6 ) P(V 5 ). The fiber of Z 0 over y is Z 0,y := M X ∩ P 5 y = CGr(2, V 5 ) ∩ P 5 y . Since the cone CGr(2, V 5 ) ⊂ P(C ⊕ 2 V 5 ) has codimension 3 and degree 5, the intersection CGr(2, V 5 ) ∩ P 5 y has dimension at least 2 and degree at most 5 (and if the dimension is 2, the degree is 5). Furthermore, P 5 Since X contains no divisors of degrees less than 10, we have dim(Z 0,y ) ≤ 3 and, moreover, if dim(Z 0,y ) = 3, any irreducible 3-dimensional component Z 0,y has even degree ([DK19, Corollary 3.5]). By Lemma 2.3, its image in Gr(2, V 5 ) must be a hyperplane section of Gr(2, V 4 ) and Lemma 2.7 gives a contradiction. Therefore Z 0,y is a surface. This argument also proves the inclusion Since the surface Z 0,y is a dimensionally transverse linear section of CGr(2, V 5 ), we obtain from Lemma 2.4 a resolution 0 → O P 5 y (−5) → O P 5 y (−3) ⊕5 → O P 5 y (−2) ⊕5 → O P 5 y → O Z 0,y → 0. It follows that the Hilbert polynomial of Z 0,y is given by (5.3). Since it is independent of y, the family of surfaces Z 0 is flat over S 0 . Finally, since Π 0 ⊂ M X and Π 0 ⊂ P 5 y by construction, we obtain Π 0 ⊂ Z 0,y .
Applying to the family Z 0 → S 0 the Hilbert closure construction from Definition 4.2, we obtain the following result.

Lemma 5.2. There is a subscheme
A is a flat family of surfaces with Hilbert polynomial (5.3) containing the plane Π 0 . Moreover, we have By Proposition 5.1, the schemes Z 0 and Z have pure dimension 4. The main result of this section is the following theorem (recall from Propositions 2.5 and 2.6 that the abelian groups H 1 ( Y ≥2 A(X) , Z) and H 5 (X, Z) are both free of rank 20 and from Theorem 4.4 that the abelian variety Alb( Y ≥2 A(X) ) is endowed with a canonical principal polarization when Y ≥3 A(X) = ∅).

Theorem 5.3. Let X be a smooth GM fivefold and let
A(X) = ∅, the Abel-Jacobi map

The boundary of the family
Let X be a smooth GM fivefold. To prove Theorem 5.3, we study the family of surfaces Z described in Lemma 5.2 over the boundary Y ≥2 A S 0 . We assume in this section that X is general (in particular it is ordinary), so that the curve Y ≥2 A,V 5 is smooth. Consider the Hilbert scheme F 2 σ (X) of σ -planes on X; we identify it with the smooth connected curve Y ≥2 A,V 5 via the isomorphism (2.19). For y ∈ Y ≥2 A,V 5 , we denote by Π y ⊂ X the corresponding σ -plane. We denote by y 0 ∈ Y ≥2 A,V 5 the point such that Π y 0 = Π 0 is the plane chosen in Section 5.1. Set [v 0 ] := π A (y 0 ). By Lemma 2.1(b), we have, for an appropriate hyperplane V 4 ⊂ V 5 , . This is a hyperplane section of the smooth curve Y 2 A,V 5 hence is finite; the induced double coverings of this set parameterizes σ -planes on X that intersect Π 0 .
Denote by Q the pullback of the family of quadrics Q along the map Y ≥2 Recall that a section S 0 → F 5 Π 0 ( Q 0 / S 0 ) was constructed in Section 5.1. Since the surface Y ≥2 A is smooth and the Hilbert scheme F 5 A , this section extends to an open subset (5.7) A which contains a general point of the curve S 0,V 5 := S 0+ S 0 ⊂ Y ≥2 A,V 5 . We denote by P 5 + ⊂ P( 2 V 5 ) × S 0+ the corresponding family of 5-spaces and, for y ∈ S 0+ , by P 5 +y ⊂ P( 2 V 5 ) the corresponding linear 5-space. By definition, we have Π 0 ⊂ P 5 +y for each y ∈ S 0,V 5 .
Lemma 5.4. For each point y ∈ S 0,V 5 , we have A,V 5 and let W 6 ⊂ 2 V 5 be the 6-dimensional subspace corresponding to the 5-space P 5 y ⊂ P( 2 V 5 ). By definition, we have Since Gr(2, V 5 /v) is a smooth 4-dimensional quadric, the maximal dimension of a linear subspace that it contains is 2, hence the subspace is at least 3-dimensional. We claim that P(W y ) is contained in X.
Let {w i } be a basis of W y , let q v be an equation of Q v , and consider a line A be the corresponding morphism that takes the closed point to [v]. Since S 0+ is étale over Y ≥2 A,V 5 , the morphism can be lifted to a morphism that takes the closed point to y. This implies that there are vectors w i in 2 V 5 such that the subspace in 2 V 5 generated by w i + w i is isotropic for the quadratic form q v + q v , where q v is an equation of Q v . We have therefore Note that q v (w i , w j ) = q v (w j , w i ) = 0, since w i , w j ∈ v∧V 5 = Ker(q v ) for all i, j. It follows that q v (w i , w j ) = 0 for all i, j, hence W y is isotropic for q v , that is, P(W y ) ⊂ Q v . Since P(W y ) ⊂ P(v ∧ V 5 ) ⊂ Gr(2, V 5 ), we conclude that thus proving the claim.
Since P(W y ) ⊂ P(v ∧ V 5 ) ∩ X and X contains no linear 3-spaces ([DK19, Theorem 4.2]), it follows that dim(W y ) = 3 and P(W y ) is a σ -plane on X. Moreover, the induced map A,V 5 . Therefore, replacing if necessary the isomorphism (2.19) by its composition with the involution of the double covering, we may assume that the above map S 0,V 5 → F 2 σ (X) coincides with the embedding S 0,V 5 → Y ≥2 A,V 5 , hence P(W y ) = Π y for all y ∈ S 0,V 5 . This means that there is an inclusion Π y ⊂ P 5 +y for all y ∈ S 0,V 5 . Since Π 0 ⊂ P 5 +y by definition, the right side of (5.8) is contained in the left side. Finally, since Π 0 ∩ Π y = ∅ by definition of S 0+ (since planes intersecting Π 0 are parameterized by the double cover of the subscheme Y ≥2 A,V 4 ), the inclusion is an equality.
Proposition 5.5. Let X be a general GM fivefold. The map Z 0,V 5 → S 0,V 5 is a flat family of 3-dimensional cubic scrolls P 1 × P 2 .
The vertex [ 2 V 2 ] = [v 0 ∧ v] of the cone does not belong to P 5 +y : if it did, Π 0 and Π y would intersect at the point [v 0 ∧ v] and this would contradict the definition of S 0+ . Therefore, the fiber of Z 0,V 5 over y is isomorphic to the 3-dimensional cubic scroll P(V 2 ) × P(V 5 /V 2 ).
Propositions 5.1 and 5.5 show that all the components of Z 0 and Z 0,V 5 have dimension 4. They are components of the scheme Z 0+ . We also consider the Hilbert closure , constructed as in Definition 4.2.

A relation between the subschemes
Let X be a general GM fivefold. In (5.4) and (5.10), we have constructed subschemes Z ⊂ X × Y ≥2 A and Z F ⊂ M X × F 2 σ (X). The proof of Theorem 5.3 is based on a relation between the schemes Z ∩ (X × S 0,V 5 ) and Z F ∩ (X × S 0,V 5 ), where the curve S 0,V 5 is considered as a subscheme of both the surface Y ≥2 A and the curve F 2 σ (X). Consider the commutative diagram (5.11) where L 2 σ (X) ⊂ X × F 2 σ (X) is the universal family of σ -planes, i : X → M X is the embedding, andσ is the isomorphism (2.19).
Proof. The left side of (5.12) can be rewritten as By Proposition 5.5, the morphism Z 0,V 5 → S 0,V 5 is a flat family of smooth 3-dimensional cubic scrolls. Since X contains no such threefolds ([DK19, Corollary 3.5]), it contains no fibers of Z 0,V 5 . Therefore, the morphism is a flat family of surfaces whose fiber over y ∈ S 0,V 5 is the dimensionally transverse intersection S y := Z 0,V 5 ,y ∩ Q 0 of the smooth 3-dimensional cubic scroll Z 0,V 5 ,y P 2 × P 1 with any non-Plücker quadric Q 0 containing X. Such an intersection is a surface of class 2f 2 + 2f 1 in Z 0,V 5 ,y , where f i is the preimage of the hyperplane class on P i under the projection Z 0,V 5 ,y P 2 × P 1 → P i . By (5.8) and (5.9), the planes Π 0 and Π y are contained in the scroll Z 0,V 5 ,y . Since they are also contained in X, they are contained in the quadric Q 0 . It follows that they are components of S y , each of class f 1 . Therefore, where S y ⊂ Z 0,V 5 ,y is a surface of class 2f 2 , that is, the product of a conic in P 2 with P 1 . In particular, it has degree 4 and contains no planes. Since Z y ⊂ Z 0,V 5 ,y is a surface of degree 5 that contains Π 0 , we have Z y = S y + Π 0 for all y ∈ S 0,V 5 ; this proves (5.12).

Abel-Jacobi maps
Let X be a smooth ordinary GM fivefold with associated Lagrangian A. Assume Y 3 A = ∅, so that Y ≥2 A is a smooth surface and the curve Y ≥2 A,V 5 , hence also the Hilbert scheme F 2 σ (X) of σ -planes in X, is a smooth curve. Let L 2 σ (X) ⊂ X × F 2 σ (X) denote the universal family of σ -planes on X. Consider the Abel-Jacobi maps AJ Z : H 1 ( Y ≥2 A , Z) → H 5 (X, Z) and AJ L 2 σ (X) : H 1 (F 2 σ (X), Z) → H 5 (X, Z) and recall the isomorphismσ : Proposition 5.7. Let X be a smooth ordinary GM fivefold with associated Lagrangian A. Assume that Y 3 A = ∅ and F σ 2 (X) is smooth. The composition of maps is equal to the map − AJ L 2 σ (X) . Proof. Analogous to the proof of Proposition 4.15.
The above proposition connects the Abel-Jacobi maps AJ Z and AJ L 2 σ (X) . The next lemma uses the Clemens-Tyurin argument (Section 3.3) to show that the latter is surjective.
Lemma 5.8. Let X be a general GM fivefold. The Abel-Jacobi map Proof. Let Y be a general GM sixfold and let X ⊂ Y be a general hyperplane section, so that X is a general GM fivefold. Set F Y := F 2 σ (Y ), the Hilbert scheme of σ -planes contained in Y . We check that the assumptions of Proposition 3.3 (with m = 2) are satisfied.
Assumption ( Applying Proposition 3.3, we deduce the surjectivity of AJ L 2 σ (X) . Combining the above results, we can now prove Theorem 5.3.
Proof of Theorem 5.3. Assume first that the GM fivefold X is general. A combination of Proposition 5.7 and Lemma 5.8 proves that the map AJ Z is surjective. By Propositions 2.5 and 2.6, its source and target are free abelian groups of rank 20. Therefore, the Abel-Jacobi map is an isomorphism.
Since the Abel-Jacobi map is defined by the cohomology class of an algebraic cycle, it preserves the Hodge structures, hence induces an isomorphism of the corresponding abelian varieties: the Albanese variety of Y ≥2 A(X) and the intermediate Jacobian of X.
A(X) was defined in Section 5.1 for all smooth X and all σ -planes Π 0 ⊂ X and since this definition works in families, these two statements follow by continuity for any X such that Y 3 A(X) = ∅. It remains to prove that the isomorphism (5.5) respects the principal polarizations. For X very general, the Picard number of Jac(X) is 1 by Corollary 3.6, hence any two principal polarizations on Jac(X) coincide. This proves the claim for very general X and, by continuity, for any smooth X such that Y ≥2 A(X) is also smooth.

Simplicity argument
We give an alternative argument for the isomorphism (5.5) for a smooth GM fivefold X, based on a simplicity result of independent interest, analogous to the one proved in Proposition 3.4.
Let S be a smooth connected projective surface and let  : C → S be a smooth (irreducible) ample curve. By the Lefschetz theorem, the morphism  * : H 1 (C, Z) → H 1 (S, Z) is surjective, hence the induced morphsim is surjective with connected kernel. We denote this kernel by K(C, S).
Consider now a connected double étale cover π : S → S and set C := π −1 (C), a smooth ample curve on the surface S.
where K( C, C), P ( C, C) (the Prym variety of the double cover C → C), and P ( S, S) are the neutral components of the respective kernels of the vertical maps π * induced by π.
Proof. The surjectivity of the maps Jac( C) → Jac(C) and Alb( S) → Alb(S) is obvious. The only thing we have to prove is the surjectivity of the map K( C, S) → K(C, S) or, equivalently, the surjectivity of the map P ( C, C) → P ( S, S). On the level of cotangent spaces, the surjectivity of this second map corresponds to the injectivity of the restriction H 1 (S, η) → H 1 (C, η| C ), where η is the line bundle of order 2 on S corresponding to the double étale covering π. Its kernel is controlled by H 1 (S, η(−C)), which vanishes by Kodaira vanishing and Serre duality because η(C) is ample on S. This proves the injectivity of the morphism H 1 (S, η) → H 1 (C, η| C ), hence the lemma.
The next statement is the main result of this section. The definition of "trivial endomorphism ring" can be found in Section 3.4.
Theorem 5.10. Let S ⊂ P N be a smooth connected projective surface and let π : S → S be a connected double étale cover. Let H ⊂ P N be a very general hyperplane and set C := S ∩ H. With the notation above, the endomorphism ring of the abelian variety K( C, C) is trivial.
Proof. We use the notation of Section 3.4. Choose a Lefschetz pencil f : S P 1 of hyperplane sections of S. The connected double étale cover π : S → S induces for each t ∈ P 1 a connected double étale cover π t : C t → C t between fibers. Denote by  t : C t → S and t : C t → S the embeddings.
For t ∈ P 1 {t 1 , . . . , t r }, the involution τ of S attached to π acts on each summand of the orthogonal direct sum decomposition 3) and it preserves the symplectic form given by cup-product. The τ-invariant subspaces are . The isogeny class of the abelian variety K(C t , S) defined in (5.13) is obtained from the Hodge structure of H 1 (C t , Q) van , hence its endomorphism ring is trivial by Proposition 3.4. Therefore, to study the isogeny class of the neutral component K( C t , C t ) of the kernel of the surjection we need to study the rational Hodge structure on the τ-antiinvariant subspace H 1 ( C t , Q) − van . For each i ∈ {1, . . . , r}, the curve C t i has two nodes over the node of C t i , hence there are two disjoint vanishing cycles δ i and δ i = τ * (δ i ). Since the vanishing cycles span the vector space H 1 ( C t , Q) van , the cycles δ 1 − δ 1 , . . . , δ r − δ r span the antiinvariant subspace H 1 ( C t , Q) − van . The image of the monodromy representationρ : π 1 (P 1 {t 1 , . . . , t r }, t) −→ Sp(H 1 ( C t , Q)) consists of automorphisms that are τ-equivariant and, reasoning as in the proof of [Voi03, Proposition 3.23], we see that, up to changing signs, the classes δ 1 − δ 1 , . . . , δ r − δ r are all in the same monodromy orbit. Moreover, as in the proof of Proposition 3.4, there is for each i ∈ {1, . . . , r} an element of π 1 (P 1 {t 1 , . . . , t r }, 0) that acts on H 1 ( C t , Q) by One then deduces from that and [PS03, Lemma 4] that the monodromy action on H 1 ( C t , Q) − van is big; it follows that the Zariski closure of the monodromy group for K( C t , C t ) is the full symplectic group Sp(H 1 ( C t , Q) − van ). As in the proof of [PS03, Theorem 17], for t ∈ P 1 very general, any endomorphism of K( C t , C t ) intertwines every element of the monodromy group, hence every element of the symplectic group. It must therefore be a multiple of the identity.
The endomorphism ring of the abelian variety K( C t , C t ) is therefore trivial.
We now apply the theorem to GM fivefolds. Let X be a general GM fivefold with Lagrangian data set (V 6 , V 5 , A). Our starting point is again the surjectivity, proved in Lemma 5.8, of the Abel-Jacobi map AJ L 2 σ (X) : H 1 (F 2 σ (X), Z) −→ H 5 (X, Z) associated with the Hilbert scheme F 2 σ (X) that parametrizes σ -planes contained in X. This Hilbert scheme is isomorphic to the smooth curve Y ≥2 A,V 5 (Lemma 2.10) defined as the inverse image by the double cover The surjectivity of the map AJ L 2 σ (X) is therefore equivalent to the connectedness of the kernel of the induced surjective morphism (5.14) Φ : Jac( Y ≥2 A,V 5 ) −→ Jac(X) between Jacobians. By Lemma 2.10 and Proposition 2.6, the dimension of this kernel is A,V 5 ) − dim(Jac(X)) = 161 − 10 = 151.
Corollary 5.11. Let X be a smooth GM fivefold with Lagrangian data set (V 6 , V 5 , A). Assume that the surface Y ≥2 A and the curve Y ≥2 A,V 5 are smooth. The morphism Φ from (5.14) then factors as where the left arrow is the Albanese map Jac( Y ≥2 Proof. Since Alb(Y ≥2 A ) = 0 (Proposition 2.5), the diagram (5.13) reads (5.15) , and P A,V 5 is the Prym variety of the double covering Y ≥2 A,V 5 → Y ≥2 A,V 5 . The genus of the curve Y ≥2 A,V 5 is 81 by (2.20), hence the dimension of the variety P A,V 5 is 80. Also, the dimension of Alb( Y ≥2 A ) is 10 by Proposition 2.5. Therefore, dim(K V 5 ) = 70. When V 5 is a very general hyperplane in V 6 , the abelian varieties Jac(Y ≥2 A,V 5 ) and K V 5 are simple by Proposition 3.4 and Theorem 5.10, hence they are the only two simple factors of the abelian variety K V 5 . Since Jac(X) has dimension 10, the abelian variety K V 5 must therefore be contained in the kernel of Φ.
In other words, the composition K V 5 → Jac( Y ≥2 A,V 5 ) Φ Jac(X) vanishes for V 5 very general. By continuity, it vanishes for all hyperplanes V 5 such that Y ≥2 A,V 5 is smooth. The kernel of Φ, being connected of dimension 151, must then be equal to K V 5 , which implies the corollary.
Note that this argument cannot be applied to GM threefolds, because the corresponding hyperplane sections Y ≥2 A,V 5 are very far from being general.

Period maps
In this section, we prove Theorems 1.1 and 1.3. We will use the description (2.8) of the coarse moduli space M GM n of smooth GM varieties of dimension n. Let (6.1) M EPW ndv = LGr ndv ( 3 V 6 )// PGL(V 6 ) be the coarse quasiprojective moduli space of EPW sextics defined by Lagrangian subspaces A with no decomposable vectors (see (2.3) for the definition). This is an open subset of the coarse moduli space M EPW of EPW sextics defined in (1.6). We denote by r the involution of M EPW ndv defined by r([A]) = ([A ⊥ ]). The morphism p n was defined in (2.9).  We now use these results to prove Theorem 1.3. and (LGr ndv ( 3 V 6 )×P(V ∨ 6 ))// PGL(V 6 ) → M EPW ndv is generically a P 5 -fibration (the fiber over any point [A] is isomorphic to P(V ∨ 6 )// PGL(V 6 ) A ). The inclusion (6.3) is an open embedding for n = 5, a closed embedding for n = 3, and (LGr ndv ( 3 V 6 ) × P(V ∨ 6 ))// PGL(V 6 ) = M GM 5 M GM 3 by (2.10). This property is reminiscent of the Satake compactification.
We can also complete the proof of Theorem 1.1.
Proof of Theorem 1.1. By Proposition 6.2, the morphism℘ defines for each Lagrangian A with no decomposable vectors a principal polarization on Alb( Y ≥2 A ) such that (1.2) holds; this proves the first part of the theorem.
By Lemma 6.1, the isomorphism (1.4) of principally polarized abelian varieties holds for all smooth GM varieties X of dimension n ∈ {3, 5}; this proves the last part of the theorem.