Gushel-Mukai varieties: linear spaces and periods
aa r X i v : . [ m a t h . AG ] J un GUSHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS
OLIVIER DEBARRE AND ALEXANDER KUZNETSOV
Abstract.
Beauville and Donagi proved in 1985 that the primitive middle cohomology of asmooth complex cubic fourfold and the primitive second cohomology of its variety of lines, asmooth hyperk¨ahler fourfold, are isomorphic as polarized integral Hodge structures. We proveanalogous statements for smooth complex Gushel–Mukai varieties of dimension 4 (resp. 6), i.e.,smooth dimensionally transverse intersections of the cone over the Grassmannian Gr (2 , Gr (2 ,
5) and a quadric). The associatedhyperk¨ahler fourfold is in both cases a smooth double cover of a hypersurface in P called anEPW sextic. Introduction
We continue in this article our investigation of Gushel–Mukai (GM) varieties startedin [DK1]. We discuss linear subspaces contained in smooth complex GM varieties and theirrelation to Eisenbud–Popescu–Walter (EPW) stratifications. These results are applied to thecomputation of the period map for GM varieties of dimension 4 or 6.We work over the field of complex numbers. A smooth
Gushel–Mukai variety is ([DK1,Definition 2.1]) a smooth dimensionally transverse intersection
CGr (2 , V ) ∩ P ( W ) ∩ Q of the cone over the Grassmannian Gr (2 , V ) of 2-dimensional subspaces in a fixed 5-dimensionalvector space V , with a linear subspace P ( W ) and a quadric Q . This class of varieties includesall smooth prime Fano varieties X of dimension n ≥
3, coindex 3, and degree 10 (i.e., suchthat there is an ample class H with Pic( X ) = Z H , K X = − ( n − H , and H n = 10; see [DK1,Theorem 2.16]).One can naturally associate with any smooth GM variety of dimension n a triple ( V , V , A ),called a Lagrangian data , where V is a 6-dimensional vector space containing V as a hyperplaneand the subspace A ⊂ V V is Lagrangian with respect to the symplectic structure on V V given by wedge product. Moreover, P ( A ) ∩ Gr (3 , V ) = ∅ in P ( V V ) when n ≥ A has no decomposable vectors ).Conversely, given a Lagrangian data ( V , V , A ) with no decomposable vectors in A , onecan construct two smooth GM varieties of respective dimensions n = 5 − ℓ and n = 6 − ℓ (where ℓ := dim( A ∩ V V ) ≤ V , V , A ) ([DK1, Theorem 3.10and Proposition 3.13]; see Section 2.1 for more details).Given a Lagrangian subspace A ⊂ V V , we define three chains of subschemes Y ≥ A ⊂ Y ≥ A ⊂ Y ≥ A ⊂ P ( V ) , Y ≥ A ⊥ ⊂ Y ≥ A ⊥ ⊂ Y ≥ A ⊥ ⊂ P ( V ∨ ) , Mathematics Subject Classification. Z ≥ A ⊂ Z ≥ A ⊂ Z ≥ A ⊂ Z ≥ A ⊂ Gr (3 , V ) , called Eisenbud–Popescu–Walter (EPW) stratifications (see Section 2.2). The first two were ex-tensively studied by O’Grady ([O1, O2, O3, O4, O5, O6]) and the third in [IKKR]. If A hasno decomposable vectors, the strata Y A := Y ≥ A ⊂ P ( V ) , Y A ⊥ := Y ≥ A ⊥ ⊂ P ( V ∨ ) , and Z A := Z ≥ A ⊂ Gr (3 , V ) , are hypersurfaces of respective degrees 6, 6, and 4, called the EPW sextic , the dual EPW sextic ,and the
EPW quartic associated with A . Moreover, there are canonical double coverings e Y A → Y A , e Y A ⊥ → Y A ⊥ , and e Z ≥ A → Z ≥ A , called the double EPW sextic , the double dual EPW sextic , and the EPW cube associated with A ,respectively. In general (more precisely, when Y ≥ A = ∅ , Y ≥ A ⊥ = ∅ , and Z ≥ A = ∅ ), these arehyperk¨ahler manifolds which are deformation equivalent to the Hilbert square or cube of a K3surface.We showed in [DK1] that these EPW stratifications control many geometrical propertiesof GM varieties. For instance, smooth GM varieties of dimension 3 or 4 are birationally iso-morphic if their associated EPW sextics are isomorphic ([DK1, Theorems 4.7 and 4.15]). Inthis article, we describe the Hilbert schemes of linear spaces contained in smooth GM varietiesin terms of their EPW stratifications and relate the Hodge structures of smooth GM varietiesof dimension 4 or 6 to those of their associated double EPW sextics.Let X be a smooth GM variety. We denote by F k ( X ) the Hilbert scheme of linearlyembedded projective k -spaces in X . The scheme F ( X ) has two connected components F σ ( X )and F τ ( X ) corresponding to the two types of projective planes in Gr (2 , V ). We construct maps F ( X ) → P ( V ) , F σ ( X ) → P ( V ) , F τ ( X ) → Gr (3 , V ) , F ( X ) → P ( V )and describe them in terms of the EPW varieties defined by the Lagrangian A associatedwith X (Theorems 4.2, 4.3, 4.5, and 4.7). We prove in particular the following results.If X is a smooth GM sixfold with associated Lagrangian A such that Y ≥ A = ∅ , thescheme F σ ( X ) has dimension 4 and the above map F σ ( X ) → P ( V ) factors as F σ ( X ) → e Y A × P ( V ) P ( V ) → P ( V ) , where the first map is a locally trivial (in the ´etale topology) P -bundle (Theorem 4.3(a)).If X is a smooth general (with explicit generality assumptions) GM fourfold, F ( X ) hasdimension 3 and the map F ( X ) → P ( V ) factors as F ( X ) → e Y A × P ( V ) P ( V ) → P ( V ) , where the first map is a small resolution of singularities (a contraction of two rational curves;see Theorem 4.7(c)).Consequently, the universal plane L σ ( X ) in the sixfold case and the universal line L ( X )in the fourfold case give correspondences L σ ( X ) y y ssssssss % % ▲▲▲▲▲▲▲▲ X e Y A and L ( X ) y y ssssssss % % ❑❑❑❑❑❑❑ X e Y A USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 3 between X and its associated double EPW sextic e Y A . We use them to construct, in dimensions n = 4 or 6, isomorphisms H n ( X ; Z ) ≃ H ( e Y A ; Z ) of polarized integral Hodge structures (up to Tate twists; see Theorem 5.1 for precise state-ments) between the vanishing middle cohomology of X (defined in (5)) and the primitivesecond cohomology of e Y A (defined in (6)).This isomorphism is the main result of this article. It implies that the period point ofa smooth GM variety of dimension 4 or 6 (defined as the class of its vanishing cohomologyHodge structure in the appropriate period space) with associated Lagrangian data ( V , V , A )depends only on the PGL( V )-orbit of A and not on V .More precisely, the period maps from the moduli stacks of GM varieties of dimension 4or 6 factor through the period map from the moduli stack of double EPW sextics—the periodspaces being the same. The first map in this factorization is a fibration with well understoodfibers (see [DK1, Theorem 3.25]). Since double EPW sextics, when smooth, are hyperk¨ahlermanifolds, the second map is an open embedding by Verbitsky’s Torelli Theorem.The article is organized as follows.In Section 2, we recall some of the results from [DK1] about the geometry of smooth GMvarieties and their relation to EPW varieties. In Section 3, we discuss the singular cohomology ofGM varieties. In Section 4, we describe the Hilbert schemes F k ( X ) for smooth GM varieties X .In Section 5, we prove an isomorphism between the vanishing Hodge structure of a generalGM variety of dimension 4 or 6 and the primitive Hodge structure of the associated doubleEPW sextic. We also define the period point of a GM variety and show that it coincides withthe period point of the associated double EPW sextic. In Appendix A, we discuss the naturaldouble coverings arising from the Stein factorizations of relative Hilbert schemes of quadricfibrations. In Appendix B, we discuss a resolution of the structure sheaf of an EPW surface Y ≥ A in P ( V ) and compute some cohomology spaces related to its ideal sheaf.We are grateful to Grzegorz and Micha l Kapustka, Giovanni Mongardi, Kieran O’Grady,and Kristian Ranestad for interesting exchanges. We would also like to thank the referee forher/his careful reading of our article.2. Geometry of Gushel–Mukai varieties
Gushel–Mukai varieties.
We work over the field of complex numbers. A smooth Gushel–Mukai (GM) variety of dimension n is ([DK1, Definition 2.1 and Proposition 2.28]) a smoothdimensionally transverse intersection(1) X = CGr (2 , V ) ∩ P ( W ) ∩ Q, where V is a vector space of dimension 5, CGr (2 , V ) ⊂ P ( C ⊕ V V ) is the cone (with vertex ν := [ C ]) over the Grassmannian of 2-dimensional subspaces in V , W ⊂ C ⊕ V V is a vectorsubspace of dimension n + 5, and Q ⊂ P ( W ) is a quadratic hypersurface.Being smooth, X does not contain the vertex ν , hence the linear projection from ν definesa regular map γ X : X → Gr (2 , V )called the Gushel map of X . We denote by U X the pullback to X of the tautological rank-2subbundle on the Grassmannian. It comes with an embedding U X ֒ → V ⊗ O X . O. DEBARRE AND A. KUZNETSOV
Following [DK1], we associate with every smooth GM variety X as in (1) the intersection M X := CGr (2 , V ) ∩ P ( W ) . This is a variety of dimension n + 1 with finite singular locus ([DK1, Proposition 2.22]).If the linear space P ( W ) does not contain the vertex ν , the variety M X is itself a dimen-sionally transverse section of Gr (2 , V ) by the image of the linear projection P ( W ) → P ( V V )from ν . It is smooth if n ≥ X is its intersection with a quadratic hypersurface. TheseGM varieties are called ordinary .If P ( W ) contains ν , the variety M X is itself a cone with vertex ν over the smoothdimensionally transverse linear section M ′ X = Gr (2 , V ) ∩ P ( W ′ ) ⊂ P ( V V ) , where W ′ = W/ C ⊂ V V , and X is a double cover of M ′ X branched along the smooth GMvariety X ′ = M ′ X ∩ Q of dimension n −
1. These GM varieties are called special .A GM variety X ⊂ P ( W ) is an intersection of quadrics. Following [DK1], we denoteby V the 6-dimensional space of quadratic equations of X . The space V can be naturallyidentified with the space of Pl¨ucker quadrics cutting out CGr (2 , V ) in P ( C ⊕ V V ), hencealso with the space of quadrics in P ( W ) cutting out the subvariety M X . This gives a canonicalembedding V ⊂ V which identifies V with a hyperplane in V called the Pl¨ucker hyperplane .The corresponding point p X ∈ P ( V ∨ ) in the dual projective space is called the Pl¨ucker point .2.2.
EPW sextics and quartics.
Let X be a smooth GM variety of dimension n . As ex-plained in [DK1, Theorem 3.10], one can associate with X a subspace A ⊂ V V which isLagrangian for the det( V )-valued symplectic form given by wedge product. Together with thepair V ⊂ V defined above, it forms a triple ( V , V , A ) called the Lagrangian data of X .The Lagrangian space A has no decomposable vectors (i.e., P ( A ) ∩ Gr (3 , V ) = ∅ )when n ≥ A ∩ V V has dimension 5 − n if X is ordinary, 6 − n if X is special ([DK1, Proposition 3.13]).Conversely, given a Lagrangian data ( V , V , A ) such that A has no decomposable vectors,we have ℓ := dim( A ∩ V V ) ≤ • an ordinary smooth GM variety X ord ( V , V , A ) of dimension 5 − ℓ , • a special smooth GM variety X spe ( V , V , A ) of dimension 6 − ℓ ,unique up to isomorphism, whose associated Lagrangian data is ( V , V , A ).Given a Lagrangian subspace A ⊂ V V , one can construct interesting varieties that playan important role for the geometry of the associated GM varieties. Following O’Grady, onedefines for all integers ℓ ≥ Y ≥ ℓA = { [ U ] ∈ P ( V ) | dim( A ∩ ( U ∧ V V )) ≥ ℓ } ⊂ P ( V ) ,Y ≥ ℓA ⊥ = { [ U ] ∈ P ( V ∨ ) | dim( A ∩ V U ) ≥ ℓ } ⊂ P ( V ∨ ) , and set Y ℓA := Y ≥ ℓA r Y ≥ ℓ +1 A and Y ℓA ⊥ := Y ≥ ℓA ⊥ r Y ≥ ℓ +1 A ⊥ . Assume that A has no decomposable vectors. Then, Y A := Y ≥ A ⊂ P ( V ) and Y A ⊥ := Y ≥ A ⊥ ⊂ P ( V ∨ ) USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 5 are normal integral sextic hypersurfaces, called
EPW sextics ; the singular locus of Y A is theintegral surface Y ≥ A , the singular locus of Y ≥ A is the finite set Y ≥ A (empty for A general), Y ≥ A = ∅ ([DK1, Proposition B.2]), and analogous properties hold for Y ≥ ℓA ⊥ . One can rewritethe dimensions of the GM varieties X associated with a Lagrangian data ( V , V , A ) as follows:if the Pl¨ucker point p X is in Y ℓA ⊥ , we havedim( X ord ( V , V , A )) = 5 − ℓ and dim( X spe ( V , V , A )) = 6 − ℓ. Still under the assumption that A contains no decomposable vectors, O’Grady constructsin [O3, Section 1.2] a canonical double cover(2) f A : e Y A −→ Y A branched over the integral surface Y ≥ A . When the finite set Y ≥ A is empty, e Y A is a smoothhyperk¨ahler fourfold ([O1, Theorem 1.1(2)]).The hypersurfaces Y A and Y A ⊥ are mutually projectively dual and the duality is realized,inside the flag variety Fl (1 , V ) := { ( U , U ) ∈ P ( V ) × P ( V ∨ ) | U ⊂ U ⊂ V } , by thecorrespondence b Y A := { ( U , U ) ∈ Fl (1 , V ) | A ∩ ( U ∧ V U ) = 0 } ([DK1, Proposition B.3]) with its birational projections b Y A pr Y, v v v v ❧❧❧❧❧❧❧❧❧❧❧ pr Y, ) ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙ P ( V ) ⊃ Y A Y A ⊥ ⊂ P ( V ∨ )(these projections were denoted by p and q in [DK1, Proposition B.3]; we change the notationto pr Y, and pr Y, in this article, but we will switch back to p and q in Appendix B).We will need the following result. Lemma 2.1.
Assume that A has no decomposable vectors. If E ⊂ b Y A is the exceptional divisorof the map pr Y, , we have two inclusions Y ≥ A ⊥ ⊂ pr Y, ( E ) ⊂ ( Y ≥ A ) ∨ ∩ Y A ⊥ , where ( Y ≥ A ) ∨ ⊂ P ( V ∨ ) is the projective dual of Y ≥ A .Proof. Since Y ≥ A is smooth at points of Y A , its projective dual is (cid:0) Y ≥ A (cid:1) ∨ = [ v ∈ Y A h v ∧ ξ ∧ ξ , v ∧ ξ ∧ ξ , v ∧ ξ ∧ ξ i , where we write A ∩ ( v ∧ V V ) = h v ∧ ξ , v ∧ ξ i for some ξ , ξ ∈ V V , and identify V V with V ∨ . Indeed, a vector v ′ ∈ V is tangent to Y A at v if one has( v + tv ′ ) ∧ ( ξ i + tξ ′ i ) = a i + ta ′ i (mod t )for some ξ ′ i ∈ V V and a ′ i ∈ A , for i ∈ { , } . Since A is Lagrangian, this implies, for i, j ∈ { , } , 0 = ( v ∧ ξ i ) ∧ ( v ∧ ξ ′ j + v ′ ∧ ξ j ) = − v ′ ∧ ( v ∧ ξ i ∧ ξ j ) . This means that the embedded tangent space to Y A at v is contained in the orthogonal tothe subspace of V ∨ generated by v ∧ ξ i ∧ ξ j . Since the former, modulo v , is 2-dimensional and O. DEBARRE AND A. KUZNETSOV the latter is 3-dimensional, the tangent space coincides with this orthogonal, hence the abovedescription of the dual variety.On the other hand, by the argument in the proof of [DK1, Proposition B.3], one haspr Y, ( E ) = [ v ∈ Y ≥ A [ ξ v ∧ ξ ∧ ξ, where the second union is taken over all v ∧ ξ ∈ A ∩ ( v ∧ V V ). In particular, we obtain theinclusion pr Y, ( E ) ⊂ ( Y A ) ∨ .For the second inclusion, since Y ≥ A ⊥ is an integral surface ([DK1, Theorem B.2]), it isenough to show that E intersects the general fiber C of the map E ′ = pr − Y, ( Y ≥ A ⊥ ) → Y ≥ A ⊥ . Thisfiber is mapped by pr Y, to a conic in Y A ([DK1, Proposition B.3]), hence H · C = 2, where H is the pullback of the hyperplane class of Y A . On the other hand, if H ′ is the hyperplane classof Y A ⊥ , then H ′ · C = 0. But E is linearly equivalent to 5 H − H ′ ([DK1, proof of Lemma B.5]),hence E · C = 10, hence E intersects C non-trivially. This finishes the proof of the lemma. (cid:3) Given a Lagrangian subspace A ⊂ V V , one can also define the closed subschemes Z ≥ ℓA := { U ⊂ V | dim( A ∩ ( V U ∧ V )) ≥ ℓ } ⊂ Gr (3 , V ) . The complements Z ℓA := Z ≥ ℓA r Z ≥ ℓ +1 A form a stratification of Gr (3 , V ). If A has no decom-posable vectors, Z A := Z ≥ A is a normal integral hypersurface in Gr (3 , V ) cut out by a quartichypersurface in P ( V V ). We call Z A an EPW quartic . The singular locus of Z A is then theintegral variety Z ≥ A of dimension 6, the singular locus of Z ≥ A is the integral variety Z ≥ A ofdimension 3, the singular locus of Z ≥ A is the finite set Z ≥ A (empty for A general), and Z ≥ A = ∅ ([IKKR, Proposition 2.6]).Moreover, there is a canonical double cover e Z ≥ A → Z ≥ A branched over Z ≥ A , and when Z ≥ A is empty, e Z ≥ A is a smooth hyperk¨ahler sixfold ([IKKR, Theorem 1.1]).The hypersurfaces Z A ⊂ Gr (3 , V ) and Z A ⊥ ⊂ Gr (3 , V ∨ ) coincide under the natural iden-tification Gr (3 , V ) ≃ Gr (3 , V ∨ ). They are related to the EPW sextics via the correspondence b Z A := { ( U , U ) ∈ Fl (3 , V ) | A ∩ ( V U ∧ U ) = 0 } with its projections b Z A pr Z, { { ✈✈✈✈✈✈ pr Z, $ $ ❏❏❏❏❏❏❏ Z A Y A ⊥ . Lemma 2.2.
Assume that the Lagrangian A contains no decomposable vectors. The map pr Z, is dominant; over Y A ⊥ , it is smooth and its fibers are -dimensional quadrics. The map pr Z, is birational onto a divisor in Z A containing Z ≥ A .Proof. Let [ U ] be a point of Y A ⊥ and let a be a generator of the one-dimensional space A ∩ V U .The 2-form on U corresponding to a ∈ V U via the isomorphism V U ≃ V U ∨ has rank 4(because a is not decomposable). The fiber pr − Z, ([ U ]) parameterizes all 3-dimensional sub-spaces U of U which are isotropic for the 2-form a . Since a has rank 4, it is a smooth3-dimensional quadric. USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 7
Analogously, let [ U ] be a point of Z A and let a be a generator of the one-dimensionalspace A ∩ ( V U ∧ V ). Let ¯ a be the image of a in the space V U ⊗ ( V /U ) ≃ Hom( U ⊥ , V U ).Over Z A , this defines a morphism of rank-3 vector bundles with fibers U ⊥ and V U . Overits degeneracy locus (which a divisor in Z A ), the projection pr Z, is an isomorphism (andpr − Z, ([ U ]) is the unique hyperplane U ⊂ V such that U ⊥ ⊂ U ⊥ is the kernel of ¯ a ). Toprove Z ≥ A ⊂ pr Z, ( b Z A ), note that if dim( A ∩ ( V U ∧ V )) ≥
2, we have a pencil of maps U ⊥ → V U ; some maps in this pencil are degenerate and their kernels give points in thepreimage of [ U ]. (cid:3) For any hyperplane V ⊂ V , we set Y ℓA,V := Y ℓA × P ( V ) P ( V ) = Y ℓA ∩ P ( V ) ,Z ℓA,V := Z ℓA × Gr (3 ,V ) Gr (3 , V ) = Z ℓA ∩ Gr (3 , V ) , e Y A,V := e Y A × P ( V ) P ( V ) = f − A ( Y A,V ) , and similarly for Y ≥ ℓA,V and others. These varieties will play an important role for the geometryof the associated GM varieties. We let f A,V : e Y A,V → Y A,V be the morphism induced by restriction of the double cover f A : e Y A → Y A .We will need the following simple observation. Lemma 2.3.
Let ( V , V , A ) be a Lagrangian data with no decomposable vectors in A . If [ U ] ∈ Z A,V , then ( U , V ) ∈ b Z A . In particular, if A ∩ V V = 0 , we have Z ≥ A,V = ∅ .Proof. Assume that U ⊂ V defines a point of Z A,V . In other words, dim( A ∩ ( V U ∧ V )) ≥ U ⊂ V . Since V U ∧ V has codimension 3 in V U ∧ V , we have A ∩ ( V U ∧ V ) = 0.This means ( U , V ) ∈ b Z A . Since V U ∧ V ⊂ V V , this contradicts A ∩ V V = 0. (cid:3) The quadric fibrations.
In [DK1, Section 4], we defined two quadric fibrations associ-ated with a smooth GM variety X of dimension n . The first quadric fibration is the map ρ : P X ( U X ) → P ( V )induced by the tautological embedding U X ֒ → V ⊗ O X . It is flat over the complement of theunion Y ≥ n − A,V ∪ Σ ( X ), where Σ ( X ) is the kernel locus (3) Σ ( X ) := pr Y, (pr − Y, ( p X )) ⊂ Y A,V ⊂ P ( V ) . If A has no decomposable vectors, the map pr Y, : pr − Y, ( p X ) → Y A,V is a closed embedding([DK1, Proposition B.3]). So, if p X ∈ Y ℓA ⊥ , the variety Σ ( X ) is isomorphic to P ℓ − embeddedvia the second Veronese embedding.The fibers of ρ can be described as follows. Lemma 2.4 ([DK1, Proposition 4.5]) . Let X be a smooth GM variety of dimension n ≥ ,with associated Lagrangian data ( V , V , A ) . For every v ∈ P ( V ) , we have (a) if v ∈ Y ℓA,V r Σ ( X ) , the fiber ρ − ( v ) is a quadric in P n − of corank ℓ ; (b) if v ∈ Y ℓA,V ∩ Σ ( X ) , the fiber ρ − ( v ) is a quadric in P n − of corank ℓ − . O. DEBARRE AND A. KUZNETSOV
Since the corank of a quadric does not exceed the linear dimension of its span, we have ℓ ≤ n − v / ∈ Σ ( X ), and ℓ ≤ n + 1 for v ∈ Σ ( X ). This implies Y A,V ⊂ Σ ( X ) for n = 3.The second quadric fibration is the map ρ : P X ( V / U X ) → Gr (3 , V )induced by the natural embedding ( V / U X ) ⊗ V U X ֒ → V V ⊗ O X . It is flat over the com-plement of the union Z ≥ n − A,V ∪ Σ ( X ), where Σ ( X ) is the isotropic locus (4) Σ ( X ) := pr Z, (pr − Z, ( p X )) ⊂ Z A,V ⊂ Gr (3 , V ) . By Lemma 2.3, we have Z A,V ⊂ Σ ( X ).In contrast with the case of the kernel locus, the map pr Z, : pr − Z, ( p X ) → Z A,V is nolonger an embedding: its fiber over a point U is the projective space P ( A ∩ ( V U ∧ V )) andwe setΣ ≥ k ( X ) := { U ∈ Σ ( X ) | dim( A ∩ ( V U ∧ V )) ≥ k } and Σ k ( X ) := Σ ≥ k ( X ) r Σ ≥ k +12 ( X ) , so that Σ ( X ) = Σ ≥ ( X ). Note that Σ ≥ ( X ) is empty if A has no decomposable vectors.The fibers of ρ can be described as follows. Lemma 2.5 ([DK1, Proposition 4.10]) . Let X be a smooth GM variety of dimension n ≥ ,with associated Lagrangian data ( V , V , A ) . For every U ∈ Gr (3 , V ) , we have (a) if U ∈ Z ℓA,V r Σ ( X ) , the fiber ρ − ( U ) is a quadric in P n − of corank ℓ ; (b) if U ∈ Z ℓA,V ∩ Σ ( X ) , the fiber ρ − ( U ) is a quadric in P n − of corank ℓ − ; (c) if U ∈ Z ℓA,V ∩ Σ ( X ) , the fiber ρ − ( U ) is a quadric in P n − of corank ℓ − . Lemmas 2.5 and 2.6 will be essential for the descriptions of the schemes of linear spacescontained in GM varieties.
Lemma 2.6.
Let A ⊂ V V be a Lagrangian subspace with no decomposable vectors, let V ⊂ V be a hyperplane, and let X = X ord ( V , V , A ) be the corresponding ordinary GM variety, ofdimension n := 5 − dim( A ∩ V V ) . If n ≥ , (a) Y ≥ A,V is a curve which is smooth if and only if Y A,V = ∅ and the Pl¨ucker point p X doesnot lie on the projective dual variety of Y ≥ A ; (b) Y A,V is a normal integral threefold and Sing( Y A,V ) = Y ≥ A,V ∪ Σ ( X ) , Sing( e Y A,V ) = Sing( Y ≥ A,V ) ∪ f − A,V (Σ ( X )) . Proof. (a) The integral surface Y ≥ A is not contained in a hyperplane ([DK1, Lemma B.6]),hence its hyperplane section Y ≥ A,V is a curve. The statement about smoothness follows fromthe definition of projective duality.(b) Since Y A is an integral sextic, we have dim( Y A,V ) = 3. If a point P ∈ Y A,V r Y ≥ A,V is singular, the tangent space to Y A at P coincides with P ( V ). Therefore, ( P, p X ) ∈ b Y A ,hence P ∈ Σ ( X ) = pr Y, (pr − Y, ( p X )). On the other hand, all points of Y ≥ A,V are singularon Y A,V , since Y ≥ A = Sing( Y A ). This gives the required description of Sing( Y A,V ).Over Y A,V , the map f A,V is ´etale, hence the singular locus of e Y A,V over Y A,V is equalto f − A,V (Σ ( X )). On the other hand, one checks that along the ramification locus f − A,V ( Y ≥ A,V ), USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 9 the double sextic e Y A,V is smooth if and only if Y ≥ A,V is. This gives the required descriptionof Sing( e Y A,V ).Finally, Y A,V is normal and integral because it is a hypersurface in P with 1-dimensionalsingular locus ( Y ≥ A,V has dimension 1 by part (a) and dim(Σ ( X )) = (5 − n ) − − n ≤ n ≥ (cid:3) Cohomology of smooth GM varieties
Hodge numbers.
Recall that the Hodge diamond of Gr (2 , V ) is
10 00 1 00 0 0 00 0 2 0 00 0 0 0 0 00 0 0 2 0 0 00 0 0 0 0 00 0 2 0 00 0 0 00 1 00 01
The abelian group H • ( Gr (2 , V ); Z ) is free with basis the Schubert classes σ i,j ∈ H i + j ) ( Gr (2 , V ) , Z ) , ≥ i ≥ j ≥ . We write σ i for σ i, ; thus σ is the hyperplane class, σ , = c ( U ), and σ i = c i ( V / U ), where U is the tautological rank-2 subbundle and V / U the tautological rank-3 quotient bundle.We compute the Hodge numbers of smooth GM varieties. Proposition 3.1.
The Hodge diamond of a smooth complex GM variety of dimension n is ( n = 1) ( n = 2) ( n = 3) ( n = 4) ( n = 5) ( n = 6)
16 61 10 01 20 10 01 10 00 1 00 10 10 00 1 00 01 10 00 1 00 0 0 00 1 22 1 00 0 0 00 1 00 01 10 00 1 00 0 0 00 0 2 0 00 0 10 10 0 00 0 2 0 00 0 0 00 1 00 01 10 00 1 00 0 0 00 0 2 0 00 0 0 0 0 00 0 1 22 1 0 00 0 0 0 0 00 0 2 0 00 0 0 00 1 00 01
Proof.
When n = 1, the Hodge numbers are those of a curve of genus 6. When n = 2, theHodge numbers are those of a K3 surface.Assume 3 ≤ n ≤
5. Since the Hodge numbers of smooth complex varieties are deformationinvariant, we may assume that the GM variety X is ordinary. It is then a smooth dimensionallytransverse intersection of (ample) hypersurfaces in G := Gr (2 , V ) and the Lefschetz HyperplaneTheorem (see Lemma 3.2) implies that the Hodge numbers of X of degree < n are those of G .Moreover, h n, ( X ) = 0 because X is a Fano variety.When n = 3, the missing Hodge number h , ( X ) was computed in [L]. When n = 4, theHodge diamond was computed in [IM, Lemma 4.1]. When n = 6, it was computed in [DK2,Corollary 4.4]. When n = 5, the missing Hodge numbers h , ( X ) and h , ( X ) were obtained byNagel using a computer (see the introduction of [N]). We now present our own computation. To compute h , ( X ), we assume that X is an ordinary fivefold. Consider the exact se-quences 0 → O X ( − → Ω G | X → Ω X → → Ω G ( − → Ω G → Ω G | X → . The sheaf Ω G ( −
2) is acyclic (by Bott’s theorem) and so is O X ( −
2) (by Kodaira vanishing),hence h i ( X, Ω X ) = h i ( X, Ω G | X ) = h i ( G, Ω G ) and h ,i ( X ) = h ,i ( G ). In particular, we obtain h , ( X ) = 0.To compute h , ( X ), we assume that X is a special fivefold, i.e., is a double covering ofa smooth hyperplane section M ′ X of G branched along a smooth GM fourfold X ′ . Using thisdouble covering, we compute Euler characteristics χ top ( X ) = 2 χ top ( M ′ X ) − χ top ( X ′ ) . Since X ′ is a GM fourfold, we have χ top ( X ′ ) = 1 + 1 + 24 + 1 + 1 = 28. On the other hand,the inclusion M ′ X ⊂ G induces isomorphisms H k ( G ; Z ) ≃ H k ( M ′ X ; Z ) for all k ∈ { , . . . , } . Inparticular, χ top ( M ′ X ) = 8, hence χ top ( X ) = −
12. Since χ top ( X ) = 1+1+2 − h , ( X )+2+1+1,we obtain h , ( X ) = 10. This finishes the proof of the proposition. (cid:3) Integral cohomology.
We now prove that the integral cohomology groups of smoothGM varieties are torsion-free. We start with a classical lemma.
Lemma 3.2 (Lefschetz) . Let X be a dimensionally transverse intersection of dimension n ofample hypersurfaces in a smooth projective variety M . (a) The induced map H k ( M ; Z ) ∼ → H k ( X ; Z ) is bijective for k < n , injective for k = n . (b) The induced map H k ( X ; Z ) ∼ → H k ( M ; Z ) is bijective for k < n , surjective for k = n . (c) If X is moreover smooth and H • ( M ; Z ) is torsion-free, so is H • ( X ; Z ) .Proof. Parts (a) and (b) are the Lefschetz Hyperplane Theorem and follow from the fact that M r X is the union of dim( M ) − n smooth affine open subsets ([Di, Chapter 5, Theorem (2.6)]).For (c), since X is smooth, the Poincar´e duality isomorphisms H k ( X ; Z ) ≃ H n − k ( X ; Z )and (a) together with (b) imply that the integral homology and cohomology groups of X aretorsion-free in all degrees except perhaps n . By the Universal Coefficient Theorem, the torsionsubgroup of H n ( X ; Z ) is isomorphic to the torsion subgroup of H n − ( X ; Z ), which is 0 by (b),hence all integral cohomology groups of X are torsion-free. (cid:3) A similar result holds for cyclic covers (this is the main theorem of [C]; see also theremarks at the very end of the article).
Lemma 3.3.
Let γ : X → M be a cyclic cover between smooth projective varieties of dimen-sion n whose branch locus is a smooth ample divisor on M . (a) The induced map γ ∗ : H k ( M ; Z ) ∼ → H k ( X ; Z ) is bijective for k < n , injective for k = n . (b) The induced map γ ∗ : H k ( X ; Z ) ∼ → H k ( M ; Z ) is bijective for k < n , surjective for k = n . (c) If H • ( M ; Z ) is torsion-free, so is H • ( X ; Z ) . We now describe the integral cohomology groups of smooth GM varieties.
Proposition 3.4.
Let X be a smooth GM variety of dimension n . (a) The group H • ( X ; Z ) is torsion-free. USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 11 (b)
The map γ ∗ ,kX : H k ( Gr (2 , V ); Z ) → H k ( X ; Z ) is bijective for k < n and injective for k = n .Proof. When X is ordinary, it is a dimensionally transverse intersection of (ample) hypersur-faces in Gr (2 , V ), hence Lemma 3.2 implies both parts (a) and (b) of the proposition.When X is special, its Gushel map factors as γ X : X γ −−→ M ′ X ֒ → Gr (2 , V ), where γ is a double cover branched along an ample divisor, and M ′ X is a dimensionally transverseintersection of (ample) hypersurfaces in Gr (2 , V ). Both parts (a) and (b) are then consequencesof Lemma 3.2 (applied to M ′ X ⊂ Gr (2 , V )) and Lemma 3.3 (applied to the double cover γ ). (cid:3) Corollary 3.5.
Let X be a smooth GM variety of dimension n . If n ≥ , the degree of anyhypersurface in X is divisible by . If n ≥ , the degree of any subvariety of codimension in X is even.Proof. We use Proposition 3.4(b). Let Y ⊂ X be a subvariety of codimension c . If c = 1, theclass of Y in H ( X ; Z ) is a multiple of the class γ ∗ X σ , which has degree 10.If c = 2 (and n ≥ Y in H ( X ; Z ) is an integral combination of γ ∗ X σ ,which has degree 6, and γ ∗ X σ , , which has degree 4. The degree of Y is therefore even. (cid:3) We will need the following computation, which was already used in [DIM].
Lemma 3.6.
Let X be a smooth ordinary GM fourfold and let Q ⊂ X be its σ -quadric, i.e.,the intersection of X with the 3-space Π := P ( v ∧ V ) ⊂ M X , where v ∈ P ( V ) is the uniquepoint in the kernel locus Σ ( X ) defined by (3) . Then [ Q ] = γ ∗ X ( σ − σ , ) ∈ H ( X ; Z ) .Proof. Let γ M X be the inclusion M X ֒ → Gr (2 , V ). By the Lefschetz Theorem (Lemma 3.2), themap γ ∗ M X : H ( Gr (2 , V ); Z ) → H ( M X ; Z ) is an isomorphism. Therefore, there exist integers a and b such that [Π] = γ ∗ M X ( a σ + b σ , ), hence [ Q ] = γ ∗ X ( a σ + b σ , ). Since the class of Πin H ( Gr (2 , V ); Z ) is σ , Gysin’s formula and Schubert calculus give σ = γ M X ∗ ([Π]) = γ M X ∗ γ ∗ M X ( a σ + b σ , ) = ( a σ + b σ , ) · σ = a ( σ + σ , ) + b σ , in H ( Gr (2 , V ); Z ). This implies a = 1 and a + b = 0, hence the lemma. (cid:3) The following lemma is also useful; we keep the notation of Lemma 3.6.
Lemma 3.7.
Let X be a smooth ordinary GM fourfold. The restriction of the bundle U X tothe quadric Q splits as O Q ⊕ O Q ( − .Proof. Since Q ⊂ Π, it is enough to show U X | Π ≃ O Π ⊕ O Π ( − P ( v ∧ V )parameterizes all two-dimensional subspaces in V that contain v . Consequently, we have aninjection of vector bundles O Π ֒ → U X | Π given by the vector v . Its cokernel is a line bundleisomorphic to det( U X | Π ) ≃ O Π ( − → O Π → U X | Π → O Π ( − → . It remains to note that Ext ( O Π ( − , O Π ) = H (Π , O Π (1)) = 0 since Π ≃ P . (cid:3) Middle cohomology lattices of smooth GM varieties of dimension 4 or 6.
Let X be a smooth GM variety of even dimension n with Gushel map γ X : X → Gr (2 , V ). The abeliangroup H n ( X ; Z ) is torsion-free (Proposition 3.4) and, endowed with the intersection form, itis, by Poincar´e duality, a unimodular lattice. We set h := γ ∗ X σ ∈ H ( X ; Z ) and(5) H n ( X ; Z ) := { x ∈ H n ( X ; Z ) | x · h = 0 } ,H n ( X ; Z ) := { x ∈ H n ( X ; Z ) | x · γ ∗ X ( H n ( Gr (2 , V ); Z )) = 0 } . These sublattices of H n ( X ; Z ) are called the primitive and the vanishing lattices of X . Lemma 3.8.
For every n , we have an injection H n ( X ; Z ) ⊂ H n ( X ; Z ) , and for n = 2 and n = 6 , we have an equality H n ( X ; Z ) = H n ( X ; Z ) .Proof. Since h is pulled back from Gr (2 , V ), we have ( x · h ) · γ ∗ X ( H n − ( Gr (2 , V ); Z )) = 0 forevery x ∈ H n ( X ; Z ) . By Lemma 3.2, the map γ ∗ X : H n − ( Gr (2 , V ); Z ) → H n − ( X ; Z ) is anisomorphism, hence ( x · h ) · H n − ( X ; Z ) = 0. We conclude x · h = 0 by Poincar´e duality.Since H ( Gr (2 , V ); Z ) = Z σ , the definitions of H ( X ; Z ) and H ( X ; Z ) are the samefor n = 2. Furthermore, the product H ( Gr (2 , V ); Z ) · σ −−−→ H ( Gr (2 , V ); Z ) is an isomorphismby Schubert calculus, hence for n = 6, the definitions are equivalent. (cid:3) Given a Lagrangian subspace A ⊂ V V with no decomposable vectors and such that Y ≥ A = ∅ , the fourfold e Y A introduced in (2) is a hyperk¨ahler manifold which is a deformationof the symmetric square of a K3 surface ([O1, Theorem 1.1(2)]). In particular, the group H • ( e Y A ; Z ) is torsion-free ([M, Theorem 1]).We denote by ˜ h ∈ H ( e Y A ; Z ) the pullback by f A of the hyperplane class on Y A ⊂ P ( V )and define the primitive cohomology(6) H ( e Y A ; Z ) := { y ∈ H ( e Y A ; Z ) | y · ˜ h = 0 } . We consider H n ( X, Z ) and H ( e Y A , Z ) as polarized Hodge structures via the intersectionpairing on the first and the Beauville–Bogomolov quadratic form q B on the second. Recallthat q B can be defined by ([B, Theorem 5(c)])(7) ∀ y ∈ H ( e Y A ; Z ) q B ( y ) = y · ˜ h . This form makes H ( e Y A ; Z ) into a lattice of rank 22.Given a lattice L and a non-zero integer m , we denote by L ( m ) the lattice L with thebilinear form multiplied by m . The discriminant of L is the finite abelian group D ( L ) := L ∨ /L. As usual, we denote by • I the odd lattice Z with intersection form (1), • I r,s the odd lattice I ⊕ r ⊕ I ( − ⊕ s , • U the even hyperbolic lattice Z with intersection form (cid:0) (cid:1) , • E the unique positive definite, even, unimodular lattice of rank 8.The following three lattices are important in this article(8) Γ := I , , Γ := E ( − ⊕ ⊕ U ⊕ , Λ := E ⊕ ⊕ U ⊕ ⊕ I , (2) . Proposition 3.9.
Let X be a smooth GM variety of dimension n = 4 or . There are isomor-phisms of lattices H n ( X ; Z ) ≃ Γ n and H n ( X ; Z ) ≃ Λ(( − n/ ) . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 13
Proof.
When n = 4, the lattices H ( X ; Z ) and H ( X ; Z ) are described in [DIM, Proposi-tion 5.1] (although the proof that these groups are torsion-free is missing).When n = 6, the class c ( X ) = 4 γ ∗ X σ is divisible by 2, hence the Stiefel–Whitneyclass w ( X ), which is its image in H ( X ; Z / Z ), vanishes. Since w ( X ) = 0 (as for any complexcompact manifold) and we are in dimension 6, the (unimodular) lattice H ( X ; Z ) is even ([HBJ,p. 115]). Since its signature is (4 ,
20) by Proposition 3.1, it is therefore isomorphic to Γ .The intersection form on the sublattice γ ∗ X ( H ( Gr (2 , V ); Z )) ⊂ H ( X ; Z ) has matrix (cid:0) (cid:1) in the Schubert basis ( γ ∗ X σ , , γ ∗ X σ ). This sublattice is moreover primitive: if not, itssaturation is unimodular, even, and positive definite of rank 2, which is absurd. By [Ni, Propo-sition 1.6.1], the discriminant group of its orthogonal H ( X ; Z ) is therefore isomorphic tothe discriminant group of γ ∗ X ( H ( Gr (2 , V ); Z )), which is ( Z / Z ) . The lattice H ( X ; Z ) ismoreover even and has signature (2 , − (cid:3) As a lattice, H ( e Y A , Z ) is also isomorphic to Λ( −
1) ([O4, (4.1.3)]). In Section 5, we willshow that the polarized Hodge structures on H n ( X, Z ) and H ( e Y A , Z ) are isomorphic (up toa twist). The isomorphism will be given by a correspondence constructed in the next section.4. Linear spaces on Gushel–Mukai varieties
Linear spaces and their types.
Let X be a smooth GM variety with its canonicalembedding X ⊂ P ( W ). We let F k ( X ) be the Hilbert scheme which parameterizes linearlyembedded P k in X , i.e., the closed subscheme of Gr ( k + 1 , W ) of linear subspaces W k +1 ⊂ W such that P ( W k +1 ) ⊂ X .The composition of the Gushel map γ X : X → Gr (2 , V ) with the Pl¨ucker embedding Gr (2 , V ) ⊂ P ( V V ) is induced by the linear projection W ⊂ C ⊕ V V → V V from thevertex ν of the cone CGr (2 , V ). Since ν / ∈ X , the Gushel map embeds P ( W k +1 ) linearlyinto Gr (2 , V ).We recall the description of linear subspaces contained in Gr (2 , V ). Any such subspacesits in a maximal linear subspace and there are two types of those. First, for every 1-dimensionalsubspace U ⊂ V , there is a projective 3-space P ( V /U ) ≃ P ( U ∧ V ) ⊂ Gr (2 , V ) . Second, for every 3-dimensional vector subspace U ⊂ V , there is a projective plane P ( V U ) ≃ Gr (2 , U ) ⊂ Gr (2 , V ) . We will say that a linear subspace P ⊂ Gr (2 , V ) is • a σ -space if it is contained in P ( U ∧ V ) for some U ⊂ V ; • a τ -space if it is contained in P ( V U ) for some U ⊂ V ; • a mixed space if it is both a σ - and a τ -space.In Gr (2 , V ), there are no projective 4-spaces and every projective 3-space is a σ -space. Forany distinct U ′ , U ′′ ⊂ V , the intersection P ( U ′ ∧ V ) ∩ P ( U ′′ ∧ V ) is the point [ U ′ ∧ U ′′ ]. Hence,for every projective 3-space P ⊂ Gr (2 , V ), there is a unique U ⊂ V such that P = P ( U ∧ V ).This defines a map σ : F ( X ) → P ( V ) . If U U , we have P ( U ∧ V ) ∩ P ( V U ) = ∅ . If instead U ⊂ U , the intersection P ( U ∧ V ) ∩ P ( V U ) = P ( U ∧ U ) is a line. Therefore, projective planes in Gr (2 , V ) are neverof mixed type and we have a decomposition into connected components F ( X ) = F σ ( X ) ⊔ F τ ( X ) , where F σ ( X ) is the subscheme of σ -planes and F τ ( X ) the subscheme of τ -planes. Again, thereis a map σ : F σ ( X ) → P ( V )taking a σ -plane P to the unique U ⊂ V such that P ⊂ P ( U ∧ V ). Analogously, for anydistinct subspaces U ′ , U ′′ ⊂ V , the intersection Gr (2 , U ′ ) ∩ Gr (2 , U ′′ ) = Gr (2 , U ′ ∩ U ′′ ) is eitherempty (if dim( U ′ ∩ U ′′ ) = 1), or a point (if dim( U ′ ∩ U ′′ ) = 2). Therefore, for any τ -plane P ⊂ Gr (2 , V ), there is a unique subspace U ⊂ V such that P = P ( V U ). This defines a map τ : F τ ( X ) → Gr (3 , V ) . Finally, any line on Gr (2 , V ) is a mixed space and there are maps σ : F ( X ) → P ( V ) and τ : F ( X ) → Gr (3 , V ) . The following proposition is crucial for our study of the schemes F k ( X ). It describes F k ( X )in terms of the relative Hilbert schemes Hilb P k which parameterize linearly embedded P k inthe fibers of the first and second quadratic fibrations (defined in Section 2.3). Proposition 4.1.
Let X be a smooth GM variety of dimension n ≥ , with associated La-grangian data ( V , V , A ) . The maps σ : F ( X ) → P ( V ) , σ : F σ ( X ) → P ( V ) , σ : F ( X ) → P ( V ) lift to isomorphisms with the following relative Hilbert schemes for the first quadric fibration F ( X ) ≃ Hilb P ( P X ( U X ) / P ( V )) ,F σ ( X ) ≃ Hilb P ( P X ( U X ) / P ( V )) ,F ( X ) ≃ Hilb P ( P X ( U X ) / P ( V )) . Analogously, the maps τ : F ( X ) → Gr (3 , V ) and τ : F τ ( X ) → Gr (3 , V ) lift to isomorphismswith the following relative Hilbert schemes for the second quadric fibration F ( X ) ≃ Hilb P ( P X ( V / U X ) / Gr (3 , V )) ,F τ ( X ) ≃ Hilb P ( P X ( V / U X ) / Gr (3 , V )) . Proof.
Let L σk ( X ) ⊂ X × F σk ( X ) be the universal family of k -dimensional σ -spaces. The map σ : F σk ( X ) → P ( V ) induces a map L σk ( X ) → X × P ( V ) that takes a pair ( x, P ), where P ⊂ X is a σ -space of dimension k and x ∈ P is a point, to ( x, σ ( P )). By definition of a σ -space, if σ ( P ) = U ⊂ V , the space P parameterizes 2-dimensional subspaces U ⊂ V suchthat U ⊂ U . Therefore, U is contained in the 2-space corresponding to the point x . In otherwords, ( x, σ ( P )) ∈ P X ( U X ). This means that we have a commutative diagram(9) X L σk ( X ) q o o p / / (cid:15) (cid:15) ✤✤✤ F σk ( X ) σ (cid:15) (cid:15) X P X ( U X ) π o o ρ / / P ( V ) . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 15
Thus, the fibers of L σk ( X ) over F σk ( X ) embed into the fibers of ρ and this embedding is linearon the fibers. This means that the map σ lifts to a map F σk ( X ) → Hilb P k ( P X ( U X ) / P ( V )).Similarly, the projection via π : P X ( U X ) → X of any P k contained in the fiber of ρ isa P k contained in X . This defines a map Hilb P k ( P X ( U X ) / P ( V )) → F σk ( X ).It is straightforward to see that the maps we constructed are mutually inverse and thusgive the isomorphisms of the first part of the proposition.The second part is proved analogously. (cid:3) In the next sections, we use this proposition and Lemmas 2.4 and 2.5 to describe theHilbert schemes F k ( X ).4.2. Projective 3-spaces on GM varieties.
As we already mentioned, smooth GM varietiescontain no linear spaces of dimension 4 and higher. The situation with projective 3-spaces isalso quite simple.
Theorem 4.2.
Let X be a smooth GM variety of dimension n , with associated Lagrangiandata ( V , V , A ) . If n ≤ , we have F ( X ) = ∅ . If n = 6 , there is an ´etale double covering F ( X ) → Y A,V ; in particular, F ( X ) is finite and is empty for X general.Proof. By Proposition 4.1, F ( X ) is the Hilbert scheme of projective 3-spaces in the fibers ofthe first quadric fibration ρ : P X ( U X ) → P ( V ). On the other hand, by Lemma 2.4, the fiber Q v = ρ − ( v ) over a point v ∈ P ( V ) is either a quadric in P n − if v / ∈ Σ ( X ), or a quadricin P n − if v ∈ Σ ( X ). Such a quadric contains a P only in the following cases: • n = 6, v / ∈ Σ ( X ), and Q v ⊂ P is a quadric of corank ≥ • n = 6, v ∈ Σ ( X ), and Q v ⊂ P is a quadric of corank ≥ • n = 5, v / ∈ Σ ( X ), and Q v ⊂ P is a quadric of corank ≥ • n = 5, v ∈ Σ ( X ), and Q v ⊂ P is a quadric of corank ≥ • n = 4, v ∈ Σ ( X ), and Q v ⊂ P is a quadric of corank ≥ Y A = ∅ (we could also invokeCorollary 3.5), and neither does the second case, because Σ ( X ) = ∅ for n = 6.In the first case, we have v ∈ Y ≥ A,V . Since Y A = ∅ , the quadric Q v has rank 2, henceis a union of two distinct 3-spaces, and the Hilbert scheme of 3-spaces in Q v consists of tworeduced points, hence is smooth. Therefore, the map F ( X ) → Y A,V is an ´etale double cover.The finiteness of F ( X ) and its emptyness for general X follow from the same propertiesof Y A . (cid:3) Planes on GM varieties.
Similar arguments provide descriptions of the schemes ofplanes. We start with σ -planes and consider the map σ : F σ ( X ) → P ( V ). Theorem 4.3.
Let X be a smooth GM variety of dimension n , with associated Lagrangiandata ( V , V , A ) . (a) If n = 6 and Y A,V = ∅ , the map σ factors as F σ ( X ) ˜ σ −−→ e Y A,V f A,V −−−−→ Y A,V ֒ −→ P ( V ) , where ˜ σ is a P -bundle. If n = 6 and Y A,V = ∅ , the scheme F σ ( X ) has one component isomorphic to a gener-ically P -fibration over e Y A,V and, for each point of Y A,V , one pair of irreducible componentsisomorphic to P .In particular, F σ ( X ) is smooth if and only if the Pl¨ucker point p X lies away from theprojective dual ( Y ≥ A ) ∨ ⊂ P ( V ∨ ) and Y A,V = ∅ . (b) If n = 5 and X is ordinary, or special with p X / ∈ pr Y, ( E ) , the map σ factors as F σ ( X ) ∼ −→ e Y ≥ A,V −→ Y ≥ A,V ֒ −→ P ( V ) , where e Y ≥ A,V → Y ≥ A,V is a double covering of the curve Y ≥ A,V branched along Y A,V .If n = 5 and X is special with p X ∈ pr Y, ( E ) , the scheme F σ ( X ) is the union of a doublecover e Y ≥ A,V and one double or two reduced components ( depending on whether the kernel point Σ ( X ) is in Y A,V or in Y A,V ) isomorphic to P and contracted by the map σ onto Σ ( X ) . (c) If n = 4 , the map σ factors as F σ ( X ) ˜ σ −−→ Y A,V ֒ −→ P ( V ) , where ˜ σ is an isomorphism over Y A,V r Σ ( X ) and a double cover over Y A,V ∩ Σ ( X ) . Inparticular, F σ ( X ) is finite and is empty if and only if Y A,V = ∅ . (d) If n ≤ , we have F σ ( X ) = ∅ .Remark . The double cover e Y ≥ A,V → Y ≥ A,V appearing in part (b) of the theorem is describedin Proposition A.2. We expect it to be the base change to P ( V ) of a natural double coverof Y ≥ A branched along Y A and analogous to O’Grady’s double cover f A : e Y A → Y A . Proof of the theorem.
In each case, by Proposition 4.1, the scheme F σ ( X ) is isomorphic to therelative Hilbert scheme of planes in the fibers of the first quadric fibration ρ : P X ( U X ) → P ( V ). The rest follows from the description of its fibers Q v := ρ − ( v ) in Lemma 2.4.(a) We assume n = 6. The locus Σ ( X ) is then empty hence, for any v ∈ P ( V ), the fiber Q v is a quadric in P . If it is non-degenerate, it contains no planes hence the map σ factorsthrough Y A,V .If the corank of Q v is 1, there are two families of planes on Q v , each parameterized by P .Hence, over Y A,V , the map σ factors as a P -fibration followed by a double covering.If the corank of Q v is 2, there is only one family of planes on Q v parameterized by P .Over Y A,V , the map σ is therefore a P -fibration.Finally, if the corank of Q v is 3, we have Q v = P ∪ P (intersecting along a plane), henceplanes on Q v are parameterized by P ∪ P (dual spaces, intersecting in a point). It followsthat F σ ( X ) has two irreducible components isomorphic to P over each point of Y A,V and acomponent that dominates Y A,V . Considering the Stein factorization of the map σ restrictedto this (main) component of F σ ( X ), we see that it is the composition of a P -bundle (awayof the preimage of Y A,V ) and a double cover of Y A,V branched along Y ≥ A,V . This P -bundle is´etale locally trivial and its Brauer class is given by the sheaf of even parts of Clifford algebras([K, Lemma 4.2]).To show that this double cover is isomorphic to e Y A,V (the base change to P ( V ) of thedouble EPW sextic), we compute, using Proposition A.2, the pushforward of O F σ ( X ) to P ( V ). USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 17
We have S = P ( V ), m = 5, E = O S ⊕ O S ( − ⊗ ( V / O S ( − ≃ O S ⊕ T S ( −
2) (where T S isthe tangent bundle), and L = O S . By Lemma 2.4, the degeneracy loci of the first quadraticfibration are D c = Y ≥ cA,V . Moreover, by [DK1, Proposition 4.5 and Lemma C.6], the cokernelsheaf C for the family of quadrics is isomorphic to Coker( O ( − ⊗ V ( V / O ( − → A ∨ ⊗ O S ),hence the double cover can be written asSpec( O Y A,V ⊕ C ( − . But this is precisely the base change to P ( V ) of the double EPW sextic ([O1, Section 4]).The statement about smoothness follows from the above description and Lemma 2.6(b).(b) We assume n = 5. If v / ∈ Σ ( X ), the fiber Q v is a quadric in P . It contains aplane if and only if its corank is at least 2. Therefore, we have two planes over each pointof Y A,V r Σ ( X ) and one double plane over each point of Y A,V r Σ ( X ). On the other hand,if v ∈ Σ ( X ), the fiber Q v is a quadric in P . It contains planes if and only if it is degenerateand these planes are then parameterized by P ⊔ P if the corank is 1 (i.e., if v ∈ Y A,V ) or bya double P if the corank is 2 (i.e., if v ∈ Y A,V ).It follows that if X is ordinary (hence Σ ( X ) = ∅ ) or if X is special and p X / ∈ pr Y, ( E )(so that, by (3), the kernel point Σ ( X ) is not on Y ≥ A,V ), there is a double cover F σ ( X ) → Y ≥ A,V branched along Y A,V , while if X is special and p X ∈ pr Y, ( E ), we have extra component(s)in F σ ( X ) as described in the statement of the theorem.(c) We assume n = 4. If v / ∈ Σ ( X ), the fiber Q v is a quadric in P . It contains a plane(and is then equal to it) if and only if its corank is 3. Hence F σ ( X ) contains Y A,V r Σ ( X ).On the other hand, if v ∈ Σ ( X ), the fiber Q v is a quadric in P . It contains a plane if andonly if its corank is 2 (and then it contains two planes). Hence F σ ( X ) contains two points foreach point of Y A,V ∩ Σ ( X ). We conclude using the fact that Y A is finite (Section 2.2).Statement (d) follows from Corollary 3.5. (cid:3) Using the second quadric fibration, we describe the scheme F τ ( X ). Theorem 4.5.
Let X be a smooth GM variety of dimension n , with associated Lagrangiandata ( V , V , A ) . (a) If n = 6 , the map τ factors as F τ ( X ) ˜ τ −−→ Z ≥ A,V ֒ −→ Gr (3 , V ) , where ˜ τ is a double covering branched along Z ≥ A,V . (b) If n = 5 , the map τ : F τ ( X ) → Gr (3 , V ) factors as F τ ( X ) ˜ τ −−→ Z ≥ A,V ֒ −→ Gr (3 , V ) , where ˜ τ is an isomorphism over Z ≥ A,V r Σ ( X ) and a double cover over Z ≥ A,V ∩ Σ ( X ) , branchedalong Z A,V . (c) If n = 4 , the map τ factors as F τ ( X ) ˜ τ −−→ Z A,V ֒ −→ Gr (3 , V ) , where ˜ τ is ´etale, is an isomorphism over Z A,V r Σ ( X ) , and is a double cover over Z A,V ∩ Σ ( X ) . In particular, F τ ( X ) is finite and it is empty if and only if Z A,V = ∅ . (d) If n ≤ , we have F τ ( X ) = ∅ .Remark . The double cover F τ ( X ) → Z ≥ A,V which appears in part (a) of the theorem isdescribed in Proposition A.2. We expect it to be the base change to Gr (3 , V ) of the doublecover e Z ≥ A → Z ≥ A constructed for general A in [IKKR]. Proof of the theorem.
In all cases, by Proposition 4.1, the scheme F τ ( X ) is isomorphic to therelative Hilbert scheme of planes in the fibers of the second quadric fibration ρ : P X ( V / U X ) → Gr (3 , V ). The rest follows from the description of its fibers Q U := ρ − ( U ) in Lemma 2.5.(a) We assume n = 6. We have Σ ( X ) = ∅ and, by Lemma 2.3, Z A,V = ∅ . For any point U ∈ Gr (3 , V ), the fiber Q U is a quadric in P . It contains a plane only if its corank is atleast 2; therefore, the map τ factors through Z ≥ A,V ⊂ Gr (3 , V ). The quadric Q U is the unionof two planes if U ∈ Z A,V and a double plane if U ∈ Z A,V , so the map τ : F τ ( X ) → Z ≥ A,V isa double cover branched along Z A,V .(b) We assume n = 5. If U / ∈ Σ ( X ), the fiber Q U is a quadric in P . It contains aplane (and is then equal to it) only if its corank is 3, hence the map τ factors through Z ≥ A,V and is an isomorphism over Z ≥ A,V r Σ ( X ). On the other hand, if U ∈ Σ ( X ), the fiber Q U is a quadric in P and it contains a plane only when the quadric has corank at least 2, hencewhen U ∈ Z ≥ A,V . More precisely, if U ∈ Z A,V ∩ Σ ( X ), the quadric Q U is the union of twoplanes and the fiber of τ is two points; if U ∈ Z A,V (by Lemma 2.3 and (4), it is then alsoautomatically in Σ ( X )), the quadric Q U is a double plane and the fiber of τ is a point. Thisproves the required statement.(c) We assume n = 4. If U / ∈ Σ ( X ), the fiber Q U is a quadric in P and so nevercontains a plane. If U ∈ Σ ( X ), the fiber Q U is a quadric in P , so it contains a plane (andis then equal to it) if and only if its corank is 3. This gives one point of F τ ( X ) over each pointof Z A,V ∩ Σ ( X ). If U ∈ Σ ( X ), the fiber Q U is a quadric in P , so it contains a plane if andonly if its corank is at least 2. This gives two points of F τ ( X ) over each point of Z A,V ∩ Σ ( X ).We conclude using the fact that Z A is finite (Section 2.2).Statement (d) follows from Corollary 3.5. (cid:3) Lines on GM varieties.
We now consider the scheme F ( X ) of lines contained in X . Theorem 4.7.
Let X be a smooth GM variety of dimension n , with associated Lagrangiandata ( V , V , A ) . (a) If n = 6 , the map σ : F ( X ) → P ( V ) is dominant with general fiber isomorphic to P . (b) If n = 5 , the map σ : F ( X ) → P ( V ) factors as F ( X ) ˜ σ −−→ ^ P ( V ) −→ P ( V ) , where ^ P ( V ) → P ( V ) is the double cover branched along the sextic hypersurface Y A,V ⊂ P ( V ) and ˜ σ is a P -bundle over the complement of the preimage of Y ≥ A,V ∪ Σ ( X ) . (c) If n = 4 , the map σ factors as F ( X ) ˜ σ −−→ e Y A,V f A,V −−−−→ Y A,V ֒ −→ P ( V ) , USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 19 where ˜ σ is birational. The non-trivial fibers of ˜ σ are P over each point of Y A,V r Σ ( X ) , P ∪ P over each point of Y A,V ∩ Σ ( X ) , and P over each point of f − A,V (Σ ( X ) r Y A,V ) . (d) If n = 3 , the map σ : F ( X ) → P ( V ) factors as F ( X ) ˜ σ −−→ Y ≥ A,V ֒ −→ P ( V ) , where ˜ σ is an isomorphism over Y ≥ A,V r Σ ( X ) and a double cover over Y ≥ A,V ∩ Σ ( X ) , branchedalong Y A,V ∩ Σ ( X ) . In Proposition 5.3, we will make part (c) more precise by showing that for an ordinaryGM fourfold X , the scheme F ( X ) is a small resolution of e Y A,V , under some explicit generalityassumptions. Proof.
In all cases, by Proposition 4.1, the scheme F ( X ) is isomorphic to the relative Hilbertscheme of lines in the fibers of the first quadric fibration ρ : P X ( U X ) → P ( V ). We now checkthat the rest follows from the description of its fibers Q v := ρ − ( v ) in Lemma 2.4.(a) We assume n = 6. We have Σ ( X ) = ∅ and all fibers of ρ are quadrics in P .Any such quadric contain a line, hence the map σ : F ( X ) → P ( V ) is dominant. Moreover, if v ∈ P ( V ) r Y A,V , the quadric ρ − ( v ) is smooth, hence lines on it are parameterized by P .(b) We assume n = 5. If v / ∈ Σ ( X ), the fiber Q v is a quadric in P . If v / ∈ Y A,V , thequadric Q v is non-degenerate, hence the family of lines on Q v is parameterized by the unionof two P . If v ∈ Y A,V the quadric Q v has corank 1 and lines on Q v are parametrized by asingle P . Therefore, the Stein factorization of the map σ : F ( X ) → P ( V ) is a compositionof a generically P -bundle with a double cover of P ( V ) branched along Y A,V , as claimed. TheBrauer class of this P -bundle is again given by the sheaf of even parts of Clifford algebras.(c) We assume n = 4. If v / ∈ Σ ( X ), the fiber Q v is a conic in P . If v / ∈ Y A,V , theconic Q v is non-degenerate, hence contains no lines. Therefore, the map σ factors through Y A,V .If v ∈ Y A,V , the conic Q v is the union of two lines, hence the map σ is ´etale of degree 2over Y A,V r Σ ( X ). If v ∈ Y A,V , the conic Q v is a double line, hence the above double coveris branched along Y A,V r Σ ( X ). Finally, if v ∈ Y A,V , the fiber Q v is the whole plane, hencelines on Q v are parameterized by the dual plane.To show that this double cover is isomorphic to e Y A,V , we compute, using Proposition A.1,the pushforward of O F ( X ) to P ( V ). We have S = P ( V ) r Σ ( X ), m = 3, L = O S , and thebundle E fits into exact sequence0 → E → O S ⊕ T S ( − → O ⊕ S → . (where T S is the tangent bundle). By Lemma 2.4, the degeneracy loci of the first quadraticfibration are D c = Y ≥ cA,V r Σ ( X ). Moreover, by [DK1, Proposition 4.5 and Lemma C.6], thecokernel sheaf C for the family of quadrics is isomorphic to Coker( O ( − ⊗ V ( V / O ( − → A ∨ ⊗ O S ), hence again the double cover can be written as Spec( O Y A,V r Σ ( X ) ⊕ C ( − P ( V ) r Σ ( X ) of the definition of the double EPW sextic (see [O1,Section 4]).If v ∈ Σ ( X ), the fiber Q v is a quadric in P . If v ∈ Y A,X , the quadric Q v is non-degenerateand lines on Q v are parameterized by P ⊔ P ; over each of the two points of f − A,V ( v ), thefiber of σ is P . If v ∈ Y A,X , the quadric has corank 1 and lines on Q v are parameterized by P . Finally, if v ∈ Y A,X , the quadric Q v has corank 2, Q v = P ∪ P , and lines on Q v areparameterized by P ∪ P .(d) We assume n = 3. If v / ∈ Σ ( X ), the fiber Q v is a quadric in P . It contains no linesunless its corank is 2 (in which case it is itself a P ). Thus, the map σ factors through Y ≥ A,V andis an isomorphism over Y A,V r Σ ( X ). If v ∈ Σ ( X ), the fiber Q v is a conic in P . If v ∈ Y A,V ,it is a union of two lines, hence the map ˜ σ : F ( X ) → Y ≥ A,V is of degree 2 over Y A,V ∩ Σ ( X ).Finally, if v ∈ Y A,V (it is then automatically in Σ ( X ); see the discussion after Lemma 2.4),the fiber Q v is a double line, hence the map ˜ σ is branched over this locus. (cid:3) Regarding items (a) and (b) in the theorem, it is possible to describe the fibers of σ over Y A,V (resp. over the preimage of Y ≥ A,V ∪ Σ ( X )). We leave this as an exercise for theinterested reader. It is also possible to describe the scheme F ( X ) by using the map τ .Finally, one can use a similar approach to describe the Hilbert schemes of quadrics inGM varieties, and even the Hilbert schemes parameterizing cubic subvarieties (twisted cubiccurves, cubic scrolls, and so on), but the description becomes more and more involved.5. Periods of GM varieties
In this section, we relate the periods of GM varieties of dimension 4 or 6 to those of theirassociated EPW sextic. The following theorem is the main result of this article.
Theorem 5.1.
Let X be a smooth GM variety of dimension n = 4 or , with associatedLagrangian data ( V , V , A ) . Assume that the double EPW sextic e Y A is smooth ( i.e., Y ≥ A = ∅ ) .There is an isomorphism of polarized Hodge structures H n ( X ; Z ) ≃ H ( e Y A ; Z ) (( − n/ − ) , where ( − is the Tate twist. In Sections 5.1 and 5.2, we prove Theorem 5.1 for X general of dimension 4 (Theo-rem 5.12) or 6 (Theorem 5.19). In Section 5.4, we define period points and maps and use themto deduce Theorem 5.1 in full generality.5.1. Periods of GM fourfolds.
Our aim in this section is to prove Theorem 5.1 for a smoothGM fourfold X , with associated Lagrangian data ( V , V , A ). We will construct an explicitisomorphism when X is general. More precisely, we assume(10) p X / ∈ ( Y ≥ A ) ∨ and Y ≥ A = ∅ . Note that Y ≥ A ⊥ ⊂ ( Y ≥ A ) ∨ by Lemma 2.1, hence for a GM fourfold X satisfying (10), we have p X ∈ Y A ⊥ , i.e., X is ordinary. We will use this observation further on.Note also that ( Y ≥ A ) ∨ does not contain Y A ⊥ , since Y ∨ A ⊥ = Y A is not equal to Y ≥ A ; therefore,the choice of p X satisfying (10) is possible. By Lemma 2.6(a), assumption (10) implies(11) Y ≥ A,V is a smooth curve and Y ≥ A,V = ∅ . Since X is ordinary, Σ ( X ) is a point, which we denote by v . Moreover, p X / ∈ pr Y, ( E ) byLemma 2.1, hence we have, by (3),(12) v ∈ Y A,V . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 21
We have Sing( Y A,V ) = { v } ∪ Y ≥ A,V by Lemma 2.6(b). Furthermore, (12) and Lemma 2.4(b)imply that(13) Q := π ( ρ − ( v ))is a smooth quadric surface contained in X , with span Π := P ( v ∧ V ) ⊂ M X .The Hilbert scheme F ( X ) of lines on X was described in Theorem 4.7(c). Under ourgenerality assumption, this description takes the following simple form. Corollary 5.2.
Under assumption (10) , the map ˜ σ : F ( X ) → e Y A,V is an isomorphism overthe complement of the two points of f − A,V ( v ) and the fiber of ˜ σ over each of these two pointsis P . We now prove that the threefold F ( X ) is smooth. Proposition 5.3.
Let X be a smooth ordinary GM fourfold, with associated Lagrangian data ( V , V , A ) , such that (10) holds. The map ˜ σ : F ( X ) → e Y A,V is then a small resolution ofsingularities of e Y A,V . In particular, F ( X ) is a smooth irreducible threefold.Proof. By Lemma 2.6(b) and (11), we have Sing( e Y A,V ) = f − A,V ( v ). Since ˜ σ is an isomorphismover the complement of f − A,V ( v ), it remains to show that F ( X ) is smooth along ˜ σ − ( f − A,V ( v )).In other words, we have to show that F ( X ) is smooth at points corresponding to lines L ⊂ X such that σ ([ L ]) = v . By deformation theory, it is enough to prove H ( L, N L/X ) = 0 for anyof these lines.By definition of the map σ , a line L with σ ([ L ]) = v lies on the 2-dimensional quadric Q = ρ − ( v ), which is smooth by Lemma 2.4. Therefore, there is an exact sequence0 → N L/Q → N L/X → N Q /X | L → . The first term is O L since Q is a smooth quadric. It is enough to show that the last term iseither O L ⊕ O L or O L (1) ⊕ O L ( − Q is the transversal intersection of Π and a quadriccutting out X in M X , we have N Q /X ≃ N Π /M X | Q . On the other hand, since M X is a hyperplane section of Gr (2 , V ), we have an exact sequence0 → N Π /M X → N Π / Gr (2 ,V ) → O Π (1) → . Finally, one easily proves the isomorphism N Π / Gr (2 ,V ) ≃ T Π ( − → N Q /X | L → T Π ( − | L → O L (1) → . Since Π ≃ P , the middle term is O L (1) ⊕ O L ⊕ O L . It follows that N Q /X | L is either O L ⊕ O L or O L (1) ⊕ O L ( − (cid:3) Under our assumptions, the set f − A,V ( v ) consists of two points, which we denote by p ′ and p ′′ . We also denote by P ′ := ˜ σ − ( p ′ ) ⊂ F ( X ) and P ′′ := ˜ σ − ( p ′′ ) ⊂ F ( X )the non-trivial fibers of the map ˜ σ : F ( X ) → e Y A,V (each of them is isomorphic to P ). Remark . The involution τ A of the double cover f A : e Y A → Y A restricts to the involutionof the double cover f A,V : e Y A,V → Y A,V . However, it does not extend to a regular involutionof F ( X ): the small resolutions of the two singular points p ′ and p ′′ of e Y A,V are not compatiblewith this involution.Denote by ι : e Y A,V → e Y A the canonical embedding. Proposition 5.5.
Let X be a smooth ordinary GM fourfold, with associated Lagrangian data ( V , V , A ) , such that (10) holds. The restriction ι ∗ : H ( e Y A ; Z ) → H ( e Y A,V ; Z ) is an isomor-phism and the composition H ( e Y A ; Z ) ι ∗ −−→ H ( e Y A,V ; Z ) ˜ σ ∗ −−→ H ( F ( X ); Z ) induces an isomorphism of Hodge structures between H ( e Y A ; Z ) and h P ′ , P ′′ i ⊥ ⊂ H ( F ( X ); Z ) .Proof. Since (10) holds, e Y A is smooth, hence ι ∗ is an isomorphism by the Lefschetz Theorem(Lemma 3.2). Set U := e Y A,V r { p ′ , p ′′ } ≃ F ( X ) r ( P ′ ∪ P ′′ ). We have a commutative diagram · · · / / H ( { p ′ , p ′′ } ; Z ) / / ˜ σ ∗ (cid:15) (cid:15) H c ( U ; Z ) / / H ( e Y A,V ; Z ) / / ˜ σ ∗ (cid:15) (cid:15) H ( { p ′ , p ′′ } ; Z ) ˜ σ ∗ (cid:15) (cid:15) / / H c ( U ; Z ) / / · · ·· · · / / H ( P ′ ∪ P ′′ ; Z ) / / H c ( U ; Z ) / / H ( F ( X ); Z ) / / H ( P ′ ∪ P ′′ ; Z ) / / H c ( U ; Z ) / / · · · of exact sequences in cohomology with compact supports. The first column is zero and thefourth column is the inclusion 0 → Z ⊕ Z . The third column therefore extends to an exactsequence(14) 0 / / H ( e Y A,V ; Z ) ˜ σ ∗ / / H ( F ( X ); Z ) r / / Z ⊕ Z , where r ( x ) := ( x · [ P ′ ] , x · [ P ′′ ]). This proves the proposition. (cid:3) Let p : L ( X ) → F ( X ) be the universal line and let q : L ( X ) → X be the naturalmorphism. These two morphisms define a correspondence between X and F ( X ) hence a mapbetween H • ( X ; Z ) and H • ( F ( X ); Z ) which we investigate. We extend diagram (9) to a com-mutative diagram(15) L ( X ) q u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ p , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ q ′ { { ✇ ✇ ✇ ✇ ✇ q ′′ ' ' X P X ( U X ) π o o ρ (cid:15) (cid:15) P X ( U X ) × P ( V ) Y A,V (cid:15) (cid:15) ? _ o o F ( X ) ˜ σ (cid:15) (cid:15) P ( V ) Y A,V ? _ o o e Y A,V f A,V o o (cid:31) (cid:127) ι / / e Y A , where the map q ′ is defined in the same way as the dashed arrow in (9) and the map q ′′ isconstructed by the universal property of the fiber product.Recall that h stands for the hyperplane class of X and ˜ h for the hyperplane class of P ( V )and its restrictions to P ( V ) and e Y A . Lemma 5.6.
The map q ′′ is finite and birational, and q ′∗ ([ L ( X )]) = 6 ρ ∗ ˜ h in H ( P X ( U X ); Z ) .In particular, the map q is generically finite of degree . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 23
Proof.
Let ( x, v ) ∈ P X ( U X ) × P ( V ) Y A,V ⊂ P X ( U X ) ⊂ X × P ( V ). If γ X ( x ) = [ U ], this meansthat v is in P ( U ) ∩ Y A,V and that ( q ′′ ) − ( x, v ) is the Hilbert scheme of lines in ρ − ( v ) passingthrough x . But ρ − ( v ) is either a reducible conic, or a double line if v = v , or the quadric Q defined in (13) if v = v . Therefore, there is a unique line through x , unless x is a singularpoint of a singular conic or v = v , in which case there are two lines through x . Thus q ′′ isfinite and birational.It follows that the pushforward q ′∗ ([ L ( X )]) ∈ H ( P X ( U X ); Z ) is the class of the divisor P X ( U X ) × P ( V ) Y A,V , hence the pullback via ρ of the class of Y A,V , which is equal to 6˜ h . (cid:3) Geometrically, this means that for a general point x of an ordinary GM fourfold X , thereare 6 lines passing through x and contained in X . Corollary 5.7.
One has q ∗ p ∗ ˜ σ ∗ ι ∗ ˜ h = 6 c ( V / U X ) = 6 γ ∗ X σ .Proof. We need to compute the pullback of ˜ h to L ( X ) and its pushforward via q to X .We can rewrite this as π ∗ q ′∗ q ′∗ ρ ∗ ˜ h . By the projection formula and Lemma 5.6, this is equal to π ∗ ( ρ ∗ ˜ h · ρ ∗ ˜ h ) = 6 π ∗ ρ ∗ ˜ h . Since π is the projectivization of U X and ρ ∗ ˜ h is a relative hyperplaneclass, we have(16) ρ ∗ ˜ h + c ( U X ) ρ ∗ ˜ h + c ( U X ) = 0 . Therefore, ρ ∗ ˜ h = ( c ( U X ) − c ( U X )) ρ ∗ ˜ h + c ( U X ) c ( U X ) and π ∗ ρ ∗ ˜ h = c ( U X ) − c ( U X ) = c ( V / U X ) . The corollary follows. (cid:3)
Lemma 5.8.
We have q ∗ p ∗ ( P ′ ) = q ∗ p ∗ ( P ′′ ) = [ Q ] ∈ H ( X ; Z ) .Proof. The subscheme P ′ ⊔ P ′′ ⊂ F ( X ) parameterizes lines on X that are contained in thesmooth quadric surface Q . The lines in each of the components P ′ and P ′′ sweep out Q once,hence the claim. (cid:3) Let X be a smooth GM fourfold. Consider the morphism(17) α : H ( X ; Z ) −→ H ( F ( X ); Z ) , x p ∗ ( q ∗ x ) . The classes (see Lemma 3.6)(18) c ( U X ) = γ ∗ X σ , and [ Q ] = γ ∗ X ( σ − σ , ) = h − c ( U X )in H ( X ; Z ) generate the subgroup γ ∗ X ( H ( Gr (2 , V ); Z )). In the following two lemmas, wecompute their images by the map α . We assume as before that X satisfies the assumptions (10). Lemma 5.9.
We have α ( c ( U X )) = ˜ σ ∗ ι ∗ ˜ h .Proof. Consider the diagram (15). The pullback of the bundle U X to P X ( U X ) is an extension0 → O ( − ρ ∗ ˜ h ) → π ∗ U X → O ( ρ ∗ ˜ h − π ∗ h ) → π ∗ c ( U X ) = ρ ∗ ˜ h ( π ∗ h − ρ ∗ ˜ h ) . Therefore, we have q ∗ c ( U X ) = q ′∗ π ∗ c ( U X ) = q ′∗ ( ρ ∗ ˜ h ( π ∗ h − ρ ∗ ˜ h )) = p ∗ ˜ σ ∗ ι ∗ ˜ h · q ∗ h − p ∗ ˜ σ ∗ ι ∗ ˜ h and, since p is a P -bundle with relative hyperplane class q ∗ h , the pushforward by p of theright side equals ˜ σ ∗ ι ∗ ˜ h . (cid:3) Recall that the surface Q = X ∩ P ( v ∧ ( V /v )) defined in (13) is a smooth quadric. Tocompute the class α ([ Q ]) in H ( F ( X ); Z ), we need some preparation.First, the Hilbert scheme of lines on Q is F ( Q ) = P ′ ⊔ P ′′ and the corresponding uni-versal line is L ( Q ) = Q ′ ⊔ Q ′′ , where the first (resp. second) component corresponds to linesparameterized by P ′ (resp. P ′′ ) and the map L ( Q ) ⊂ L ( X ) q −→ X induces isomorphisms Q ′ ≃ Q and Q ′′ ≃ Q .Second, we have U Q := U X | Q ≃ O Q ⊕ O Q ( −
1) by Lemma 3.7, hence ρ ( P Q ( U Q )) ⊂ P ( V ) is the quadratic cone C v Q over Q ⊂ P ( V /v ) with vertex v . Set S := Y A,V ∩ C v Q ⊂ P ( V ) . This is an intersection of two distinct irreducible hypersurfaces in P ( V ) (Lemma 2.6(b)) hencea Cohen–Macaulay surface containing v . Lemma 5.10.
There is a surface R ⊂ F ( X ) such that α ([ Q ]) = [ R ] ∈ H ( F ( X ); Z ) and the map σ = f A ◦ ˜ σ : R → P ( V ) is birational onto S .Proof. We first describe q − ( Q ). Since L ( X ) is smooth of dimension 4 (Proposition 5.3) and q is dominant (Lemma 5.6), q − ( Q ) has everywhere dimension ≥
2. We see on the diagram (15)that the map q : L ( X ) → X factors through the map q ′′ : L ( X ) → P X ( U X ) × P ( V ) Y A,V ;moreover, we have q − ( Q ) = q ′′− ( P Q ( U Q ) × P ( V ) Y A,V ) . The situation is summarized in the following cartesian diagram Q _(cid:127) (cid:15) (cid:15) P Q ( U Q ) π o o _(cid:127) (cid:15) (cid:15) P Q ( U Q ) × P ( V ) Y A,V ? _ o o _(cid:127) (cid:15) (cid:15) q − ( Q ) q ′′ o o _(cid:127) (cid:15) (cid:15) X P X ( U X ) π o o P X ( U X ) × P ( V ) Y A,V ? _ o o L ( X ) . q ′′ o o The projection P Q ( U Q ) × P ( V ) Y A,V → P ( V ) factors through S and is an isomorphism overthe dense open subscheme S := S r { v } . Let ˜ S ⊂ P Q ( U Q ) × P ( V ) Y A,V be the preimageof S and set R := q ′′− ( ˜ S ) ⊂ q − ( Q ) ⊂ L ( X ) . Let ˜ S ⊂ ˜ S be the open subscheme of ˜ S parameterizing pairs ( x, v ) such that v = v and x isnot a singular point of the conic ρ − ( v ). Let S be its isomorphic image in S ⊂ S and let R be its preimage in R . By Lemma 5.6, the map q ′′ | R : R → ˜ S induces an isomorphism ofschemes R ∼ → ˜ S . Thus the map σ ◦ p : R → P ( V ) induces an isomorphism R ∼ → S ; sincewe show below that S is dense in S , it follows that R has pure dimension 2.Consider now R := R r R . By definition, its image in S is contained in S := S r S .We show below dim( S ) ≤
1, hence S is dense in S and, since the map R → S is finite(since the Hilbert scheme of lines in any conic is finite), we also have dim( R ) ≤ q − ( Q ) r R = q − ( Q ) ∩ p − ( σ − ( v )). Since σ − ( v ) = F ( Q ), we have q − ( Q ) r R = L ( Q ) × X Q = Q ′ ⊔ Q ′′ . All this shows that q − ( Q ) has pure dimension 2and that we can write q ∗ ([ Q ]) = a ′ [ Q ′ ] + a ′′ [ Q ′′ ] + [ R ] USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 25 for some integers a ′ and a ′′ . The map p contracts the surfaces Q ′ and Q ′′ onto the curves P ′ and P ′′ respectively, hence p ∗ kills their classes and we obtain α ([ Q ]) = p ∗ ( q ∗ ([ Q ])) = a ′ p ∗ ([ Q ′ ]) + a ′′ p ∗ ([ Q ′′ ]) + p ∗ ([ R ]) = p ∗ ([ R ]) . Recall that the map σ ◦ p induces an isomorphism R ∼ → S . Setting R := p ( R ) ⊂ F ( X ) , we obtain α ([ Q ]) = [ R ] and the map σ : R → P ( V ) is birational onto S = S .It remains to prove dim( S ) ≤
1. Let E ⊂ W ⊗ O S be the rank-3 vector bundle over S with fiber at a point v of S the linear span of the conic ρ − ( v ) (see the proof of Theorem 4.7(c)for its description). Consider the line subbundle L ֒ → E whose fiber over a point v ∈ S whichis the image of a point ( x, v ) ∈ ˜ S is given by the point x . By definition of S , this is a singularpoint of the conic, hence the quadratic forms on E induces a quadratic form on the quotientbundle E / L . Its discriminant locus is a divisor in S which, by construction, is contained inthe corank-2 locus of ρ .Assume by contradiction dim( S ) ≥
2. The dimension of this locus is then ≥
1. Since,away of v , it coincides with the smooth irreducible curve Y A,V , this curve is contained in S ,hence in the quadric C v Q . This contradicts Corollary B.6. (cid:3) Corollary 5.11.
One has ι ∗ ˜ σ ∗ α ( h ) = 3˜ h .Proof. By (18), we have ι ∗ ˜ σ ∗ α ( h ) = ι ∗ ˜ σ ∗ α ([ Q ]) + 2 ι ∗ ˜ σ ∗ α ( c ( U X )) = ι ∗ ˜ σ ∗ ([ R ]) + 2 ι ∗ ˜ σ ∗ ˜ σ ∗ ι ∗ ˜ h (we use Lemma 5.9 and Lemma 5.10 in the last equality). By the projection formula, the secondsummand equals 2˜ h , so it remains to show that the first summand ι ∗ ˜ σ ∗ ([ R ]) equals ˜ h . Since F ( X ) × P ( V ) S is birational to the double cover e Y A,V × P ( V ) S of S and contains a surface R that maps to S birationally, we have F ( X ) × P ( V ) S = R ∪ τ A ( R ) , where τ A is the birational involution of F ( X ) induced by the involution of e Y A . Since S is cutout in Y A,V by a quadric, we have ι ∗ ˜ σ ∗ ([ R ]) + ι ∗ ˜ σ ∗ ([ τ A ( R )]) = 2 ι ∗ ι ∗ ˜ h = 2˜ h . The two summands on the left side are interchanged by the involution τ A of the double cover f A : e Y A → Y A . We would like to show that they are equal. For that, let us first assume that X ,hence also A , is very general. The vector space H , ( e Y A ) ∩ H ( e Y A ; Q ) then has rank 2, generatedby ˜ h and c ( e Y A ) ([O5, Proposition 3.2]) and both of these classes are τ A -invariant. Since ι ∗ ˜ σ ∗ ([ R ]) and ι ∗ ˜ σ ∗ ([ τ A ( R )]) both belong to this space, they are also τ A -invariant, hence equal.Finally, since H ( e Y A ; Z ) is torsion-free ([M, Theorem 1]), they are both equal to ˜ h .Going back to the general case where X only satisfies (10), we note that the class ι ∗ ˜ σ ∗ ([ R ]) − ˜ h ∈ H ( e Y A ; Z ) depends continuously on X and is zero for X very general, aswe showed above; therefore, it is zero for all X . (cid:3) We are now ready to prove Theorem 5.1 for general GM fourfolds. Recall that in (15),the map p : L ( X ) → F ( X ) is a P -fibration for which q ∗ h is a relative hyperplane class.Therefore, L ( X ) is isomorphic to the projectivization of a rank-2 vector bundle. We denoteby c and c its Chern classes, so that(19) q ∗ h + p ∗ c · q ∗ h + p ∗ c = 0 . The primitive and vanishing cohomology subgroups H ( X ; Z ) ⊂ H ( X ; Z ) ⊂ H ( X ; Z ) and H ( e Y A ; Z ) ⊂ H ( e Y A ; Z ) were defined in (5) and (6), and the map α was defined in (17). Theorem 5.12.
Let X be a smooth ordinary GM fourfold, with associated Lagrangian data ( V , V , A ) , such that assumption (10) holds. We have (20) ∀ x ∈ H ( X ; Z ) α ( x ) · α ( h ) = − x . In particular, the restriction α : H ( X ; Z ) → H ( F ( X ); Z ) of α is injective. Furthermore, itinduces an isomorphism β : H ( X ; Z ) ∼ −→ H ( e Y A ; Z ) compatible with the Beauville–Bogomolov form ∀ x ∈ H ( X ; Z ) q B ( β ( x )) = − x , hence an isomorphism H ( X ; Z ) ( − ∼ → H ( e Y A ; Z ) of polarized Hodge structures.Proof. We follow the argument from [BD]. Since p is a P -bundle and q ∗ h a relative hyperplaneclass, we may write ∀ x ∈ H ( X ; Z ) q ∗ x = p ∗ x · q ∗ h + p ∗ x , where x i ∈ H i ( F ( X ); Z ) for i ∈ { , } . We have then α ( x ) = x . To see what primitivity of x means, we compute, using (19), q ∗ ( x · h ) = p ∗ x · q ∗ h + p ∗ x · q ∗ h = p ∗ ( x − x · c ) · q ∗ h − p ∗ ( x · c ) . Thus x · h = 0 implies x · c = 0 , x = x · c , and we can rewrite q ∗ x = p ∗ ( α ( x )) · ( q ∗ h + p ∗ c ) . Taking squares and using (19), we obtain q ∗ x = p ∗ ( α ( x ) ) · ( q ∗ h + 2 p ∗ c · q ∗ h + p ∗ c ) = p ∗ ( α ( x ) ) · ( p ∗ c · q ∗ h + p ∗ ( c − c )) = α ( x ) · c . On the other hand, since, by Lemma 5.6, the degree of q is 6, we have q ∗ x = 6 x . We obtain α ( h ) = − c from (19) and all this proves (20).This relation implies α ( x ) α ( x ) α ( h ) = − x x for all x , x ∈ H ( X ; Z ) and theinjectivity of α follows from the non-degeneracy of the intersection pairing on H ( X ; Z ) .If x ∈ H ( X ; Z ) , we have, by adjunction and Lemma 5.8, α ( x ) · [ P ′ ] = x · q ∗ p ∗ [ P ′ ] = x · [ Q ] = 0because, by (18), the class [ Q ] is in γ ∗ X H ( Gr (2 , V ); Z ). Similarly, we have α ( x ) · [ P ′′ ] = 0.From (14), we obtain that α maps H ( X ; Z ) into the subgroup ˜ σ ∗ ι ∗ H ( e Y A ; Z ) of H ( F ( X ); Z ).By Proposition 5.5, this defines an injective map β : H ( X ; Z ) → H ( e Y A ; Z ) such that ∀ x ∈ H ( X ; Z ) α ( x ) = ˜ σ ∗ ( ι ∗ ( β ( x ))) . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 27
It remains to show that the image of β is the primitive cohomology H ( e Y A ; Z ) and that β iscompatible with the Beauville–Bogomolov form. Keeping the assumption x ∈ H ( X ; Z ) andusing Corollary 5.7, we have0 = x · c ( V / U X ) = x · q ∗ p ∗ ˜ σ ∗ ι ∗ (˜ h ) = α ( x ) · ˜ σ ∗ ι ∗ (˜ h ) == ˜ σ ∗ ι ∗ ( β ( x )) · ˜ σ ∗ ι ∗ (˜ h ) = ι ∗ ( β ( x )) · ι ∗ (˜ h ) = β ( x ) · ι ∗ ι ∗ (˜ h ) = β ( x ) · ˜ h . This proves β ( x ) ∈ H ( e Y A ; Z ) by the definition (6) of the primitive cohomology group.For the compatibility with the Beauville–Bogomolov form q B , we observe − x = 16 α ( x ) α ( h ) = 16 ˜ σ ∗ ι ∗ β ( x ) α ( h ) = 16 β ( x ) · ι ∗ ˜ σ ∗ α ( h ) = 16 β ( x ) · h = q B ( β ( x )) . The first equality is (20), the second is the definition of β , the third follows from adjunction,the fourth is Corollary 5.11, and the last is (7).Finally, the lattices H ( X ; Z ) and H ( e Y A ; Z ) both have rank 22 and same discriminantgroup ( Z / Z ) ([DIM, Proposition 5.1] or Proposition 3.9 for H ( X ; Z ) , and [O2, (1.0.9)]for H ( e Y A ; Z ) ), hence the injective anti-isometry β is a bijection. (cid:3) Periods of GM sixfolds.
Our aim in this section is to prove Theorem 5.1 for a smoothGM sixfold X , with associated Lagrangian data ( V , V , A ). Again, we will provide an explicitisomorphism for general X , namely for those satisfying the same assumption (10)—whichimplies (11). Since X is a sixfold, Σ ( X ) is empty and, in contrast with the fourfold case,Lemma 2.6(b) says that Y A,V is smooth away from Y ≥ A,V , and e Y A,V is a smooth threefold.The scheme of σ -planes on X was described in Theorem 4.3(a). Under our generalityassumption, this description takes the following simple form. Corollary 5.13.
Let X be a smooth GM sixfold, with associated Lagrangian data ( V , V , A ) ,such that assumption (10) holds. Then, F σ ( X ) is a smooth fourfold and the map ˜ σ : F σ ( X ) → e Y A,V is a P -fibration. We denote by P y ≃ P the fiber of ˜ σ over a point y ∈ e Y A,V . These fibers all have the samecohomology class which we denote by [ P ] ∈ H ( F σ ( X ); Z ). As before, we let ι : e Y A,V → e Y A bethe canonical embedding. Proposition 5.14.
Let X be a smooth GM sixfold, with associated Lagrangian data ( V , V , A ) ,such that (10) holds. The composition H ( e Y A ; Z ) ι ∗ −−→ H ( e Y A,V ; Z ) ˜ σ ∗ −−→ H ( F σ ( X ); Z ) induces an isomorphism of Hodge structures between H ( e Y A ; Z ) and [ P ] ⊥ ⊂ H ( F σ ( X ); Z ) .Proof. The map ι ∗ is an isomorphism by Lemma 3.2. We then use the fact that ˜ σ is a P -fibration with fiber class [ P ]. (cid:3) Let p : L σ ( X ) → F σ ( X ) be the universal σ -plane and let q : L σ ( X ) → X be the naturalmorphism. The analogue of the diagram (15) is(21) L σ ( X ) q t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ p , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ q ′ z z ✉ ✉ ✉ ✉ ✉ q ′′ ( ( X P X ( U X ) π o o ρ (cid:15) (cid:15) P X ( U X ) × P ( V ) Y A,V (cid:15) (cid:15) ? _ o o F σ ( X ) ˜ σ (cid:15) (cid:15) P ( V ) Y A,V ? _ o o e Y A,V f A,V o o (cid:31) (cid:127) ι / / e Y A , where ρ is a fibration in 3-dimensional quadrics, and the dashed and dotted arrows are con-structed in the same way. Lemma 5.15.
The map q ′′ is generically finite of degree and q ′∗ ([ L σ ( X )]) = 12 ρ ∗ ˜ h . Inparticular, the map q is generically finite of degree .Proof. Take a point ( x, v ) ∈ P X ( U X ) × P ( V ) Y A,V ⊂ P X ( U X ) ⊂ X × P ( V ). If γ X ( x ) = [ U ],we have v ∈ P ( U ) ∩ Y A,V and ( q ′′ ) − ( x, v ) is the Hilbert scheme of planes in ρ − ( v ) passingthrough x . But ρ − ( v ) is either a cone over P × P or a cone over a conic with vertex a line.In the first case, there is a unique plane of each type through x , unless x is the vertex of thecone, and in the second case, there is a unique plane (with multiplicity 2) through any pointof the cone not lying on the vertex. Thus, q ′′ is generically finite of degree 2.It follows that the class q ′∗ ([ L σ ( X )]) ∈ H ( P X ( U X ); Z ) is twice the class of the fiberproduct P X ( U X ) × P ( V ) Y A,V , hence twice the pullback via ρ of the class of Y A,V , which isequal to 6 ρ ∗ ˜ h . (cid:3) Geometrically, this means that for a general point x of a GM sixfold X , there are 12planes passing through x and contained in X .For every v ∈ P ( V ), set Q v := π ( ρ − ( v )) = γ − X ( P ( v ∧ V )) ⊂ X. This is a 3-dimensional quadric.
Lemma 5.16.
Let y ∈ e Y A,V and set v := f A,V ( y ) ∈ Y A,V ⊂ P ( V ) . We have q ∗ p ∗ ([ P ]) =[ Q v ] = γ ∗ X σ ∈ H ( X ; Z ) .Proof. By the proof of Theorem 4.3, the line P y parameterizes planes (of one of the two possibletypes) on the singular quadric Q v . Since these planes cover the whole quadric and there is aunique such plane through any smooth point of Q v , the map q : p − ( P y ) → Q v is birational,hence q ∗ p ∗ ([ P ]) = [ Q v ]. By definition of Q v , we have [ Q v ] = γ ∗ X ([ P ( v ∧ V )]) = γ ∗ X σ . (cid:3) Corollary 5.17.
For any u ∈ P ( V ) , we have p ∗ q ∗ [ Q u ] · [ P ] = 2 .Proof. By adjunction, it is enough to prove [ Q u ] · [ Q v ] = 2 for any distinct u, v ∈ P ( V ).Since the quadrics are preimages of the spaces P ( u ∧ V ) and P ( v ∧ V ) under the doublecovering γ X : X → Gr (2 , V ), it is enough to show that the intersection of those spaces is 1,i.e., that σ = 1, which follows from Schubert calculus. (cid:3) Lemma 5.18.
The class q ∗ p ∗ ( p ∗ q ∗ [ Q u ] · ˜ σ ∗ ι ∗ ˜ h ) ∈ H ( X ; Z ) is contained in the subgroup γ ∗ X ( H ( Gr (2 , V ); Z )) of H ( X ; Z ) . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 29
Proof.
Using diagram (21), we can rewrite the class in question as q ∗ p ∗ ( p ∗ q ∗ [ Q u ] · ˜ σ ∗ ι ∗ ˜ h ) = π ∗ ( q ′∗ ( p ∗ p ∗ q ∗ [ Q u ]) · ρ ∗ ˜ h ) . The class q ′∗ ( p ∗ p ∗ q ∗ [ Q u ]) belongs to H ( P X ( U X ); Z ) ≃ H ( X ; Z ) ⊕ H ( X ; Z ) · ρ ∗ ˜ h , hence, byProposition 3.4(b), it can be written as a linear combination of the classes π ∗ h , π ∗ c ( U X ), and π ∗ h · ρ ∗ ˜ h . It is thus enough to show that each of these classes multiplied by ρ ∗ ˜ h and pushedforward to X is in the required subgroup. From (16) and the equality c ( U X ) = h , one easilyobtains π ∗ ( π ∗ h · ρ ∗ ˜ h ) = h , π ∗ ( π ∗ c ( U X ) · ρ ∗ ˜ h ) = h · c ( U X ), and π ∗ ( π ∗ h · ρ ∗ ˜ h ) = h − h · c ( U X ) . This proves the lemma. (cid:3)
Consider the morphism α : H ( X ; Z ) −→ H ( F σ ( X ); Z ) , x p ∗ ( q ∗ x ) . The map p is a P -fibration for which q ∗ h is a relative hyperplane class. Therefore, L σ ( X )is isomorphic to the projectivization of a rank-3 vector bundle. We denote by c , c , and c itsChern classes, so that(22) q ∗ h + p ∗ c · q ∗ h + p ∗ c · q ∗ h + p ∗ c = 0 . Multiplying by q ∗ h , one obtains(23) q ∗ h + p ∗ ( c − c ) · q ∗ h + p ∗ ( c − c c ) · q ∗ h − p ∗ ( c c ) = 0 . The primitive and vanishing cohomology subgroups H ( X ; Z ) = H ( X ; Z ) ⊂ H ( X ; Z )(they are equal by Lemma 3.8) and H ( e Y A ; Z ) ⊂ H ( e Y A ; Z ) were defined in (5) and (6). Theorem 5.19.
Let X be a smooth GM sixfold, with associated Lagrangian data ( V , V , A ) ,such that assumption (10) holds. We have (24) ∀ x ∈ H ( X ; Z ) α ( x ) · c = 12 x . In particular, the restriction α : H ( X ; Z ) → H ( F σ ( X ); Z ) is injective. Furthermore, itinduces an isomorphism β : H ( X ; Z ) ∼ −→ H ( e Y A ; Z ) compatible with the Beauville–Bogomolov form ∀ x ∈ H ( X ; Z ) q B ( β ( x )) = x , i.e., an isomorphism of polarized Hodge structures.Proof. We use the same argument as in the proof of Theorem 5.12. Since p is a P -bundle and q ∗ h is a relative hyperplane section, we may write ∀ x ∈ H ( X ; Z ) q ∗ x = p ∗ x · q ∗ h + p ∗ x · q ∗ h + p ∗ x , where x i ∈ H i ( F σ ( X ); Z ) for i ∈ { , , } . We have then α ( x ) = x . To see what the primitivityof x means, we compute, using (22), q ∗ ( x · h ) = p ∗ x · q ∗ h + p ∗ x · q ∗ h + p ∗ x · q ∗ h = p ∗ ( x − x · c ) · q ∗ h + p ∗ ( x − x · c ) · q ∗ h − p ∗ ( x · c ) . Thus, the condition x · h = 0 implies x · c = 0 , x = x · c , x = x · c , and we can rewrite q ∗ x = p ∗ ( α ( x )) · ( q ∗ h + p ∗ c · q ∗ h + p ∗ c ) . Taking squares and using (22) and (23), we obtain( q ∗ x ) = p ∗ ( α ( x ) ) · ( q ∗ h + 2 p ∗ c · q ∗ h + p ∗ ( c + 2 c ) · q ∗ h + 2 p ∗ ( c c ) · q ∗ h + p ∗ c )= p ∗ ( α ( x ) ) · ( p ∗ c · q ∗ h + p ∗ ( c c − c ) · q ∗ h + p ∗ ( c − c c ))= α ( x ) · c . On the other hand, by Lemma 5.15, we have ( q ∗ x ) = 12 x . This proves (24). The injectivityof α then follows as in the proof of Theorem 5.12.Since x ∈ H ( X ; Z ) , we have α ( x ) · [ P ] = x · q ∗ p ∗ [ P ] = x · [ Q v ] = x · γ ∗ X σ = 0 . A combination of this equality with Proposition 5.14 shows that there is an injective map β : H ( X ; Z ) → H ( e Y A ; Z ) such that α ( x ) = ˜ σ ∗ ( ι ∗ ( β ( x ))) . It remains to show that the image of β is in the primitive cohomology H ( e Y A ; Z ) and that β is compatible with the Beauville–Bogomolov form.Let x ∈ H ( X ; Z ) . Set m := β ( x ) · ˜ h = β ( x ) · ι ∗ ι ∗ ˜ h = ι ∗ β ( x ) · ι ∗ ˜ h . Then α ( x ) · ˜ σ ∗ ι ∗ ˜ h = ˜ σ ∗ ι ∗ β ( x ) · ˜ σ ∗ ι ∗ ˜ h = m [ P ] . Multiplying this by p ∗ q ∗ [ Q u ] and using Corollary 5.17, we obtain α ( x ) · ( p ∗ q ∗ [ Q u ] · ˜ σ ∗ ι ∗ ˜ h ) = 2 m. By adjunction, this is equal to x · q ∗ p ∗ ( p ∗ q ∗ [ Q u ] · ˜ σ ∗ ι ∗ ˜ h ) and by Lemma 5.18, the latter is zerosince x is in the vanishing cohomology; therefore, m = 0. This proves β ( x ) ∈ H ( e Y A ; Z ) .To show compatibility with the Beauville–Bogomolov form, we observe12 x = α ( x ) c = ˜ σ ∗ ι ∗ β ( x ) c = β ( x ) ι ∗ ˜ σ ∗ c . On the other hand, by Proposition 5.23 below, we have ι ∗ ˜ σ ∗ c = 6˜ h , hence, by (7), x = 112 β ( x ) h = 12 β ( x ) ˜ h = q B ( β ( x )) . Finally, the lattices H ( X ; Z ) and H ( e Y A ; Z ) have same rank 22 and same discriminantgroup ( Z / Z ) (Proposition 3.9 for H ( X ; Z ) and [O2, (1.0.9)] for H ( e Y A ; Z ) ), hence theinjective isometry β is a bijection. (cid:3) Nearby Lagrangians.
Our aim here is to prove Proposition 5.23, which was used in theproof of Theorem 5.19. This section was inspired by [F, Section 6].We start with some preparation. Let A and A be Lagrangian subspaces in a symplecticvector space V , such that the intersection B := A ∩ A has codimension 2 in both A and A . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 31
Lemma 5.20. If codim A ( B ) = codim A ( B ) = 2 , the Lagrangian subspaces A ⊂ V such that codim A ( A ∩ A ) = codim A ( A ∩ A ) ≤ are parameterized by the line P ( A /B ) ≃ P ( A /B ) ≃ P . Moreover, if A ′ , A ′′ ⊂ V are twodistinct such subspaces, A ′ ∩ A ′′ = B .Proof. Since B = A ∩ A , we have B ⊥ = A + A ⊂ V . The vector space B ⊥ /B is symplecticof dimension 4 and for any Lagrangian subspace A ⊂ V ,¯ A := ( A ∩ B ⊥ ) / ( A ∩ B )is a Lagrangian subspace in B ⊥ /B (called the B -isotropic reduction of A ). Since the subspaces¯ A i = A i /B do not intersect, they give a Lagrangian direct sum decomposition B ⊥ /B = ¯ A ⊕ ¯ A . Assume now codim A i ( A ∩ A i ) ≤
1. Note that A ∩ A = A ∩ A (since otherwise A ∩ A wouldbe at most 1-codimensional), hence A = ( A ∩ A ) + ( A ∩ A ). This implies A ⊂ A + A = B ⊥ ,hence B ⊂ A ⊥ = A . Thus ¯ A = A/B ⊂ B ⊥ /B and in particular, A is determined by the space¯ A as its preimage under the linear projection B ⊥ → B ⊥ /B .The conditions codim A i ( A ∩ A i ) ≤ P ( ¯ A ) intersects each skew line P ( ¯ A i ) in P ( B ⊥ /B ). Finally, the pairing between ¯ A and ¯ A induced by the symplectic formon B ⊥ /B is non-degenerate, hence for every point of P ( ¯ A ), there is a unique point in P ( ¯ A )such that the line joining them is Lagrangian. Thus, the set of ¯ A (and hence the set of A aswell) is parameterized by either of the lines P ( ¯ A ) or P ( ¯ A ).It follows from the above description that the lines P ( ¯ A ) form one connected componentof the scheme of lines on a smooth quadric in P ( B ⊥ /B ) (the lines P ( ¯ A ) and P ( ¯ A ) beingin the other component). In particular, two such distinct lines do not intersect, hence theirpreimages in P ( B ⊥ ) intersect along P ( B ). This proves the second statement. (cid:3) Assume now that the symplectic vector space V is V V . Let B ⊂ V V be an isotropicsubspace of dimension 8 (hence of codimension 2 in any Lagrangian containing it). Set Y B := { v ∈ P ( V ) | B ∩ ( v ∧ V V ) = 0 } and Y B,V := Y B ∩ P ( V ) . Remark . A parameter count shows dim( Y B ) ≤ B . In fact, this is even truefor a general B inside a given Lagrangian subspace which contains no decomposable vectors.Let A , A ⊂ V V be Lagrangian subspaces with no decomposable vectors such that B := A ∩ A has codimension 2 in each of them. Consider the family { A p } p ∈ P of Lagrangiansubspaces discussed in Lemma 5.20 and set Y ≥ A ,A := { ( v, p ) ∈ P ( V ) × P | v ∈ Y ≥ A p } ⊂ P ( V ) × P . We denote by pr : Y ≥ A ,A → P ( V ) the first projection and set Y ≥ A ,A ; V := Y ≥ A ,A × P ( V ) P ( V ). Lemma 5.22.
Let A , A ⊂ V V be Lagrangian subspaces with no decomposable vectors suchthat B = A ∩ A has codimension in each of them. If dim( Y B ) ≤ , we have Y A ∩ Y A = pr( Y ≥ A ,A ) and the map pr : Y ≥ A ,A → Y A ∩ Y A is an isomorphism over a dense open subset of Y A ∩ Y A .Moreover, for any V ⊂ V such that dim( Y B,V ) ≤ , the map pr : Y ≥ A ,A ; V → Y A ,V ∩ Y A ,V is again an isomorphism over a dense open subset. Proof.
Let us prove pr( Y ≥ A ,A ) ⊂ Y A ∩ Y A . If A p is any of the Lagrangian spaces in thefamily and v ∈ Y ≥ A p , we have dim( A p ∩ ( v ∧ V V )) ≥
2. But codim A i ( A p ∩ A i ) ≤
1, hence A i ∩ ( v ∧ V V ) = 0, so v ∈ Y A i both for i = 1 and i = 2.Since Y ≥ A ,A is proper, it remains to show that pr is an isomorphism over a dense opensubset. Since Y A and Y A are distinct hypersurfaces in P ( V ) = P , any irreducible componentof their intersection has dimension at least 3, so it is enough to show that the map pr : Y ≥ A ,A → Y A ∩ Y A is an isomorphism over the complement of Y B .Let v ∈ ( Y A ∩ Y A ) r Y B . We first show that v is a smooth point of Y A i . Assume to thecontrary v ∈ Y ≥ A . Then, dim( A ∩ ( v ∧ V V )) ≥ B ∩ ( v ∧ V V ) = 0. It follows that A = B ⊕ ( A ∩ ( v ∧ V V ))(and in particular, the second summand is 2-dimensional). On the other hand, take any non-zero a ∈ A ∩ ( v ∧ V V ). Then, a is orthogonal to both summands in the above equation (sincethe first summand is contained in A and the second in v ∧ V V ). Therefore, a ∈ A ⊥ = A ,hence a ∈ A ∩ A = B and v ∈ Y B , a contradiction. The same argument works for A insteadof A .We now know that both spaces A i ∩ ( v ∧ V V ) are one-dimensional. If a and a are gen-erators, their projections to B ⊥ /B are linearly independent (otherwise, v ∈ Y B ). Furthermore, A := B ⊕ h a , a i is a Lagrangian subspace in V V and its intersections with A and A are both 9-dimensional.Therefore, A = A p for some p ∈ P and, since h a , a i ⊂ A ∩ ( v ∧ V V ), we obtain ( v, p ) ∈ Y ≥ A ,A and v ∈ pr( Y ≥ A ,A ).Now let ( v, p ) ∈ Y ≥ A ,A with v / ∈ Y B . The space A p intersects v ∧ V V away from B ,hence, by the second part of Lemma 5.20, p is uniquely determined by v . This means that themap pr is an isomorphism over the complement of Y B .Finally, if a hyperplane V ⊂ V satisfies dim( Y B,V ) ≤
1, the subset ( Y A ,V ∩ Y A ,V ) r Y B,V is dense open in Y A ,V ∩ Y A ,V and the map pr is an isomorphism over it. (cid:3) Let now X be a smooth special GM sixfold such that (10) holds, with Lagrangian subspace A ⊂ V V and Pl¨ucker hyperplane V ⊂ V . Since A ∩ V V = 0, the canonical projection V V → V V / V V ≃ V V induces an isomorphism A ≃ V V .Recall that L σ ( X ) = P F σ ( X ) ( P ), where P is a rank-3 vector bundle on F σ ( X ) andthe map q in (21) is induced by an embedding of vector bundles P → ( C ⊕ V V ) ⊗ O F σ ( X ) .The composition of the above embedding with the projection to V V ⊗ O F σ ( X ) is still amonomorphism of vector bundles (since planes on X do not pass through the vertex of thecone CGr (2 , V )).The vector bundle P ∨ is globally generated by its space V V ≃ V V ∨ of global sec-tions; therefore, for ω general in P ( V V ), the zero-locus in F σ ( X ) of ω viewed as an elementof H ( F σ ( X ) , P ∨ ) has dimension 1 and the set of ω such that this dimension jumps hascodimension 2 or more. Thus, for a general choice of a line P ⊂ P ( V V ), the zero-locus is1-dimensional for every point ω ∈ P . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 33
Choose a general codimension-2 subspace B ⊂ A such that • X B := X × P ( V V ) P ( B ) is a smooth special fourfold , • dim( Y B,V ) ≤ , • for any ω ∈ B ⊥ ∩ V V the zero-locus of ω in F σ ( X ) is 1-dimensional.Note that dim( B ⊥ ∩ V V ) = 2 (the dimension is obviously at least 2 and it is at most 2 since A ∩ V V = 0). By [DK1, Proposition 3.14(a)], the Lagrangian subspace of the fourfold X B is A := B ⊕ ( B ⊥ ∩ V V ) . Each ω ∈ B ⊥ ∩ V V determines a hyperplane in P ( V V ) containing P ( B ). We denote by X ω the corresponding hyperplane section of X . For all ω , we have inclusions X B ⊂ X ω ⊂ X and every X ω is a special GM fivefold which is smooth for general ω . We set e D ( B ) := { (Π , ω ) | Π ⊂ X ω , ω ∈ P ( B ⊥ ∩ V V ) } ⊂ F σ ( X ) × P . Let pr : e D ( B ) → F σ ( X ) be the projection. It gives a birational map e D ( B ) → D ( B ), where D ( B ) ⊂ F σ ( X ) is the degeneracy locus of the morphism ( B ⊥ ∩ V V ) ⊗ O F σ ( X ) → P ∨ . Itsatisfies [ D ( B )] = c ( P )in H ( F σ ( X ); Z ). Proposition 5.23.
We have ι ∗ ˜ σ ∗ c ( P ) = 6˜ h in H ( e Y A ; Z ) .Proof. The above discussion shows that we need to describe ι ∗ ˜ σ ∗ [ D ( B )]. Let P ⊂ P be theopen subset of those ω such that X ω is a smooth hyperplane section of X , let e D ω ⊂ e D ( B ) bethe fiber of e D ( B ) over ω ∈ P , and let e D ⊂ e D ( B ) be the preimage of P .Choose any ω ∈ P and let A ω := A ( X ω ) be the Lagrangian subspace associated with X ω .By [DK1, Proposition 3.14(a)], we havedim( A ω ∩ A ) = dim( A ω ∩ A ) = 9 . This shows that the pencil P is the same as the pencil of Lemma 5.20.By Theorem 4.3(b), the Stein factorization of the map σ : F σ ( X ω ) → P ( V ) is the doublecovering e Y ≥ A ω ,V of Y ≥ A ω ,V ⊂ P ( V ). This means that the Stein factorization of the map σ : e D → P ( V ) × P is a double covering of the subscheme Y ≥ A ,A ; V × P P . By Lemma 5.22, its projection to P ( V ) is birational onto Y A ,V ∩ Y A ,V .This means that the composition pr ◦ σ : e D ( B ) → P ( V ) is generically finite of degree 2onto Y A ,V ∩ Y A ,V . Since it factors through e Y A ,V , the induced map ˜ σ : e D ( B ) → e Y A ,V is eitherbirational onto e Y A ,V × P ( V ) Y A ,V , or generically surjective of degree 2 onto a section of thedouble cover e Y A ,V × P ( V ) Y A ,V → Y A ,V ∩ Y A ,V . In the first case, we have ˜ σ ∗ [ D ( B )] = 6 ι ∗ ˜ h ,hence ι ∗ ˜ σ ∗ [ D ( B )] = 6˜ h . In the second case, we have ι ∗ ˜ σ ∗ ([ D ( B )]) + τ ∗ A (cid:0) ι ∗ ˜ σ ∗ ([ D ( B )]) (cid:1) = 12˜ h ,where τ ∗ A is the action on H ( e Y A ; Z ) of the involution of the double covering f A . In thesecond case, the same arguments used at the end of the proof of Corollary 5.11 show that wealso have ι ∗ ˜ σ ∗ [ D ( B )] = 6˜ h . (cid:3) Period points and period maps.
In this section, we discuss period points and periodmaps for smooth GM varieties of dimensions 4 or 6 and for double EPW sextics. We use thenotation of Section 3.3. In particular, we consider the lattices Γ , Γ , and Λ defined by (8).Consider the automorphism group O (Λ) and the stable orthogonal group e O (Λ) ⊂ O (Λ)of automorphisms of Λ which act trivially on its discriminant group D (Λ) = Λ ∨ / Λ. It hasindex 2 in O (Λ).Another description of e O (Λ) will be important. Consider the even lattice Γ . By choosingvectors e and e with square 2 in the first and second copies of U in Γ , we obtain a primitiveembedding of the lattice I , (2) into Γ . Furthermore, the group O (Γ ) acts transitively on theset of such embeddings ([J]). The orthogonal sublattice h e , e i ⊥ ⊂ Γ is isomorphic to Λ( − , Z / Z ) ).The subgroup O (Γ ) h e ,e i ⊂ O (Γ ) stabilizing the sublattice h e , e i preserves the or-thogonal Λ( − O (Γ ) h e ,e i → O (Λ) which is surjective and the stablegroup e O (Λ) is the isomorphic image under this map of the subgroup O (Γ ) e ,e ⊂ O (Γ ) h e ,e i of elements stabilizing both e and e .Analogously, in the lattice Γ , there are vectors e and e generating a sublattice iso-morphic to I , (2) such that e + e is characteristic in Γ . Again by [J], the group O (Γ ) actstransitively on the set of such embeddings, the orthogonal h e , e i ⊥ is isomorphic to Λ, andthere are morphisms O (Γ ) h e ,e i ։ O (Λ) and O (Γ ) e ,e ∼ → e O (Λ) (see [DIM, Section 5.1] fordetails).The groups e O (Λ) and O (Λ) act properly and discontinuously on the complex variety(25) Ω := { ω ∈ P (Λ ⊗ C ) | ω · ω = 0 , ω · ¯ ω < } . The quotient D := e O (Λ) \ Ωis a quasi-projective 20-dimensional variety. It has a canonical involution r D , associated withthe further degree-2 quotient D → O (Λ) \ Ω. Proposition 5.24.
Let X be a smooth GM variety of dimension n = 4 or . The one-dimensional subspace H n/ ,n/ − ( X ) ⊂ H n ( X, C ) gives rise to a well defined point in D . This point is called the period point of X and will be denoted by ℘ ( X ). Proof.
Assume first n = 4. The abelian group H ( Gr (2 , V ); Z ) is generated by the Schubertclasses σ , and σ . By [DIM, Section 5.1], there exists an isometry φ : H ( X ; Z ) ∼ → Γ , calleda marking of X , such that(26) φ − ( e ) = γ ∗ X σ , and φ − ( e ) = γ ∗ X σ − γ ∗ X σ , , where e , e ∈ Γ were defined above. Any two markings differ by the action of an element of thegroup O (Γ ) e ,e ≃ e O (Λ). The marking carries the vanishing cohomology lattice H ( X ; Z ) (defined in (5)) onto the orthogonal h e , e i ⊥ ≃ Λ. Its complexification φ C : H ( X ; C ) ∼ → Γ ⊗ C takes the one-dimensional subspace H , ( X ) (see Proposition 3.1), which is orthogonal to γ ∗ X H ( Gr (2 , V ); C ), to a point in the manifold Ω defined in (25). The equivalence class of thispoint in the quotient D = e O (Λ) \ Ω is well defined.
USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 35
The situation when n = 6 is similar: the abelian group H ( Gr (2 , V ); Z ) is generatedby σ , and σ , there exists a marking φ : H ( X ; Z ) ∼ → Γ such that(27) φ − ( e ) = γ ∗ X σ , and φ − ( e ) = γ ∗ X σ , where again e , e ∈ Γ were defined above, and any two markings differ by the action of anelement of the group O (Γ ) e ,e ≃ e O (Λ). The marking carries the vanishing cohomology lattice H ( X ; Z ) (defined in (5)) onto the orthogonal h e , e i ⊥ ≃ Λ( − φ C takes the one-dimensional subspace H , ( X ) (see Proposition 3.1) to a point in the samedomain Ω (note that the anti-isometry property of φ C is compensated by the change in signin the Hodge–Riemann relations for a (4 , , D is well defined. (cid:3) Remark . If, in the above construction of the period point, we replace the conditions (26)and (27) by similar conditions with e and e exchanged, we obtain a new period point whichis r D ( ℘ ( X )) (this is because there is an element of O (Γ n ) h e ,e i which exchanges e and e , andthe image of this isometry by the surjection O (Γ n ) h e ,e i ։ O (Λ) is not in e O (Λ)).An analogous construction can be made in another situation: if e Y A is a smooth doubleEPW sextic, the one-dimensional subspace H , ( e Y A ) ⊂ H ( e Y A ; C ) gives rise to a period point ℘ epw ( e Y A ) in the same variety D (see [O4, Section 4.2]). This period point may also be definedfor all Lagrangian subspaces A with no decomposable vectors (i.e., even when Y ≥ A = ∅ ; see[O4, Section 5.1]). The main result of [O2] is ℘ epw ( e Y A ⊥ ) = r D ( ℘ epw ( e Y A )). Lemma 5.26.
For any smooth GM variety X of dimension or , with associated La-grangian A ( X ) satisfying (10) , one has either ℘ ( X ) = ℘ epw ( e Y A ( X ) ) or ℘ ( X ) = r D ( ℘ epw ( e Y A ( X ) )) .Proof. Consider first the case of fourfolds. Choose markings φ for X , and ψ for e Y A ( X ) , andconsider the commutative diagram H ( X ; Z ) ( − φ (cid:15) (cid:15) β / / H ( e Y A ( X ) ; Z ) ψ (cid:15) (cid:15) Λ ψ ◦ β ◦ φ − / / Λ , where β is the isomorphism of Theorem 5.12. Since β is compatible with polarizations, thebottom map g := ψ ◦ β ◦ φ − is in O (Λ). Since β is a morphism of Hodge structures, we have g ( φ C ( H , ( X ))) = ψ C ( H , ( e Y A ( X ) ) . If g ∈ e O (Λ), we have ℘ ( X ) = ℘ epw ( e Y A ( X ) ); otherwise ℘ ( X ) = r D ( ℘ epw ( e Y A ( X ) )).For sixfolds, we use the same argument with the isomorphism β of Theorem 5.19. (cid:3) Proposition 5.27.
Either for any smooth GM variety X of dimension resp. whoseassociated double EPW sextic e Y A ( X ) is smooth, one has ℘ ( X ) = ℘ epw ( e Y A ( X ) ) , or for any suchvariety X , one has ℘ ( X ) = r D ( ℘ epw ( e Y A ( X ) )) .Proof. Let X → S be a smooth family of GM varieties of dimension 4 (resp. 6) over anirreducible base S , such that every GM variety of dimension 4 (resp. 6) is isomorphic to somefiber of that family (see the proof of [KP, Proposition 3.4] for a construction of such a family). It is classical that the period point construction defines a period map ℘ S : S → D which isalgebraic.By [DK3], we have a family of Lagrangian data ( V , V , A ), where V is a rank-6 vectorbundle on S , V ⊂ V is a rank-5 vector subbundle, and A ⊂ V V is a Lagrangian subbundle.We choose an open covering ( S α ) of S such that these vector bundles are all trivial on each S α .Refining further the covering and applying [O3, Proposition 3.1], we construct, for each α , afamily f Y α → S α of (possibly singular) double EPW sextics. These families define period mapswhich fit together to define an algebraic map ℘ epw S : S → D , where S ⊂ S is the dense opensubset where the double EPW sextics are smooth.Since D is separated, the sets S := { s ∈ S | ℘ S ( s ) = ℘ epw S ( s ) } and S := { s ∈ S | ℘ S ( s ) = r D ◦ ℘ epw S ( s ) } are closed in S . By Lemma 5.26, the dense subset S ⊂ S corresponding to smooth GMfourfolds (resp. sixfolds) satisfying (10) is the union of its closed subsets S ∩ S and S ∩ S .Since S is irreducible, one of them, say S i , is S . This means that S i contains S , henceits closure S , and proves the lemma. (cid:3) To go from one of the possibilities of the proposition to the other, it suffices to change theconvention defining the period point ℘ ( X ) (and the period map) as explained in Remark 5.25.We may therefore assume that(28) ℘ ( X ) = ℘ epw ( e Y A ( X ) )holds for any smooth GM fourfold or sixfold with smooth e Y A ( X ) . This implies Theorem 5.1 infull generality.We end this section with some consequences of (28) based on results from [DK1] and [DIM]. Remark . In [DK1, Section 3.6], we said that smooth GM varieties of thesame dimension are period partners if they are constructed from the same Lagrangian subspace A ⊂ V V (with no decomposable vectors) but possibly different hyperplanes V ⊂ V . ByTheorem 5.1, period partners of dimensions 4 or 6 have the same period point.Conversely, since double EPW sextics have the same period point if and only if theyare isomorphic ([O4, Theorem 1.3]), smooth GM fourfolds (or sixfolds) are period partners ifand only if they have the same period point. By [DK1, Theorem 3.25], isomorphism classesof period partners of a GM fourfold are parametrized by Y A ⊥ ⊔ Y A ⊥ , modulo the finite groupAut( Y A ⊥ ) (for A general, Y ≥ A ⊥ is empty and Y A ⊥ ⊔ Y A ⊥ = Y A ⊥ ). Similarly, isomorphism classesof period partners of a GM sixfold are parameterized by P ( V ∨ ) r Y A ⊥ , modulo Aut( Y A ⊥ ). Remark . Pretending that smooth GM varieties have coarsemoduli spaces (see [DK3]), we go, following [DIM], through some geometrically defined sub-varieties of these moduli spaces and discuss, using the period map, their relation with somenatural divisors in the period domain D . We use the notation introduced in [DIM]. • Smooth GM fourfolds containing σ -planes ([DIM, Section 7.1]). They form a co-dimension-2 family X σ -planes whose period points cover a divisor D ′′ ⊂ D . A smooth GMfourfold X contains a σ -plane if and only if Y ≥ A,V = ∅ (Theorem 4.3(c)). In particular, Y ≥ A = ∅ ; this means that A is in the O’Grady divisor ∆ ([O4, (2.2.3)]) and implies ℘ epw (∆) = D ′′ . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 37 If A is general in ∆, the set Y ≥ A is just one point v ([O6, Section 5.4]). The condition Y ≥ A,V = ∅ is then equivalent to v ∈ V , i.e., to p X ∈ v ⊥ . Thus the fiber of the period map X σ -planes → D ′′ is equal to the hyperplane section of Y A ⊥ ⊔ Y A ⊥ defined by v ⊥ (moduloautomorphisms). This fiber was also described in [DIM, Section 7.1] as a P -bundle overa degree-10 K3 surface. • Smooth GM fourfolds containing τ -quadratic surfaces ([DIM, Section 7.3]). Theyform a codimension-1 family X τ -quadrics whose period points cover the divisor D ′ = r D ( D ′′ ) ⊂ D . A general fiber of the period map X τ -quadrics → D ′ is, on the one hand,isomorphic to Y A ⊥ (modulo automorphisms), and, on the other hand, birationally isomor-phic to the quotient by an involution of the symmetric square of a K3 surface ([DIM,Section 7.3]). This fits with [O3, Corollary 3.12 and Theorem 4.15]: a desingularizationof e Y A ⊥ is the symmetric square of a K3 surface. • Smooth GM fourfolds containing a cubic scroll ([DIM, Section 7.4]). They form acodimension-1 family which contains the 3-codimensional family of smooth GM fourfoldscontaining a τ -plane (called a ρ -plane in [DIM]) and the period points of both familiescover an r D -invariant divisor D ⊂ D . By Theorem 4.5(c), the condition to contain a τ -plane implies Z ≥ A = ∅ . The divisor D is therefore contained in the closure of the imageby the period map ℘ epw of the locus of Lagrangians subspaces A such that Z ≥ A = ∅ ; sincethis locus is an irreducible divisor ([IKKR, Lemma 3.6]), they are equal. • Singular GM fourfolds ([DIM, Section 7.6]). The O’Grady divisor Σ (see [O4]) corre-sponds to Lagrangian subspaces A containing decomposable vectors. The correspondingperiod points (under a suitable extension of the period map ℘ epw discussed in [O4]) fillout a divisor S ⋆ ⊂ D ([O4, (4.3.3) and Proposition 4.12]); this is the r D -stable divisor D of [DIM], which corresponds to periods of nodal GM fourfolds ([DIM, Section 7.6]). • Smooth GM sixfolds containing a P . By Theorem 4.2, a smooth GM sixfold containsa P if and only if Y ≥ A,V = ∅ . In particular, A is in ∆ and the period point is in D ′′ .As above, when A is general in ∆, one has Y ≥ A = { v } and the condition Y ≥ A,V = ∅ isequivalent to p X ∈ v ⊥ . Thus the fiber of the period map is equal to the hyperplane sectionof P ( V ∨ ) r Y A ⊥ defined by v ⊥ (modulo automorphisms) and the codimension of the familyof GM sixfolds containing a P is 2. Appendix A. Linear spaces on families of quadrics
Let S be a base scheme which we assume to be Cohen–Macaulay and irreducible. Let E be a vector bundle on S of rank m and let L ⊂ S E ∨ be a line subbundle. Consider theprojectivization pr : P S ( E ) → S and the relative line bundle O (1) on P S ( E ). Let Q ⊂ P S ( E )be the family of quadrics defined as the zero-locus of the section of the line bundle pr ∗ L ∨ ⊗ O (2)corresponding to the morphism L ֒ → S E ∨ via the isomorphism H ( P S ( E ) , pr ∗ L ∨ ⊗ O (2)) ≃ H ( S, L ∨ ⊗ S E ∨ ) ≃ Hom( L , S E ∨ ) . We denote by D c ( Q ) ⊂ S the corank- c degeneracy locus of the induced map E ⊗ L → E ∨ ofvector bundles and by C the cokernel sheaf of this map; it is supported on D ( Q ).In this appendix, we discuss the relative Hilbert scheme F k ( Q ) := Hilb P k ( Q /S ). Weconcentrate on the cases k ∈ { , } (i.e., on the Hilbert schemes of lines and planes) anddescribe the Stein factorization of the canonical morphism ϕ : F k ( Q ) → S . Note that F k ( Q )is a subscheme in the relative Grassmannian π : Gr S ( k + 1 , E ) → S . We denote by U thetautological subbundle of rank k + 1 on Gr S ( k + 1 , E ). Proposition A.1.
Assume D m − ( Q ) = S . We have a resolution → L ⊗ (det( U )) ⊗ → L ⊗ S U ⊗ det( U ) → L ⊗ S U → O Gr S (2 , E ) → O F ( Q ) → on Gr S (2 , E ) . Moreover, the pushforward to S of O F ( Q ) is given as follows • if m = 3 , then ϕ ∗ O F ( Q ) ≃ O D ( Q ) ⊕ ( C ⊗ L ⊗ det( E )) ; • if m = 4 , then ϕ ∗ O F ( Q ) ≃ O S ⊕ ( L ⊗ det( E )) ; • if m ≥ , then ϕ ∗ O F ( Q ) ≃ O S .Proof. Since F ( Q ) is the zero-locus of a section of the rank-3 vector bundle L ∨ ⊗ S U ∨ on Gr S (2 , E ), its codimension is at most 3. On the other hand, for a quadric of rank r in P m − ,the dimension of the Hilbert scheme of lines is equal to 2 m − r ≥ m − r ≤
2. Stratifying F ( Q ) by the preimages of the subsets S r D m − ( Q ) and D m − ( Q ), wesee that the codimension of the first stratum is 3 and the codimension of the second stratumis codim( D m − ( Q )) + 2. Since D m − ( Q ) = S , the codimension of F ( Q ) is at least 3. Since Gr S (2 , E ) is Cohen–Macaulay, the section of L ∨ ⊗ S U ∨ defining F ( Q ) is regular and theKoszul complex provides a resolution of its structure sheaf. A standard description of theexterior powers of a symmetric square ([W, Proposition 2.3.9]) gives the above explicit form.For the second part, we apply the Borel–Bott–Weil Theorem to compute the derivedpushforwards to S of the terms of the Koszul complex. The result is R • π ∗ O Gr S (2 , E ) ≃ O S ,R i π ∗ S U ≃ ( det( E ) ⊗ E ∨ m = 3 and i = 1 , otherwise ,R i π ∗ S U ⊗ det( U ) ≃ det( E ) ⊗ E det( E )0 if m = 3 and i = 1 , if m = 4 and i = 2 , otherwise ,R i π ∗ (det( U )) ⊗ ≃ ( det( E ) m = 3 and i = 2 , otherwise . Therefore, the pushforward of the Koszul complex for m = 3 gives an exact sequence0 → ( L ⊗ det( E ) ) ⊕ ( L ⊗ det( E ) ⊗ E ) α −−→ ( L ⊗ det( E ) ⊗ E ∨ ) ⊕ O S → ϕ ∗ O F ( Q ) → . The map α is the direct sum of a twist of the map α : L ⊗ E → E ∨ and of a twist of itsdeterminant α : L ⊗ det( E ) → O S . The cokernel of α is C and the cokernel of α is thestructure sheaf of the degeneracy locus D ( Q ). This gives the result for m = 3.For m = 4, the pushforward of the Koszul complex gives ϕ ∗ O F ( Q )) = O S ⊕ ( L ⊗ det( E )),and for m ≥
5, just ϕ ∗ O F ( Q )) = O S . (cid:3) For k = 2, the computation is analogous, but more complicated, since the Koszul complexis longer. We denote by Σ a,b,c U the Schur functor of the rank-3 tautological subbundle U on Gr S (3 , E ) corresponding to the highest weight ( a, b, c ) of the group GL . We also consider thecomposition L ⊗ E ⊗ E ∨ −→ E ∨ ⊗ E ∨ −→ V E ∨ , where the first map is given by the family of quadrics and the second is canonical. Denoteby C its cokernel sheaf; it is supported on the degeneracy locus D ( Q ). USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 39
Proposition A.2.
Assume D m − ( Q ) = S and codim( D m − ( Q )) ≥ . There is a resolution → L ⊗ Σ , , U → L ⊗ Σ , , U → L ⊗ Σ , , U → ( L ⊗ Σ , , U ) ⊕ ( L ⊗ Σ , , U ) → L ⊗ Σ , , U → L ⊗ S U → O Gr S (3 , E ) → O F ( Q ) → on Gr S (3 , E ) . Moreover, the pushforward to S of O F ( Q ) is given as follows: • if m = 4 , we have ϕ ∗ O F ( Q ) ≃ O D ( Q ) ⊕ ( C ⊗ L ⊗ det( E )) ; • if m = 5 , we have ϕ ∗ O F ( Q ) ≃ O D ( Q ) ⊕ ( C ⊗ L ⊗ det( E )) ; • if m = 6 , we have ϕ ∗ O F ( Q ) ≃ O S ⊕ ( L ⊗ det( E )) ; • if m ≥ , we have ϕ ∗ O F ( Q ) ≃ O S .Proof. By definition, F ( Q ) is the zero-locus of a section of the rank-6 vector bundle L ∨ ⊗ S U ∨ on Gr S (3 , E ), so its codimension is at most 6. On the other hand, for a quadric of rank r in P m − , the dimension of the Hilbert scheme of planes is equal to 3 m −
15 for r ≥
5, 3 m − ≥ r ≥
3, and 3 m −
12 for r ≤
2. Thus, stratifying F ( Q ) by the subsets S r D m − ( Q ), D m − ( Q ) r D m − ( Q ), and D m − ( Q ), we see that under our assumption, the codimensionis 6, the section of L ∨ ⊗ S U ∨ defining F ( Q ) is regular, and the Koszul complex provides aresolution of its structure sheaf. A standard description of the exterior powers of a symmetricsquare ([W, Proposition 2.3.9]) gives the above explicit form.For the second part, we apply the Borel–Bott–Weil Theorem to compute the derivedpushforwards to S of the terms of the Koszul complex. The result is the following R • π ∗ O Gr S (2 , E ) ≃ O S ,R i π ∗ S U ≃ ( det( E ) ⊗ V E ∨ m = 4 and i = 1 , otherwise ,R i π ∗ Σ , , U ≃ det( E ) ⊗ (( E ⊗ E ∨ ) / O S )det( E ) ⊗ E ∨ m = 4 and i = 1 , if m = 5 and i = 2 , otherwise ,R i π ∗ Σ , , U ≃ ( det( E ) ⊗ S E ∨ m = 4 and i = 2 , otherwise ,R i π ∗ Σ , , U ≃ det( E ) ⊗ S E det( E ) ⊗ E det( E )0 if m = 4 and i = 1 , if m = 5 and i = 2 , if m = 6 and i = 3 , otherwise ,R i π ∗ Σ , , U ≃ ( det( E ) ⊗ (( E ⊗ E ∨ ) / O S )0 if m = 4 and i = 2 , otherwise ,R i π ∗ Σ , , U ≃ det( E ) ⊗ V E det( E ) m = 4 and i = 2 , if m = 5 and i = 4 , otherwise , R i π ∗ Σ , , U ≃ ( det( E ) m = 4 and i = 3 , otherwise . Therefore, the pushforward of the Koszul complex for m = 4 gives the exact sequence · · · → ( L ⊗ det( E ) ⊗ S E ∨ ) ⊕ ( L ⊗ det( E ) ⊗ (( E ⊗ E ∨ ) / O S )) α −−→ ( L ⊗ det( E ) ⊗ V E ∨ ) ⊕ O S → ϕ ∗ O F ( Q ) → . The map α is the direct sum of a twist of the map α : L ⊗ (( E ⊗ E ∨ ) / O S ) → V E ∨ and ofthe exterior cube of the family of quadrics α : L ⊗ S ( V E ) → O S . The cokernel of α is C ,and that of α is O D ( Q ) . This gives the result for m = 4.For m = 5, the pushforward of the Koszul complex gives0 → ( L ⊗ det( E ) ) ⊕ ( L ⊗ det( E ) ⊗ E ) α −−→ ( L ⊗ det( E ) ⊗ E ∨ ) ⊕ O S → ϕ ∗ O F ( Q ) → . The map α is described as in Proposition A.1 and gives the result for m = 5. For m = 6, thepushforward of the Koszul complex gives ϕ ∗ O F ( Q )) = O S ⊕ ( L ⊗ det( E )), and, for m ≥ ϕ ∗ O F ( Q )) = O S . (cid:3) Appendix B. Resolutions of EPW surfaces
In this appendix, we discuss a resolution of the structure sheaf of an EPW surface Y ≥ A in P ( V ) and compute some cohomology spaces related to its ideal sheaf. We use freely thenotation and results of [DK1, Appendix B], especially those introduced in Proposition B.3. Inparticular, we set b Y A := { ( v, V ) ∈ Fl (1 , V ) | A ∩ ( v ∧ V V ) = 0 } and(29) b Y ′ A := { ( a, v, V ) ∈ P ( A ) × Fl (1 , V ) | a ∈ P ( A ∩ ( v ∧ V V )) } . When A contains no decomposable vectors, the projection b Y ′ A → Fl (1 , V ) induces an iso-morphism b Y ′ A ∼ → b Y A . We denote by H and H ′ the hyperplane classes of P ( V ) and P ( V ∨ ) andby p : b Y A → Y A and q : b Y A → Y A ⊥ the projections (we switch back from the notation pr Y, and pr Y, used in the main body of the article to the notation used in [DK1]). We also denoteby H A the pullback of the hyperplane class of P ( A ) to b Y A via the map b Y A → P ( A ) providedby the identification of b Y A with b Y ′ A (when A contains no decomposable vectors). We begin witha simple lemma. Lemma B.1. If A contains no decomposable vectors, there is an isomorphism p ∗ O b Y A ∼ → O Y A and R > p ∗ O b Y A = 0 .Proof. By definition, b Y A = b Y ′ A is the zero-locus of the composition O P ( A ) ( − H A ) ⊠ O P ( V ) → A ⊗ O P ( A ) × P ( V ) → V V ⊗ O P ( A ) × P ( V ) → O P ( A ) ⊠ V T P ( V ) ( − H )on P ( A ) × P ( V ), hence equals the zero-locus of the corresponding section of the vector bundle O P ( A ) ( H A ) ⊠ V T P ( V ) ( − H ). Since the codimension of b Y A in P ( A ) × P ( V ) equals the rank ofthat vector bundle, we have a Koszul resolution(30) · · · → O P ( A ) ( − H A ) ⊠ V ( V T P ( V ) )( − H ) → O P ( A ) ( − H A ) ⊠ V T P ( V ) ( − H ) → O P ( A ) × P ( V ) → O b Y A → . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 41
Pushing it forward to P ( V ), we obtain an exact sequence0 → det( V T P ( V ) ( − H )) → O P ( V ) → p ∗ O b Y A → O P ( V ) ( − H ), hence p ∗ O b Y A is the structure sheaf of a sextic hypersurface, whichclearly coincides with Y A . (cid:3) The crucial observation on which the results of this appendix are based is the following.
Lemma B.2. If A contains no decomposable vectors, there is a linear equivalence of divisors (31) 2 H A ≡ lin H + H ′ . Proof.
The definition (29) implies that the image of the tautological embedding O b Y A ( − H A ) ֒ → A ⊗ O b Y A is contained in the kernel of the composition A ⊗ O b Y A → V V ⊗ O b Y A → p ∗ ( V T P ( V ) ( − H )) , itself identified with the kernel of the morphism p ∗ ( V T P ( V ) ( − H )) → ( V V /A ) ⊗ O b Y A . Thuswe have a natural embedding O b Y A ( − H A ) ֒ → p ∗ ( V T P ( V ) ( − H ))of vector bundles: over a point ( a, v, V ) of b Y A with a = v ∧ η , it is given by η ∈ V ( V /v ) ⊂ V ( V /v ). Its wedge square is a map O b Y A ( − H A ) → p ∗ ( V T P ( V ) ( − H )) ≃ p ∗ (Ω P ( V ) )that sends a point ( a, v, V ) of b Y A , with a = v ∧ η , to η ∧ η ∈ V ( V /v ) ⊂ V ( V /v ) whichis non-zero since a is indecomposable. This map is therefore an embedding of vector bundles.Twisting it by O ( H ) and composing with the canonical embedding gives a map O b Y A ( H − H A ) ֒ → p ∗ (Ω P ( V ) ( H )) ֒ → V ∨ ⊗ O b Y A which defines a map b Y A → P ( V ∨ ), ( a, v, V ) v ∧ η ∧ η . In the proof of [DK1, Proposition B.3],it was shown that this map is the projection q : b Y A → Y A ⊥ . Therefore, we have an isomorphismof line bundles O b Y A ( H − H A ) ≃ O b Y A ( − H ′ ), hence (31). (cid:3) This allows us to find a simple resolution of the ideal sheaf I Y ≥ A , Y A of the EPW sur-face Y ≥ A in the EPW sextic Y A . Lemma B.3. If A contains no decomposable vectors, there is an exact sequence (32) 0 → V ( V T P ( V ) )( − H ) → A ∨ ⊗ V T P ( V ) ( − H ) → S A ∨ ⊗ O P ( V ) ( − H ) → I Y ≥ A , Y A → of sheaves on P ( V ) .Proof. Denote by E the exceptional divisor of the birational morphism p : b Y A → Y A . We showfirst that E coincides with the scheme-theoretic preimage of the EPW surface Y ≥ A . For this,recall that Y ≥ A is by definition the corank-2 degeneracy locus of the composition A ⊗ O P ( V ) → V V ⊗ O P ( V ) → V T P ( V ) ( − H ) . When pulled back to b Y A , it extends to a complex O b Y A ( − H A ) ֒ → A ⊗ O b Y A → p ∗ ( V T P ( V ) ( − H )) ։ O b Y A ( H A ) , hence the preimage of Y ≥ A is the degeneracy locus of the induced map( A ⊗ O b Y A ) / O b Y A ( − H A ) → Ker( p ∗ ( V T P ( V ) ( − H )) ։ O b Y A ( H A )) . This is a morphism between two vector bundles of rank 9, hence the preimage of Y ≥ A is theCartier divisor in b Y A defined by a section of the line bundledet(Ker( p ∗ ( V T P ( V ) ( − H )) ։ O b Y A ( H A ))) ⊗ det(( A ⊗ O b Y A ) / O b Y A ( − H A )) ∨ ≃ O b Y A (6 H − H A ) . But 6 H − H A ≡ lin H − H − H ′ ≡ lin H − H ′ and this is linearly equivalent to E by a computationin [DK1] (a paragraph before Lemma B.6). All global sections of the line bundle O b Y A ( E ) areproportional (since E is the exceptional divisor of a birational morphism), hence the scheme-theoretic preimage of Y ≥ A equals E .Since E = p − ( Y ≥ A ), there is an embedding of schemes p ( E ) ⊂ Y ≥ A . On the otherhand, p ( E ) and Y ≥ A coincide set-theoretically ([DK1, Proposition B.3]) and the scheme Y ≥ A is reduced and normal ([DK1, Theorem B.2]), hence p ( E ) = Y ≥ A . Since the fibers of the map p : E → Y ≥ A are connected, it also follows that there is an isomorphism p ∗ O E ∼ → O Y ≥ A .We now compute p ∗ O E . We use the linear equivalence 6 H − H A ≡ lin E shown above andcompute the derived pushforward of the line bundle O b Y A ( − E ) ≃ O b Y A (2 H A − H ). Twistingthe Koszul resolution (30) by O b Y A (2 H A − H ) and pushing forward to P ( V ), we obtain anexact sequence0 → V ( V T P ( V ) )( − H ) → A ∨ ⊗ V T P ( V ) ( − H ) → S A ∨ ⊗ O P ( V ) ( − H ) → p ∗ O b Y A ( − E ) → , and vanishing of higher pushforwards. Using this and Lemma B.1 and applying pushforwardto the standard exact sequence 0 → O b Y A ( − E ) → O b Y A → O E , we obtain an exact sequence0 → p ∗ O b Y A ( − E ) → O Y A → p ∗ O E → . The right term is isomorphic to O Y ≥ A , as we have shown above, hence the left term is I Y ≥ A , Y A .This proves the lemma. (cid:3) Remark
B.4 . The equality E = p − ( Y ≥ A ) shown in the proof means that b Y A is the blowupof Y A along Y ≥ A .The sequence (32) can be merged with the standard resolution of Y A to give an exactsequence(33) 0 → V ( V T P ( V ) )( − H ) → A ∨ ⊗ V T P ( V ) ( − H ) → ( S A ∨ ⊕ C ) ⊗ O P ( V ) ( − H ) → O P ( V ) → O Y ≥ A → Y ≥ A . USHEL–MUKAI VARIETIES: LINEAR SPACES AND PERIODS 43
Corollary B.5. If A contains no decomposable vectors, the following table computes the co-homology spaces for some twists of the sheaf O Y ≥ A t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 H ( Y ≥ A , O Y ≥ A ( tH )) V A H ( Y ≥ A , O Y ≥ A ( tH )) 0 0 V V ∨ A ∨ H ( Y ≥ A , O Y ≥ A ( tH )) C V ∨ S V ∨ S V ∨ S V ∨ S V ∨ /V S V ∨ / ( S A ∨ ⊕ C ) Moreover, H ( P ( V ) , I Y ≥ A , P ( V ) (2 H )) = H ( P ( V ) , I Y ≥ A , P ( V ) ( H )) = 0 .Proof. It consists of a straightforward computation using (33) and the Borel–Bott–Weil theo-rem. (cid:3)
In Section 5, we used the following simple consequence of these computations.
Corollary B.6. If A contains no decomposable vectors, the curve Y ≥ A,V ⊂ P ( V ) is not con-tained in a quadric.Proof. We have an exact sequence0 → I Y ≥ A , P ( V ) ( − H ) → I Y ≥ A , P ( V ) → I Y ≥ A,V , P ( V ) → P ( V ). The cohomology sequence of its twist by O P ( V ) (2 H ) gives an exactsequence H ( P ( V ) , I Y ≥ A , P ( V ) (2 H )) → H ( P ( V ) , I Y ≥ A,V , P ( V ) (2 H )) → H ( P ( V ) , I Y ≥ A , P ( V ) ( H )) . By Corollary B.5, the spaces at both ends vanish, hence so does the middle space. (cid:3)
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Univ Paris Diderot, ´Ecole normale sup´erieure, PSL Research University,CNRS, D´epartement Math´ematiques et Applications45 rue d’Ulm, 75230 Paris cedex 05, France
E-mail address : [email protected] Algebraic Geometry Section, Steklov Mathematical Institute,8 Gubkin str., Moscow 119991, RussiaThe Poncelet Laboratory, Independent University of MoscowLaboratory of Algebraic Geometry, National Research University Higher Schoolof Economics, Russian Federation
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