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On subgroups of an automorphism group of an irreducible symplectic manifold

Let X be an irreducible symplectic manifold and L a nef line bundle on X which is isotropic with respect to the Beauville-Bogomolov quadratic form. It is known that a subgroup Aut(X,L) of an automorphism group of X which fix L is almost abelian. We give a formula of the rank of Aut(X,L) in terms of MBM divisors. We also prove that the nef cone of X cut out MBM classes, which is a generalization of Kovac's structure theorem of nef cones of K3 surfaces


Introduction
We start with recalling definitions of a Neron-Severi group, an ample cone and a nef cone. We denote by NS R (X) the R-vector space generated by NS(X). An ample cone Amp(X) of X is the cone in NS R (X) defined by Amp(X) := NS R (X) ∩ K (X), where K (X) is the Kähler cone of X. The closure of Amp(X) in NS R (X) is said to be nef cone and denoted by Nef(X).
We also recall the definition of irreducible symplectic manifolds. Definition 1.2. A compact Kähler manifold X is said to be irreducible symplectic if X has the following three properties: (1) X is simply connected; (2) X carries a holomorphic symplectic form and; (3) dim H 0 (X, Ω 2 X ) = 1. A K3 surface has above three properties and gives a plain example of irreducible symplectic manifolds. It is expected that K3 surfaces and irreducible symplectic manifolds share many geometric properties. One of the biggest geometric features of K3 surfaces is Global Torelli theorem, which was obtained in [8], [12] and [17]. Theorem 1.1 (Global Torelli Theorem for projective K3 surfaces). Let X and X ′ be projective K3 surfaces. Assume that there exists an isometry φ : H 2 (X ′ , Z) → H 2 (X, Z) with respect to the cup products. If φ respects the Hodge structure and φ(Amp(X ′ )) ∩ Amp(X) = ∅, there exists an automorphism Φ such that the induced morphism Φ * on cohomologies coincides with φ.
A higher dimensional analogue of Global Torelli Theorem was obtained by Verbitsky in [21,Theorem 1.18]. After introducing the definition of monodromy groups, we state it in a form suitable for use in this paper according to [13,Theorem 1.3 (2)]. Definition 1.3. Let X be an irreducible symplectic manifold. We denote by q X the Beauville-Bogomolov quadratic form on H 2 (X, Z). Let O(H 2 (X, Z), q X ) be an isometry group with respect to q X . Let us consider smooth morphisms X → (B, o) such that (B, o) is an analytic space with the reference point o and the fibre at o is isomorphic to X. We note that B may have any kind of singularities. For such a smooth morphism, we have a natural representation π 1 (B, o) → O(H 2 (X, Z)). A subgroup Mon(X) of O(H 2 (X, Z), q X ) is the subgroup generated by the images of all such representations. Theorem 1.2 (Global Torelli Theorem for projective irreducible symplectic manifolds). Let X be a projective irreducible symplectic manifold. Assume that there exists an element φ of Mon(X) which respects the Hodge structures and φ(Amp(X))∩ Amp(X) = ∅. Then there exists an automorphism Φ of X such that the induced automorphism Φ * on H 2 (X, Z) coincides with φ.
By the above theorem, we will have automorphisms of projective irreducible symplectic manifolds if we construct elements of Mon(X) which satisfies the assumptions of Theorem 1.2. In this note, we will construct such elements of Mon(X) and give three applications. The first application concerns with the structure of nef cones of projective irreducible symplectic manifolds. By [10] and [7], we have the following structure theorem of a Kähler cone of an irreducible symplectic manifold. Theorem 1.3. Let X be an irreducible symplectic manifold and N (X) the set of rational curves on X. We define the positive cone C (X) in H 1,1 (X, R) by where q X is the Beauville-Bogomolov form and κ is a Kähler class. Then We denote by C NS (X) the intersection of NS R (X) and C (X). A nef cone Nef(X) of an irreducible symplectic manifold X can be described as follows: where stands for the closure in NS R (X). On the other hand, Kovacs gave a description of an effective cone of a K3 surface in [11], whose dual cone with respect to the cup product is a nef cone. Theorem 1.4 ( [11, Corollary 1] ). Let X be a projective K3 surface whose Picard number is greater than two. We denote by N (X) the set of (−2)-curves on X. If N (X) = ∅, then an effective cone Eff(X) of X coincides with C NS (X). If N (X) = ∅, then If we consider the dual statement of the above theorem, we find that Nef(X) coincides with C NS (X) if N (X) = ∅. If N (X) = ∅, It is a natural question whether a nef cone of an irreducible symplectic manifold has a similar structure. We give a positive answer of this question. To state our result, we recall monodromy birationally minimal classes, which is introduced in [1, Definition 1.13]. Definition 1.4 (Monodromy Birationally Minimal Class). Let X be an irreducible symplectic manifold. A cohomology class e of H 1,1 (X, R) ∩ H 2 (X, Q) is said to be Monodromy birationally minimal if there exists an element γ of Mon(X) such that γ(e) ⊥ ∩ K (X) is an open set of γ(e) ⊥ . We denote by MBM(X) the set of Monodromy birationally minimal classes of X.
By the above remark, we can restate the structure of the nef cone of a K3 surface in the following form. If MBM(X) = ∅, then Nef(X) coincides with C NS (X). If MBM(X) = ∅, there exists a subset N (X) of MBM(X) such that Now we state the first application. Theorem 1.5. Let X be a projective irreducible symplectic manifold whose Picard number is greater than two. If MBM(X) = ∅, then Nef(X) coincides with C NS (X). If MBM(X) = ∅, there exists a subset N (X) of MBM(X) such that where q X is the Beauville-Bogomolov quadratic form.
Remark 1.2. If X is a K3 surface, N (X) coincides with the set of smooth rational curves. In the above theorem, N (X) seems to coincide with the set of rational cohomology class corresponding to smooth rational curves.
The second application concerns with the rank of a subgroup of an automorphism group of an irreducible symplectic manifold which fixes a line bundle. We recall the definition of an almost abelian group according to [14]. Definition 1.5. A group G is said to be almost abelian of rank r if G has a normal subgroup G (0) such that |G : G (0) | < ∞ and G (0) sits in the following exact sequence. Let W R be a sub linear space in generated by c 1 (L) and c 1 (L) ⊥ ∩ MBM(X) • . We denote by Aut(X, L) the subgroup of Aut(X) defined by Aut(X, L) := {g ∈ Aut(X)|g * L ∼ = L} Then Aut(X, L) is almost abelian of rank dim NS R (X) − dim W R − 1. Remark 1.3. Let X be a K3 surface. Assume that X admits an elliptic fibration π : X → P 1 . We denote by L the pull back of the tautological bundle of P 1 . For a point t of P 1 , we let n t be the number of irreducible components of the fibre at t. In this case, c 1 (L) ⊥ ∩ MBM(X) • consists of irreducible components of reducible singular fibres of π. Since cohomology classes of irreducible components of a reducible singular fibre has only one relation in H 2 (X, Z), dim W R = 1 + t∈P 1 (n t − 1) and Shioda-Tate formula in [18], [20] and [19] asserts that the rank of Mordell-Weil group of π coincides with dim NS R (X) − dim W R − 1. Since Mordell-Weil group of π can be considered as a subgroup of Aut(X, L), Theorem 1.6 can be considered as a generalization of Shioda-Tate formula.
By Theorem 1.6, the rank of Aut(X, L) is less than or equal dim H 2 (X, R) − 2. It is a natural question whether this bound is sharp. The third application is that the bound is attained after deforming the pair (X, L). Definition 1.6. Let X and X ′ be compact Kähler manifolds. We also let L and L ′ be line bundles on X and X ′ , respectively. Two pairs (X, L) and (X ′ , L ′ ) are deformation equivalent if there exists a smooth morphism π : X → B over an analytic space B and a line bundle L on X which has the following two properties: (1) There exist two points p and p ′ of B such that ι : π −1 (p) ∼ = X and ι ′ : Theorem 1.7. Let X be an irreducible symplectic manifold whose second Betti number is greater than five and L an isotropic line bundle with respect to Beauville-Bogomolov quadratic form. Then there exists an irreducible symplectic manifold X ′ and a line bundle L ′ on X ′ such that (X, L) is deformation equivalent to (X, L) in the sense of Definition 1.6 and the rank of This note is organized as follows. In section 1, We will construct special elements of O(H 2 (X, Z)), which is a key of the proof of Theorems 1.5, 1.6 and 1.7. In sections 2,3, and 4, we give a proof of Theorem 1.5, 1.6 and 1.7, respectively.

Acknowledgement
The author would like to express his gratitude for Professors Shigeru Mukai, Yoshinori Namikawa, Kieji Oguiso and Hisanori Ohashi.

Construction of elements of the monodromy group
We recall a standard properties of isometry group of a lattice due to the step 2 of the proof of [4, Proposition 3.2].
Proposition 2.1. Let Λ be a lattice of rank n whose index is (1, n − 1). Assume that Λ contains an isotropic element ℓ. Let W be a negative definite sub lattice contained in ℓ ⊥ . Assume that n − rank(W ) > 2.
(2) We will use the same notation as in the proof of part (1). For an element x of C (Λ R ), we have the following expression.
Corollary 2.1. Let X be a projective symplectic manifold. Assume that there exists an element ℓ of NS(X) which is isotropic with respect to Beauville-Bogomolov quadratic form. We denote by Mon(X) the monodromy group of X and by n the Picard number of X. Let W be a negative definite sublattice of NS(X) which is contained in ℓ ⊥ . Assume that n− rank(W ) > 2. Then Mon(X) contains a subgroup Γ which has the following three properties: (1) Γ is isomorphic to Z n−rankW −2 ; (2) The action of Γ respects the Hodge structure of H 2 (X, Z) and Γ acts on the transcendental lattice of H 2 (X, Z) trivially; (3) For every element g of Γ, g(ℓ) = ℓ and g(w) = w for all elements of W and; (4) For every element g of Γ and every element x of C NS (X), lim m→∞ g m x = ℓ in P(NS R (X)).
Proof. By Proposition 2.1, we have a subgroupΓ of O(NS(X)) which has the following three properties: (1)Γ is isomorphic to Z n−rankW −2 ; (2) For every element g ofΓ, g(ℓ) = ℓ and g(w) = w for all elements of W and; (3) For every element g ofΓ and every element x of C NS (X), lim m→∞ g m x = ℓ in P(NS R (X)).
3. Proof of Theorem 1.5 Before starting to prove Theorem 1.5, we prepare Lemma 3.1. Proof. Let e be an element of MBM(X) such that e ⊥ ∩ Nef(X) is an open set of e ⊥ . We choose a 2-plane H in NS R (X) as H contains e, H ∩ U = ∅ and H is defined over NS Q (X). The restriction Nef(X) ∩ H is generated by two rays ℓ 1 and ℓ 2 . Since H ∩ U = ∅, we may assume that q X (ℓ 1 ) = 0, where q X is the Beauville-Bogomolov quadratic form of X. Let π : X → Def(X) be a Kuranishi family of X. We define the subset Ω in P(H 2 (X, C)) by Ω := {x ∈ P(H 2 (X, C))|q X (x) = 0, q X (x +x) > 0}.

Proof of Theorem 1.6
Proof. First we will prove that Aut(X, L) is an almost abelian group and its rank is at most dim NS R (X)−dim W R −1. Let Γ be the image of the natural representation ρ : Aut(X, L) → O(NS(X)). By [15,Corollary 2.7], the kernel of ρ is finite. Hence it is enough to prove that Γ is an almost abelian group of rank at most dim NS R (X) − dim W R − 1 by [15,Proposition 9.3 (2)]. Let us consider the natural projection r : c 1 (L) ⊥ → c 1 (L) ⊥ /Zc 1 (L).
We define the latticeW bȳ We choose elements {e 1 , . . . , e k } of c 1 (L) ⊥ ∩ MBM(X) • as their residue classes give a generator ofW . We note that k = rank(W ) = dim W R − 1. Let W be the sub lattice of NS(X) generated by {e 1 , . . . , e k }. Then there exists a natural isomorphism W ∼ =W . Since the induced bilinear form on c 1 (L) ⊥ /Rc 1 (L) is negative definite, W is negative definite and hence W also is. Let W ⊥ be the orthogonal lattice of W with respect to the Beauville-Bogomolov quadratic form. Since Γ preserves c 1 (L) and c 1 (L) ⊥ ∩ MBM(X) • , Γ preserves W and W ⊥ . We consider the following homomorphism and the projection is finite and the kernel of µ 2 is finite. Hence the kernel of Γ → µ 2 • µ 1 (Γ) is finite. Therefore it is enough to prove that µ 2 • µ 1 (Γ) is almost abelian of rank at most by [15,Proposition 9.3 (2)]. Since W ⊥ is a lattice whose index is (1, n−rank(W )−1) and Γ preserves c 1 (L), µ 2 • µ 1 (Γ) is an almost abelian group whose rank is at most n − rank(W ) − 2 by [14, Proposition 2.9] and we are done. Next we will prove that Since W is negative definite and contained in c 1 (L) ⊥ , we have a subgroup Γ 0 of Mon(X) by Corollary 2.1. We note that Γ 0 is isomorphic to Z dim NSR(X)−rank(W )−2 . Let g be an element of Γ 0 . By [1,Theorem 1.19] and [13,Lemma 5.17], g(Amp(X)) coincides with a connected component of C NS (X) \ e∈MBM(X) e ⊥ whose closure contains c 1 (L). Hence, if g(Amp(X)) = Amp(X), there exists an element e of c 1 (L) ⊥ ∩ MBM(X) • such that g(Amp(X)) ⊂ e >0 and Amp(X) ⊂ e <0 . By the definition of W and Corollary 2.1, g fixes c 1 (L) and all elements of c 1 (L) ⊥ ∩MBM(X) • . Hence there are no such elements in c 1 (L) ⊥ ∩ MBM(X) • . Therefore g(Amp(X)) = Amp(X) and there exists an element Φ of Aut(X, L) such that Φ * = g by Theorem 1.2. This implies that Γ 0 is a subgroup of Aut(X, L) and the rank of Aut(X, L) coincides with dim NS R (X) − rank(W ) − 2.
5. Proof of Theorem 1.7 Lemma 5.1. Let Λ be a lattice whose index is (2, rank(Λ) − 2). We fix a positive integer N and define We denote by Gr ++ (2, Λ R ) the open set of Grassmanian Gr(2, Λ R ) which consists of positive 2-planes in Λ R . Let Gr ++ (2, Λ R ) • be a subset of Gr ++ (2, Λ R ) defined by Proof. Let w be an element of Λ R such that w, w > 0. It is enough to prove that the subset V of w ⊥ defined by is open in w ⊥ . It is obvious that V = ∅. We choose an element v of V . For a positive number ǫ, we consider the subset of w ⊥ defined by Since the signature of the bilinear form on w ⊥ is (1, dim w ⊥ − 1), the above set is compact. Hence the set is finite. Therefore min x∈ΛN ∩w ⊥ | v, x | is positive and we are done.
We recall the definition of marked irreducible symplectic manifolds, their moduli and the global period map.
Definition 5.1. Let Λ be a lattice whose index is (3, rank(Λ) − 3). A marked irreducible symplectic manifold (X, ϕ) is a pair of an irreducible symplectic manifold X and an isometry ϕ : H 2 (X, Z) → Λ. Two marked irreducible symplectic manifold (X, ϕ) and (X ′ , ϕ ′ ) are isomorphic if there exists an isomorphism Φ : X ∼ = X ′ such that ϕ ′ = ϕ • Φ * , where Φ * is the induced isometry H 2 (X ′ , Z) → H 2 (X, Z). A moduli space of marked irreducible symplectic manifold M Λ is the set of isomorphic classes of marked irreducible symplectic manifolds. We define the global period map P : M Λ → P(Λ C ) by The following lemma is well-known for specialist and we add it for readers convenience. Proof. Let X → Def(X) be the Kuranishi family of X. By [5, Théorème 5], we have a holomorphic morphism p X : Def(X) → Ω Λ . If (X ′ , ϕ ′ ) is another marked irreducible symplectic manifold which is isomorphic to (X, ϕ) in the sense of Definition 5.2, p X ′ : Def(X ′ ) → Ω Λ can be patched p X by universality of the Kuranishi space. Hence M Λ carries a structure of a complex manifold and P is holomorphic. Since p X is locally isomorphic, the image of P is an open set of Ω Λ .
The following Proposition is the punch line of the proof of Theorem 1.7.a Proposition 5.1. Let X be an irreducible symplectic manifold whose Betti number is greater than five and L a line bundle on X with q X (c 1 (L)) = 0, where q X is the Beauville-Bogomolov quadratic form. Then there exists an irreducible symplectic manifold X ′ and a line bundle L ′ which has the following four properties: (1) The pairs (X, L) and (X ′ , L ′ ) are deformation equivalent in the sense of Definition 1.6; (2) The line bundle L ′ is nef; (3) The Picard number of X ′ is equal to dim H 2 (X ′ , R) − 2 and; (4) The intersection c 1 (L ′ ) ⊥ ∩ MBM(X ′ ) is empty.
We will define two subsets of Ω Λ,ℓ ⊥ . The first one is defined by We choose an element t of Ω Λ,ℓ ⊥ . There exist sequences a m and b m in Λ Q such that lim m→∞ a m = Re(t), lim Since t ∈ Ω Λ,ℓ ⊥ , Re(t), Re(t) > 0, Im(t), Im(t) > 0 and Re(t), Im(t) = 0. Hence we may assume that a m , a m > 0 and b m , b m > 0 for all m. Moreover we may assume that lim m→∞ a m , b m = 0. We define other sequences c m and d m in Λ Q by Then a m + √ −1d m ∈ Ω max Λ,ℓ ⊥ . By definition, lim m→∞ a m + √ −1d m = t and we are done.