Effectivity of Semi-positive Line Bundles

We review work by Campana–Oguiso–Peternell (J Differ Geom 85(3):397–424, 2010) and Verbitsky (Geom Funct Anal 19(5):1481–1493, 2010) showing that a semi-positive line bundle on a hyperkähler manifold admits at least one non-trivial section. This is modest but tangible evidence towards the SYZ conjecture for hyperkähler manifolds.

where I(h) denotes the multiplier ideal sheaf.
(ii) Due to [21,Thm. 2], for q > n the morphism H q (X, L ⊗ I(h)) / / H q (X, L) induced by the inclusion L ⊗ I(h) ⊂ L is the zero map for any nef line bundle L on a compact hyperkähler manifold X of dimension 2n. In fact, according to another result of Verbitsky [24,Thm. 1.6], one has H q (X, L) = 0, q > n, for any nef and, more generally, for any pseudo-effective line bundle L.

Finiteness of Non-polar Hypersurfaces
An integral hypersurface Y ⊂ X of a compact complex manifold is called polar if there exists a meromorphic function f ∈ K(X) that has a pole along Y , i.e. Y is contained in the pole divisor (f ) ∞ of f . On a projective manifold, every integral hypersurface is polar. However, for general non-projective manifolds this fails, but the following result was proved by Fischer-Forster [8] and in the case that K(X) = C by Krasnov [14].

Proposition 2.3.
A compact connected complex manifold X contains at most finitely many integral hypersurfaces Y ⊂ X that are not polar. More precisely, the number of non-polar hypersurfaces is bounded by h 1,1 (X) + dim(X) − h 1,0 (X).
where K X is the sheaf of rational (meromorphic) functions and K(X) = H 0 (X, K X ) is the function field of X. In particular, if K(X) = C, then for any vector bundle E one has h 0 (X, E) rk(E).
Proof. Suppose there exist sections s 1 , . . . , s r+1 ∈ H 0 (X, E ⊗ K X ) linearly independent over K(X). Then there is a proper closed analytic subset such that all sections s i are holomorphic on its open complement U ⊂ X and such that (after renumbering) the sections s 1 , . . . , s r span the subspace s 1 (x), . . . , s r+1 (x) ⊂ E(x) of constant (maximal) dimension r at every point x ∈ U . In particular, on U we can write ( * ) s r+1 = r i=1 a i · s i for certain holomorphic functions a i ∈ O X (U ). It suffices to check that the a i are meromorphic functions which is a local question. Thus, we may think of the s i as vectors s i = (s i j ) j=1,...,r of meromorphic functions and view (a i ) as a solution of the system of linear equations ( * ). Expressing (a i ) in terms of the adjoint matrix and the vector (s r+1 j ) proves that all a i are indeed meromorphic.
Proof of Proposition 2.3. We shall follow [14] and assume K(X) = C. This is the only case that will be needed for Corollary 2.5 and its application later on. For the general case we refer to [8].
Applying d log, the sheaf of complexified Cartier divisors K * X /O * X ⊗ Z C is identified with the quotient of Ω 1 X ⊂ Ω 1 X,log , where the latter sheaf is by definition locally generated by all holomorphic one-forms and logarithmic one-forms d log f with f a local section of K * X . Taking cohomology yields a long exact sequence Since H 0 (X, Ω 1 X,log ) ⊂ H 0 (X, Ω 1 X ⊗K X ) and since we assume K(X) = C, the lemma . Corollary 2.5. If a compact complex manifold X contains infinitely many integral hypersurfaces, then its algebraic dimension satisfies a(X) > 0, i.e. K(X) = C. (ii) There exist infinitely many integral hypersurfaces Y ⊂ X, and in particular

Sections of Twists of Vector Bundles
Proof. Let F ⊂ E be the coherent subsheaf of E generated by all homomorphisms L −m i / / E. By assumption, F is non-trivial, so F is a torsion free sheaf of positive rank r:=rk(F ) > 0. Furthermore, we may assume that there exist integers n 1 , . . . , n r−1 (among the m i ) and an unbounded subsequence m j of m i for which there exist injections Taking determinants yields non-trivial global sections s j ∈ H 0 (X, M ⊗ L m j ), where M := det(F ) and m j := n i + m j . This proves (i).
Let us turn to (ii). There is nothing to prove in the case that X is projective or weaker that K(X) = C. So we assume that X contains only finitely many integral for an unbounded sequence m j and, in particular, L is Q-effective.
A priori, X could contain only one integral hypersurface Y ⊂ X and all sections of powers of L m are of the form s m for some s ∈ H 0 (X, L) with Z(s) = Y . In other words, the above result only ensures the existence of one non-trivial section of the line bundles L m up to passing to powers, which is not enough to make progress on Conjecture 1.2.

Cones on Hyperkähler Manifolds
We recall some basic notations and facts concerning the various cones relevant for the arguments below.
The positive cone C X ⊂ H 1,1 (X, R) of a compact hyperkähler manifold X is the connected component of the open set of all classes α ∈ H 1,1 (X, R) with q(α) > 0 that contains a positive class. It contains the Kähler cone K X ⊂ C X of all Kähler classes as an open subcone. The closure of the Kähler cone K X ⊂ C X , the nef cone, is the set of all classes α ∈ C X with C α 0 for all rational curves C ⊂ X, cf. [12,Prop.3.2], and the open Kähler cone K X ⊂ C X is the set of all classes α ∈ C X with The birational Kähler cone BK X is by definition the union K X of all Kähler cones of birational compact hyperkähler manifolds X . Here, we use that any birational correspondence X ∼ X induces a natural Hodge isometry H 2 (X, Z) ∼ = H 2 (X , Z), cf. [11,Prop.25.14]. Clearly, BK X ⊂ C X and according to [11,Prop.28.7] its closure BK X ⊂ C X , the modified nef cone, is the set of all classes α ∈ C X with q(α, D) 0 for all uniruled divisors. By linearity, it suffices to test this for prime exceptional divisors, i.e. irreducible divisors D ⊂ X with q(D) < 0. Alternatively, BK X can be described as the dual of the pseudo-effective cone E X of all classes α ∈ H 1,1 (X, R) that can be represented by a positive current, see [12,Cor. 4.6]. In particular, all effective divisors D ⊂ X define classes in E X . Note that in particular q(L, D) 0 for any nef line bundle L and any effective divisor D ⊂ X. 394 F. Anella and D. Huybrechts Vol. 90 (2022) According to Boucksom [3,Thm. 4.8], any pseudo-effective class α ∈ H 1,1 (X, R) admits a Zariski decomposition α = P (α) + N (α), where P (α) ∈ BK X and N (α) is the class of an exceptional R-divisor, i.e. N (α) = a i D i with D i ⊂ X irreducible divisors such that the matrix (q(D i , D j )) is negative definite. Furthermore, P (α) and N (α) are orthogonal, i.e. q(P (α), N(α)) = 0. We shall need the Zariski decomposition only for divisor classes α ∈ H 1,1 (X, Z) and in this case P (α) and N (α) are in fact rational.

Stability of the Tangent Bundle
Due to existence of a Kähler-Einstein metric in each Kähler class, the tangent bundle T X of a compact hyperkähler manifold X is μ-stable with respect to any Kähler class ω ∈ K X . In fact, stability holds with respect to all ω in the interior of BK X , see also Sect. 5.1. Proof. In the projective case, the assertion is a consequence of a general result due to Campana-Peternell [6, Thm. 0.1] showing that any torsion free quotient (Ω 1 X ) ⊗q / / / / F has a pseudo-effective determinant det(F) unless X is uniruled. Verbitsky [25] gives an alternative argument relying on the observation that all tensor powers Ω ⊗N X of the cotangent bundle are μ-semistable with respect to any class in the birational Kähler cone. More precisely, let α ∈ BK X be a class corresponding to a Kähler class ω ∈ K X on some birational model X of X. Since X and X are isomorphic in codimension one, the inclusion M ⊂ Ω ⊗N X carries over to an inclusion M ⊂ Ω ⊗N X . To conclude use the stability of Ω X , which proves q(α, M ) = q(ω , M ) 0. Hence, q(α, M * ) 0 for all α ∈ BK X , i.e. M * is pseudoeffective.

Proofs
In this section we present two proofs of the main theorem. The original of Campana-Oguiso-Peternell [5] applies only to the case that the hyperkähler manifold is nonprojective. Verbitsky [25] showed how to combine the original approach with Boucksom's Zariski decomposition to also cover the algebraic case. In the next section we will sketch a different approach that reduces the projective case to the non-projective one.

Non-algebraic Case
We follow the arguments in [5].
Proof. Assume L is a non-trivial nef line bundle on a non-projective compact hyperkähler manifold X of dimension 2n and assume q(L) = 0. Suppose H 0 (X, L m ) = 0 for all m > 0. The Riemann-Roch formula [10,11] simply states χ(X, L m ) = n+1. Thus, there exists an even number q > 0 and an unbounded sequence m i of positive integers such that H q (X, L m i ) = 0. By virtue of Proposition 2.1 this implies H 0 (X, Ω 2n−q X ⊗ L m i ) = 0. Combining Corollary 2.5 and Proposition 2.6, we conclude that L is Q-effective, in which case we are done, or K(X) = C. For example, the former case holds if ρ(X) = 1. In the latter case, the algebraic reduction [22,Ch. 3] provides us with a diagramX Here,X is a compact complex manifold, B is smooth and projective of dimension at least one, and π is birational. The pull-backf * H of a very ample line bundle H on B can be written asf * H ∼ = π * M ⊗ O(−E) with E ⊂X effective, in fact π-exceptional but possibly trivial, and M ∈ Pic(X). This yields inclusions Since dim(B) 1, this shows that M is non-trivial and effective. In fact, as H is very ample, we may assume that M admits two linearly independent sections with distinct zero divisors D 1 , D 2 ⊂ X without common irreducible components.
According to [3,Prop. 4.2], for any two such divisors D 1 , D 2 ⊂ X we have q(D 1 , D 2 ) 0. Indeed, up to a positive scalar q(D 1 , D 2 ) = D 1 ∩D 2 (σσ) n−1 0, since (σσ) n−1 is a positive form. Applied to our situation this yields q(M ) 0. On the other hand, since X is assumed to be non-projective, the projectivity criterion [10,Thm. 3.11] implies q(M ) 0. Therefore, q(M ) = 0. However, as the form q restricted to H 1,1 (X, R) satisfies the Hodge index theorem, every line bundle M on X with q(M ) = 0 is a rational multiple of L. As L was assumed semi-positive (hence, nef) and M is effective, M is a positive rational multiple of L. Therefore, L is Q-effective.

Algebraic Case
In fact, the following arguments taken from [25] apply also to non-algebraic hyperkähler manifolds and thus subsume the original proof in [5] Proof. The first part of the proof is identical to the one in the non-algebraic case. 0, see Sect. 2.4. Therefore, q(M, L) = 0. In the case that ρ(X) = 2, we can conclude already that M is a rational multiple of L and that, therefore, L is Q-effective.
For ρ(X) > 2, we consider the Zariski decomposition of the pseudo-effective line bundle M * as P + N with P contained in the closure of the birational Kähler 396 F. Anella and D. Huybrechts Vol. 90 (2022) cone and N exceptional effective. In particular, q(P ) 0 with q(P, L) > 0 unless P is a rational multiple of L and q(N ) < 0 unless N = 0. Then, 0 = q(L, M ) = q(L, P )+q(L, N ) with both summands non-negative and, therefore, both zero. Thus, the Zariski decomposition of M * is of the form λL + N , with λ ∈ Q 0 . On the other hand, M ⊗L m i is effective for an unbounded sequence of positive integers m i . Hence, (m i − λ)L can be written as the sum of the two effective divisors M ⊗ L m i and N . Therefore, L itself is Q-effective.

Semi-positivity Under Deformations
We will now show that alternatively the proof in the algebraic case can be reduced via deformation to the non-algebraic case. The techniques are potentially relevant to make progress on Conjecture 1.2.
First recall that for a smooth proper family X / / Δ of complex manifolds with central fibre X = X 0 of Kähler type, all nearby fibres X t are of Kähler type as well, i.e. for all t after shrinking Δ to an open neighbourhood of 0 ∈ Δ. More precisely, if the Kähler class on X stays of type (1, 1) on the nearby fibres, then it is Kähler there as well. This classical result is due to Kodaira and Spencer [15]. Note that since the Kähler property is a combination of the open condition that a real (1, 1)form ω is positive and the closed condition dω = 0, this is a priori not clear. In the case of closed semi-positive forms ω, the corresponding statement fails. Similarly, if α ∈ H 1,1 (X, R) is a nef class that stays of type (1, 1) on all the fibres X t , as a class on X t it need not be nef, see [18] for an example.
Proof. We skip the proof. This is a point-wise statement which boils down to linear algebra.

Open Questions
Besides the two conjectures stated in the introduction, there are a number of related questions that seem approachable.

Stability of the Tangent Bundle
Due to the existence of a hyperkähler (and hence Kähler-Einstein) metric on a hyperkähler manifold X, the tangent bundle T X is μ-stable. In fact, T X is μ-stable with respect to every Kähler class and, as explained in Sect. 2.5, with respect to the generic class in the birational Kähler cone.
Question 5.1. Is the tangent bundle T X of a hyperkähler manifold μ-stable with respect to any class in the positive cone?
This would subsume Proposition 2.7 and would allow one to conclude the stronger statement that the line bundle M constructed in the proof of Proposition 2.6 and used in the two proofs in Sect. 3 is contained in the closure of the positive cone.

Elliptic and Parabolic Hyperkähler Manifolds
The paper [5] discusses the possibilities for the algebraic dimension a(X) = trdegK(X) of a compact hyperkähler manifold and how the algebraic dimension is related to the intersection form on the Néron-Severi group. We only touch upon one aspect here.
Question 5.2. Assume X is a compact hyperkähler manifold of algebraic dimension zero, i.e. K(X) = C. Is the Beauville-Bogomolov form q on NS(X) ∼ = H 1,1 (X, Z) negative definite?
Following [5], X is called elliptic if q is negative definite on NS(X). It is known that elliptic hyperkähler manifolds satisfy K(X) = C. The above question is the converse.
Similarly, X is called parabolic if q on NS(X) is semi-negative definite with one isotropic direction and hyperbolic if it has signature (1, ρ(X) − 1). By the Hodge index theorem and the projectivity criterion for hyperkähler manifolds, the latter is equivalent to X being projective. The analogue of Question 5.2 in the parabolic case is the conjecture that X is parabolic if and only if a(X) = n. According to [5,Thm. 3.6], any non-algebraic compact hyperkähler manifold satisfies a(X) n, so that the cases 0 < a(X) < n would need to be excluded.
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