A Nonvanishing Conjecture for Cotangent Bundles

In this paper we study the positivity of the cotangent bundle of projective manifolds. We conjecture that the cotangent bundle is pseudoeffective if and only the manifold has non-zero symmetric differentials. We confirm this conjecture for most projective surfaces that are not of general type.

1. Introduction 1.A. Main result.A central part in the minimal model program in algebraic geometry is the so-called nonvanishing conjecture: given a projective manifold or, more generally, a variety with klt singularities, X, whose canonical class K X is pseudoeffective, one has H 0 (X, O X (mK X )) = 0 for some positive integer m.This conjecture has been proven some time ago in dimension at most three, but is wide open in higher dimensions.In analogy to the nonvanishing conjecture, one might ask for 1.1.Conjecture.Let X be a normal projective variety with klt singularities.Let 1 ≤ q ≤ dim X.Then Ω

[q]
X , the sheaf of reflexive holomorphic differentials in degree q, is pseudoeffective, (see Definition 3.5), if and only for some positive integer m one has H 0 (X, S [m] Ω [q] X ) = 0.In the case q = dim X, this is of course the nonvanishing conjecture stated above.The only general result confirming Conjecture 1.1 is given in [HP19, Thm.1.6]:Suppose X is klt and smooth in codimension two with K X ≡ 0. If Ω [1| X is pseudoeffective, there is a quasi-étale cover X → X such that q( X) > 0. In particular one has H 0 (X, S [m] Ω [1] X ) = 0 for some positive integer m ([Ane18, Prop.2.2], see also Lemma 4.6).While the pseudoeffectivity of K X is equivalent to the non-uniruledness of the manifold, we do not know many examples where Ω q X is pseudoeffective, but not big.We expect that this property is actually quite restrictive, our Theorem 1.2 confirms this intuition in the first non-trivial case.In this paper we are mainly interested in the case q = 1.Already for smooth surfaces, Conjecture 1.1 is delicate.In this case we can assume without loss of generality that X is minimal, see Proposition 4.1.By surface classification, see Corollary 4.14, the problem starts with κ(X) = 1.We basically settle this case: 1.2.Theorem.Let f : X → B be a (minimal) smooth elliptic surface with κ(X) = 1 such that Ω 1 X is pseudoeffective.Suppose one of the following.a) f is not isotrivial b) f is isotrivial and the general fiber does not have complex multiplication c) The tautological class on P(Ω 1 X ) is nef in codimension one.Then q(X) > 0 (see Definition 2.2), so there is a positive integer m such that Note that each of the cases requires a different proof, Theorem 1.2 is obtained as the union of Corollary 5.5, Corollary 6.8 and Corollary 6.14.
In general, the pseudoeffectivity of Ω [1] X does not imply q(X) > 0: a smooth complete intersection surface X ⊂ P N is simply connected, so q(X) = 0.However, if N ≥ 4 and the multidegrees are sufficiently high, the cotangent bundle Ω 1 X is ample [Bro14].For surfaces of general type, Conjecture 1.1 is open.If c 2 1 (X) > c 2 (X), then by Bogomolov's vanishing theorem but already the boundary case c 2 1 (X) = c 2 (X) is unclear.In higher dimension, things get worse due to the singularities of minimal models.For example, we know Conjecture 1.1 for terminal threefolds with numerically trivial canonical class, but we cannot deduce easily Conjecture 1.1 for smooth threefolds X with κ(X) = 0, although X has a terminal minimal model as above.We would finally like to point out the connection to a question posed by H.Esnault, see [BKT13]: let X be a projective (or compact Kähler) manifold whose fundamental group π 1 (X) is infinite.Does there exist a positive integer m such that H 0 (X, S m Ω 1 X ) = 0? An intermediate step might be to prove that Ω 1 X is pseudoeffective.Brunebarbe-Klingler-Totaro confirm Esnault's conjecture if there is a representation π 1 (X) → GL(N, C) with infinite image.The key point of their proof is to show that in many cases, the cotangent bundle Ω 1 X is even big.Theorem 1.2 fits in the framework of Esnault's conjecture: it is well-known to experts that if the cotangent bundle of an elliptic surface is not pseudoeffective, then π 1 (X) is finite (see Appendix A for a proof).In view of Theorem 1.2, the two properties should actually be equivalent (which is actually the case up to the exceptional isotrivial case of Theorem 1.2) and imply the existence of symmetric differentials.
1.B.Strategy of the proof.Let X be a smooth projective surface such that K X is nef and c 1 (K X ) 2 = 0. Then K X is semiample, so we have the Iitaka fibration f : X → B such that the general fibre F is elliptic and mK X ≃ f * A with A an ample divisor.Assume now that Ω X := Ω1 X is pseudoeffective, then one expects that there exists a pseudoeffective subsheaf of Ω X that is induced by a pull-back from the base B. Let f * Ω B → Ω X be the cotangent map, and denote by the saturation.If f * Ω B (D) is pseudoeffective, Proposition 5.2 shows that q(X) > 0.
Thus the main issue in Theorem 1.2 is to show that f * Ω B (D) is pseudoeffective.A natural approach is to show that the sheaf Ω X → ω X/B (−D) is not pseudoeffective if f is not almost smooth 1 .However, by a theorem of Brunella [Bru06], the line bundle ω X/B (−D) is always pseudoeffective!This leads us to considering the more refined quotient sheaf where Z has support in the singular points of the reduction of the fibres.The basic idea of the proof of Theorem 1.2 is to show that the torsion-free sheaf I Z ⊗ ω X/B (−D) is not strongly pseudoeffective (see Definition 3.7), although its bidual is a pseudoeffective line bundle.Thanks to a result of Demailly-Peternell-Schneider [DPS94] this idea leads immediately to the result in the non-isotrivial case, see Section 5.For an isotrivial fibration this approach only yields the weaker statement appearing as part c) of the main theorem, see Subsection 6.D. On the other hand we know that X is birational to a quotient (C × E)/G, so we aim to compute explicitly the spaces of global sections following a strategy introduced by Sakai [Sak79].Apart from the technical setup, the main difficulty is to understand the local obstruction near the fixed points of the group action.For A 1 -singularities this information is provided by [BTVA19, Prop.3.2.].We expect that a similar description of the local obstruction for klt singularities would allow to handle the case when the elliptic curve has complex multiplication.
1.C.Structure of the paper.In Section 3 we introduce a positivity notion ("strongly pseudoeffective") that is adapted for this type of torsion-free sheaf, and present material on pseudoeffective torsion free sheaves which will be used in later sections.
Section 4 is concerned with some general results on varieties with pseudoeffective cotangent sheaves.In particular, generalised Kodaira dimensions are introduced and a relation to the MRC fibration is studied.The last two sections are devoted to the proof of Theorem 1.2.Section 5 gives the general setup and settles the case that the elliptic fibration is not isotrivial.The surprisingly difficult isotrivial case finally is studied in Section 6.

Basic notations
We work over the complex numbers, for general definitions we refer to [Har77].We use the terminology of [Deb01] and [KM98] for birational geometry and notions from the minimal model program and [Laz04] for notions of positivity.Manifolds and varieties will always be supposed to be irreducible and reduced.
2.1.Notation.Let X be a normal complex variety.As usual, Ω 1 X denotes the sheaf of Kähler differentials, and we set If X is klt and µ : X → X a is desingularization, then by [GKKP11, Thm.1.4],
A finite surjective map γ : X ′ → X between normal varieties is quasi-étale if its ramification divisor is empty (or equivalently, by purity of the branch locus, γ is étale over the smooth locus of X).

Definition.
Let X be a normal projective variety with klt singularities.Then, as usual, X ) is the irregularity of X. Further, we denote by q(X) the maximal irregularity q( X), where X → X is any quasi-étale cover.
While the irregularity q(X) is a birational invariant of projective varieties with klt singularities, this is not the case for q(X): 2.3.Example.Let τ : E 1 → P 1 be a (hyper)elliptic curve, and denote by i E1 the involution induced by the double cover.Let E 2 be an elliptic curve, and denote by i E2 the involution defined by z → −z.The surface is normal and has A 1 -singularities in the branch points of the quasi-étale map E 1 × E 2 → X ′ .The projection on the first factor induces an isotrivial elliptic fibration f ′ : that has exactly 2g(E 1 ) + 2 singular fibres, all of them are multiple fibres of multiplicity 2 such that the reduction is isomorphic to Denote by µ : X → X ′ the minimal resolution, then the induced elliptic fibration f : X → P 1 is relatively minimal, isotrivial and has exactly 2g(E 1 ) + 2 singular fibres, all of them of type I * 0 (in Kodaira's terminology, see [BHPVdV04, V, Table 3]).In the classical case where E 1 is an elliptic curve, the surface X is a K3 surface of Kummer type.In particular we have q(X) = 0. Remark.The reduction of any fibre of π is a projective space, and π is locally trivial if and only if F is locally free [AT82, p.27].For locally free sheaves, the following definition of pseudoeffectivity is now in common.

Definition.
Let X be a projective variety, and let F be a locally free sheaf on X. Denote by π : P(F ) → X the projectivisation, and by ζ the tautological class on P(F ).We say that F is pseudoeffective if ζ is a pseudoeffective Cartier divisor class.
Remark.By [Dru18, Lemma 2.7], the locally free sheaf F is pseudoeffective if and only if for some ample Cartier divisor H on X and for all c > 0 there exist numbers j ∈ N and i ∈ N such that i > cj and We will use the following lemma, which will be generalised below.
3.4.Lemma.Let f : X → Y be a surjective morphism of projective varieties and F a locally free sheaf on Y .Then F is pseudoeffective if and only if f * (F ) is pseudoeffective.
Remark.Note that the statement applies in particular to the normalisation, so for locally free sheaves pseudoeffectivity can be verified on the normalisation.
Proof.Recall the pull-back formula for the tautological classes where p : P(f * (F )) = P(F ) × Y X → P(F ) is the canonical projection.Thus we are reduced to the case where F has rank one, which is immediate by [Laz04, Thm.2.2.26, Prop.2.2.43].

3.B.
Strongly pseudoeffective torsion-free sheaves.For the purpose of this paper it is not sufficient to discuss the positivity of locally free sheaves, in fact we will need the more subtle positivity properties of torsion-free sheaves.It will suffice to consider normal varieties.We first recall the definition of pseudoeffectivity for reflexive sheaves from [Dru18] and [HP19, Defn.2.1].
3.5.Definition.Let X be a normal projective variety, and let F be a reflexive sheaf on X.Then F is pseudoeffective if for some ample Cartier divisor H on X and for all c > 0 there exist numbers j ∈ N and i ∈ N such that i > cj and Remark.An equivalent definition using an adapted resolution of singularities of P(F ) is given in [HP19].
3.6.Example.Our definition of pseudoeffectivity is less restrictive than [BDPP13, Defn.7.1]: if G ⊂ F is a pseudoeffective reflexive subsheaf, then F is pseudoeffective.In particular if F = L ⊕ H where L is pseudoeffective and H an antiample reflexive sheaf, then F is pseudoeffective in the sense of Definition 3.5, but not in the sense of [BDPP13,Defn.7.1].Definition 3.5 makes also sense for torsion-free sheaves, but would not be very useful: by definition a torsion-free sheaf would be pseudoeffective if and only if its bidual is pseudoeffective.The following definition takes this difference into account: 3.7.Definition.Let X be a normal projective variety, and let F be a torsion free sheaf on X.We say that F is strongly pseudoeffective if for some ample Cartier divisor H on X and for all c > 0 there exist numbers j ∈ N and i ∈ N such that i > cj and H 0 (X, (S i F )/Tor ⊗ O X (jH)) = 0.
3.8.Remark.For locally free sheaves, the Definitions 3.3, 3.5, 3.7 obviously coincide.Even for reflexive sheaves, Definition 3.7 is more restrictive than Definition 3.5: in general (S i F )/Tor is not reflexive and has less global sections than its bidual, so we might have H 0 (X, (S i F )/Tor ⊗ O X (jH)) = 0, although F is pseudoeffective in the sense of Definition 3.5.We thank C. Gachet for the following example: be the double cover of the A 1 -singularity, we identify the polynomial ring of X to its image C[u 2 , v 2 , uv] in C[u, v].The invariant elements under the involution are exactly the odd polynomials, i.e. the polynomials that can be written as uf Remembering that this ring is actually the function ring of the A 1 -singularity, we see that (C[u, v] Z2 ) ⊗2 is isomorphic to the maximal ideal defining the origin.Let now q : A → X be the quotient of an abelian surface under the involution z → −z, so X is the singular Kummer surface.Since S [2] Ω X is globally generated, the sheaf of reflexive differentials Ω [1] X is pseudoeffective.Let us see that it is not strongly pseudoeffective: we have Ω X ≃ F ⊕F , where F is the sheaf of Z 2 -invariants for the natural action on Ω A ≃ O A dz 1 ⊕ O A dz 2 .It is immediate to see that, near the fixed points, the Z 2 -action on O A dz l identifies to the action j in the paragraph above.Thus, using the local computation, we see that Xsing if i even, Xsing ⊗ F if i odd.Combined with Example 3.12 this shows that F is not strongly pseudoeffective.
In general it is not clear if one can check strong pseudoeffectivity by looking at a tautological class on (a modification of) the projectivisation.However there is a natural construction in a special case: 3.10.Setup.Let F be a torsion-free sheaf on a normal projective variety X such that F ≃ I Z ⊗ E where I Z is an ideal sheaf and E is a locally free sheaf.Let µ : X → X be the blow-up of the ideal sheaf I Z , then X is a (not necessarily normal) variety [Har77, II, Prop.7.16].We denote by Note also that if Z is locally generated by a regular sequence, one has S i I Z ≃ I i Z for all i ≥ 0 (e.g.[BC18, Prop.2.2.8]).In particular the blowup Bl IZ (X) coincides with the projectivisation P(I Z ).
3.11.Lemma.In the situation of Setup 3.10, the torsion-free sheaf F is strongly pseudoeffective if and only if the locally free sheaf O X (1) ⊗ µ * E on the variety X is pseudoeffective.
Proof.Let H be an ample Cartier divisor on X.By the projection formula and (1) one has for all i ≥ 0. Moreover we know by [Mic64] that for all i ≥ 0. Now we apply [Dru18, Lemma 2.7] (see [HLS20, Lemma 2.2.] for the case where the divisor is only big).
3.12.Example.Let X be a normal projective variety, and let I Z ⊂ O X an ideal sheaf.Then I Z is not strongly pseudoeffective by Lemma 3.11.On the other hand let Z ⊂ P 2 be a point, then where F is the strict transform of a line through Z.

Corollary.
In the situation of Setup 3.10, suppose that the ideal sheaf I Z is locally generated by a regular sequence (e.g. if Z is a locally complete intersection scheme).Then the following statements are equivalent: a) The sheaf F is strongly pseudoeffective; b) The locally free sheaf O X (1) ⊗ µ * E on the blow-up Proof.The equivalence between a) and b) is shown in Lemma 3.11.
Denote by π : P(µ * E) → X the projectivisation, and by Since I Z is locally generated by a regular sequence, we have S i I Z ≃ I i Z for all i ≥ 0 (e.g.[BC18, Prop.2.2.8]).Thus for all i ≥ 0 we have . Thus b) and c) are equivalent.

3.C.
Generalised Kodaira-Iitaka dimensions and functoriality.In this subsection we introduce a Kodaira-Iitaka dimension for reflexive sheaves and establish functoriality.
3.14.Definition.Let X be a normal projective variety, and let F be a reflexive sheaf on X.Let π : P → X be a desingularization of the normalization of the unique component P ′ (F ) of P(F ) dominating X such that the preimage of the singular locus of X and of the singular locus of the sheaf F is a divisor in P .Let ζ be a tautological class on P , [HP19, Defn.2.2].Then we define κ(X, F ) = κ(P, ζ).
Remark.If π : P → X denotes the projection, then for all positive numbers m, and therefore the definition is independent on the choices made.We will use Definition 3.14 in Section 4 to introduece a generalised Kodaira dimenion of X (Definition 4.5).The next result generalizes Lemma 3.4 for finite morphisms.
3.15.Lemma.Let f : X → X be a finite morphism of normal projective varieties.Let F be a reflexive sheaf on X.Then the following holds: Proof.Let π : P → X be the projective manifold from Definition 3.14, and ζ a tautological class on P .By the construction of ζ (see [HP19, Defn.2.2]), we have for all positive integers m.We introduce the fibre product Let σ : P → P be a desingularization, which is an isomorphism outside the singular locus of P and set ζ := σ * p * 1 (ζ), where p 1 : P → P denotes the projection.Let π : P → X be the projection and π := π • σ.
Claim: There exists a π-exceptional divisor D on P , such that for all m ∈ N. Proof of the claim: By [Nak04, III.5.10.3] there exists π-exceptional divisor D on is reflexive for all m ∈ N. Let X 0 ⊂ X be the locus where X is smooth and F is locally free.Since X is normal and F is reflexive, the complement of X 0 has codimension at least two.Set now Thus by flat base change [Har77, III, Prop.9.3] we have an isomorphism [m] (F ) are isomorphic over X 0 .Since they are both reflexive, they are isomorphic : indeed the complement of X 0 has codimension at least two, since f is finite.For the same reason we have Let now µ : P → P ′ and p ′ 1 : P ′ → P be the Stein factorisation of the generically finite morphism p 1 • σ, i.e. µ is birational onto the normal variety for all m ∈ N.This shows that , where we use again the inclusion , by [Uen75, Thm 5.13], the statement follows.
In Section 4 this will be applied to sheaves of reflexive differentials, Corollary prop-cover2.

3.D.
Pseudoeffective sheaves on fibered surfaces.The results of this section will be relevant to the study of elliptic surfaces.Let us recall the Zariski decomposition on surfaces [Bau09, Thm.], [Laz04, Thm.2.3.19]: • Let D be an effective Q-divisor on a smooth surface.Then there exist uniquely determined effective Q-divisor P and N with such that P is nef, the divisor N = a j N j is zero or has negative definite intersection matrix and P • N j = 0 for all j.
• The same statement holds if D is pseudoeffective, except that in this case P is not necessarily effective.
The following lemma is well-known to experts, we give the details in order to prepare the proof of its singular version in Lemma 3.17.Proof.Note that the statements are invariant under taking multiples.
Up to replacing L by some multiple, we can assume that P and N are Cartier divisors.Since N is effective and we see that N has support in the fibres of f and ) is locally free of rank one, and we have where E is an effective divisor E, supported in fibers of f .Since P 2 ≥ 0 and E 2 ≤ 0 [BHPVdV04, III, Lemma 8.2], it follows E 2 = 0. Hence by (ibid) there exists a number k such that kE = f * (H) with some effective divisor H. Thus, again up to replacing L by some multiple, we have O X (P ) = f * M for some line bundle M on B.
If M ≡ 0, it is ample on B, so .
3.17.Lemma.Let X be an irreducible reduced projective surface, and let f : X → B be a fibration over a smooth curve B such that the general fibre F is smooth.Let L be a pseudoeffective line bundle on X such that L| F ≃ O F .a) There exists m ∈ N and a numerically trivial line bundle M on B such that for some numerically trivial line bundle M , then h 0 ( X, L⊗k ) > 1 for some k ∈ N.
3.18.Remark.We will frequently apply the lemma in the case where B is a rational curve.In this case one obtains h 0 ( X, L⊗m ) > 0 for some m ∈ N.
Since the dimension of the space of global sections is not necessarily invariant under normalisation, the statement requires some work: Proof.Note that the statement is invariant under taking multiples.
Let µ : X → X be the composition of normalisation and desingularisation, set is pseudoeffective, and we denote by the Zariski decomposition.By Lemma 3.16, a) we have, up to taking multiples, that O X (P ) ≃ f * M for some line bundle on M .Thus we see that where N is a Cartier divisor on X such that N ≡ µ * N .By Lemma 3.16, b),c) it is sufficient to show that we can choose N to be effective.
Proof of the claim.Since is locally free of rank one, hence for some m ≫ 0, we have Thus we can fix an effective Cartier divisor Up to taking multiples and replacing E B by a linearly equivalent divisor, we can also suppose that supp(f * E B ) ⊂ X nons .Now observe that is also a Zariski decomposition and the negative part of the Zariski decomposition is unique in the numerical equivalence class, we finally obtain N = µ * R.

Corollary.
Let X be a smooth projective surface, and let f : X → B a fibration over a smooth rational curve B. Let Z ⊂ X be a local complete intersection of codimension 2. Let L be a line bundle on X such that L F ≃ O F for the general fiber F of f .Then I Z ⊗L is strongly pseudoeffective if and only if κ(X, I Z ⊗L) ≥ 0, i.e., if there exists a positive integer m such that Proof.One direction being obvious, so assume that I Z ⊗ L is pseudoeffective.Let µ : X → X be the blow-up of X along Z and denote by E the exceptional divisor.Since Z does not surject onto B, the general fibre of f • µ is smooth.By Corollary 3.13 the the line bundle O X (1) ⊗ µ * (L) ≃ O X (−E) ⊗ µ * (L) is pseudoeffective.By Remark 3.18 we see that there exists some m ∈ N such that Since Z is a local complete intersection, we have µ * (O X (−mE)) = I m Z .Thus we conclude by the projection formula.

Pseudoeffective cotangent sheaves and the Nonvanishing Conjecture
In this section we gather some basic facts on the behaviour of pseudoeffective cotangent sheaves under birational maps and finite covers.We also establish a relation with the MRC fibration of the variety.
X is pseudoeffective, so does Ω [q] X .b) Suppose that X is smooth.Then the converse also holds.
Proof.a) We choose ample divisors Ĥ on X and H on X such that with E an effective divisor supported on the exceptional locus of µ.By assumption, for all c > 0, there are numbers i and j with i > cj such that X , we conclude.b) Suppose that X is smooth.By a), we may assume X to be smooth as well.Then all the involved sheaves are locally free, in particular X , and the claim follows.
4.2.Example.Assertion 4.1,b) fails in general if X is singular, even if X has only canonical singularities.In fact, the paper [GKP16] constructs a normal projective surface X with the following properties.
• X has only ADE singularities ; • the minimal desingularization X is rationally connected ; X .Another example is a K3 surface of Kummer type, see Example 2.3.

Corollary.
Let X be a normal projective variety with klt singularities, and let X X ′ be a composition of divisorial contractions and flips. If X is pseudoeffective, so does Proof.By Proposition 4.1,a) it suffices to treat the case of a flip µ : X X ′ .Since a flip is an isomorphism in codimension two, one has for all i, j ∈ N. Thus the condition in Definition 3.5 holds for a big Q-Cartier divisor, which is sufficient (see [HLS20, Lemma 2.2]).
X .If f is quasi-étale, the converse also holds.
Proof.Over the smooth locus of X we have a injective morphism X .Since the complement of f −1 (X nons ) has codimension at least two, the morphism extends to X ). which gives the first claim.
Assume now that f is quasi-étale and that Ω is pseudoeffective.Now we conclude by Lemma 3.15.
At this point we introduce generalised Kodaira dimension: 4.5.Definition.Let X be a normal projective variety with klt singularities and 1 ≤ q ≤ n.Then we define κ q (X) = κ(X, Ω [q] X ).
Proof.This follows from Lemma 3.15, since for a quasi-étale morphism X .
4.7.Remark.Let µ : X → X be a birational morphism of normal projective varieties with klt singularities.Then κ q ( X) ≤ κ q (X) with equality if X is smooth.The same inequality holds if µ : X X is a composition of divisorial contractions and flips.
Although Conjecture 1.1 can be formulated for any p, we are mainly interested in the case p = 1.We next confirm the conjecture for p = 1 in case K X ≡ 0. 4.8.Proposition.Let X be a normal projective variety with klt singularities such that K X ≡ 0. Assume that X is smooth in codimension two, e.g., X has terminal singularities.Then the following are equivalent: X is pseudoeffective ; b) we have q(X) > 0, i.e., there exists a quasi-étale cover X → X such that for some positive integer m, i.e., κ 1 (X) ≥ 0.
We will now discuss the relation between the pseudoeffectivity of Ω q X and the MRC fibration.First, the rational connectedness criterion given in [CDP15] can be stated as follows.
4.9.Theorem.Let X be a projective manifold of dimension n.Then X is rationally connected if and only if for all 1 ≤ q ≤ n the vector bundle Ω q X is not pseudoeffective.
Proof.If X is rationally connected, it is dominated by very free rational curves [Kol96, IV.3.9].It is then straightforward to verify the vanishing condition in Definition 3.5.Assume that Ω q X is not pseudoeffective for all 1 ≤ q ≤ n.Let F ⊂ Ω q X be an invertible subsheaf, then F is not pseudoeffective (see Example 3.6).Hence by [CDP15, Thm.1.1],X is rationally connected.Theorem 4.9 can be generalised as follows.
4.10.Theorem.Let X be a projective variety of dimension n.Fix some r ∈ {1, . . ., n} and assume that Ω [q] X is not pseudoeffective for all r ≤ q ≤ n.Then X is uniruled, and the base Z of the MRC fibration satifies dim Z ≤ r − 1.
Proof.By Proposition 4.1 we can assume without loss of generality that X is smooth.Since K X = Ω n X is not pseudoeffective, the manifold X is uniruled by [BDPP13].Hence we consider the MRC fibration f : X Z.
Up to replacing Z by a resolution and X by a blow-up, we may assume, by Proposition 4.1, that Z is smooth and f is a morphism.
Arguing by contradiction we suppose that d := dim Z ≥ r.By [GHS03], the variety Z is not uniruled, hence K Z is pseudoeffective.Choose H Z ample on Z and set H X = f * (H Z ).Since K Z is pseudoeffective, for all c > 0 there exist integers i and j with i > cj such that X is pseudoeffective and d ≥ r, a contradiction to our assumption.
If X is smooth, the converse to Theorem 4.10 is also true: 4.11.Proposition.Let X be a uniruled projective manifold of dimension n, and let f : X Z be the MRC fibration.If d = dim Z, then for every Ω q X is not pseudoeffective for all d + 1 ≤ q ≤ n.
Proof.Let F be a general fiber of f .Then F is rationally connected of dimension dim F = n − d.Let C ⊂ F be a general very free rational curve [Kol96, IV.3.9],so Thus for every q ≥ d + 1, the exterior power q T X | C is ample.Hence Ω q X is not pseudoeffective.
For smooth varieties, the MRC-fibration should allow to reduce Conjecture 1.1 to non-uniruled varieties: 4.12.Conjecture.Let X be a uniruled projective manifold, and let f : X Z be the MRC fibration to the projective manifold Z.Let 1 ≤ q ≤ n.Then Ω q X is pseudoeffective if and only if Ω q Z is pseudoeffective.
Note that, by Proposition 4.1, we may assume f holomorphic.Then one direction is clear: if Ω q Z is pseudoeffective, then so does f * (Ω q Z ) by Lemma 3.4.Hence Ω q X is pseudoeffective, see Example 3.6.Vice versa assume that Ω q X is pseudoeffective.Applying [BCHM10, Cor.1.3.2],f factors into a sequence of divisorial contractions and flips, ending with a Mori fiber space f ′ : X ′ → Z of relative Picard number one.By Corollary 4.3, we may therefore assume that f is a Mori contraction, but now X may have terminal singularities instead of being smooth.4.13.Proposition.Conjecture 4.12 is true in dimension three.
Proof.As just noticed it suffices to treat Mori contractions f : X → Z where X has terminal singularities.Since Z is not uniruled, we may assume that dim Z = 2; otherwise Z is a curve of genus g ≥ 1 and there is nothing to prove.Further, since K Z is pseudoeffective, only the case q = 1 needs to be treated.Now f is a generically a conic bundle [AW97, 4.1].More precisely, the singular locus of X being finite, there is a finite set B = f (Sing(X)) in Z such that, setting Z 0 = Z \ B and X 0 = f −1 (Z 0 ), the map f 0 : X 0 → Z 0 is a conic bundle with only finitely many non-reduced fibers.Furthermore, f −1 (B) is one-dimensional.
Since −K X is relatively ample and relatively globally generated on X 0 , we can choose a very ample Cartier divisor H on Z such that −K X/Z + f * H =: A is ample and satisfies H 0 (X, O X (A)) = 0.In particular there is an injection We claim that for every c > 1 there exist positive integers k, j such that k ≥ cj such that Since f −1 (B) has codimension at least two, this shows that f * Ω Z is pseudoeffective.
Thus Ω Z is pseudoeffective by Lemma 3.4.
Proof of the claim.Since Ω X is pseudoeffective, there exist positive integers i, j such that i ≥ 2cj such that We consider the canonical exact sequence Since X 0 → Z 0 is a conic bundle, we know that df cannot vanish along a divisor D.
Thus Ω X/Z is torsion free and the singular locus of Ω X/Z is at most one-dimensional.Thus we get for some k ∈ {0, . . ., i}.Since ω X0/Z0 ⊗ O X (jA) has negative degree on the fibres of f if i − k > j we see that k ∈ {i − j, . . ., j}.Note that since i ≥ 2cj and c > 1 this implies that k ≥ j.Using the morphism (3) obtain Since i − k ≤ j we finally obtain the claim.
Remark.The key point of the proof above is that the morphism df does not vanish along a divisor D 0 .In higher dimension, since the total space of the Mori fibre space is not necessarily smooth, this might very well happen.Then these type of arguments only show that f * (Ω q (D) is pseudoeffective where D has support inside the support of D 0 .At least in dimension three we also see that where X Z is the MRC fibration of a uniruled smooth threefold X and Z is smooth.
Proof.If κ(X) = −∞, the surface X is uniruled.Since Ω X is pseudoeffective, by Proposition 4.11 the base of the MRC fibration is a curve of genus at least one.Thus we have q(X) > 0. If κ(X) = 0, let X min be the minimal model of X.Then Ω Xmin is pseudoeffective by Proposition 4.1.Thus by Proposition 4.8 one has H 0 (X min , S m Ω 1 Xmin )) = 0 for some positive integer m.Since X min is smooth, we have an isomorphism In the next two sections we will deal with surfaces X of Kodaira dimension κ(X) = 1.

Elliptic surfaces: general set-up and the non-isotrivial case
We are starting here to study elliptic fibrations f : X → B with κ(X) = 1 towards Conjecture 1.1.If f is almost smooth, i.e., the only singular fibers are multiples of elliptic curves, then c 2 (X) = e(X) = 0 [BHPVdV04, III, Prop.11.4].Thus by Noether's formula χ(X, O X ) ≤ 0, and therefore q(X) > 0, so there is nothing to prove.We first fix notations.
5.1.Setup.Let X be a smooth projective surface, and let f : X → B be an elliptic fibration onto a smooth curve B. We set so D is an effective divisor having support exactly on the irreducible components of a fibre that are not reduced.The exact sequence where Z is a local complete intersection scheme of codimension two whose support coincides with the singular points of the reduction (f * b) red of the fibres [Ser96, Prop.3.1(iii)].
We denote by π : P(Ω X ) → X the projectivisation, and by ζ → P(Ω X ) the tautological class.We set (6) Since I Z is a local complete intersection of codimension two, the projectivisation coincides with the blow-up of the ideal sheaf I Z (see Setup 3.10).In particular Y is a prime divisor in P(Ω X ) and Denote by K ⊂ B the finite set of points such that the fibre f * b is not multiple and not reduced.The divisor D can be decomposed as where the F i are the reductions of multiple f -fibres and is simply the remainder, i.e. the part of D coming from non-multiple, non-reduced fibres.It follows from Kodaira's classification [BHPVdV04, V, Sect.7, Table 3] that the support of D 0 does not contain any fibre, so the intersection matrix of D 0 is negative definite by Zariski's lemma [BHPVdV04, III, Lemma 8.2].It is now straightforward to check that (8) is the Zariski decomposition of D (see Subsection 3.D) with P = s i=1 (m i − 1)F i .If f is relatively minimal, the canonical bundle formula [BHPVdV04, V, Thm.12.1, Prop.12.2] holds: and deg(R Finally assume that f is relatively minimal and B = P 1 .Then (9) implies that (10) where a = χ(X, O X ) and F is a general fiber.

Proposition.
In the situation of Setup 5.1, suppose that f * Ω B (D) is pseudoeffective.Then we have q(X) > 0.
Proof.The statement is trivial if g(B) ≥ 1, so assume B ≃ P 1 .We follow the philosophy of [Cam04, Sect.3.5].By Remark 3.18 one has κ(f * Ω B (D)) ≥ 0. Choose a positive integer m and a non-zero section s ∈ H 0 (X, (f * Ω B (D)) ⊗m ).Let E be the divisor defined by s.Note that E is supported on fibers of f and that where F is a general fiber of f .By the discussion in Setup 5.1 we know that the nef part of D is represented by s i=1 (m i − 1)F i ≡ λF where the F i are the reductions of multiple f -fibres.Since which is a positive current.By (7) this implies the statement.
By Proposition 5.2 and 4.6, we conclude 5.5.Corollary.Let X be a smooth projective surface, and let f : X → B be a non-isotrivial elliptic fibration Then the following are equivalent: In summary, Conjecture 1.1 holds for non-isotrivial elliptic fibrations.

Elliptic surfaces: the isotrivial case
In this section we treat isotrivial elliptic fibrations and prove parts (b) and (c) of Theorem 1.2.The standard isotrivial fibration f R factors through its relative minimal model.Since we assumed that f is relatively minimal and the relative minimal model of an elliptic surface is unique, we have a birational morphism µ : R → X such that f R = f • µ.We summarise the construction in a commutative diagram: In general the birational map µ is not an isomorphism [Ser96, (2.4)], but as shown in Lemma 6.3, it is an isomorphism unless E has complex multiplication.
6.1.Assumption.In the whole subsection we assume that f : X → B is a relatively minimal, isotrivial elliptic fibration that is standard, i.e. the birational map µ (cf.Notation 6.A) is an isomorphism.For simplicity of the exposition we thus identify X = R.
The following lemmas are well-known, we include them for lack of reference: 6.2.Lemma.Using the Notation 6.A, let x ∈ C be a point with non-trivial stabiliser G x on C. then G x acts either by translation or as the involution z → −z on the elliptic curve x × E.
In particular all the singular fibers of f are either multiple elliptic curves or curves of type I * 0 in Kodaira's classification, see e.g., [BHPVdV04, V, Sect.7, Table 3].
Proof.We follow the description of the singularities of If G b,e = Z 4 , the group acts as z → iz on the elliptic curve C/Z ⊕ iZ.This action fixes the origin and 1+i 2 , so we obtain two singularities of type A 1,4 in the quotient.The points 1 2 and i 2 have non-trivial stabiliser, but are in the same orbit, so we obtain one singular point of type A 1,2 in the quotient.The minimal resolution of R → (C × E)/G thus yields a fibre with a central component of multiplicity 4, two components of multiplicity 1 and self-intersection −4 and one component of multiplicity 2 and self-intersection −2.This fibre is the log-resolution of a fibre of Kodaira's type III, but it is not relatively minimal.If finally G b,e = Z 6 , then we see in a similar fashion that the action of ζ on the elliptic curve C/Z ⊕ ζZ (with ζ = e i 2π 3 ) leads to a log resolution of the configuration obtained for type IV and the action of −ζ leads to a log resolution of the configuration obtained for type II.The contradiction is then as before.
6.3.Lemma.Let f : X → B be a minimal isotrivial elliptic fibration as in Notation 6.A. Assume that if x ∈ C is a point with non-trivial stabiliser G x on C, then G x acts either by translation or as the involution z → −z on the elliptic curve x × E. Then f : X → B is standard.In particular, if E does not have complex multiplication, then f : X → B is standard.
Proof.Under our assumptions, the singular fibers of p C • λ = f • µ are either multiple elliptic curves or of type I * 0 .But then µ must be an isomorphism, due to Kodaira's classification, applied to f .We have the following crucial where R t ⊂ R is the ramification corresponding to points y ∈ C such that G y ≃ Z my acts by translation on y × E (see Lemma 6.2).If y ∈ C is a point with this property, every point in its orbit G.y has the same property.Since the orbit has length d my , this induces a ramification divisor of order d my−1 my .Now note that the corresponding f -fibre over ȳ = ψ(y) is multiple elliptic with multiplicity m y , so it defines a component of D that is numerically equivalent to my−1 my F where F is a general fibre of f .Thus we see that Suppose now that this is not the case: then we have 6.5.Construction.We again use the Notation 6.A.Let Ā be an ample Cartier divisor on (C × E)/G, and set A X = λ * Ā.Then by definition (and [HLS20, Lemma 2.2]) the vector bundle Ω X is not pseudoeffective if and only if there exists a c > 0 such that for all i, j ∈ N such that i > cj one has Denote by D ⊂ X the exceptional locus of λ, then we have morphisms where the last isomorphism is due to the fact that S i Ω C×E is reflexive, since C × E is smooth.Finally let A C be an ample divisor on C and A E an ample divisor of degree one on E. Set A := p * C (A C ) ⊗ p * E (A E ) with the canonical projections p C : C × E → C and p E : C × E → E. Then for some l ∈ N sufficiently high we have an inclusion so for all i, j ∈ N we obtain an inclusion 6.6.Definition.We say that a η ∈ H 0 (C × E, S i Ω C×E ⊗ O C×E (jA)) induces a holomorphic symmetric differential with values in A X on X if it is in the image of an inclusion Φ in (12).
Remarks.The definition is a slight abuse of terminology, since the inclusion Φ in (12) is only defined for j divisible by l.Since in the definition of pseudoeffectivity we can always replace i and j by il and jl, we will, for the simplicity of notation, ignore this point.Note also that the chain of inclusions does not use that C is proper, so the terminology also applies to an analytic open subset ∆ × E ⊂ C × E with some subgroup G x ⊂ G acting on ∆ × E.

6.C.
The nonvanishing conjecture for standard isotrivial fibrations.The goal of this subsection is to prove the following 6.7.Theorem.Let X be a smooth projective surface that admits relatively minimal, isotrivial elliptic fibration f : X → B that is standard (see Subsection 6.A).If f * Ω B (D) is not pseudoeffective, then Ω X is not pseudoeffective.
By Proposition 5.2 and 4.6 we thus obtain: 6.8.Corollary.Let f : X → B be a standard isotrivial fibration; e.g., f is an isotrivial elliptic fibration such that the general fiber does not have complex multiplication.Then the following are equivalent: a) Ω 1 X is pseudoeffective ; b) we have q(X) > 0 ; c) we have H 0 (X, S m Ω 1 X )) = 0 for some positive integer m The proof of Theorem 6.7 is done by showing that if i ≫ j, a non-zero does not induce a holomorphic symmetric differential on X.In other words, Φ = 0. Since the details are somewhat technical, let us first recall and reprove a result of Sakai.
6.9.Example.[Sak79, §4, (D)] Let C be a hyperelliptic curve, and let τ = (i C , i E ) be the involution on C × E defined by the hyperelliptic involution i C on C and the map i E : E → E, z → −z on an elliptic curve E. The minimal resolution X → (C × E)/ τ , has an isotrivial fibration f : X → P 1 such that all the singular fibres are of type I * 0 , in particular it is relatively minimal and standard.Then one has H 0 (X, ) for some l ∈ 0, . . ., i. Assume that α ⊗ β induces a holomorphic symmetric differential on X. Arguing by contradiction we assume that α = 0 and β = 0.If (x, 0) is a fixed point of τ , choose local coordinates z 1 on C and z 2 on E such that locally near (x, 0), the involution is given by (z 1 , z 2 ) → (−z 1 , −z 2 ).In these local coordinates we write α = f α (z 1 )dz l 1 and β = f β (z 2 )dz i−l 2 .For a general point in the exceptional divisor over the point (x, 0), we can choose local coordinates (u, v) on X such that z 1 = u 2 , z 2 = uv.In these coordinates the exceptional divisor is given by u = 0. Substituting (z 1 , z 2 ) by these coordinates we see that α ⊗ β induces the meromorphic symmetric differential on X.Looking at the term of du i we see that the differential is holomorphic along the exceptional divisor if and only if and only if α ⊗ β vanishes with order at least i in the fixed point.Since 0 = β ∈ H 0 (E, ω ⊗i−l

E
) ≃ H 0 (E, O E ) does not vanish, this shows that α ∈ H 0 (C, ω ⊗l C ) vanishes with order i in x.Since the involution i C has 2g(C) + 2 fixed points, we obtain that α vanishes along a divisor of degree at least i ) has dimension one for every i − l one can reduce the general case to rank one tensors, so the statement follows.This settles the proof of the example.
In the proof of Theorem 6.7 we have h 0 (E, ω ⊗i−l E ⊗O E (jA E )) > 1, so the symmetric differentials are not global rank one tensors.Somewhat surprisingly, this leads to a much weaker local obstruction (cf.[BTVA19, Prop.3.2]),i.e., the vanishing order of the symmetric differential in a fixed point can be strictly smaller than i.We will improve this local estimate by taking into account that the vanishing order along E is bounded by j deg A E .
6.10.The local obstruction -setup.Let us describe the local obstruction for a holomorphic symmetric differential on C × E to induce a holomorphic symmetric differential on X: using the notation of Lemma 6.4, fix a point x ∈ Z ⊂ C, and let τ x be the generator of G x ≃ Z 2 .Let 0 ∈ E be a fixed point of τ x , for this local computation we choose A E := 0 to be the corresponding ample divisor of degree one2 .Let x ∈ ∆ ⊂ C be a small disc and choose a local coordinate z 1 such that the action of G x is given by z 1 → −z 1 (in particular x = 0).We have where z 1 is a local coordinate on ∆ and C{z 1 } denotes the algebra of convergent power series in z 1 .Since E is an elliptic curve, we have h 0 (E, O E (jA E )) = j, and we may choose a basis of sections s j,0 , . . ., s j,j−2 , s j,j such that s j,k vanishes with order exactly k in the neutral element 0 of the elliptic curve.In particular we have a C-basis of H 0 (∆ × E, O ∆×E (jA)) Note that by construction z n−k 1 s j,k vanishes with order exactly n in (0, 0).Since E is an elliptic curve, its cotangent bundle is trivial and we denote by dz 2 a global generator of Ω E .Let now be the space of symmetric differentials with values in O ∆×E (jA).Then we can decompose In the setup just introduced, the following holds : 6.11.Lemma.Let S be the minimal resolution of (∆ × E)/G x .If ω induces a holomorphic symmetric differential with values in A S on S (see Definition 6.6), then for all n ∈ N the form ω n induces a holomorphic symmetric differential with values in A S .Moreover, if n < i, there exists a differential Remark.Since G x ≃ Z 2 acts locally as (z 1 , z 2 ) → (−z 1 , −z 2 ), one has n = i mod 2 [BTVA19, Sect.3], so i+n 2 and i−n 2 are positive integers.
Proof.The property of being holomorphic is local, so we can just apply the proof of [BTVA19,Prop.3.2].
Proof of Theorem 6.7.We use the notation of Subsection 6.A and Construction 6.5.Denote by Z ⊂ C the set of points in x ∈ C such that the stabiliser G x = τ x acts as the involution z → −z on the elliptic curve x × E. Since f is standard and f * Ω B (D) is not pseudoeffective, we know by Lemma 6.4 that Z has at least 2g − 1 element where g = g(C).For every x ∈ Z we fix a point x ′ ∈ (x × E) such that (x, x ′ ) is a fixed point of τ x .Fix some rational ǫ > 0 such that 2g−2 2g−1 + ǫ < 1 and fix a positive integer N such that ( 2g − 2 2g − 1 + ǫ)N ∈ N.
Assume now that η ∈ H 0 (C × E, S i Ω C×E ⊗ O C×E (jA)) induces a holomorphic symmetric differential with values in A X on X (see Definition 6.6).Then for every x ∈ Z, the restriction to some neighbourhood ∆ × E induces a holomorphic symmetric differential on the minimal resolution S of (∆ × E)/G x .Thus by Corollary 6.12 we have for every i ∈ N that Thus we get an inclusion We claim that this last vector space is zero for i ≥ Yet by Lemma 6.4 we know that Z has at least 2g − 1 elements, hence Since deg(ω mN C ⊗ M * ) = mN (2g − 2), we obtain a contradiction to (13).
6.D. Isotrivial fibrations and the Zariski decomposition.Let f : X → P 1 be an isotrivial elliptic fibration over a rational curve, and assume that Ω X is pseudoeffective.Since the proof of Theorem 6.7 is a bit tedious, we present here a more conceptual approach based on the ideas of Subsection 5.B.The considerations of this section are independent of whether f is standard or not.We use the notations of the setup 5.1.Let ζ be the tautological class of P(Ω X ).
If the elliptic fibration f is not almost smooth, we would like to show that the subvariety Y := P(I Z ⊗ ω X/B (−D)) ⊂ P(Ω X ) defined by f is in the negative part of the Zariski decomposition of ζ.If f is isotrivial, the restriction P(Ω X | F ) over a general fibre F is isomorphic to P 1 × F , so the proof of Proposition 5.4 does not apply.We therefore have to use some global information to explicitly compute the restriction ζ| Y .6.13.Proposition.In the situation of Setup 5.1, assume that f is isotrivial, relatively minimal and not almost smooth.Then ζ| Y is not pseudoeffective.

3 .
Pseudoeffective sheaves 3.1.Notation.Let G be a coherent sheaf on a variety X, and let T ⊂ G its torsion subsheaf.Then we denote by G/Tor the quotient G/T .Furthermore, we set S [m] (G) := (S m G) * * .3.A.Projectivization of sheaves.3.2.Definition.Let F be a coherent sheaf on a variety X.Then we denote by π : P(F ) → X the projectivisation of F in the sense of [AT82, II, §2, Sect.2].We denote by ζ P(F ) (or ζ when no confusion is possible) the Cartier divisor class associated to the tautological line bundle O P(F ) (1).
is a pseudoeffective Weil divisor class (cf.[Nak04, II, Defn.5.5]).Setting D = (p ′ 1 ) * µ * ( D), we have an inclusion of Weil divisors µ * ( D) ⊂ (p ′ 1 ) * D. Thus we have an inclusion of Weil divisor classes is a chain of isomorphisms for all m ∈ N. Hence the reflexive sheaf F is pseudoeffective by [HP19, Lemma 2.3].Proof of the second statement: As for the first statement, the inequality κ( X, f [ * ] (F )) ≥ κ(X, F ) is immediate.Let us show the other inequality: by the claim we know that ζ + D is a tautological class on P .Thus by assumption, one has κ( P , ζ

3 .
16. Lemma.Let X be a smooth projective surface, and let f : X → B be a fibration over a smooth curve B. Let L be a pseudoeffective line bundle on X such that L F ≃ O F for the general fiber F of f .Let D be a Cartier divisor such that L ≃ O X (D), and let D = P + N its Zariski decomposition.Then the following holds: a) Up to taking multiples, one has O X (P ) ≃ f * M with M a nef line bundle.b) If P ≡ 0, one has κ(L) ≥ 1. c) If P ≡ 0, there exists m ∈ N and a numerically trivial line bundle M on B such that κ(X, L ⊗m ⊗ f * M * ) = 0.
6.A. Notation.In the situation of Setup 5.1, assume that f : X → B is relatively minimal and isotrivial.Denote by E the elliptic curve such that a general f -fibre is isomorphic to E. By [Ser96, Sect.2] there exists a smooth curve C and a finite group G acting diagonally on the product C × E such that X is birational to the quotient (C × E)/G and the fibration f corresponds to the fibration (C × E)/G → C/G ≃ B induced by projection on the first factor.More precisely, denote by q : C × E → (C × E)/G the quotient map, and by pC : (C × E)/G → C/G the map induced by the projection p C .Denote by λ : R → (C × E)/G the minimal resolution of singularities, then the exceptional divisors are Hirzebruch-Jung strings [Ser96, 2.0.2] and the singular fibres of f R := pC • λ : R → B are described in [Ser96, Thm.2.1].Following Serrano, we call f R the standard model of the isotrivial fibration f .
2.0.2]: the group G acts on C ×E, let b ∈ B a point with non-trivial stabiliser G b ⊂ G.The group G b is cyclic [FK80, III.7.7,Cor.].If it acts freely on E, the quotient (B ′ ×E)/G is smooth near the fibre which is multiple elliptic.Assume now that there exists a point e ∈ E with non-trivial stabiliser G b,e ⊂ G b .Then G b,e ⊂ Aut(E, e).In view of [Har77, IV, Cor.4.7] this strongly limits the possibilities: If G b,e = Z 2 , the group acts as z → −z on E and we are done.

6. 4 .
Lemma.Using the Notation 6.A, let Z denote the set of points in x ∈ C such that G x acts as the involution z → −z on the elliptic curve x × E. If f * Ω B (D) is not pseudoeffective, then Z has at least 2g(C) − 1 elements.Proof.Since f * Ω B (D) is not pseudoeffective, we have B ≃ P 1 .The action of G on the curve C defines a Galois cover ψ : C → C/G ≃ P 1 of degree d = |G|.By the Hurwitz formula we have 2g(C) − 2 = deg K C = −2d + deg R, where R is the ramification divisor of ψ.If x ∈ Z, then it is a point with stabiliser G x ≃ Z 2 , hence ψ ramifies with order 2 in x.Thus deg R x = 1 and )), and let ω = n∈N ω n be the decomposition such that ω n ∈ V n .Finally let M := z 1 dz 2 − s 1,1 dz 1 be the holomorphic 1-form with values in A giving in local coordinates the form z 1 dz 2 − z 2 dz 1 .