Rational curves on fibered varieties

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one covered by rational curves contracted by the fibration. We then focus on the case of varieties with numerically trivial canonical bundle and we discuss several consequences of this result.


Introduction
Finding rational curves in a projective variety X is useful to understand the geometry of X because these curves are strongly related to many invariants. Rational curves on Calabi-Yau varieties are particularly useful but the existence of such curves in full generality on these varieties is proven only in dimension two by Bogomolov-Mumford [MM83]. On K3 surfaces there are rational curves in any ample linear series. This leads to define Beauville-Voisin class as the zero-cycle class of a point on a rational curve [BV04]. In higher dimension doing this is more difficult because it is hard to find an ample divisor H i → X with i * (CH 0 (H)) = Z, and we do not expect that there is a divisor with CH 0 (H) = Z, e.g. H rational. Let us briefly give some other motivations. A rational morphism to a manifold without rational curves is everywhere defined. Finding a rational curve on a variety implies that it is not hyperbolic in the sense of Kobayashi. This list can be made much longer.
The experience with minimal model program suggests us that even if one is mainly interested in smooth varieties, the natural setting is to allow at least log terminal singularities. The aim of this paper is to extend the results proven in [DFM16] in a singular setting tipical of minimal model program. We use some techniques that lead us to prove some new results also in the smooth case.
The core of this paper is the following result.
Theorem 0.1. Let X be a normal variety of dimension n with log terminal singularities and vanishing augmented irregularity. Suppose that there exists a surjective morphism φ from X to a variety B of dimension n − 1. If there exists a cartier divisor L on B such that φ * L ∼ K X , then there exists a subvariety of codimension one in X that is covered by rational curves contracted by φ.
An important consequence of this theorem is the case with numerically trivial canonical bundle.
Theorem 0.2. Let X be a normal variety of dimension n with log-terminal singularities and numerically trivial canonical bundle and vanishing augmented irregularity. Suppose that there exists a surjective morphism φ freom X to a variety B with dimension n − 1. Then there exists a subvariety of codimension one in X that is covered by rational curves contracted by φ.
In the case of varieties with trivial canonical bundle we study what happens in the Beauville-Bogomolov decomposition. In Section 2 we study the case of a fibration on a curve. Let us state a more readable consequence of the main result of Section 2.
Theorem 0.3. Let X be a smooth variety with vanishing augmented irregularity and numerically trivial canonical bundle. Suppose that there exists a fibration in abelian varieties to a curve. Then X does contain a rational curve.
Acknowledgements. The author warmly thanks Raffaele Carbone for the continuous helpful discussion during the writing of this paper. The author would also like to thank his advisor, Simone Diverio, for the continuous help provided during this work.
1. Elliptic fiber spaces 1.1. Definitions e notations. In this paper every variety will be an irreducible projective variety over the complex number. The variety X will be always normal and of dimension n ≥ 2. The notations and standard properties about singularities that are used in this article can be found for example in [KM98]. For the reader's convenience we recall some definitions that will be used in this paper.
Definition 1.1. A morphism f : X → Y between normal varieties is called quasi-étale if f is quasi-finite andétale in codimension one.
Remark 1.2. A quasi-étale morphism to a smooth variety is globallyétale by standard argument of purity of branch locus.
The augmented irregularity of Y is the following, not necessarily finite, positive integer For any variety it holds the inequalityq(Y ) ≥ q(Y ). In general the equality does not hold. Moreover it may happen that this supremum is not achieved; in this case the augmented irregularity is infinite. This happens also in dimension one as we see in the following example.
Example 1.4. The behavior of the augmented irregularity for curves is easy to describe using Riemann-Hurwitz formula. The augmented irregularity of a genus zero curve is zero. Indeed a P 1 is simply connected and it is regular. Any finiteétale cover of a genus one curve is again a genus one curve by Riemann-Hurwitz formula, soq(C) = 1. A curve C with g(C) ≥ 2 has a cover of degree d from a curve C ′ of genus g(C ′ ) = d · (g(C) − 1) + 1. Indeed its fundamental group is Z 2g that has subgroups of index d arbitrary large. This subgroup corresponds to anétale coverC of degree d, whose genus is given by Riemann-Hurwitz formula and equals g(C) = d(g − 1) + 1. So we can find anétale cover of C with arbitrary large irregularity. Hencẽ q(C) = ∞.
Definition 1.5. A fibration is a morphism between normal varieties with connected fibers. An elliptic fiber space is a fibration such that the general fiber is a smooth genus one curve.
Remark 1.7. The points in the image of the singular locus of f i.e. f (sing(f )) 1 , are the singular values of f .
One can associate to any elliptic curve a complex number called its jinvariant. This association is modular, which means that an elliptic family f : Y → B comes with a rational map j : B C called j-function that is at least defined over the smooth values of f . For some standard facts about the j-function of an elliptic family the reference can be found in [Kod63].
Remark 1.8. Consider the following two different definitions of isotriviality for a flat family. One can ask that two general fibers are isomorphic, or that the smooth fibers are isomorphic. In the general setting the first definition is strictly more general than the second. An example of this situation is given by a degeneration of an Hirzebruch surface F n into an F m with m > n, [Ser06, See Example 1.2.11(iii)]. For elliptic fibrations these two definitions coincide. Indeed a smooth degeneration of an elliptic curve is again elliptic by Kodaira's table [BHPVdV04]. Since the j-invariant is constant on a dense subset of the base it is constant. We can conclude that every smooth fiber is a smooth elliptic curve with the same j-invariant, so the smooth fibers are isomorphic.
1.2. Fischer-Grauert Theorem. A well-known theorem proved by Fischer and Grauert [FG65] tells us that a proper holomorphic submersion with isomorphic fibers is locally a product in the complex topology. This means that given a proper holomorphic submersion f : X → B between complex manifolds such that for any t, s ∈ B the fibers X t and X s are isomorphic, then for any p ∈ B there exists a neighborhood U p ⊂ B open in complex topology such that the family X| Up ≃ X p × U p splits in a product over the base. The same statement does not hold in the Zariski topology as we can see in the following example.
Example 1.9. Let f : X ′ → X be any finite unramified (henceétale) morphism between varieties of degree d > 1. For example f can be a finite unramified morphism of degree d from a smooth curve of genus d(g − 1) to a smooth curve of genus g. For any p ∈ X, the fiber over p is a scheme given by d distinct reduced points. In particular any two fibers are isomorphic. However for any U ⊆ X open in the Zariski topology, the preimage U ′ := f −1 (U ) is a non-empty Zariski-open subset of X ′ . In particular since U ′ is connected it is not isomorphic to the product between d points and U that has d connected components.
For the general philosophy on the relation between complex topology andétale topology one can expect that the same statement of Fischer-Grauert Theorem holds for theétale topology. Since we were unable to find a reference on this subject, for the reader's convenience we prove some statements that will be useful for what follows. We hope to address a more careful analysis on this problem in a forthcoming paper.
Proposition 1.10. Let Y → B a smooth proper morphism between normal quasi-projective varieties such that for any t ∈ B the variety Y t is a smooth curve of genus g ≥ 1. Suppose moreover that for any s, t ∈ B the curves Y t and Y s are isomorphic, then there exists a finiteétale morphismB → B such that the pullback Y × BB ≃ Y t ×B is a product.
Proof. Fix a point 0 ∈ B. By GAGA's principle we can consider B and Y 0 as complex manifolds, in this way we can study the monodromy around zero as follows. Fix an integer number n greater than three and consider the action of the fundamental group of the base on the first cohomology group of the central fiber with coefficient in Z n Since Y 0 is a complete curve of genus g the group H 1 (Y 0 , Z n ) ≃ Z 2g n is finite. This implies that Aut(H 1 (Y 0 , Z n )) is finite and hence Ker(φ) π 1 (B, 0) is a normal subgroup of finite index of the fundamental group of the base. By the standard correspondence between subgroup of index d of π 1 (B, 0) and etale cover of B of degree d, the subgroup Ker(φ) corresponds to a finité etale coverB of B. Moreover the action of π 1 (B,0) is trivial on the first cohomology group with coefficients in Z n of the pullback family Y × BB . This construction, called J n -rigidification, is useful because for n ≥ 3 there are no automorphisms of a curve with positive genus acting in a trivial way on H 1 (C, Z n ). In particular there exists a fine moduli space with an universal family U g,n → M g,n (see for example [Bea96]). The classifying morphism B → M g,n is constant because the morphism B → M g is constant (this morphism is constant since all fibers are isomorphic). It follows that there is a pullback diagram / / M g,n and since the classifying morphismB → M g,n is constant, the variety Y × B B is isomorphic to the productB × Y 0 .
In this paper we need the previous result only for elliptic fibrations. Let us recall that for elliptic fibration we mean just a morphism between normal varieties such that the generic fiber is a smooth genus one curve, in particular the morphism may not have sections. Since the previous proof use many topological tools, we give another more algebraic proof of the following statement, that is essentially Proposition 1.10 for curves with genus one.
Proposition 1.11. Let Y → B a smooth projective morphism between normal varieties such that for any t ∈ B the variety Y t is the same curve of genus g = 1, i.e. a smooth isotrivial family of elliptic curves. Then there exists a finiteétale morphismB → B such that the pullback Y × BB ≃ Y t ×B is a product.
To prove this proposition we need two results.
Lemma 1.12. If f : Y → B is an isotrivial family with a section of smooth elliptic curves, then there exists a finiteétale mapB → B such that the pullback family Y × BB is isomorphic to the trivial family.
This result is [Har10, Corollary 26.5]. The big difference between Proposition 1.11 and Lemma 1.12 is that in the lemma the family of elliptic curves has a section. So we have to combine this result with the following.
Lemma 1.13. Let f : Y → B be a projective morphism between normal varieties. Assume that B is smooth and f isétale locally trivial and the generic fiber F has numerically trivial canonical bundle. Then there is a finiteétale cover This lemma is stated and proved in [KL09,Lemma 17]. Finally we can give an algebraic proof of Proposition 1.11.
Proof of 1.11. We have to prove that f isétale locally trivial, i.e. for any The morphism is smooth and projective so there exists a multi-section Σ of f that isétale at p. Shrinking Σ we can suppose that the fiber product Σ × B Y → Σ is a family of smooth elliptic curves with a section and the fibers are pairwise isomorphic, so by Lemma 1.12 This proves that f isétale locally trivial. We can apply Lemma 1.13 and the proof is completed.
1.3. Proof of Theorem 0.1. Before start proving Theorem 0.1 we need a lemma that is essentially stated in [DFM16].
Lemma 1.14. Let X π → B be an elliptic fiber space. If the subvariety of singular values Z := π(sing(π)) has codimension at least two then the family π is isotrivial.
Proof. Since B is normal it is smooth in codimension one and also the subvariety Z ∪ B sing has codimension at least two. We denote B 0 := Z c ∩ B reg . All the fibers over B 0 are smooth elliptic curves for Remark 1.8. So it is well-defined the regular function that sends a point t ∈ B 0 the j-invariant of the elliptic curve over t [Kod63]. Since (B 0 ) c as codimension at least two and B is normal, the function j 0 can be extended to a function j : B → C. Since B is projective this function must be constant. This means that the fibers over B 0 are isomorphic.
For the reader's convenience we state again the theorem that we are going to prove.
Theorem 1.15. Let X be a variety with log terminal singularities and q(X) = 0. Suppose that there exists a surjective morphism φ : X → B to a variety of dimension n − 1. If there exists a Cartier divisor L on B such that φ * L ∼ K X , then there exists a subvariety of codimension one in X that is covered by rational curves contracted by φ.
Proof. The proof is divided in several steps and maybe some steps are already known to the experts.
Step 1 : the morphism φ : X → B is an elliptic fibration. We can suppose, by taking the normalization of B and passing to Stein factorization, that the morphism φ has connected fibers and the base B is normal. For dimensional reasons the generic fiber is a curve. Since X is normal X sing ⊂ X has codimension at least two, so φ(X sing ) ⊂ B has positive codimension. The restriction on the regular part of X is a morphism from a smooth variety, so there is a non-empty open subset U ⊂ B where the morphism φ : A smooth curve with trivial canonical bundle is a genus one curve and a smooth degeneration of a genus one curve has again genus one [BHPVdV04, See Section V.7], so every fiber of φ 0 : X 0 → B 0 is a genus one curve.
Step 2: the subvariety Z has codimension one in B. Suppose by contradiction every irreducible component of Z has codimension at least two. By Lemma 1.14 the family φ is isotrivial, so by Proposition 1.10 or 1.11 there exists a variety C 0 and a finiteétale cover C 0 τ → B 0 such that the pullback given by the composition α 0 := ψ −1 •τ ′ : C 0 ×E → X 0 is finiteétale because τ is the pullback of a finiteétale morphism. In particular the composition of the morphisms C 0 × E α 0 → X 0 i → X is quasi-finite andétale. By Zariski's Main Theorem [Gro67] a quasi-finite morphism is always the composition of an open immersion and a finite morphism, so there is a commutative diagram Since the subvariety X Z ∩ Exc(φ) c has dimension at most dim(Z) + 1 and we are assuming that cod B (Z) ≥ 2, the dimension of X Z is bounded by Theorem 2] the φ-exceptional locus is covered by rational curves contracted by φ (In Theorem 2 of [Kaw91] Kawamata didn't say explicitly that the rational curves are contracted by φ, however this is clear from his proof). This implies that if the exceptional locus of φ has codimension one in X, it is a uniruled subvariety of codimension one of X. This allows us to assume that cod X (X Z ) ≥ 2.
Since α is finite, also i ′ (C 0 × E) c has codimension at least two in Y . In particular since α isétale in i ′ (C 0 × E) ⊂ Y , this argument proves that α is a finite quasi-étale cover of X, so by hypothesis H 1 (Y, O Y ) = 0. By [KL09,Proposition 5.20] Y has log terminal singularities. As proved in [GKP16b, Proposition 6.9] there is an isomorphism H 0 (Y, Ω Y is a reflexive sheaf, so it is isomorphic to the sheaf of one forms on the regular part. The variety C 0 is smooth because it is a finité etale cover of B 0 , so Ω we reach a contradiction, so if there are no uniruled divisors on X then Z has codimension one in B. Step 3: restriction to a fibration onto a curve with some singular values. Let H be a very ample divisor on B such that (n − 2)H + L is globally generated. The pullback φ * H is a globally generated Cartier divisor. Moreover there is an isomorphism because φ has connected fibers. This isomorphism implies that general elements in |H| are general also in |φ * (H)|. So we can choose n − 2 general divisors D 1 , . . . , D n−2 ∈ |H| such that C := D 1 ∩ . . . ∩ D n−2 is a smooth irreducible curve in B reg not contained in Z and S := φ −1 (D 1 )∩ . . . ∩ φ −1 (D n−2 ) is a normal surface. We call again φ the morphism φ| S . Since Z has codimension one, it must intersect C. Indeed Z ·C = Z ·D 1 ·. . .·D n−2 = Z ·H n−2 > 0 because H is ample in B. This means that φ must have some singular fibers.
Step 4: the case where φ −1 (p i ) ∩ sing(S) = ∅. Let S be a minimal resolution of S S We can assume β is relatively minimal. Indeed if there are some (−1)-curves on S contracted by β, the image of such curves are again rational curves in S because they cannot be contracted to a point by minimality of the resolution. If there are (−1)-curves in the general surface S constructed above, then the union of such rational curves cover a divisor of X. Let p 1 , . . . , p k the points in C ∩ Z. The singular curves φ −1 (p i ) ⊂ S are exactly φ −1 (p i ) = ν(β −1 (p i )). Since β is a minimal elliptic fibration, by Kodaira's table [BHPVdV04, Section V.7] a fiber of β can be a smooth genus one curve, a sum of (possibly non reduced) rational curves or a non reduced genus one curve. If φ −1 (p i ) contains some singular points of S, then β −1 (p i ) = ν −1 (φ −1 (p i )) contains an exceptional divisor of ν, in particular β −1 (p i ) must be sum of rational curves. Since not every rational curve of β −1 (p i ) can be contracted in S, the curve φ −1 (p i ) = ν(β −1 (p i )) must be sum of rational curves in S.
Step 5: the case where φ −1 (p i ) ⊂ S reg . The curve φ −1 (p i ) is the central fiber of a family S 0 φ → ∆ of elliptic curves. Since φ −1 (p i ) is not smooth, by Kodaira's table it is a rational curve or a non reduced irreducible elliptic curve. We need to exclude the last possibility.
By adjunction formula the canonical bundle of S reg is base point free. Indeed K Sreg ∼ (K X + (n − 2)φ * H)| Sreg ∼ φ * (L + (n − 2)H)| Sreg the canonical bundle is the restriction of the pullback of a base point free divisor.
By [BHPVdV04,V.12.3] the canonical bundle of S reg can be computed using the formula K Sreg ∼ φ * D + (m i − 1)F i for some divisor D on the base and the sum runs over all the multiple fiber F i with multiplicity m i . The restriction of the canonical bundle of S reg to F i is base point free because K Sreg is base point free. By the above formula for any i the canonical bundle restricted to F i is Theorem 0.1 is inspired by [DFM16] where they proved a similar result in the case X is a smooth projective manifold with finite fundamental group.
Remark 1.16. For a smooth projective variety Y with finite fundamental group the augmented irregularity is trivial. Indeed a quasi-étale cover is ań etale cover for purity of branch locus. The fundamental group of anétale coverỸ of Y is a subgroup of the fundamental group of Y , so it is finite. The first Betti number of a variety with finite fundamental group is zero, so by Hodge theory also H 1 (Ỹ , OỸ ) = 0, and henceq(Y ) = 0. This remark implies that Theorem 1.15 is stronger than [DFM16, Theorem 1.1] also for smooth varieties. An interesting application of Theorem 1.15 is the following corollary.
Corollary 1.17. A klt variety X withq(X) = 0, k(X) = n − 1 and whose canonical bundle is semiample of exponent one does contain a uniruled divisor.
We conclude this section with an example where one can apply Theorem 1.15.
Example 1.18. Fix two integer numbers r ≥ 1 and d ≥ 2. Consider a smooth variety X 3,r ⊂ P 2 ×P d given by the zero locus of a bihomogeneous polynomial of bedegree (3, r). This variety has the natural projection π : X 3,r → P d . The augmented irregularity of X 3,r is zero because it is simply connected by Lefschetz hyperplane theorem. By Grothendieck-Lefschetz Theorem the Picard group of X 3,r is isomorphic to Pic(P 2 ×P d ) and by adjunction formula the canonical bundle is K X 3,r ∼ O X 3,r (0, r − d − 1). In particular K X 3,r ∼ π * O P d (r − d − 1). So we can apply Theorem 1.15: it follows that this kind of family of elliptic curves can't be everywhere smooth but it degenerates on a divisor in rational curves.
1.4. Trivial canonical bundle. The following formulation of Theorem 0.2 is a generalization of [DFM16, Corollary 1.2]. Also for smooth varieties, Theorem 1.19 seems more general than their result because in [DFM16] a Calabi-Yau variety must have finite fundamental group. Such finiteness condition is a priori stronger than the vanishing of the augmented irregularity (see Remark 1.16). However one can see as consequences of Beauville-Bogomolov decomposition for smooth varieties that this two conditions are equivalent: a smooth projective variety with numerically trivial canonical bundle and vanishing augmented irregularity has finite fundamental group. It is conjectured that the same implication holds also in the singular case, at least for varieties with mild singularities.
Theorem 1.19. Let X be a variety with log terminal singularities, K X ≡ 0 andq(X) = 0. Suppose that there exists a morphism φ : X → B whose general fiber is a curve. Then there exists a uniruled subvariety of codimension one in X that is covered by rational curve contracted by φ.
Proof. By global index one theorem [GGK17, Proposition 2.18] there is a variety X ′ with canonical singularities and a finite quasi-étale morphism α : X ′ → X such that K X ′ ∼ 0. A finite quasi-étale cover Y → X ′ is also (after the composition with α) a finite quasi-étale cover of X. This proves thatq(X ′ ) ≤q(X) and soq(X ′ ) = 0. If there is a subvariety V ⊂ X ′ of dimension n − 1 that is covered by rational curves, then also the variety α(V ) ⊂ X is covered by rational curves. Since the canonical bundle of X ′ is linearly equivalent to the trivial line bundle it is automatically the pullback of the trivial line bundle. The hypotheses of Theorem 1.15 are verified, so the theorem is proved.
Recently several different definitions of Calabi-Yau varieties appeared. To preserve the dichotomy given by Beauville-Bogomolov decomposition between irreducible symplectic varieties and Calabi-Yau varieties in the singular setting, a useful definition is given for example in [GGK17], [Dru18], [HP17] or some related papers. In particular in [HP17] they prove that exists a version of the Beauville-Bogomolov decomposition for varieties with canonical singularities and smooth in codimension two. In these definitions of Calabi-Yau varieties and irreducible symplectic varieties there are some conditions on the reflexive exterior algebra of forms, that in particular imply that such varieties must have vanishing augmented irregularity. In particular Theorem 1.19 can be applied to any product of Calabi-Yau and irreducible symplectic variety with such definition. We can be more precise. Let X be a variety with at most log terminal singularities, smooth in codimension two. Suppose moreover K X ≡ 0 and that there is a surjective morphism φ : X → B to a variety of dimension n − 1. By the above arguments and [HP17, Theorem 1.5] or [GGK17,Theorem B] there is a quasi-étale map f : A × Y → X with A an abelian variety of dimensionq(X) and q(Y ) = 0. Passing throught the Stein factorization we get an elliptic fibration α : A× Y →B. If the restriction of α to {t}× Y for generic t is a family of curves, then we can apply Theorem 1.19 and find an uniruled divisor on {t} × Y . This implies that there is also an uniruled divisor on A × Y and hence its image under f is again an uniruled subvariety of codimension one in X.
Remark 1.20. We can't expect that we can always apply Theorem 1.19 to the restriction of the fibration to {t} ×X because it may happen that α is a projection, i.e. X = E × Y → Y for some elliptic curve E.

Fibration over curves
In this section we study the dual case of an elliptic fibration: the case of a surjective morphism π : X → C to a curve. Passing through the Stein factorization we can assume π has connected fibers and since X is normal we can assume that C is smooth. So it is sufficient to study the geometry of a morphism with connected fibers onto a smooth curve. A fiber of a morphism onto a curve is a semiample divisor with numerical dimension one. So it is a priori more general to work only with a nef divisor with numerical dimension one than a fibration on a curve. The following is a possible definition of Calabi-Yau variety that we will use in this section.
Definition 2.1. A Calabi-Yau variety X is a variety smooth in codimension two with at most log terminal singularities andq(X) = 0.
Calabi-Yau variety as in Definition 2.1 means exactly that in the Beauville-Bogomolov decomposition [HP17, Theorem 1.5] the abelian variety is trivial. Let us give two definitions to make clearer the statement of the main result of this section.
Definition 2.2. For a normal variety Y smooth in codimension two the second Chern class is defined as the pushforward of c 2 (Ỹ ) for some resolutioñ Y of Y .
If a variety has singularities in codimension two then the second Chern class can be not-well defined.
Definition 2.3. Let Y be a normal variety. The numerical dimension of a nef class in x ∈ N 1 (Y ) is the maximum integer k such that x k = 0 as element in N k (X).
Finally we can give the statement of the main result of this section.
Theorem 2.4. Let X be a variety with log terminal singularities smooth in codimension two, trivial canonical class and vanishing augmented irregularity. Suppose that there exists a nef Q-divisor D with numerical dimension one such that c 2 (X) · D = 0 in N 3 (X). Then X does contain a rational curve.
Remark 2.5. By [GKP16a, Theorem 1.17], the intersection of the second Chern class with any n − 2 ample line bundles is positive.
Remark 2.6. Theorem 2.4 is a generalization of [DFM16, Theorem 1.6] also for smooth varieties. Indeed for smooth varieties with trivial canonical bundle with a fibration onto a curve with general fiber an abelian variety F , the class of F in N 1 (X) has numerical dimension one and intersect in zero the second Chern class of X, i.e. F · c 2 (X) = 0 [DFM16, Section 3].
Remark 2.7. Also if we assume X to be a Calabi-Yau manifold in the sense of [DFM16] the Theorem 2.4 is a generalization of [DFM16, Theorem 1.6] because a divisor with numerical dimension one wich intersect in zero the second Chern class of X is just conjecturally semiample.
The geometric meaning of 2.4 is clear if the divisor is also semiample. In this case the Itaka fibration associated to D is a fibration onto a curve. A general fiber F of such morphism does not intersect c 2 (X), i.e. c 2 (F ) = F · c 2 (X) = 0. If F is contained in the regular part of X, then by adjunction formula F has automatically trivial canonical bundle. This in particular implies that there is an abelian variety with a finite quasi-étale cover to F .

Preliminar results.
To prove Theorem 2.4 we need some basic results. The well-known statement for Q-divisors that a nef divisor is big if and only if it has positive top self-intersection [Laz04, Theorem 2.2.13] holds also for R-divisors. This fact is well-known to the experts but we haven't found any references in the literature. The following is an interesting consequence for variety with no rational curves and numerically trivial canonical bundle.
Proposition 2.8. Let X be a variety with log terminal singularities, numerically trivial canonical bundle and no rational curves. Then the ample cone and the big cone coincide.
Proof. Let D be any effective Q-divisor. For small positive and rational ε the pair (X, εD) is klt. Since there are no rational curves in X, the cone theorem [KM98, Theorem 3.7] tells us that εD is also nef. It follows that the effective cone is contained in the nef cone. Passing to the interior of such cones we get the thesis. Now we define two cones that help us to study nef divisors that are not ample.
Definition 2.9. The null cone N X ⊂ N 1 (X) is the set of classes of divisors D such that D n = 0. The boundary cone B X ⊂ N 1 (X) is the boundary of the nef cone.
Note that these cones are not convex cones. The following corollary explains the relation between these cones in our context. Corollary 2.10. Let X be a variety with log terminal singularities, numerically trivial canonical bundle and no rational curves. The boundary of the ample cone is contained in the null cone, i.e. B X ⊂ N X .
Proof. In the boundary of the ample cone there are nef R-divisors that by Proposition 2.8 are not big R-divisors. These R-divisors has trivial top selfintersection and so they are in the null cone.
This corollary leads us to find (many) divisors with numerical dimension n − 1 as explained in the following proposition.
Proposition 2.11. Let X be a variety with log terminal singularities and numerically trivial canonical bundle and without rational curves. Let H and D be two divisors on X that are respectively ample and nef of numerical dimension one. There is a (unique) rational number t 0 such that the Qdivisor N (D, H) = H − t 0 · D in nef and has numerical dimension n − 1.
Proof. The line in N 1 (X) for t ∈ R N t = H + t · D gives us an interesting divisor in the intersection with the null cone. This line is parallel to the extremal ray of the nef cone generated by [D]. The divisor D is nef so the line N t is contained in the nef cone for t ≥ H n nH n−1 ·D and intersect the null cone when there is the equality. The divisor in the intersection N = H − H n nH n−1 ·D D is a Q-divisor because H and D are Qdivisors and H n nH n−1 ∈ Q. The divisor N has numerical dimension n − 1 because N n−1 · D = H n−1 · D = 0 and it is not ample.
In particular this proposition implies the following corollary.
Corollary 2.12. Let X be a variety with log terminal singularities smooth in codimension two, with numerically trivial canonical bundle and with no rational curves. If c 2 (X) = 0 as element in N 2 (X) but c 2 (X) · D = 0 in N 3 (X) for some Q-divisor D with ν(D) = 1, then there exists an ample Qdivisor H such that the Q-divisor N (D, H) constructed in Proposition 2.11 satisfies c 2 (X) · N (D, H) n−2 > 0.
Proof. By Proposition 2.11 X contains a Q-divisor N of numerical dimension n−1. By the work of Miyoka [Miy87] we know that the intersection of c 2 (X) with n − 2 nef divisors is non negative since X is not uniruled. Since the ample cone is open and c 2 (X) = 0 there is an ample divisor H such that H n−2 · c 2 (X) = 0 and hence by [Miy87] H n−2 · c 2 (X) > 0. By hypothesis c 2 (X) · D = 0, so c 2 (X) · (N (D, H)) n−2 = c 2 (X) · (H − H n nH n−1 ·D D) n−2 = c 2 (X) · H n−2 > 0.
2.2. Proof of Theorem 2.4. Now the proof of Theorem 2.4 follows from Theorem 1.19, Proposition 2.11 and Corollary 2.12.
Proof of Theorem 2.4. Suppose by contradiction there are no rational curves in X. Thanks to Corollary 2.12 we can find a nef Q-divisor N such that 0 < c 2 (X) · N n−2 = 12 Td 2 (X) · N n−2 . So applying [Kol15,Theorem 10] the divisor N induces an elliptic fiber space X → B. Thus we can apply Theorem 0.2 to find rational curves in X, which gives a contradiction.
The idea of Theorem 2.4 is to find a nef divisor D in X with Itaka dimension n − 1. In the proof of Theorem 2.4 we explained that in our setting it is sufficient to find a nef Q-divisor D with numerical dimension n − 1 such that D n−2 · c 2 (X) > 0. A careful analysis in dimension three can be found in [DF14]. They work with smooth varieties but their proofs works verbatim also for varieties with log terminal singularities smooth in codimension two.