AND HOLOMORPHIC LAGRANGIAN FIBRATIONS

. Given a holomorphic Lagrangian ﬁbration of a compact hyperk¨ahler manifold, we use the diﬀerential geometry of the special K¨ahler metric that exists on the base away from the discriminant lo-cus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal rational curves admits a parallel splitting, which is nontrivial when the base is not half-dimensional projective space. Combining this with results of Voisin and Bakker-Schnell, we deduce that the base must be projective space, a result ﬁrst proved by Hwang.


Introduction
Let X 2n be a hyperkähler manifold, so X is a simply connected compact Kähler manifold with a holomorphic symplectic 2-form Ω such that H 2,0 (X) = CΩ.By Yau's Theorem every Kähler class on X contains a unique Ricci-flat Kähler metric.It was later realized by Beauville [5] that these metrics are hyperkähler, which means that they have holonomy equal to Sp(n).
Suppose that B is an irreducible normal complex analytic space with 0 < dim B < 2n, and f : X → B is a holomorphic surjective map with connected fibers.Then work of Matsushita [37] shows that necessarily dim B = n, that all irreducible components of the fibers of f are Lagrangian with respect to Ω, and the smooth fibers are tori.We call such f a holomorphic Lagrangian fibration.The following basic conjecture is widely expected to hold: Conjecture 1.1.If X is a hyperkähler manifold and f : X → B is a holomorphic Lagrangian fibration, then B ∼ = P n .This conjecture is clearly true when n = 1.The most striking result about this Conjecture is due to Hwang [27]: Theorem 1.2 (Hwang [27]).Conjecture 1.1 holds if X is projective and B is smooth.Theorem 1.2 was later extended to X Kähler and B smooth by Greb-Lehn [16].Assuming X projective and n = 2, it was proved by Ou [48] that either B is smooth (hence P 2 ) or else it has just one very specific singular point.This case was later ruled out independently by Bogomolov-Kurnosov [6] and Huybrechts-Xu [25], so the conjecture is known in this case.It is also known for some families of hyperkähler manifolds [4,10,36,41,63], but it remains open in general.
There are also a number of partial results towards Conjecture 1.1 in general, see [24] for an excellent recent overview.It is known that B must be a Kähler space (see e.g.[16,Prop. 2.2]) and Moishezon [38, §2.3], and that B is Q-factorial and has at worst klt singularities (by [38,Theorem 2.1]).It follows that B has at worst rational singularities, and hence it is projective by [45,Corollary 1.7].Again thanks to [38,Theorem 2.1] we see that B is a Fano variety with Picard number one, and in particular it is uniruled [42] and simply connected [54].The rational cohomology of B is isomorphic to the one of P n [50].It is also known that the map f is locally projective [9], so the smooth fibers are abelian varieties, and if B is smooth then the discriminant locus D ⊂ B of f has pure codimension 1 by [29,Proposition 3.1].
Our main result forms part of a new proof of Hwang's theorem, as well as Greb-Lehn's extension.In order to describe this, suppose B is not P n .Then from a result of Cho-Miyaoka-Shepherd-Barron [10], which uses Mori theory, it follows that there is a rational curve in B (not contained in D) with anticanonical degree at most n.We show that such a curve is free, and together these imply that the Grothendieck decomposition of the pullback of T B to this rational curve has some degree zero factors.Taking such rational curves with minimal anticanonical degree, we can consider the universal family U with evaluation map µ : U → B, which we may assume is a submersion over a Zariski open set B • ⊂ B (which we may assume is equal to B\D up to enlarging D), and the positive degree factors in the Grothendieck decomposition define a nontrivial holomorphic subbundle V ⊂ µ * T B • , whose rank is strictly less than n.At the same time, classical work of Freed [13] shows that on B • there is a "special Kähler metric" g SK , whose Kähler form ω SK is parallel with respect to a "special Kähler connection" ∇ SK on T R B • , which is torsion-free, flat, and d ∇ SK J = 0 (where J is the complex structure of B).Our main result is then: Theorem 1.3.In this setting, V is preserved by the pullback of the Chern connection of g SK .
We also show that the corresponding real subbundle V R ⊂ µ * T R B • is also preserved by the pullback of the special Kähler connection ∇ SK .This in turn can be interpreted as giving a nontrivial splitting of a real variation of Hodge structures (which naturally exists on B • ) when pulled back via µ.As we will discuss below, by combining Theorem 1.3 with work of Voisin [60], Hwang [26,27] and Bakker-Schnell [2], one can deduce Hwang's Theorem 1.2.
Let us first give some intuition for our approach.One of the key features of the rich geometry of special Kähler metrics is that they have nonnegative bisectional curvature.Recall here the fundamental theorem of Mori [44] and Siu-Yau [52] which states that a compact Kähler manifold with positive bisectional curvature must be isomorphic to P n .This was generalized by Mok [43] to classify compact Kähler manifolds with nonnegative bisectional curvature: their universal cover splits as a product of a Euclidean factor, of projective space, and of compact Hermitian symmetric spaces of rank 2. A large part of our arguments are motivated by trying to extend Mok's techniques to our noncompact manifold B\D with an incomplete metric with nonnegative bisectional curvature, making essential use of the special features of special Kähler metrics, which are summarized in Section 2.
To prove Theorem 1.3, thanks to a recent result of Bakker [1] we need to consider two cases: either f has maximal variation or f is isotrivial.In the first case, we prove in Section 4 a crucial rigidity result (Theorem 4.4) which shows that the bisectional curvature of ω SK vanishes when evaluated on a vector in V and a vector in its orthogonal complement.For this, we use results of Zhang and the second-named author [58] on the asymptotic behavior of ω SK near D, as well as a strictly positive lower bound for ω SK near D obtained by Gross, Zhang and the second-named author in [17,18,56].These are explained in Section 3. In Section 5 we then supplement the rigidity result by showing that the rough Laplacian of the bisectional curvature of ω SK evaluated on the same vectors vanishes as well.This result is analogous to a statement in Mok [43], although our proof is quite different.Equipped with these rigidity results, in Section 6 we adapt an argument of Mok [43] and conclude.In the isotrivial case the rigidity results are trivial because ω SK is flat, but this flatness can be effectively exploited to show again that V is preserved by the Chern connection of g SK .
In section 7 we sketch how Theorem 1.2 follows by combining Theorem 1.3 with a number of recent results in the literature.As mentioned above, we first show that the real subbundle V R ⊂ µ * T R B • which corresponds to V is preserved also by the pullback of the special Kähler connection ∇ SK .This uses again our rigidity theorem.Then we invoke an important result of Hwang [26,27], which also has a recent proof by Bakker-Schnell [2] (Theorem 7.2 below), which gives that the map µ must have connected fibers.Thus, our splitting descends to a parallel splitting of T R B • , from which we obtain a parallel real (1, 1)-form on B • which is not proportional to ω SK , which is contradiction to a result of Voisin [60].
Lastly, in Section 8 we make some comments on the obstacles that we faced when trying to extend our approach to the case when B is singular.
Remark 1.4.In the first draft of our paper, our original argument in Section 7 to construct the parallel form on B • turned out to be incomplete.After our first draft was posted to arXiv, Bakker and Schnell sent us their paper [2] with a new proof of Hwang's theorem.As mentioned above, to deduce Theorem 1.2 from Theorem 1.3 we now rely on their paper.On the other hand, without using [2], what our arguments show is that B must be P n provided that µ has connected fibers.As pointed out to us by Hwang, this result was implictly proved by Cho-Miyaoka-Shepherd-Barron [10, §7] using a different method (under the extra assumption that f has a section, which was removed by Nagai [46]).
1.1.Acknowledgments.This research was conducted during the period when the first-named author served as a Clay Research Fellow.The secondnamed author was partially supported by NSF grant DMS-2231783, and part of this work was carried out during his visit to the Center for Mathematical Sciences and Applications at Harvard University, which he would like to thank for the hospitality.We are grateful to B. Bakker, J. Cao, J. Kollár, M. Pȃun, T. Peternell, C. Schnell and J. Starr for enlightening email communications, and especially to B. Bakker, J.-M.Hwang, C. Schnell, C. Voisin and the referee for very useful comments on previous drafts.The secondnamed author would also like to thank Y. Zhang for discussions about this approach during the writing of [58].This work is dedicated to the memory of Jean-Pierre Demailly, a giant in the field of complex geometry and a dear colleague and friend, who left us much too early.

Special Kähler metrics
2.1.Notation.Let us first fix some notation.For a complex manifold B we will denote by T R B its real tangent bundle, and with T B ⊂ T C B = T R B ⊗ C its holomorphic tangent bundle (of complex tangent vectors of type (1, 0)).The dual of T B will be denoted by Ω 1  B .The complex structure will be denoted by J : T R B → T R B. We will also denote B • := B\D and X 2.2.Existence of special Kähler metrics.The paper by Freed [13], following work of Donagi-Witten [12], shows that the base of an algebraic integrable system (which in our case is B • ) admits a geometric structure called "special Kähler metric", ω SK .This means that (B • , J, ω SK ) is a Kähler manifold and there is a torsion-free flat connection ∇ SK on T R B • which makes ω SK parallel and d ∇ SK J = 0 (however, in general ∇ SK J = 0), where [13, p.33]).The Riemannian metric associated to ω SK will be denoted by g SK and its Levi-Civita/Chern connection, which in general is different from ∇ SK , will be denoted simply by ∇ (see (7.56) below for an explicit formula relating ∇ and ∇ SK ).On every sufficiently small open set U ⊂ B • we can find special holomorphic local coordinates {z j } n j=1 (whose real parts are flat Darboux coordinates) and a holomorphic map Z : U → H n into the Siegel upper half space , such that Z(y) are the periods of the torus fiber f −1 (y), and we can write It is also worth noting that special Kähler manifolds can only be complete if they are flat, by a result of Lu [35].See [58] for a description of the metric completion of (B • , ω SK ) and of its metric singularities.Special Kähler metrics have a Hodge-theoretic origin (see [23], [39]): as mentioned earlier there is a natural weight-one polarized real variation of Hodge structures R 1 f * R X • on B • , whose Hodge bundle of type (1, 0) is isomorphic to T B • (by contracting with the holomorphic symplectic form), and its Hodge metric is exactly the special Kähler metric.
In [56,17,18] it is also shown that ω SK can be written as ω B + i∂∂ϕ for some Kähler metric ω B on B and some function ϕ ∈ C ∞ (B • ) ∩ L ∞ (B).In fact, a priori there is a different special Kähler metric on B • for each chosen Kähler class [ω B ] on B, but since B is smooth Fano and of Picard number one, it follows that b 2 (B) = 1 so there is a unique choice of Kähler class up to scaling.In the following, we fix one such ω B once and for all.This way, we can unambiguously talk about "the" special Kähler metric ω SK in the following.
2.3.Curvature properties.Following Freed [13], there is a holomorphic symmetric cubic form Ξ ∈ H 0 (B • , Sym 3 T * B • ) such that, in any local holomorphic coordinate system, the curvature tensor of ω SK can be written as R ijkℓ = g pq SK Ξ ikp Ξ jℓq , and on any sufficiently small U as above we can find a holomorphic function F : U → C such that, in special holomorphic coordinates, the period matrix and the cubic form can be written as (2.2) From the curvature formula (2.1) we see in particular that ω SK has nonnegative bisectional curvature on B • : given any v, w ∈ T 1,0 B • we have where {e p } is any g SK -unitary frame.We will also use the following dichotomy, which was conjectured by Matsushita, and after progress by van Geemen-Voisin [59] it was recently proved by Bakker [1]: Theorem 2.1.Either f is isotrivial, or else f has maximal variation.This dichotomy is then reflected in the curvature properties of ω SK : In the second case, up to replacing D with a larger closed analytic subvariety we will always assume that Ric g SK > 0 on B • .
Proof.We use Bakker's Theorem 2.1.If f is isotrivial, then the local period map Z is constant, so from (2.2) we see that Ξ ≡ 0 on B • , and (2.1) shows that ω SK is flat.If f has maximal variation, then the period map Z is generically of maximal rank (equal to n), so Z is an immersion on a Zariski open subset of B • (which, up to enlarging D, we may assume is equal to B • ).Given any v ∈ T 1,0 x B • , the Ricci curvature of ω SK in the direction of v is given by Ric(v, v) = p,q |Ξ(v, e p , e q )| 2 0, and if this vanishes for some v = 0 then in special holomorphic coordinates we have that for all p, q 0 = Ξ(v, e p , e q ) = ∂ 3 F ∂v∂e p ∂e q = ∂ ∂v Z pq , so the period map is not an immersion at x, a contradiction.
Remark 2.3.The holomorphic sectional curvature of ω SK is given by where v ∈ T 1,0 B • is a unit vector (and in the last equality we use special holomorphic coordinates).The condition that ω SK has (strictly) positive holomorphic sectional curvature on B • thus means that none of the "diagonal" entries of the period matrix Z Z ij v i v j = ∂ 2 F ∂v∂v is locally constant.We expect that this always holds (up to enlarging D) when f has maximal variation.Remark 2.4.We are grateful to B. Bakker for the following observation.Let f : S → P 1 be an elliptic fibration of a K3 surface S. For n 2 let X = S [n] be the Hilbert scheme parametrizing length n subschemes of S. We obtain an induced holomorphic Lagrangian fibration f : X → (P 1 ) [n] = P n whose general fiber is isomorphic to the product of n general fibers of f , and if f has maximal variation then so does f .Since the period matrix of such a torus is diagonal, we see that the period map Z of f has Z ij = 0 for i = j.It follows that for these examples the special Kähler metric, which is not flat if f has maximal variation, nevertheless does not have strictly positive bisectional curvature on P n \D, since in local special coordinates we have Rm(e i , e i , e j , e j ) = R iijj = 0, for all i = j.Thus, to prove our main theorem, it would not be sufficient to prove a suitable noncompact version of the Mori-Siu-Yau theorem [44,52], but we must instead generalize the work of Mok [43].

Estimates on the special Kähler metric
We collect in this section two crucial estimates for the special Kähler metric ω SK , which are contained or follow from earlier work of the secondnamed author and coauthors [56,57,58].See also [8,21] for a study of the asymptotics of special Kähler metrics on Riemann surfaces.
3.1.Strict positivity.The first estimate, taken from [17,18,56,57], says that the positivity of ω SK does not degenerate as we approach D. Since this statement is valid even if B is singular, we present it in this generality.Proposition 3.1.Let X be a hyperkähler manifold, f : X → B a holomorphic Lagrangian fibration with B a normal analytic variety.Let ω B be a smooth Kähler metric on B (in the sense of analytic spaces) and ω SK the special Kähler metric on B • cohomologous to ω B .Then there is C > 0 such that on B • we have Proof.Fix a Kähler metric ω X on X and for t 0 let ω t be the hyperkähler metric on X cohomologous to f * ω B + e −t ω X .Then the Schwarz Lemma [56, Lemma 3.1] (using also [57, Proof of Theorem 3.2] in the case when B is singular) gives on X • (with C independent of t 0), and thanks to [17, Theorem 1.1], [22] and [18, Theorem 1.2] we know that as t → ∞ we have locally uniformly on X • (and even locally smoothly), so we conclude that on X • , and since f is a submersion over B • this is equivalent to If B has quotient singularities (which is expected to hold in general [24,Remark 1.11]) then we can replace ω B with an orbifold Kähler metric ω orb , and a similar argument gives the stronger bound 3.2.Ricci curvature bounds near D. From now on, we return to our standing assumption that B is smooth.The second crucial estimate is a bound for the Ricci curvature of ω SK .We have seen in the previous section that ω SK has nonnegative Ricci curvature on B • .In fact, as shown in [53,56] (see also [58,Prop. 4.1]), we have where ω WP is the Weil-Petersson form of the family of abelian varieties f : X • → B • (pullback of the Weil-Petersson metric on the moduli space via the moduli map).Concretely, on B • we have , where c > 0 and σ is a holomorphic symplectic form on X, and to obtain ω WP it suffices to take −i∂∂ log of the fiber integral in (3.4) divided by the local Euclidean volume form.
Recall that the discriminant locus D ⊂ B is a closed analytic subvariety of pure codimension 1, see [29,Proposition 3.1].Let x ∈ D be any smooth point of D, and choose an open neighborhood U of x with local holomorphic coordinates centered at x such that D ∩ U = {z 1 = 0}.Thus, at points of D ∩ U , the vectors ∂ ∂z 2 , . . ., ∂ ∂zn are tangent to D, while ∂ ∂z 1 is transversal.The main claim is the following: for some constant C > 0.
Proof.We will use freely the arguments in [58, §4.3] (these are stated for X projective hyperkähler, but all arguments there go through for general X hyperkähler using that Lagrangian fibrations are locally projective [9]).By the Monodromy Theorem, there is m ∈ N >0 such that the eigenvalues of the monodromy operator T (acting on H 1 (f −1 (y), Z) for some fixed basepoint y ∈ U \D) are mth roots of unity.We may assume without loss that in our coordinates U is the unit polydisc, and letting Ũ be the unit polydisc with coordinates (t 1 , . . ., t n ), we define the branched covering q : Ũ → U, q(t 1 , . . ., t n ) = (t m 1 , t 2 , . . ., t n ).Then after pulling back to Ũ , the monodromy operator T becomes unipotent, with (T − Id) 2 = 0. Thanks to the argument in [58, P. 774], we can find holomorphic functions w 1 , . . ., w n on Ũ , which are special holomorphic coordinates on Ũ ∩ {t 1 = 0} (but need not form a coordinate system at points on {t 1 = 0}, and they may even vanish there), such that, on Ũ ∩ {t 1 = 0}, we can write where Z jk (t) is the local period map pulled back to Ũ .Thus, if we denote by dV E the Euclidean volume form on Ũ given by the coordinates t 1 , . . .t n , we have and since det is holomorphic and nonzero on Ũ ∩ {t 1 = 0}, we get To estimate this, following [58,Lemma 4.3] we use Schmid's Nilpotent Orbit Theorem [49] and see there are b jk ∈ Q and a holomorphic map Q from Ũ to the space of symmetric n × n complex matrices, such that on Ũ ∩ {t 1 = 0} we have for some branch of log.Thus, and furthermore (see [58,Lemma 4.3]) there is C > 0 such that on Ũ ∩ {t 1 = 0} we have and so the inverse matrix of ImZ(t), whose entries will be denoted by (ImZ(t)) pq , satisfies Differentiating the determinant gives First we take j 2, and differentiating (3.8) gives ImQ qs (t) C, using (3.10) and the fact that Q is holomorphic on all of Ũ .As for the t 1 direction, differentiating (3.8) we have Going back to (3.7), this shows that on Ũ ∩ {t 1 = 0}) we have and so on U ∩ {z 1 = 0} we have for j 2, as desired.
Remark 3.4.We expect that the sharp bound in (3.6) in general is of the form [62] when dim B = 1.One may be able to show this by proving an asymptotic expansion for the fiber integral in (3.4) which can be differentiated term-by-term, as in [3,55].

Rational curves and rigidity
Recall that B is a Fano manifold, hence uniruled.Let ν : P 1 → B be a rational curve (i.e. a nonconstant holomorphic map) whose image is not contained in D. Our first result of this section shows that ν is a free rational curve, in the terminology of Mori Theory, cf.[34].
4.1.Freeness of the rational curve.By Grothendieck's Theorem, the vector bundle ν * T B splits and so we can write for some integers a i , which we order by a 1 • • • a n .Dualizing, we have (4.12) and B with the Hermitian metric h SK induced by the special Kähler metric ω SK .Lemma 4.1.We have a n 0.
Proof.This argument was suggested to us by M. Pȃun.Consider the nontrivial section v ∈ H 0 (P 1 , ν * Ω 1 B ⊗ O(a n )) which corresponds to the quotient morphism ν * T B → O(a n ).Equip L := O(a n ) with a smooth metric h L on P 1 , and equip ν * Ω 1 B with the smooth metric ν * h SK on P 1 \ν −1 (D) which is the pullback of the metric induced by ω SK .Thus, the curvature of ν * h SK is Griffiths nonpositive on P 1 \ν −1 (D), since ω SK has nonnegative bisectional curvature on B • and dualization reverses the sign of Griffiths positivity (see e.g.[11,§VII.6]).Equip then Differentiating log |v| 2 h on P 1 \ν −1 (D) we have the well-known identity of (1, 1)-forms on Since R h L is a smooth form on P 1 , we see that log |v| 2 h is quasi-psh on P 1 \ν −1 (D), and using (3.3) we see that sup where h B is the smooth metric on Ω 1 B induced by ω B .Thus log |v| 2 h is bounded above, hence by the Grauert-Remmert extension theorem [14] the inequality R h L + i∂∂ log |v| 2 h 0 extends over the singularities to all of P 1 (in the weak sense).Integrating this over P 1 and using Stokes thus gives as desired.
Lemma 4.1 says that every rational curve in B which is not contained in D is free, and by Mori Theory it deforms to cover a Zariski dense subset of B (see e.g.[34]).
The pullback morphism for some 0 ℓ n − 1, where a 1 a 2 • • • a n−ℓ 1, a 1 2, and Recall now a result by Cho-Miyaoka-Shepherd-Barron [10, Cor.0.4 (11)], which uses Mori theory: Theorem 4.2.Let B be a uniruled projective manifold, D an effective divisor, and suppose that, for any rational curve ν : P 1 → B which is not contained in D, we have the inequality If, in our setting, for all rational curves ν : P 1 → B not contained in D we have ℓ = 0, i.e. a i > 0 for all i, then since a 1 2 it would follow that −K B • ν(P 1 ) = n i=1 a i n + 1 and so B would be isomorphic to P n .In other words, if B ∼ = P n then there exists a rational curve ν 0 : P 1 → B not contained in D which has ℓ 1, i.e. there are some trivial factors O ⊕ℓ in the splitting (4.11).We may also assume that the anticanonical degree q := −K B • ν 0 (P 1 ) is as small as possible among all rational curves not contained in D (and satisfies 2 q n), and we will call these minimal degree rational curves, which is consistent with the standard terminology, e.g. in [28].By Lemma 4.1, this rational curve ν 0 is free and so it deforms to cover a Zariski dense subset of B. Let K be the irreducible component of the space of rational curves in B (see [34,§II.2]) which contains ν 0 , which we fix once and for all.From Mori Theory (see [34] and [28, §3]) we have that K is a quasiprojective variety equipped with a universal P 1 -bundle ρ : U → K and an evaluation map µ : U → B. For any t ∈ K we will also write is the rational curve corresponding to t, and will denote the morphism to B.
Furthermore, the generic rational curve in K is free and not contained in D, K is smooth at such curves, and the integers a i , ℓ in the decomposition (4.15) are the same for all generic such curves.Given x ∈ B • there is some minimal degree rational curve ν in K that passes through x and is smooth at x. Thanks to [34, Prop.II.3.7],we can also assume that ν(P 1 ) intersects D only at the regular points of D (since the singularities of D have codimension at least 2 in B), and that these intersections are transverse.The evaluation morphism µ : U → B is a submersion over a Zariski open subset of B, which up to enlarging D we may assume equals B • .Thus, if we define U is a submersion.The metric g SK on T B • induces by pullback a metric µ * T B • over U • , which we will denote by the same symbol, and similarly for the connections ∇ and ∇ SK , which induce pullback connections denoted in the same way.

Lemma 4.3.
There is a locally free sheaf V ♯ on U such that for every t ∈ K, the restriction V ♯ Ut of V ♯ to the rational curve U t equals the factor O ⊕ℓ in the splitting (4.16) for ν * t Ω 1 B .Proof.For the sake of clarity, we first define the fiber V ♯ at any point on U t ∼ = P 1 .For this, we consider ν * t Ω 1 B , which from the splitting (4.16 ) is then ℓ-dimensional, and we can find a basis of such sections which are linearly independent at all points of P 1 .The fiber of V ♯ at any point on U t is then defined as the linear span of any given basis of H 0 (U t , ν * t Ω 1 B ).To prove that this collection of ℓ-dimensional vector spaces form a locally free sheaf, consider first the locally free sheaf µ * Ω 1 B on U , and take its direct image sheaf [20, Cor.III.12.9] shows that ρ * µ * Ω 1 B is a locally free sheaf on K.We then set B , which is a locally free sheaf over U whose fibers agree with our previous description.
Our main interest will be with the restriction of V ♯ to U • , which will be denoted with the same notation.This is a holomorphic vector bundle over U • , which is naturally a subbundle of µ * Ω 1 B • .We then define a holomorphic subbundle V ⊂ µ * T B • over U • as the annihilator of V ♯ , namely For any t ∈ K we have that the restriction of V to U t equals the factor t T B, it follows that the tangent direction to the image of ν t at any point on this curve (which is a line in T B • ) when pulled back via µ lies in the fiber of V over U t .
We then define a smooth complex subbundle N ⊂ µ * T B • over U • as the g SK -orthogonal complement of V, and N ♯ ⊂ µ * Ω 1 B • as its annihilator (or equivalently as the g SK -orthogonal complement of V ♯ ), so that over U • we have the splittings are not yet known to be holomorphic (we will prove this later on).Note also that the (complex antilinear) smooth isomorphism • , defined by the metric g SK (by "lowering the index" and conjugating) maps N isomorphically onto V ♯ .4.2.The rigidity theorem.We have the following rigidity statement: Theorem 4.4.Given a rational curve U t for some t ∈ K, with morphism ν t : P 1 → B, and given a section u ∈ H 0 (P 1 , V ♯ Ut ), let ν * t h SK be the smooth metric on ν * t Ω 1 B over P 1 \ν −1 t (D) induced by g SK , and let R ν * t h SK be its curvature.Then we have: (a) Let ζ be the smooth section of N Ut over P 1 \ν −1 t (D) which corresponds to u under (4.19), and let α be a tangent vector to ν t (P 1 ).Then at any point on ν t (P 1 )∩B • the curvature tensor of g SK satisfies Proof.(a) Equip V ♯ Ut with the smooth metric h on Since ω SK has nonnegative bisectional curvature, the induced metric on Ω 1 B (and hence also the one on ν * t Ω 1 B ) is Griffiths nonpositively curved, and since curvature decreases in subbundles, the metric h is also Griffiths nonpositively curved.
As in (4.14), on P 1 \ν −1 t (D) we have Thus log |u| 2 h is psh on P 1 \ν −1 t (D), and again using (3.3) we see that sup where h B is the smooth metric on Ω 1 B induced by ω B .Thus log |u| 2 h is bounded above, and by the Grauert-Remmert extension theorem [14] it extends to a global psh function on P 1 , which is therefore constant.
Thus |u| 2 h is a nonzero constant, and from (4.25) we deduce that (4.26) R h (u), u h = 0, on P 1 \ν −1 t (D).But using again the curvature decreasing property, we have 0 and so (c) Given v ∈ H 0 (P 1 , V Ut ) and a point x ∈ ν t (P 1 ) ∩ B • , we can find a holomorphic family {ν s } s∈∆ of rational curves in K that pass through x, with tangent vectors α s at x (with ∆ ⊂ K a small disc in some chart centered at our original point t ∈ K), and such that d ds s=t α s = v(x).Let w = d ds ν s be the first-order deformation (holomorphic) vector field on this family.When restricted to each U s , w is a section of and since w(x) = 0, it must be a section of the i O(a i ) factors, namely a section of V Us .Pick a smooth family U of 1-forms on this family, i.e. a C ∞ section of the relative cotangent bundle, with u s := U | Us ∈ H 0 (P 1 , V ♯ Us ), and with u t = u.Then by definition along ν s we have ι w U Us ≡ 0, for all s ∈ ∆, and so along ν t we have which vanishes at x since w(x) = 0.
We now use this to prove (4.24), which by (2.1) implies (4.23).For this, let ζ s , s ∈ ∆, be the smooth section of N Us over P 1 \ν −1 s (D) which maps to u s under (4.19), and recall that from (4.22) at x we have Ξ x (α s , ζ s , β) = 0, for all s ∈ ∆.Taking d ds s=t of this, we get (4.28) Now at x we have that L w ζ is the vector that maps to L w U under (4.19), since at x the metric g SK does not get differentiated as it does not depend on s.Since we have shown that (L w U )(x) = 0, we deduce that (L w ζ)(x) = 0, and so (4.24) follows from (4.28).
(d) Given a section u ∈ H 0 (P 1 , ν * t V ♯ ), an analogous computation as in (a) gives and so we conclude that ∇u = 0 on P 1 \ν −1 t (D).(e) This is a direct consequence of part (d) and [32,Prop. 1.4.18].
so we can view v and ζ also as tangent vectors in B • .With this in mind, we have the following useful corollary: Proof.Let t ∈ K be such that the corresponding rational curve U t contains x, and as usual denoted by ν t : P 1 → B the corresponding morphism.Since V Ut ∼ = ⊕ i O(a i ), a i > 0, is a globally generated vector bundle, we can find a global section V ∈ H 0 (P 1 , V Ut ) such that V (x) = v.Let then u ∈ V ♯ x be the covector which is the image of ζ under (4.19).Since V ♯ Ut ∼ = O ⊕ℓ is a trivial vector bundle, we can find a global section U ∈ H 0 (P 1 , V ♯ Ut ) such that U (x) = u.Then Theorem 4.4 (c) applies to U and V , and (4.31), (4.32) follow from (4.23), (4.24).
5. The Ricci curvature in the direction of N Given x ∈ U • and vectors v ∈ V x , ζ ∈ N x (which we can also view as tangent vectors in T µ(x) B • using (4.30)),Corollary 4.5 shows that at µ(x) the Riemann curvature tensor of g SK satisfies R vvζζ = 0.
As customary, we define the "rough Laplacian" of the Riemann curvature tensor of g SK , evaluated on v, ζ by where {e i } is a local unitary frame.
The following is the main result of this section: Let t ∈ K be such that the corresponding rational curve U t contains x, and as usual denoted by ν t : P 1 → B the corresponding morphism.As in the proof of Corollary 4.5, we can extend v to a section v ∈ H 0 (P 1 , V Ut ) and we can find a section u ∈ H 0 (P 1 , V ♯ Ut ) such that the image of u under (4.19) is a smooth section ζ ∈ N | Ut over P 1 \ν −1 t (D) which extends the given vector ζ.The Ricci curvature R ζζ along this curve and evaluated at ζ will also be denoted by Ric g SK (u, u), which is a smooth function on P 1 \ν −1 t (D).We wish to show that Ric g SK (u, u) is a constant function on P 1 \ν −1 (D).We will proceed in steps.

Subharmonicity of Ric g SK (u, u).
To start, we prove the following: It is clear that given any compact K ⋐ B • there is some 0 < s K ≪ 1 such that g s is a Kähler metric on K for 0 s s K .
Standard direct computations (cf.[43, P.185]) show that given any x ∈ B • and two nonzero (1, 0) tangent vectors v, ζ at x, we have the evolution equation at x and s = 0 for the bisectional curvature of g s evaluated along v and ζ where, as in Mok [43], we define Equation (5.34) is identical to the corresponding evolution of the bisectional curvature in the directions v, ζ along the Kähler-Ricci flow, see [43].Thanks to the crucial Lemma 5.3 below, we see that Equip ν * t Ω 1 B over the compact set ν −1 t (K) with the Hermitian metric h s induced by g s .At any point y ∈ ν −1 t (K) for 0 s s K , using the argument in (4.25), we have (5.36)i∂∂ log |u| 2 hs + R hs (u), u hs |u| 2 hs 0.
We know from Theorem 4.4 (d), that u is parallel with respect to h 0 (the metric induced by g SK ), hence (assuming without loss that u is nontrivial) we can scale and assume without loss that |u| 2 h 0 ≡ 1 on P 1 \ν −1 t (D).On the other hand, from Theorem 4.4 (a), we know that (4.20) holds, and so Thus the LHS of (5.36) vanishes at y for s = 0 and is nonnegative for 0 s s K , hence at y we have and writing u = u j dz j and |u| 2 hs = u i u j g ij s , observe that Furthermore, we can write but the last two terms vanish since using (2.1) and ( 4.24), we can write and putting these all together gives (5.38) using (5.35).Since K ⋐ B • is arbitrary, this shows that the function Ric g SK (u, u) is subharmonic on P 1 \ν −1 t (D).We used the following lemma, which is the analog of "condition (♯)" in Mok, but the proof here is substantially easier: Lemma 5.3.In the setting of Theorem 5.1, at µ(x) we have 31) we then see that at µ(x) we have and so Ξ vζβ (x) = 0 for all β ∈ T µ(x) B • , and furthermore for all µ, ν ∈ Now take the definition of ∆R and use (2.1) and the fact that Ξ is holomorphic to get For the F (R) term, from its definition we see that at µ(x) we have As in Mok [43, (7)], if we pick {e µ } a unitary basis of eigenvectors of the Hermitian form H v (µ, ν) = R vvµν , then in this basis we see that (5.39) Proof.Since the function Ric g SK (u, u) on P 1 \ν −1 t (D) is subharmonic by Proposition 5.2, it suffices to show that it is bounded.
Recall that, using (4.30), our sections v, ζ can be viewed as vector fields along ν t (P 1 ) ∩ B • .Our first claim is that for every y ∈ ν t (P 1 ) ∩ B • and local sections v of V and ζ of N near y, we have (5.40)R vζ = 0.
Indeed, recall from (2.1) that where {e p } is a g SK -unitary frame at our point y.Since µ * T B • = V ⊕ N , we may choose the frame so that e j ∈ V for 1 j n − ℓ, and e j ∈ N for n − ℓ + 1 j n.Recalling from (4.32) that Ξ uvw = 0 whenever u ∈ V and v ∈ N , we see that Ξ vpq = 0 except possibly when 1 p, q n − ℓ, so that Ξ vpq Ξ ζpq = 0, since Ξ ζpq = 0 when 1 p, q n − ℓ, proving our claim.
Recall that, as explained earlier, we may assume that ν t (P 1 ) intersects D only at regular points of D and that these intersections are transverse.To prove the boundedness of Ric g SK (u, u) it suffices to prove near any of the finitely many points in ν −1 t (D).Let y be such a point, and choose an open neighborhood U of z = ν t (y) in B with local holomorphic coordinates centered at z such that D ∩ U = {z 1 = 0} and ν t (P Thanks to Proposition 3.3 we know that on V we have Using (3.3), together with the fact that u is a holomorphic section on all of P 1 , we see that (5.43) sup In our coordinates we can write and the function ζ 1 is equal to dz 1 , u g SK .From (5.43) we see that and from this and (5.41) we see that on V we have Since ∂ 1 is the tangent vector to ν t (P 1 ), it belongs to V Ut .On the other hand ζ belongs to N Ut , hence (5.40) (restricted to V ) gives and since Ric g SK 0 on {z 1 = 0}, Cauchy-Schwarz together with (5.41) and (5.45) give C, and using again that Ric g SK 0, together with (5.45) we can estimate as desired.
Indeed, Proposition 5.4 shows that the function Ric g SK (u, u) on P 1 \ν −1 t (D) is constant, hence going back to (5.38) and recalling (5.35) and (5.34) shows that Recalling Lemma 5.3, we see that (5.46) holds.To finally deduce from (5.46) that the last equality in (5.33) holds, it suffices to plug in the fact that F (R) vvζζ = 0 into (5.39), and see that as desired.
6. Constructing a parallel subbundle of µ * T B • Recall that above we have constructed a decomposition µ * T B • = V ⊕ N over U • , where V is a nontrivial holomorphic subbundle, which is not equal to µ * T B • whenever B ∼ = P n .
The following is then our main theorem (Theorem 1.3): Theorem 6.1.The holomorphic subbundle V ⊂ µ * T B • over U • is preserved by ∇, the pullback of the Levi-Civita connection of ω SK .
Recall that by Theorem 2.1 f is either of maximal variation or isotrivial.The proof of Theorem 6.1 will be quite different in these two cases.
Observe that after Theorem 6.1 is proved, it follows that the orthogonal complement N ⊂ µ * T B • is also a holomorphic subbundle, preserved by ∇, see e.g.[32,Prop. 1.4.18], and the same holds for their duals V ♯ , N ♯ ⊂ µ * Ω 1 B • .6.1.Maximal Variation Case.In this section we give the proof of Theorem 6.1 in the case when f has maximal variation.Recall from Corollary 2.2 that in this case g SK has positive Ricci curvature on B • .
We work at a point x ∈ U • .Let v be a local holomorphic section of V near x, and let γ : (−ε, ε) → U • be a smooth curve with γ(0) = x, γ(0) = η = 0.The goal of Theorem 6.1 is then to show that ∇ η v ∈ V. Using the decomposition µ * T B • = V ⊕ N , we can write (the minus sign is only to match the notation in Mok [43]), so we wish to show that ζ = 0.The following argument is a modification of a result of Mok [43, Proposition 3.1'], specifically of equation ( 21) on p.211: Proposition 6.2.At µ(x) we have Here and in the following we are again using (4.30) to view ζ, ζ ′ also as tangent vectors in T µ(x) B • .Also, since ∇ is the pullback connection, when taking , and so we see that ξ(0) = ξ, ζ(0) = ζ.Given an arbitrary ζ ′ ∈ N x , let χ(t) be the parallel transport of ζ ′ along γ, so that χ(0) = ζ ′ and ∇ γ(t) χ(t) = 0. We can also write where O(t 3 ) denotes a vector-valued function of length bounded above by Ct 3 .Recalling (2.1), we can express the curvature tensor in terms of Ξ, and since Ξ(v, ζ ′ , β) = Ξ(ξ, ζ ′ , β) = 0 for all ζ ′ ∈ N x and all β ∈ T µ(x) B • (by Corollary 4.5), many terms in this expansion vanish.Using furthermore that R vζβδ = 0 for all β, δ ∈ T µ(x) B • (by Theorem 5.1), the expression finally reduces to Defining (similarly to Mok) and since the bisectional curvature is nonnegative, we have R vvθθ 0, and so At this point notice that (6.49) using that β(t), χ(t) are parallel along γ and that ∇ is a pullback connection.
On the other hand we claim that at x we have ∇ 2 ww R vvζ ′ ζ ′ 0, for all real tangent vectors w at x. Indeed, pick a curve in U • passing through x and tangent to w, and let ṽ(t), ζ′ (t) be the parallel transport of v, ζ ′ along this curve, then R ṽ(t)ṽ(t) ζ′ (t) ζ′ (t) 0, and R vvζ ′ ζ ′ = 0 by Corollary 4.5, and so as claimed.But recall that Theorem 5.1 showed that ∆R vvζ ′ ζ ′ = 0, and since this is an average of terms of the form ∇ 2 ww R vvζ ′ ζ ′ as µ * (w) varies among all g SK -unit tangent vectors at µ(x), we see that necessarily ∇ 2 ww R vvζ ′ ζ ′ = 0 for all w.Using (6.48) and (6.49) we get |Ξ(ζ, e p , e q )| 2 = 0.
Since we assume f of maximal variation, Ric g SK > 0 on B • , and so ζ = 0.This concludes the proof of Theorem 6.1 when f has maximal variation.
6.2.Isotrivial Case.In this section we give the proof of Theorem 6.1 in the case when f is isotrivial, and so ω SK is flat by Corollary 2.2.We wish to show that the subbundle V ⊂ µ * T B • is parallel under ∇, and by duality this is equivalent to showing that Recall that ρ : U → K is a P 1 -bundle.Thus, given x ∈ U • and v ∈ T x U • , we can decompose T x U • as the direct sum of the tangent line to the vertical P 1 direction and a complementary subspace, and thus write v = v 1 +v 2 , where v 1 is tangent to a rational curve U t (for some t ∈ K) that contains x (which on U t corresponds to a point y ∈ P 1 ) and v 2 is transverse to U t .The rational curve morphism will be as usual denoted by ν t : P 1 → B. We may also assume that ν t (P 1 ) intersects D only at regular points of D. Since ν t is free, we can deform it in a 1-parameter family π : P 1 × ∆ → B, with s ∈ ∆ (where ∆ ⊂ K is a small disc in some chart centered at t ∈ K), such that ν s := π(•, s) : P 1 → B are rational curves in K which are also not contained in D and such that the first order deformation vector ∂ ∂s s=t ν s ∈ H 0 (P 1 , ν * t T B) agrees with v 2 at x. Up to shrinking ∆, we have a natural inclusion σ : and contains the point (y, 0).
For every s ∈ ∆, σ * V ♯ P 1 ×{s} is a trivial vector bundle of rank ℓ over ν s , which over P 1 \ν −1 s (D) is equipped with the metric induced by ω SK .For each s ∈ ∆ we can then choose a global holomorphic frame u 1 (s), . . ., u ℓ (s) ∈ H 0 (P 1 , σ * V ♯ P 1 ×{s} ), smoothly dependent on s ∈ ∆.Thanks to Theorem 4.4 (d), each u i (s) is parallel (with respect to the connection induced by ω SK ) over P 1 \ν −1 s (D).Varying s, these sections define a smooth frame u 1 , . . ., u ℓ of σ * V ♯ over P 1 × ∆, which is parallel when restricted to each (P 1 × {s}) ∩ π −1 (B • ).Fix now any 1 i ℓ, and recall that (6.51) , where N ♯ is the annihilator of N ⊂ µ * T B • .Let P be the g SK -orthogonal projection onto the σ * N ♯ factor, which is defined on π −1 (B • ) and consider Proof.We work at an arbitrary point in P 1 \ν −1 t (D), let W be any local holomorphic vector field near our point which is tangent to the P 1 factor.Since the splitting ν is preserved by ∇ (by Theorem 4.4 (e)), and since g SK is flat, we have Now, since u i is parallel along the rational curve ν s (P 1 )\D for all s ∈ C, we have that ∇ W u i vanishes identically on U × ∆ and so is parallel, it is in particular holomorphic over P 1 \ν −1 t (D).The following Lemma then implies that ι * (P(∇ V u i )) extends to a holomorphic section of ν * t Ω 1 B over P 1 : Proposition 6.4.Let w ∈ H 0 (P 1 \ν −1 t (D), ν * t Ω 1 B ) be a holomorphic section which is parallel with respect to ∇ (the Chern connection induced by ω SK ).Then w extends to a holomorphic section of ν * t Ω 1 B over all of P 1 .Proof.Since w is parallel, its pointwise length |w| 2 ν * t g SK is constant on P 1 \ν −1 t (D).Recall that from (3.3) we have that (6.52) is a finite subset of P 1 , we consider the extension problem of w across each of these points, so let y ∈ ν −1 t (D) be one of them.Recall that D is regular at the point x = ν t (y), and we can choose local holomorphic coordinates z 1 , . . .z n on a chart U centered at x such that D ∩ U = {z 1 = 0}.The volume form ω n SK is given by a fiber integration as in (3.4), and its asymptotic behavior near D is studied in [19, Theorem 2.1] (see also [7,18] for the case when dim B = 1 and [31], [55] for dim B arbitrary) where it is shown that (6.53) on U ∩ {z 1 = 0}, for some C > 0 and γ ∈ (0, 1].Combining (6.52) and (6.53) gives the crude bound (6.54) see also [58, (2.1) and Theorem 3.4] and [19,Theorem 1.1] for sharper and more general such bounds.Passing to the dual metric on Ω 1 B and pulling back via ν t , (6.54) implies that on the punctured neighborhood ν −1 t (U ∩{z 1 = 0}) of y in P 1 we have and from this we see that |w| 2 is a trivial bundle over ν −1 t (U ), we can represent w locally as an n-tuple of holomorphic functions on ν −1 t (U ∩{z 1 = 0}), and since these functions are in L 2 , they extend holomorphically across the point y (see e.g.[47, Proposition 1.14]), which gives us the desired extension of w.
At this point we have shown that ι * (P(∇ V u i )) gives a holomorphic section w ∈ H 0 (P 1 , ν * t Ω 1 B ). Recalling the splitting (4.12), we see that w must be a section of the factor O ⊕ℓ , i.e. a section of V ♯ Ut .Since it is also a section of N ♯ Ut , it must be identically zero.This shows that ι * (P(∇ V u i )) = 0, and so ι * (∇ V u i ) ∈ V ♯ Ut , and so (6.50) is established.This concludes the proof of Theorem 6.1 when f is isotrivial.

Obtaining a parallel (1, 1)-form and Hwang's Theorem
In this section we show how to combine our main theorem 1.3 with results of Voisin [60], Hwang [26,27] and Bakker-Schnell [2] to deduce Theorem 1.2.The key step is the following: Theorem 7.1.Suppose that B ∼ = P n .Then there is a nontrivial real (1, 1)form ψ on B • with ∇ SK ψ = 0 and ψ not proportional to ω SK .
First, we show that Theorem 1.2 follows from this (we do not need to assume that X is projective): Proof of Theorem 1.2.Suppose for a contradiction that B ∼ = P n .Then by Theorem 7.1 the 2-forms ω SK and ψ on B • are both ∇ SK -parallel and not proportional, and thus they give us a 2-dimensional space of global sections of the local system R 2 f * R X • over B • .However, as observed by Voisin [60, Lemma 5.5], a result of Matsushita [40] together with Deligne's invariant cycles theorem show that this space of sections is always 1-dimensional, a contradiction.
Since B ∼ = P n , we know that V ⊂ µ * T B • is a nontrivial proper holomorphic subbundle over U • , which by Theorem 6.1 is preserved by ∇.As mentioned after Theorem 6.1, the g and analogously for N R .The bundle V R is isomorphic to V via the usual inverse map T R B → T B given by u → u−iJ(u) 2 (and similarly for N R ), and on U • we have a splitting Consider now the Stein factorization of µ : U → B, given by where U → Z has connected fibers and p : Z → B is finite.Define also Z • := p −1 (B • ).To complete the proof of Theorem 7.1, we will then need the following theorem which is implicit in the work of Hwang [27], and also appears in the recent work of Bakker-Schnell ([2, Proposition 3.2 and proof of Theorem 1.1]) relying on ideas of Hwang [26,27]: Theorem 7.2.Suppose the splitting (7.55) is preserved by ∇ SK , then p : Z → B is an isomorphism.
We can now give the proof of Theorem 7.1: Proof of Theorem 7.1.Since B ∼ = P n , we have the nontrivial splitting (7.55).By definition, V R is preserved by J, and since V is preserved by ∇ (and ∇J = 0), it follows that V R is also preserved by ∇.We claim that V R is preserved by ∇ SK .To see this, recall that Freed shows in [13, (1.29)] that the special Kähler connection on T R B is given by (7.56) where as usual ∇ is the Levi-Civita connection of g SK and A ∈ Λ 1,0 Hom(T B • , T B • ) is given by (7.57)A ℓ ij = √ −1g kℓ SK Ξ ijk , and the same holds for the pullback connection on U • .Given a local section α of V and a local (1, 0) vector field v ∈ T U , we wish to show that Since we know that ∇ v+v (α + α) ∈ V R , it suffices to check that and so it suffices to see that or equivalently that g SK (A v (α), ζ) = 0 for all local sections ζ of N .But from (7.57) we see that which vanishes by Theorem 4.4 (c).This concludes the proof that V R is preserved by ∇ SK .An analogous argument shows that N R is also preserved by ∇ SK , and so the splitting (7.55) is preserved by ∇ SK .Applying Theorem 7.2 we see that p : Z • → B • is an isomorphism, so we may assume that µ : U • → B • has connected fibers.The vector bundle µ * T R B • is trivial when restricted to these fibers, and its subbundles V R , N R restricted to a fiber are preserved by the pullback connection ∇ SK (which when restricted to the fiber is a trivial connection), and so V R and N R are pullbacks of vector bundles on B • (denoted by the same notation), which are subbundles of T R B • and are still preserved by ∇ SK .We then define a (1, 1)-form ψ on B • by projecting ω SK onto V R .Since ∇ SK ω SK = 0 and V R is preserved by ∇ SK , it follows that ∇ SK ψ = 0 (and also ∇ψ = 0 for the same reason), and since V R is a nontrivial proper subbundle of T R B • , we see that ψ is nonzero and not proportional to ω SK , and we are done.

Comments about the case when B is singular
It is tempting to ask whether our method can be used to prove that B ∼ = P n even when B is singular.As mentioned in the Introduction, this is currently known only for n 2 [6,25,48].In general, it is known that B is a normal projective variety, with at worst klt singularities, which is Fano with Picard number one.The natural generalization of our approach (following [51], who generalized Mori's Theorem [44] to the singular setting) would be to consider a functorial resolution of singularities π : B → B and to show that we must have B ∼ = P n , which forces B ∼ = P n as well.In this setting, B is a uniruled projective manifold and D = π −1 (D) is a divisor, so many of our arguments above can be repeated on B• := B\ D, which carries a special Kähler metric ω SK .The fact that π is functorial gives us a morphism µ : π * T B → T B which is an isomorphism on B• , where T B = Hom(Ω 1 B , O B ) is the reflexive tangent sheaf.Given a rational curve ν : P 1 → B, which is not contained in D, pulling back µ via ν we obtain a sheaf injection A := (π • ν) [ * ] T B → ν * T B, between these vector bundles on P 1 (which both split as a direct sum of line bundles which should have nonnegative degrees).Here we use the standard reflexive pullback notation (π • ν) [ * ] T B := (ν * π * T B) * * .Using Theorem 4.2, if B ∼ = P n then ν * T B contains a nontrivial O factor, hence so does A. To implement our strategy, one would need a rigidity statement like in Theorem 4.4 for either one of these trivial summands, and a crucial ingredient of the proof of the rigidity statement is that sections of the dual of the relevant bundle should have bounded norm (with respect to the pullback of ω SK ).The first fundamental issue is that it is not clear to us how to show that sections of ν * Ω 1 B or of A * have bounded norm.The key ingredient for this when B is smooth was the estimate (3.3), but when B is singular this by itself is not sufficient to prove boundedness.
What can be shown using results in [15] is rather that sections of the reflexive pullback (π • ν) [ * ] Ω [1] B have bounded norm, but in general the Grothendieck decomposition of this vector bundle is different from those of ν * Ω 1 B and A * , and it may happen that these have some nontrivial O factor but (π • ν) [ * ] Ω [1] B does not, which invalidates our approach.This undesirable phenomenon can only happen when the generic rational curve (of the type that we are considering) when projected down to B always passes through some singular point of B. This however seems unavoidable in general, as finding low-degree rational curves in normal Fano varieties that can be deformed to avoid the singularities is a very delicate problem in algebraic geometry, see e.g.[30,33,61].

Corollary 2 . 2 .
Either ω SK is flat on B • , or else ω SK has positive Ricci curvature on a Zariski open subset of B • .