A Positivity Property of Ample Vector Bundles Christophe Mourougane, Shigeharu Takayama

A positivity property of ample vector bundles Christophe Mourougane, Shigeharu Takayama

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Christophe Mourougane, Shigeharu Takayama. A positivity property of ample vector bundles. 2005. hal-00004912v1

HAL Id: hal-00004912 https://hal.archives-ouvertes.fr/hal-00004912v1 Preprint submitted on 16 May 2005 (v1), last revised 19 Nov 2005 (v2)

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Y ude,w hwta for that show we bundles, s of P ( E laoaini an in ollaboration ezrs(..may (i.e. zeros re ) O h ait of variety the , c vanishing uce E hnfrall for Then ( r.If ure. e o all for hen − 1) othe to k E the , is 2. AMPLENESSANDPOSITIVITY We refer to [12] or [16] for basics about ample vector bundles and to [9] or [4, chapter VII] for basics about positive vector bundles. All vector bundles are assumed to be holomorphic. A vector bundle E on a compact complex manifold X is said to be semi-ample if for some positive integer k, its symmetric power SkE is generated by its global sections. Associated to E, we have π : P(E) X the variety of rank one quotients of E together with its tautological quotient line bundle (1)→. The semi-ampleness of E is rephrased that for every x X, every section s of (k) OE ∈ x OE over the fiber P(Ex) extends to a global section of E(k) over P(E). This in particular implies that (k) is generated by its global sections. A vectorO bundle E is said to be ample if its associated line OE bundle E(1) is ample on P(E). This in particular implies the existence of an integer k such that for every xO X, every section s of (k) over the first infinitesimal neighborhood of the fiber P(E ) ∈ x OE x extends to a global section of E(k) over P(E). A vector bundle E is said toO be Griffiths positive, if it can be endowed with a smooth hermitian metric h such that for all x X and all non-zero decomposable tensors v e TXx Ex, the curvature term Θ(E, h)(v, v∈)e, e is positive, where Θ(E, h) ∞ (X, Herm⊗ (E∈)) is the⊗ curvature h ih ∈ C1,1 of the Chern connection of (E, h). Recall the formula ∇E,h Θ(E, h)ξ,ξ ξ, ξ √ 1∂ ξ 2 ∂ ξ 2 (2.1) h i = √ 1∂∂ log ξ 2 + h∇E,h ∇E,h i − || || ∧ || || ξ 2 − − || || ξ 2 − ξ 4 || || || || || || √ 1∂∂ log ξ 2 ≥ − − || || for a nowhere zero local holomorphic section ξ of a holomorphic vector bundle E equipped with a smooth hermitian metric h. The last two terms give the norm at x of the fundamental form of the inclusion ξ E. A continuous hermitian metric h on a vector bundle b : E X is said to be OX ⊂ → Griffiths positive, if there exists a smooth positive real (1, 1)-form ωX on X such that in the sense of currents h( ξ) √ 1∂h(ξ) ∂h(ξ) √ 1∂∂ log h(ξ)+ ∇ − ∧ b⋆ω , − − h(ξ) − h(ξ)2 ≥ X where h is seen as a continuous quadratic function on the total space E X 0 . At the points where the metric h is smooth, these two notions of Griffiths positivity co¨ınci− de.×{ } The theory of resolution of the ∂-equation with L2-estimates for example shows that Griffiths positivity and ampleness are equivalent for line bundles. This implies through the curvature computation of E(1) that Griffiths positive vector bundles are ample. The converse is a problem raised by Grif- fiths,O and solved positively on curves by Umemura [21] using the concept of stability (see also [3]). Notation and Assumption. In the rest of this paper, we will use the following notations. We let E X be a holomorphic vector bundle of rank r > 1 on a compact connected complex manifold → r−1 X, and π : P(E) X be the associated P -bundle with the line bundle E(1). We assume E is semi-ample at least,→ and we take and fix an arbitrary positive integer k such thatO SkE is generated by its global sections.

3. CYCLIC COVERS

A reference for this part is [6, 3]. By our assumption at the end of 2, E(k) is generated by its global sections. Then Bertini’s§ theorem (see for example [11, page 137])§ O insures that a generic P P section s of E(k) over (E) is transverse to the zero section (i.e. ds|Ds : T (E)|Ds E(k)|Ds is O 2 → O surjective), and defines a smooth divisor Ds := (s = 0). Let p = p : Y P(E) s s → be the cyclic covering of P(E) obtained by taking the k-th root out of Ds. We intend to study the morphism π p : Y X. The space ◦ s → Y := l (1)/lk = s(p(l)) s { ∈ OE } is a smooth hypersurface of the total space (1), and the map p is a finite cover totally ramified OE along the zero locus Ds of s. The space Ys may also be described as the spectrum Spec s of the algebra A +∞ i=0 E( i) s := ⋆ ⊕⋆ O⋆ − A (l sˇ(l ) , l E( k)) ×s − ∈ O − where sˇ is the sheaf inclusion ( k) = P . The direct image of the structure sheaf OE − → OE O (E) OYs is hence p = k−1 ( i). Simply note that no negative power ( i) has non-zero ⋆OYs As ≃ ⊕i=0 OE − OE − sections on the fibers of π to infer that the direct image (π p)⋆ Ys is X . This shows that the fibers of π p are connected and that the covering space Y is therefore◦ O connected.O One can check by a ◦ s local computation that the morphism π p is smooth over the set X Σs of points x X where P ◦ − ∈ s|P(Ex) Γ( (Ex), E(k)) is transverse to the zero section. We shall call Σs the discriminant locus of π p∈. O We◦ will in fact work with the top degree direct image of the structure sheaf. Because p is a finite morphism, the spectral sequence of composition of direct image functors reduces to the following: r−1(π p) = r−1π 0p R ◦ ⋆OYs R ⋆ R ⋆OYs = r−1π k−1 ( i) R ⋆ ⊕i=0 OE − k−1 ⋆ (3.1) = π ωP (i) ⊕i=0 ⋆ (E)/X ⊗ OE k−1 ⋆ ⋆ = i=0 π⋆ ( E(i r) π det E) ⊕ O − ⊗ ⋆ = k−1 Si−rE det E . ⊕i=r ⊗ Here we have used Serre duality on the fibers of the smooth morphism π with relative dualizing sheaf ⋆ ωP = ( r) π det E. (E)/X OE − ⊗ 4. THE HODGE METRIC We recall the basics on geometric variations of Hodge structures and Griffiths’s computations ([10, theorem 6.2]) of the curvature the Hodge metric (see also [20, 7] and [22, chapter 10]). We also recall the method of Fujita [7]. Let f : Y B be a projective and§ surjective morphism of complex manifolds having connected fibers. We fix→ an ample line bundle on Y . 4.1. The Hodge metric. Here we assume that f : Y B is smooth, in particular we regard f : Y B as a smooth family of polarized complex projective→ manifolds of dimension n. Fix a non- → d negative integer d. The local system f⋆C can be realized as the sheaf of germs of the flat sections dR d of the holomorphic vector bundle HC associated with the locally free sheaf ( f⋆C) B endowed Hd 1 Hd R ⊗ O with the flat holomorphic connection : C ΩB C, the Gauss-Manin connection. By ∇ → ⊗ p,d−p semi-continuity and Hodge decomposition, the vector spaces H (Yb, C) (b B) have constant p,d−∈p d dimension. By elliptic theory they hence form a differentiable sub-bundle H of HC. Denote by p i,d−i d p F the differentiable sub-bundle i≥pH of HC. By a theorem of Griffiths, the F have natural ⊕ 3 d structure of holomorphic sub-vector bundles of HC and they satisfy the transversality condition for the Fp 1 Fp−1 Ep Fp Fp+1 Gauss-Manin connection ΩB . A relative Dolbeault theorem identifies := / ∇ ⊂ ⊗ d−p p with the holomorphic vector bundle associated with the locally free sheaf f⋆ΩY/B. R p We now recall the construction of the hodge metric on the primitive part of E . We fix a family ηb 2 d (b B) of polarizations given by a section of f Z. The bilinear form on the fibers of HC given by ∈ R ⋆ − d(d 1) n−d S(c ,c ):=( 1) 2 η c c 1 2 − b ∧ 1 ∧ 2 ZYb is non-degenerate (hence defines a pseudo-metric) on the primitive part Pd := Ker(ηn−d+1 : Hd H2n−d+2) (which is also a differentiable sub-bundle of Hd). The differentiable subbundles Hp,d→−p ′ ′ and Hp ,d−p are orthogonal unless p + p′ = d and h(c):=(√ 1)p−qS(c, c) − Hp,d−p Hp,d−p Pd Fp Fp Pd defines a positive definite metric on prim := . We set prim := . Those bundle also have natural holomorphic structures and satisf∩ y the transversality condition∩ for the in- Ep Fp Fp+1 duced Gauss-Manin connection. We also set prim := prim/ prim. The fiber-wise isomorphism of Ep p,d−p C d C Ep ( prim)b with Hprim (Yb, ) H (Yb, ) enables to equip the holomorphic vector bundle prim with a smooth positive definite hermitian⊂ metric, called the Hodge metric. We need some definitions in order to express the curvature of the corresponding Chern connection. p Ep 1 Ep−1 Fp Denote by : ΩB the B-linear map built by first lifting to applying the Gauss- ∇ → ⊗ Ep−O1 1,0 Fp Hd Fp Manin connection and projecting to . The second fundamental form in ∞ (B,Hom( , C/ )) of the sequence C p d d p 0 F HC HC/F 0 → → → → d with respect to the Gauss-Manin connection (or equivalently with the flat metric on HC) actually p Ep 1 Ep−1 induces : ΩB . Formulae for the curvature of quotient hermitian holomorphic vector bundles∇ then→ lead to⊗ Ep Ep Theorem 4.1. [10, theorem 5.2] The curvature Θ( prim) of the holomorphic vector bundle prim endowed with its Hodge metric is given by p p p+1 p+1 Θ(Ep )(V, V )σ, σ = σ, σ ( )⋆σ, ( )⋆σ h prim iHodge h∇V ∇V iHodge −h ∇V ∇V iHodge where V is a local vector field on B and σ a local section of Ep. We now apply this result in the case of the family of the cyclic covers π p : Y X obtained ◦ s → by taking the k-th root of a section s of E(k). We have to restrict the study over Zariski open sets 0 −1 0 O 0 0 Ys := (p π) (X Σs) and X := X Σs so that π p : Ys X becomes a smooth family. ◦ 0 − − ◦ → Then, since vanishes, the above theorem implies the following ∇ 0 0 r−1 k−1 i−r ⋆ E E 0 0 Corollary 4.2. The vector bundle prim = = (π p)⋆ Ys /X = i=r (S E det E)|X0 with the Hodge metric is Griffiths semi-negative. R ◦ O ⊕ ⊗ 4.2. Singularities of the Hodge metric. We now deal with the general case, namely f : Y B may not be smooth. In this section we will give rough descriptions of the singularities. Detailed→ results need explicit form of the isomorphisms in (3.1) and will be given in 5.3. We therefore assume that the base B is one dimensional. § The Hodge metric on the direct image of the relative canonical sheaf is described as follows. Let b ∈ B be a point and let (U, t) be a local coordinate centered at b = t =0 . A section ω Γ(U, f⋆KY/B) 4 { } ∈ ⋆ ⋆ – when regarded as a section in Γ(U, f⋆Hom(f KB,KY )) and applied to f dt – gives a section of K on f −1(U) which we denote by ω dt. If ϕ Γ(Y ,K ) fulfills the relation ω dt = ϕ f ⋆dt Y · b ∈ b Yb · b ∧ over Yb (which amounts to saying that in the differentiable trivialization Y|U Yb U, the section ω is sent to ϕ ), then the Hodge norm at b B of the section ω is ≃ × b ∈ − 2 n n(n 1) ω =(√ 1) ( 1) 2 ϕ ϕ , || ||Hodge − − b ∧ b ZYb here n = dim Y 1. Fujita checked that in this setting in case dim B = 1 the Hodge metric on − f⋆KY/B is bounded from below by a positive quantity and hence that the only possible singularities of the Hodge metric on f⋆KY/B are poles (see [7, lemma 1.12]). We just give the typical example which occurs for a local model of our cyclic covers (for some positive integer m). π p : Y = (t, z, l) C3/lk = t + zm X = t C ◦ s { ∈ } → { ∈ } (t, z, l) t 7→ The cotangent bundle Ω1 is generated by dt,dz,dl subject to the relation klk−1dl dt mzm−1dz = Ys − − 0. If ω dt is written as η(z, l)dz dl for a holomorphic function η(z, l), then ϕ0 may be chosen · −1 1−k ∧ to be ϕ0 = k l η(z, l)dz with a pole of order k 1 on the fiber over t = 0 and no singularities elsewhere. −

Now, note that over smooth fibers of f, the Serre dual of an orthonormal basis (ωi) of (Γ(Yb,KYb ),Hodge) is (ω ), which is an orthonormal basis of (H0,n(Y , C),Hodge). Hence the Hodge metric on r−1(π i b R ◦ p)⋆ Ys may acquire zeros at the points x over which the section s|P(Ex) is not transverse to the zero section.O This can also be inferred from the formula in 5.3 which furthermore proves the continuity of the metric on r−1(π p) . § R ◦ ⋆OYs 5. EXPLICIT ISOMORPHISM We make explicit the isomorphisms in (3.1) in terms of Dolbeault isomorphism and Serre duality for metrized vector bundles. This enables us to describe the Hodge metric. We keep the notation and assumption for E X made at the end of 2. We furthermore fix a reference hermitian metric g on E and then naturally→ on E⋆ and on (i). § OE 5.1. Calculus lemma. To make Serre duality on a projective space explicit, we pose an elementary r−1 calculus lemma. We consider a projective space P with a homogeneous coordinates [a]=(a1 : . . . : ar) and with the Fubini-Study Kähler form Ω. Let zj = aj/a1 (2 j r) be a standard local coordinate. We can write the volume form as ≤ ≤ r−1 Ωr−1 √ 1∂∂ log a 2 dz dz dz dz dz dz − | | 2 3 r 2 3 r = r−1 = ǫr ∧ ∧···∧ ∧ 2 r∧ ∧···∧ , (r 1)! (2π) (r 1)! (1 + z ) − − − − || || r−1 (r 1)(r 2) r−1 2 r 2 where we have set ǫ := (√ 1) ( 1) 2 /(2π) and z = z . r − − || || j=2 | j| Lemma 5.1. Let m be a positive integer, and let I = (i1, i2, , im)Pand J = (j1, j2, , jm) be m-tuples of integers in 1, 2, ,r . ··· ··· (1) If I and J are not{ equal··· modulo} change of order, it follows from parity reasons that r−1 ai1 aj1 ai2 aj2 aim ajm Ω 2···m =0. [a]∈Pr−1 a (r 1)! Z | | 5 − (2) If I and J are equal modulo change of order, then

2m1 2m2 2mr r−1 r a1 a2 ar Ω i=1 mi! | | | | 2m···| | = . Pr−1 a (r 1)! (r 1+ m)! Z[a]∈ | | − Q− Here the integer i appears mi times in I (hence in J), consequently 0 mi m and m = m1 + + m . One uses a convention 0!=1. ≤ ≤ ··· r 5.2. Explicit isomorphism. Since the problem : isomorphism in (3.1) is local on X, it is enough to argue on a small open neighborhood U of a fixed point x0 X. Choose a local frame (ej)1≤j≤r for E on U. A vector of E (resp. of the dual bundle E⋆) will be∈ denoted by a E (resp. a⋆ E⋆). Hence a point on P(E) will be denoted by [a⋆] P(E). ∈ ∈ i∈−r We first describe the isomorphism S E det E = π (KP (i)). The isomorphism ⊗ ∼ ⋆ (E)/X ⊗ OE Si−rE π (i r) is given over x U with a⋆ E⋆ by → ⋆OE − ∈ ∈ x i−r 0 S E (π (i r)) = H (P(E ), (i r) P ) x → ⋆OE − x x OE − | (Ex) P (Ex) E(i r)|P(Ex) f → O i−−r . 7→ [a⋆] f, a⋆ (a⋆)−i+r 7→ h i Here we denote by a⋆i−r the (i r)-fold symmetric product, and by f, a⋆i−r the duality pairing. A − h i vector of the bundle π⋆(KP(E)/X E(i)) is represented by a relative holomorphic (r 1)-from with ⊗ O ⋆ − P values in E(i) with respect to a local coordinate of the fibers. We take a point [a0] (Ex0 ) with ⋆ r O ⋆ ∈ a0 = j=1 a0jej , and let assume that a01 = 0 (this is not a special assumption). On a neighborhood ⋆ P 6 of [a0] (E), we use a standard local coordinate on the fiber: zj = aj/a1 (2 j r) for [a⋆] =P [∈ r a e⋆] P(E). There exists a canonical map q : E⋆ X 0 ≤P(E)≤which is j=1 j j ∈ − ×{ } → expressed in local coordinates around a reference point (x , a⋆), by P 0 0 q(x, a⋆)= q(x, a e⋆)=(x, [a : : a ])=(x, z ,...,z ). j j 1 ··· r 2 r The relative Euler sequence : X

ι ⋆ ⋆ p 0 ( 1) π E TP ( 1) 0 → OE − → → (E)/X ⊗ OE − → – which is built from the map q : E⋆ X 0 P(E) – is given by − ×{ }→ b a b a ∂ p x, [a⋆], b e⋆ = x, [a⋆], j 1 − 1 j a⋆ . j j a2 ∂z ⊗ j≥1 ! j≥2 1 j ! X X ⋆ ∂ ⋆ Since p(a1e )= a , the natural determinant isomorphism for the relative Euler sequence reads j ∂zj ⊗ ⋆ ⋆ det TP ( r) π det E (E)/X ⊗ OE − → ∂ ∂ ∂ (a⋆)r a e⋆ a e⋆ a e⋆ a e⋆ ∂z ∧ ∂z ∧···∧ ∂z ⊗ 7→ j j ∧ 1 2 ∧ 1 3 ∧···∧ 1 r 2 3 r j≥1 ! X = are⋆ e⋆ e⋆. 1 1 ∧ 2 ∧···∧ r By composition, we have an isomorphism i−r S E det E π KP (i) ⊗ → ⋆ (E)/X ⊗ OE ⋆ ⋆i−r r ⋆ −i f e1 e2 er [a] f, a a1dz2 dz3 dzr (a ) . ⊗ ∧ ∧···∧ 7→ 7→ h 6 i ∧ ∧···∧ ⊗ i−r Let (eI )|I|=i−r be a local frame of S E induced from (ej)1≤j≤r. Then by Lemma 5.1, the dual basis of (e e e e ) is represented through the integration along the fibers of I ⊗ 1 ∧ 1 ∧···∧ r |I|=i−r π : P(E) X, up to some positive constant multiple, by the following set of ∂-closed relative (0,r 1)-forms→ with values in ( i) : − OE − ⋆ i ⋆ ⋆i−r r dz2 dz3 dzr (a ) [a ] eI , a a1ǫr ∧ ∧···∧2 r ⋆ 2i . 7→ h i (1 + z ) ⊗ a ⋆ || || || ||g r−1 r−1 0 ˇ We secondary write the isomorphism between (π p)⋆ Ys and π⋆ ( p⋆ Ys ) in Cech R ◦ Or −R1 R O ⋆ cohomology. There is a standard Stein covering = Wj j=1 of π (U) with Wj = [a ] r W { } { ∈ P(E); a⋆ = a e⋆ E⋆, x U, a =0 , provided U is a unit ball in a local chart. Then j=1 j j ∈ x ∈ j 6 } r−1 0 r−1 −1 k−1 ˇ r−1 k−1 Pπ⋆ p⋆ Ys (U)= H (π (U), E( i)) = H ( , E( i)). R R O ⊕i=0 O − W ⊕i=0 O − We use a Stein covering p−1 = p−1(W ) r of (π p)−1(U) to compute r−1(π p) . W { j }j=1 ◦ R ◦ ⋆OYs Because the higher direct images of by p vanish, the Stein covering p−1 is acyclic for the OYs W p⋆-functor. Then, r−1(π p) (U)= Hr−1((π p)−1(U), )= Hˇ r−1(p−1 , ). R ◦ ⋆OYs ◦ OYs W OYs ˇ r−1 k−1 ˇ r−1 −1 The isomorphism between H ( , i=0 E( i)) and H (p , Ys ) is given by pull back, viewing elements of ( i) as relativeW ⊕ homogeneousO − polynomialsW of degreeO i on the total space of OE − E(1). The correspondance between Cechˇ and Dolbeault cohomologies can be made explicit using aO partition of unity as in [2, II proposition 9.8]. We find an explicit isomorphism as follows. Lemma 5.2. The following map describes the isomorphism in (3.1) : k−1Si−rE⋆ det E⋆ r−1(π p) ⊕i=r ⊗ → R ◦ ⋆OYs ⋆i−r r ⋆ ⋆ ⋆ ⋆ ⋆ i ⋆ i ⋆ eI , a a1dz2 dz3 dzr eI e1 e2 er (x, [a ], l) ǫr l , (a ) p h i ⋆ 2i ∧ ∧···∧2 r . ⊗ ∧ ∧···∧ 7→ 7→ h i a ⋆ (1 + z ) || ||g || || ! 5.3. Singularities of the Hodge metric (explicit formula). In our case: π p : Ys X, the singularities of the Hodge metric can be described more explicitly. We keep◦ the notations→ of the previous paragraph. ⋆ −k 0 In the local frame (a ) for E(k), the section s H (P(E), E(k)) is given by a local holo- O ⋆ ∈ ⋆ −k O k ⋆ morphic function σ = σ(x, z2,...,zr) as s([a ]) = σ (a ) . On Ys the equality l = s([a ]) reads k ⋆ · λ = σ, where λ = l, a is the local coordinates on the fibers of E(1). Then by Lemma 5.2, after h i ⋆ ⋆ ⋆O ⋆ integrating along the fibers of p, the Hodge metric hs(eI e1 e2 er) on a neighborhood U of x X is of the form, up to some positive constant multiple,⊗ ∧ ∧···∧ 0 ∈ ⋆i−r 2 2r 2i r−1 eI, a a1 σ k Ω x k |h i|⋆ 4|i | | | 7→ ⋆ Pr−1 a ⋆ (r 1)! Z[a ]∈ (Ex) || ||g − for x U Σ , where Ω denotes the relative Fubini-Study Kähler form with respect to π : P(E) ∈ − s → X. The local expression on Ys where σ has a k-th root may serve to check that the integrand is ⋆ homogeneous in a and that the Hodge metric hs is smooth on X Σs. The Hodge metric hs is initially defined on r−1(π p) as in 4.1. The above explicit− formula on X Σ also describes the R ◦ ⋆OYs|X−Σs § − s behavior of hs around Σs. Thanks to this description, it is possible to extend hs as a continuous (but r−1 maybe degenerate) hermitian metric on (π p)⋆ Ys by the same formula also at x U Σs. We R ◦ O r−1 k−1 i−∈r ⋆ ∩ ⋆ also call this continuous extension hs the Hodge metric on (π p)⋆ Ys = i=r S E det E . 7 R ◦ O ⊕ ⊗ 6. PROOF OF THEOREMS ;AMPLENESSANDPOSITIVITY Proof of Theorem 1.1. Let E be a semi-ample vector bundle of rank r > 1 on a compact complex manifold X. We take an arbitral positive integer k so that SkE is generated by its global sections. By the explicit expression in 5.3, Hodge metrics on k−1 (Si−rE det E)⋆ may acquire zeros § ⊕i=r ⊗ at the points x over which the section s|P(Ex) is identically zero. We now explain how the semi- ampleness assumption on the vector bundle E helps to remove those singularities of the Hodge metric. For every point x in X, we choose a generic section s of (k) over the fiber P(E ) transverse to x OE x the zero section. We extend the section sx to a global section of E(k) over P(E) and take a generic section close to this extension. It will be transverse to the zeroO section on each fiber of points in a neighborhood of x, because this is an open condition. This shows that on the compact manifold X, k−1 i−r ⋆ adding a finite number of Hodge metrics hα on i=r (S E det E) seen as the direct image r−1 ⊕ ⊗ P (π pα)⋆ Yα for different covering maps pα : Yα = Ysα (E) of degree k, we get a R ◦ O → ℓ continuous non-degenerate (i.e. positive on every non-zero vector) hermitian metric h = α=1 hα. The metric hα is continuous on X, and smooth (as a Hodge metric on a smooth family) and non- degenerate outside the discriminant locus Σ . The set of points x X where one of theP sections sα ∈ sα|P(Ex) is not transverse to the zero section is a proper Zariski closed subset of X, that we will denote by Σh := αΣsα . Next let∪ us discuss the curvature property on each direct summand Si−rE det E for i = r,...,k 1. Take a point x in X Σ , a non-zero vector ξ (Si−rE det E)⋆ ,⊗ and a nowhere zero local− 0 h 0 x0 − i−r ⋆ ∈ ⊗ holomorphic section ξ Γ(U, (S E det E) ) achieving the value ξ0 at x0 and normal at x0 for the metric h (i.e. ξ(x∈)=0). Then, the⊗ last two terms in the formula (2.1) ∇h 0

⋆ Θ((Si−rE det E) , h)ξ,ξ ξ, ξ √ 1∂ ξ 2 ∂ ξ 2 h ⊗ x0 i = √ 1∂∂ log ξ 2 + h∇h ∇h i − || || ∧ || || ξ 2 − − || ||h ξ 2 − ξ 4 || || || || || ||

r−1 vanish at x0. Now, Corollary 4.2: Griffiths curvature formula for (π pα)⋆ Yα|X−Σh asserts that 2 R ◦ O the function log ξ hα – whose complex Hessian (or Levi form) is the opposite of the curvature of || || i−r ⋆ 2 a line sub-bundle of (S E det E) – is plurisubharmonic on U. It then follows that log ξ h = log( ξ 2 ) is plurisubharmonic⊗ on U. This gives the Griffiths semi-negativityof (Si−rE det|| ||E)⋆ α || ||hα ⊗ on X Σh. Since h is continuous and Σh is an analytic subset (of zero Lebesgue measure), we can concludeP− that the dual continuous metric h⋆ on Si−rE det E is Griffiths semi-positive on the whole of X. ⊗

Remark 6.1. We can take the integer k large enough to make the rank of SkE exceed the dimension k of X. This together with the generation by global sections ensure that a generic section sX of S E has no zero on X (see for example [13, II, ex 8.2]). No fiber of π is hence contained in the divisor 0 Ds of zeros of the section s H (P(E), E(k)), which is associated with sX via the isomorphism k ∈ O S E ∼= π⋆ E(k). (Then we can also see, by a local computation, that every irreducible component of singularO fibers of π p has multiplicity one.) In the formula in 5.3, if s is not identically zero on Pr−1 over a point x◦, the right hand side integral (the Hodge norm§ at x) is not zero. Hence the Hodge metric associated to the covering Ys is non-degenerate on the whole of X. The metric h = hs is continuous on X, and smooth (as a Hodge metric on a smooth family) outside Σs. This gives a slight simplication of the proof of Theorem 1.1. 8 Proof of Theorem 1.2. We now assume that E is ample. Let us recall Legendre-type formula for a metric h = ℓ h on (Si−rE det E)⋆ gotten from different cyclic coverings Y (for a (1, 0)- α=1 α ⊗ sα form u, u 2 denotes √ 1u u): | | P − ∧ 2 2 2 ∂ log ξ 2 ∂ log ξ 2 ξ 2 ξ 2 ξ √ 1∂∂ log ξ α<β hα hβ hα hβ √ 1∂∂ log( ξ 2 )= α || ||hα − || ||hα + || || − || || || || || || . − || ||hα ξ 2 ( ξ 2 )2 α P α hα P α hα X || || || || Applying Griffiths curvature formulaP (Corollary 4.2) for individual coveringP and the formula (2.1) for a line subbundle we infer that in the right hand side, the first term is semi-positive. We need to add further Hodge metrics hα to make the second term – hence the left hand side – strictly positive. The explicit expression in 5.3 will help to translate the algebraic ampleness assumption on E into a positivity property for a well§ chosen metric on Si−rE det E. We may first assume that the chosen ⊗ local frame (ej)1≤j≤r for E is normal at x0 X for the fixed metric g on E. Hence, denoting by π⋆ the push forward of currents by π : P(E) ∈ X – in other words, the integration along the fibers of the proper submersion π –, we have →

2i ⋆i−r 2 2r k −1 r−1 ⋆ ⋆ ⋆ ⋆ eI , a a1 σ Ω ∂ log hs(eI e1 e2 er) = 2iπ⋆ |h i|⋆| 4i | | | ∂σ . ⊗ ∧ ∧···∧ a ⋆ ∧ (r 1)! || ||g − ! We now take a positive integer k so large that the map H0 P(E), (k) H0 P(E), (k) π⋆( / 2 ) OE → OE ⊗ OX Mx is surjective for every x X. By the compactness of X, we can henceforth choose enough, but ∈ 0 a finite number of sections sα H (P(E), E(k)) to ensure positivity in all the directions in the Legendre formula. This gives continuous∈ hermitianO metrics on SkE det E with Griffiths positive curvature. Using a regularization process as described in [18] these metrics⊗ may be smoothed keeping Griffiths positivity of the curvature. p Ep 1 Ep−1 Remark 6.2. Griffiths [10, proposition 2.16] showed that the operator : ΩB can 1 ∇ 1 → ⊗ be expressed as a cup product with the Kodaira-Spencer class ρ ΩB,b H (Yb,TYb) of the family f : Y B coupled with a natural pairing. In our setting, this∈ in turn⊗ can be related with the → infinitesimal displacement of the hypersurfaces Ds,x of P(Ex) given by the vanishing of the section s|P(Ex), namely (see [14, chapter 5.2 (c)]) ⋆ TX H0(D , (D ) ) δ H1(D ,TD ) → s,x O s,x |Ds,x → s,x s,x v (∂ s) ρ(v) 7→ v |Ds,x 7→ ⋆ where v is a holomorphic vector fielde lifting v on P(Ex). The map δ is the coboundary map in the long exact sequence associated with the short exact sequence for the normal bundle of the divisor Ds,x e 0 TD T P(E ) (D ) 0. → s,x → x |Ds,x → O s,x |Ds,x → Our computations make explicit the idea that E being ample, the sections s|P(Ex) move sufficiently to p make the operator : Ep Ω1 Ep−1 have non-zero contribution in the curvature formula. ∇ → B ⊗

9 Christophe Mourougane / Institut de Math´ematiques de Jussieu / Plateau 7D / 175, rue du Chevaleret / 75013 Paris / France. Shigeharu Takayama / Graduate School of Mathematical Sciences / University of Tokyo / Meguro / Tokyo 153 / Japan.

REFERENCES [1] Berndtsson, Bo. Curvature of vector bundles and subharmonicity of Bergman kernels, Preprint 2005. [2] Bott, Raoul and Tu, LoringW., Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer- Verlag, New York, (1982). [3] Campana, F. and Flenner, H., A characterization of ample vector bundles on a curve, Math. Ann., 287, (1990), 4, 571–575. [4] Demailly, Jean-Pierre, Complex analytic and algebraic geometry http://www-fourier.ujf-grenoble.fr/˜demailly/books.html [5] Demailly, Jean-Pierre, Regularization of closed positive currents and intersection theory, J. Algebraic Geom., 1, (1992), 361–409. [6] Esnault, Hélène and Viehweg, Eckart, Lectures on vanishing theorems, DMV Seminar 20,Birkhäuser Verlag,Basel, (1992). [7] Fujita, Takao, On Kahler¨ fiber spaces over curves, J. Math. Soc. Japan, 30, (1978), 4, 779–794. [8] Fulton, William, On the topology of algebraic varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, 15–46, Amer. Math. Soc., Providence, RI, (1987). [9] Griffiths, Phillip A., Hermitian differential geometry, Chern classes, and positive vector bundles. Global Analysis, Papers in Honor of K. Kodaira (Ed. D. C. Spencer and S. Iyanaga), 185–251. Princeton Univ. Press, 1969. [10] Griffiths, Phillip A., Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Inst. Hautes Etudes´ Sci. Publ. Math., 38, (1970), 125–180. [11] Griffiths, Phillip and Harris, Joseph, Principles of algebraic geometry, Wiley Classics Library, Reprint of the 1978 original, John Wiley & Sons Inc., New York, (1994). [12] Hartshorne, Robin, Ample vector bundles, Inst. Hautes Etudes´ Sci. Publ. Math., 29, (1966), 63–94. [13] Hartshorne, Robin, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, (1977). [14] Kodaira, Kunihiko, Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences] 283, Springer-Verlag, New York, (1986). [15] Kollár, János, Higher direct images of dualizing sheaves. I, Ann. of Math. (2), 123, (1986), 1, 11–42. Higher direct images of dualizing sheaves. II, Ann. of Math. (2), 124, (1986), 1, 171–202. [16] Lazarsfeld, Robert, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , 49, Springer-Verlag, Berlin, 2004. [17] Manivel, Laurent, Vanishing theorems for ample vector bundles, Invent. Math., 127, (1997), 2, 401–416. [18] Mourougane, Christophe, Images directes de fibres´ en droites adjoints, Publ. Res. Inst. Math. Sci., Kyoto University., 33, (1997), 6, 893–916. [19] Ramanujam, C. P., Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. (N.S.), 36, 1972, 41–51. [20] Schmid, Wilfried, Variation of Hodge structure: the singularities of the period mapping, Invent. Math., 22, (1973), 211–319. [21] Umemura, Hiroshi, Some results in the theory of vector bundles, Nagoya Math. J., 52, (1973), 97–128. [22] Voisin, Claire, Hodge theory and complex algebraic geometry, Cambridge Studies in Advanced Mathematics, 77, Cambridge University Press, (2003).