Compact Complex Manifolds

with Numerically Effective Cotangent Bundles

Henrik Kratz

Received: June 23, 1997

Communicated by Thomas Peternell

Abstract. We prove that a projective manifold of dimension n = 2 or 3 and Kodaira dimension 1 has a numerically effective cotangent bundle if and only if the Iitaka fibration is almost smooth, i.e. the only singular fibres are multiples of smooth elliptic curves (n = 2) resp. multiples of smooth Abelian or hyperelliptic surfaces (n = 3). In the case of a threefold which is fibred over a rational curve the proof needs an extra assumption concerning the multiplicities of the singular fibres. Furthermore, we prove the following theorem: let X be a complex manifold which is hyberbolic with respect to the Carathéodory-Reiffen-pseudometric, then any compact quotient of X has a numerically effective cotangent bundle.

1991 Mathematics Subject Classification: 32C10, 32H20

Introduction

It is a natural question in algebraic geometry to classify manifolds by positivity properties of their tangent resp. cotangent bundles. The first result of this kind was obtained by Mori who solved the Hartshorne-Frankel conjecture [Mo]: every projective n-dimensional manifold with ample tangent bundle is isomorphic to the complex projective space ℙ_{n}. A degenerate condition of ampleness is numerical effectivity. A line bundle L on a projective manifold X is called numerically effective (abbreviated “nef”) if L.C ≥ 0 for all curves C ⊂ X. A vector bundle E is said to be nef if the tautological quotient line bundle O_{ℙ(E)}(1) on ℙ(E), the projective bundle of hyperplanes in the fibres of E, is nef. Taking the Hartshorne-Frankel conjecture as a guideline, Campana and Peternell considered projective manifolds whose tangent bundles are nef and classified them in dimension 2 and 3 [CP]. For dimension 3 this has been done by Zheng [Zh] too. In general, for arbitrary compact complex manifolds the “nefness” of the tangent bundle leads to strong structural constraints [DPS]. The purpose of this paper is to investigate some aspects of manifolds X whose cotangent bundles Ω_{X}^{1} are nef. In the first part we will give a characterization of 2 and 3 dimensional manifolds with Kodaira dimension κ(X) = 1 and nef cotangent bundle. We will prove:

Theorem 1 Let X be a minimal projective manifold of dimension n = 2 or 3 with κ(X) = 1 and let π : X → C be the Iitaka fibration of X. Then the following conditions are equivalent:

(i) Ω_{X}^{1} is nef. (ii) π is almost smooth, in the sense that the only singular fibres of π are multiples of smooth elliptic curves (n = 2) resp. Abelian or hyperelliptic surfaces (n = 3).

• Exception: To prove (ii)⇒(i) in the case n = 3 and g(C) = 0 we need the assumption that ∑ \frac{m_{i}−1}{m_{i}} ≥ 2, where the m_{i} are the multiplicities of the singular fibres.

• The equivalence of (i) and (ii) holds also for compact Kähler surfaces.

This theorem generalizes a result of Fujiwara [Fu] who worked in arbitrary dimension but under the stronger assumption that Ω_{X}^{1} is semi-ample, i.e. that some power of O_{ℙ(Ω_{X}^{1})}(1) is globally generated. The implication (i) ⇒ (ii) relies on the topological constraints, namely the Chern class inequalities, which hold, when the cotangent bundle is nef. To prove (ii) ⇒ (i) we will proceed in two steps. First, we will show that the assertion is true for a smooth fibration. This follows basically from Griffiths’s theory on the variation of the Hodge structure. Then, we will study the base-change which reduces an almost smooth fibration to a smooth one and show that this process allows to carry over the “nefness” of the cotangent bundle. In fact, we will prove in any dimension that a projective manifold has a nef cotangent bundle if (a) it admits a smooth Abelian fibration over a manifold with nef cotangent bundle or (b) it admits an almost smooth Abelian fibration over a curve C such that either (i) g(C) ≥ 1 or (ii) g(C) = 0 and ∑ \frac{m_{i}−1}{m_{i}} ≥ 2. We remark that the fibres F of the Iitaka fibrations in Theorem 1 are paraAbelian varieties, i.e. there exists an unramified cover T → F where T is an Abelian variety. In view of this, we expect in any dimension that a manifold with Kodaira dimension 1 has a nef cotangent bundle if and only if the Iitaka fibration is almost smooth with para-Abelian fibres. In the second part of this paper we consider complex manifolds X which are hyperbolic with respect to the Carathéodory-Reiffen pseudometric. We will show :

Theorem 2 Let X be a complex manifold which is hyperbolic with respect to the Carathéodory-Reiffen pseudometric and let Q be a compact quotient of X with respect to a subgroup of the automorphism group of X which operates fixpointfree and properly discontinuously. Then Ω_{Q}^{1} is nef.

In particular, any compact quotient of a bounded domain G ⊂ ℂ^{n} possesses a nef cotangent bundle. Since the canonical bundle of such a quotient is ample, this yields a class of manifolds with maximal Kodaira dimension and nef cotangent bundle. To prove theorem 2 we apply the technique of singular hermitian metrics which was developed by Demailly. The Carathéodory-Reiffen pseudometric of X defines a Finsler structure on the tangent bundle of Q and this gives us a singular hermitian metric on O_{ℙ(Ω_{Q}^{1})}(1). The hyperbolicity of X guarantees that this metric is continuous and that the associated curvature current is positive. These conditions imply that O_{ℙ(Ω_{Q}^{1})}(1) is nef.

Acknowledgments: I would like to thank M. Schneider and Th. Peternell for their help and encouragement.

1 Basic definitions and properties

Let X and Y be compact complex manifolds and let L be a holomorphic line bundle on X.

Definition 1 (i) When X is projective, L is said to be nef, if L · C = ∫_{C} c_{1}(L) ≥ 0 for every curve C in X. (ii) Let X be an arbitrary compact complex manifold equipped with a hermitian metric ω. Then L is said to be nef, if for all ϵ > 0 there exists a smooth hermitian metric h_{ϵ} on L such that the associated curvature form satisfies

Ω_{h_{ϵ}}(L) ≥ −ϵ · ω.

(iii) Let E be a holomorphic vector bundle on X and ℙ(E) the projective bundle of hyperplanes in the fibres of E. Then we call E nef over X, if the tautological quotient line bundle O_{ℙ(E)}(1) is nef over ℙ(E).

We will frequently use the following propositions which are proved in [DPS].

Proposition 1 Let f : Y → X be a holomorphic map and let E be a holomorphic vector bundle over X. Then E nef implies f^{∗}E nef, and the converse is true if f is surjective and has equidimensional fibres.

Proposition 2 Let E and F be holomorphic vector bundles. Then

(i) E, F nef ⇒ E ⊗ F nef. (ii) E nef ⇒ det(E) nef.

Proposition 3 Let 0 → F → E → Q → 0 be an exact sequence of holomorphic vector bundles. Then

(i) E nef ⇒ Q nef. (ii) F, Q nef ⇒ E nef.

Proposition 1 immediately implies

Proposition 4 Let Y be a finite unramified covering of X. Then Ω_{X}^{1} is nef if and only if Ω_{Y}^{1} is nef.

A fibration of X over Y is a surjective holomorphic map π : X → Y whose fibres are connected. A point x ∈ X is said to be critical if the tangent map Dπ(x) has not maximal rank. The images π(x) ∈ Y of the critical points are the critical values of π. They form a proper analytic subset of Y, i.e. in the case, where Y is a curve, a finite subset {a_{1}, …, a_{l}}. Let y ∈ Y and let J be the ideal sheaf of y in O_{Y}. Then the fibre X_{y} is the complex subspace (π^{−1}(y), O_{X}/π^{∗}(J) · O_{X}) of X, and a fibre X_{y} is singular if and only if y is a critical value. A fibration, for which Dπ has maximal rank everywhere, is called smooth. When we consider a fibration π : X → C over a curve C, we will always assume that C is smooth. Such a fibration is said to be almost smooth, if the only singular fibres of π are multiples of smooth irreducible subvarieties. Their multiplicities will be denoted by m_{i} with 1 ≤ i ≤ l, so that the singular fibres are X_{a_{i}} = m_{i}F_{i}, where the F_{i} are smooth irreducible subvarieties.

We will denote the Kodaira dimension of X by κ(X). Let X be a projective manifold with κ(X) ≥ 1 for which a power of the canonical bundle is globally generated. Then for m big enough the m−canonical map gives us a holomorphic map π : X → Z where Z is a projective variety with dim Z = κ(X). Such a map π is called Iitaka fibration (cf. [Ue]).

2 Manifolds with κ = 1 and nef cotangent bundle

We will now prove

Theorem 3 Let X be a minimal projective manifold of dimension n = 2 or 3 with κ(X) = 1 and let π : X → C be the Iitaka fibration of X. Then the following conditions are equivalent:

(i) Ω_{X}^{1} is nef. (ii) π is almost smooth, in the sense that the only singular fibres of π are multiples of smooth elliptic curves (n = 2) resp. Abelian or hyperelliptic surfaces (n = 3).

• Exception: To prove (ii)⇒(i) in the case n = 3 and g(C) = 0 we need the assumption that ∑ \frac{m_{i}−1}{m_{i}} ≥ 2, where the m_{i} are the multiplicities of the singular fibres.

• The equivalence of (i) and (ii) holds also for compact Kähler surfaces.

Proof: (i) ⇒ (ii) If X is an n-dimensional projective manifold with Ω_{X}^{1} nef, it satisfies the Chern class inequality c_{1}(X)^{2} ≥ c_{2}(X) ≥ 0, i.e.

c_{1}(X)^{2} · H_{1} · . . . · H_{n−2} ≥ c_{2}(X) · H_{1} · . . . · H_{n−2} ≥ 0

for all ample divisors H_{i} (cf. [DPS], Thm. 2.5). For n = 2 and 3 the abundance conjecture holds which means that a power of the canonical bundle of X has to be globally generated so that we get from κ(X) = 1 that c_{1}(X)^{2} ≡ 0 and hence c_{1}(X)^{2} ≡ c_{2}(X) ≡ 0. Here ≡ denotes numerical equivalence. So for n = 2 we have an elliptic surface X whose topological Euler characteristic is e(X) = c_{2}(X) = 0. On the other hand, if π : X → C is the Iitaka fibration of X and X_{a_{i}} are the singular fibres (1 ≤ i ≤ l), we calculate e(X) = ∑ e(X_{a_{i}}) . But now the assertion follows, because e(X_{a_{i}}) ≥ 0 and e(X_{a_{i}}) = 0 if and only if the fibre X_{a_{i}} is a multiple of a smooth elliptic curve (cf. [BPV], Chap. III, Prop. 11.4). This argument remains true for a compact Kähler surface. For n = 3 we have a minimal threefold with the extremal Chern classes c_{1}(X)^{2} ≡ 3c_{2}(X) ≡ 0 and the assertion follows from [PW], Theorem 2.1. (ii) ⇒ (i) We will prove this direction by reducing it to the case of a smooth fibration.

2.1 Smooth fibrations

We will consider smooth Abelian fibrations first:

Proposition 5 Let X and Y be projective manifolds and let π : X → Y be a smooth fibration, whose fibres are Abelian varieties. Then the relative cotangent bundle Ω_{X/Y}^{1} is nef. If Ω_{Y}^{1} is nef, Ω_{X}^{1} is nef too.

Proof: (1) We claim that π^{∗}(π_{∗}Ω_{X/Y}^{1} ) = Ω_{X/Y}^{1} . For all y ∈ Y the cotangent bundle of the fibre Ω_{X_{y}}^{1} is trivial, so that π_{∗}Ω_{X/Y}^{1} is locally free of rank equal to the dimension of the fibres (cf. [Ha], Chap. III, Cor. 12.9). Moreover for all y ∈ Y we have (π_{∗}Ω_{X/Y}^{1} )_{y} ≅ H^{0}(X_{y}, Ω_{X_{y}}^{1}) and thus (π^{∗}(π_{∗}Ω_{X/Y}^{1} ))_{x} ≅ H^{0}(X_{y}, Ω_{X_{y}}^{1}) for π(x) = y . Now, the canonical homomorphism α : π^{∗}(π_{∗}Ω_{X/Y}^{1} ) → Ω_{X/Y}^{1} is described stalkwise by α_{x} : σ ↦ σ(x) with σ ∈ H^{0}(X_{y}, Ω_{X_{y}}^{1}). Since Ω_{X/Y}^{1} |_{X_{y}} is globally generated, α_{x} is surjective and hence bijective.

(2) Any smooth fibration π : X → Y of projective manifolds gives rise to a variation of the Hodge structure in its fibres X_{y} (y ∈ Y). From this Griffiths deduces [Gr], Cor. 7.8

Theorem 4 For all n ∈ {1, …, dim_{ℂ} X_{y}} the bundles R^{n}π_{∗}(O_{X}) are seminegative in the sense of Griffiths.

Now the bundle E = R^{n}π_{∗}(O_{X}) is conjugate linear to Ē = π_{∗}(Ω_{X/Y}^{n}) so that the curvature matrices with respect to unitary bases behave as

ΩĒ = Ω̄_{E} = −Ω_{E}^{t}.

Since the transposition of the curvature matrix does not change its positivity properties, the preceding theorem can equivalently be formulated as

Theorem 5 For all n ∈ {1, …, dim_{ℂ} X_{y}} the bundles π_{∗}(Ω_{X/Y}^{n}) are semipositive in the sense of Griffiths.

In particular, since semipositivity implies “nefness”, π_{∗}(Ω_{X/Y}^{n}) is nef and hence for a smooth Abelian fibration Ω_{X/Y}^{1} = π^{∗}(π_{∗}Ω_{X/Y}^{1} ) is nef too. The second assertion follows immediately from the relative cotangent sequence and Proposition 3.

Remark: Proposition 5 holds also for compact elliptic surfaces π : X → C, because for a smooth π one knows from the study of the period map that deg(π_{∗}ω_{X/C}) = 0 (cf. [BPV], Chap. III, Thm. 18.2). We have a similar proposition for smooth hyperelliptic fibrations:

Proposition 6 Let X be a projective 3-dimensional manifold and let π : X → C be a smooth fibration, whose fibres are hyperelliptic surfaces. Furthermore, let g(C) ≥ 1. Then Ω_{X}^{1} is nef.

Proof: We consider the relative Albanese factorization of π, i.e. the commutative diagram

X ⟶^{A_{π}} A(X/C)
π ↘ ↓ Alb(π)
C,

where A(X/C) is a smooth fibration over C whose fibres over a ∈ C are the Albanese tori Alb(X_{a}) of the fibres X_{a} of π. The existence of such a relative Albanese diagram is proved in [Ca]. Since the tangent bundle of a hyperelliptic surface is nef, the Albanese map A_{π} |_{X_{a}}: X_{a} → Alb(X_{a}) is a surjective submersion with smooth elliptic curves as fibres ([DPS], Prop. 3.9.). But also A_{π} is smooth: let x ∈ X, π(x) = a and A_{π}(x) = y, then both tangent directions of T A(X/C)_{y} lie in the image of DA_{π}(x). First, we can find a tangent vector v ∈ (T A(X/Y) |_{Alb(X_{a})})_{y} in the image of DA_{π}(x) |_{X_{a}} (because A_{π} |_{X_{a}} is smooth). Now let (x_{1}, x_{2}, x_{3}) be a coordinate system centered in x and let z_{1} be a coordinate centered in a, such that Dπ(x).\frac{∂}{∂x_{1}} = \frac{∂}{∂z_{1}} . Using the commutativity of the relative Albanese diagram, we get

0 ≠ Dπ(x).\frac{∂}{∂x_{1}} = DAlb(π)(y) ∘ DA_{π}(x).\frac{∂}{∂x_{1}} .

In particular, w := DA_{π}(x).\frac{∂}{∂x_{1}} ≠ 0, and since DAlb(π)(y).v = 0 the vectors v and w have to be linear independent. We can now apply Proposition 5 twice to conclude that Ω_{X}^{1} is nef: Alb(π) : A(X/C) → C is a smooth fibration of projective manifolds whose fibres are elliptic curves and by assumption g(C) ≥ 1, so that Ω_{A(X/C)}^{1} has to be nef. Since A_{π} : X → A(X/C) is a smooth elliptic fibration too, Ω_{X}^{1} is also nef.

2.2 Almost smooth fibrations

Let X be a compact complex manifold of dimension n and let π : X → C be an almost smooth fibration over a smooth curve C. As above we will denote the critical values of π by a_{1}, …, a_{l} and their multiplicities bym_{i} where 1 ≤ i ≤ l, so that the singular fibres are X_{a_{i}} = m_{i}F_{i}, where the F_{i} are smooth irreducible subvarieties. To get rid of the multiple fibres we will now perform a base change which was introduced by Kodaira for elliptic surfaces ([Kod], Thm 6.3), but may be used in this general context as well. Let m_{0} be the lowest common multiple of the multiplicities and let d be their product. Then we choose a finite covering σ : C′ → C, which has \frac{d}{m_{i}} ramification points of order m_{i} − 1 over the points a_{i} where 0 ≤ i ≤ l. Remark that we have to add one extra point a_{0} which is not contained in the set of critical values. Then the normalization of the fibre product X ×_{C} C′ gives us a smooth fibration φ : X′ → C′ and a commutative diagram (cf. [Kod], Thm 6.3)

X′ ⟶^{f} X
φ ↓  ↓ π
C′ ⟶^{σ} C  .

Here f is a finite covering which is unramified over X − π^{−1}(a_{0}), because the multiplicities of π and σ compensate each other over a_{i} (i ≥ 1), and f has \frac{d}{m_{0}} ramification divisors of order m_{0} − 1 over π^{−1}(a_{0}). Assume that we knew Ω_{X′}^{1} is nef, then we would like to carry this over to Ω_{X}^{1}. However, it is not possible to apply Proposition 4 since f is ramified. But we have the following commutative diagram with exact rows which was already used in [Fu]

0 ⟶ f^{∗}(L) ⟶ f^{∗}(Ω_{X}^{1}) ⟶ Ω_{X′/C′}^{1} ⟶ 0
↓  ↓  ∥
0 ⟶ φ^{∗}(K_{C′}) ⟶ Ω_{X′}^{1} ⟶ Ω_{X′/C′}^{1} ⟶ 0.

Let D = ∑_{i=1}^{l}(m_{i} − 1)F_{i} then L = π^{∗}(K_{C}) ⊗ O_{X}(D) is the full subbundle of Ω_{X}^{1} associated to π^{∗}(K_{C}) (cf. [Re]). To prove the commutativity of this diagram one uses basically the fact that the restriction of f to a fibre of φ is unramified. For i ≥ 1 we have π^{∗}(a_{i}) = m_{i}F_{i}. So, defining A := ∑_{i=1}^{l} \frac{(m_{i}−1)}{m_{i}} · a_{i} we get L = π^{∗}(K_{C} ⊗ O_{C}(A)). Combining the diagram and Proposition 5, we obtain

Corollary 1 Let X be a projective manifold of arbitrary dimension and let π : X → C be an almost smooth fibration, whose fibres are Abelian varieties. Assume furthermore that (i) g(C) ≥ 1 or (ii) g(C) = 0 and deg A ≥ 2. Then Ω_{X}^{1} is nef.

Proof: The process described above allows us to pass to a smooth Abelian fibration φ, for which Ω_{X′/C′}^{1} is nef by Proposition 5. Moreover the line bundle L = π^{∗}(K_{C} ⊗ A) is nef, since our assumptions guarantee that deg(K_{C} ⊗A) = 2g(C)−2+deg A ≥ 0. If L is nef, then f^{∗}(L) and f^{∗}(Ω_{X}^{1}) are nef (Proposition 3). Since f is a finite surjective map, we finally deduce from Proposition 1 that Ω_{X}^{1} is nef.

Remark: (i) The corollary holds for arbitrary compact surfaces too, because Proposition 5 remains true in that case. (ii) If S is a surface with κ(S) = 1 and π : S → ℙ_{1} is an almost smooth elliptic fibration, the condition that deg A ≥ 2 (resp. that L is nef) is automatically satisfied. We have deg(π_{∗}(ω_{S/ℙ_{1}})) = 0 and therefore π_{∗}(ω_{S/ℙ_{1}}) = O_{ℙ_{1}} (cf. [BPV]). Now the formula for the canonical bundle of an elliptic fibration yields K_{S} = π^{∗}(K_{ℙ_{1}})⊗O_{S}(D), so that L = K_{S} is nef since κ(S) = 1.

Similarly we get

Corollary 2 Let X be a projective 3-dimensional manifold with κ(X) ≥ 0 and let π : X → C be an almost smooth fibration, whose fibres are hyperelliptic surfaces. Assume furthermore that (i) g(C) ≥ 1 or (ii) g(C) = 0 and deg A ≥ 2. Then Ω_{X}^{1} is nef.

Proof: To deduce from Proposition 6 that Ω_{X′/C′}^{1} is nef as a quotient of Ω_{X′}^{1}, we have to assure that g(C′) ≥ 1. But g(C′) = 0 leads to −∞ = κ(X′) ≥ κ(X) which contradicts our assumptions.

In particular, these two corollaries yield the direction (ii)⇒ (i) in Theorem 3 which is now completely proved.

3 Quotients with nef cotangent bundle

The goal of this section is to prove that compact quotients of a manifold which is hyperbolic with respect to the Carathéodory-Reiffen pseudometric have a nef cotangent bundle. We will use the notion of singular hermitian metrics as introduced in [De1]:

Definition 2 Let L be a holomorphic line bundle over a compact complex manifold X and let θ_{α} : L |_{U_{α}} ⟶^{≃} U_{α} × ℂ be a local trivialization of L. Then a singular hermitian metric on L is given by

∥ξ∥ = |θ_{α}(ξ)| · e^{−φ_{α}(x)}, x ∈ U_{α}, ξ ∈ L_{x},

where φ_{α} ∈ L_{loc}^{1}(U_{α}) is an arbitrary real valued function, called the weight function of the metric with respect to the trivialization θ_{α}.

The curvature form of the singular metric on L is locally given by the closed (1, 1)- current c(L) = \frac{i}{π} ∂ ¯∂φ_{α}. We will write c(L) ≥ 0, if c(L) is a positive current in the sense of distribution theory, i.e. if the weight functions φ_{α} are plurisubharmonic. Remark: We will say that a singular metric is continuous (or simply that it is a continuous metric), if the weight functions φ_{α} are continuous on the trivialization sets. The main ingredient for the following arguments will be the next proposition which is independently due to Demailly, Shiffman and Tsuji (see e.g. [De2])

Proposition 7 Let L be a holomorphic line bundle on a compact complex manifold X. Then L is nef, if there exists a continuous metric with c(L) ≥ 0.

In fact the proposition is even true in the case where the Lelong numbers of the metric (which are zero everywhere for a continuous metric) are zero except for a countable set of points (cf. Thm. 4.2 in [JS]). Let E be a holomorphic vector bundle over a compact complex manifold X. As in [Rei] and [Ko] we define

Definition 3 A Finsler structure on E is a continuous function F : E → ℝ_{≥0}, so that for all η ∈ E:

(i) F (η) > 0 for η ≠ 0, (ii) F (λη) = |λ|F (η) for all λ ∈ ℂ. If we require in (i) only ≥, F is said to be a Finsler pseudostructure.

Let P (E) denote the projective bundle of lines in the fibres of E, p : P (E) → X the projection and O_{P (E)}(−1) the subbundle of p^{∗}E whose fibre over a point in P (E) is given by the complex line represented by that point. Then we have a map p̃ : O_{P (E)}(−1) → E which is biholomorphic outside the zero sections of O_{P (E)}(−1) and E. The set of all plurisubharmonic functions on a complex manifold Y will be denoted by P SH(Y).

Proposition 8 (a) Any Finsler structure F on E defines via

∥ξ∥ := F ∘ p̃(ξ), ξ ∈ O_{P (E)}(−1).

a continuous metric on O_{P (E)}(−1).

(b) If log F ∈ P SH(E\{0}), then −φ_{α} ∈ P SH(U_{α}).

Proof: (a) Let θ_{α} : O_{P (E)}(−1) |_{U_{α}} ⟶^{≃} U_{α} × ℂ be a local trivialization and let s_{α} be a local holomorphic section of O_{P (E)}(−1) |_{U_{α}} which describes the trivialization. Then the corresponding weight function is

−φ_{α}(x) = log ∥s_{α}(x)∥ = log F ∘ p̃(s_{α}(x)), x ∈ U_{α}.

The map p̃ ∘ s_{α} : U_{α} → E is clearly holomorphic. Moreover for x ∈ U_{α} we have s_{α}(x) ≠ 0, so that property (i) in the definition of Finsler structures leads to F ∘ p̃(s_{α}(x)) > 0. From this we conclude −φ_{α} ∈ C^{0}(U_{α}).

(b) If f : Y → Z is a holomorphic map between complex manifolds and the function u ∈ P SH(Z), then u ∘ f ∈ P SH(Y) (cf. [JP], Appendix, PSH 7). So, since p̃ ∘ s_{α} is holomorphic, we have −φ_{α} ∈ P SH(U_{α}).

Proposition 9 Let E → X be a holomorphic vector bundle over a compact complex manifold X. If there exists a Finsler structure F : E → ℝ_{≥0} such that log F ∈ P SH(E\{0}), then E^{∗} is nef.

Proof: To prove that E^{∗} is nef, we have to show that L := O_{P (E)}(1) ≅ O_{ℙ(E^{∗})}(1) is nef. According to Proposition 8 the Finsler structure F : E → ℝ_{≥0} induces a continuous metric on O_{P (E)}(−1) so that −φ_{α} ∈ P SH(U_{α}). For the dual bundle L = O_{P (E)}(1) equipped with the dual metric the weight functions are given by φ_{α}^{∗} = −φ_{α}, hence we have a continuous metric on L whose current is positive and the assertion follows from Proposition 7.

Let X be a connected complex manifold. A Finsler (pseudo-) structure on the tangent bundle T X is called a differential (pseudo-) metric. Any such X admits a differential pseudometric: for p ∈ X and η ∈ T X_{p} we define

γ_{X}(p, η) := sup{|Dg(p).η| : g ∈ O(X, Δ), g(p) = 0},

where Δ is the open unit disc in ℂ and O(X, Δ) the set of all holomorphic maps from X to Δ. Reiffen shows in [Rei]:

Proposition 10 The map γ_{X} : T X → ℝ_{≥0} is a differential pseudometric, which has the following invariance property. Let f : X → Y be a holomorphic map of connected complex manifolds, then

γ_{Y} (f(p), Df(p).η) ≤ γ_{X}(p, η),

in particular, for a biholomorphic map f the equality holds.

The function γ_{X} is called the Carathéodory-Reiffen pseudometric and X is said to be γ-hyperbolic, if γ_{X} is a differential metric. Examples: (i) Any bounded domain G ⊂ ℂ^{n} is γ-hyperbolic (cf. [JP], Chap. II, Prop. 2.3.2). Proposition 10 immediately implies: let i : X → Y be a holomorphic immersion and let Y be γ-hyperbolic, then X is γ-hyperbolic too. This gives us (ii) Let Y be a Stein manifold and let G̃ be a bounded domain in Y, i.e. there exists an embedding Y ↪ ℂ^{N} and a bounded domain G ⊂ ℂ^{N}, such that G̃ = Y ∩ G is connected. Then G̃ is γ-hyperbolic.

Proposition 11 Let X be a γ-hyperbolic manifold. Then the function

log γ_{X} : T X\{0} → (−∞, +∞)

is plurisubharmonic.

Proof: Since the logarithm is strictly increasing, we have

log γ_{X}(p, η) = sup{log |Dg(p).η| : g ∈ O(X, Δ), g(p) = 0}.

The tangent map of a holomorphic map is again holomorphic, so that g̃(p, η) := log |Dg(p).η| is in P SH(T X) (see [JP], Appendix, PSH 4). Hence log γ_{X} = sup_{g}{g̃} is the supremum of plurisubharmonic functions. By assumption γ_{X} is a differential metric, i.e. γ_{X} is continuous and γ_{X} : T X\{0} → ℝ_{>0}, thus log γ_{X} : T X\{0} → (−∞, ∞) is also continuous. Now we get our assertion from the following fact ([JP], Appendix, PSH 14). If a family (u_{α})_{α∈A} of plurisubharmonic functions is locally uniformly bounded from above, then the function

u_{0} := (sup_{α∈A} u_{α})^{∗}

is again plurisubharmonic, where “∗” denotes the upper semicontinuous regularization. But we don’t need to regularize log γ_{X}, since it is already continuous and this assures also that the family {g̃} is locally uniformly bounded from above. Let G be a subgroup of the automorphism group Aut(X), which operates fixpointfree and properly discontinuously on X. Then the quotient Q = X/G is a Hausdorff space which admits a unique complex structure, such that the projection π : X → Q is a holomorphic and locally biholomorphic map. We can now prove

Theorem 6 Let X be a γ-hyperbolic manifold and let Q = X/G be a compact quotient as above. Then the cotangent bundle Ω_{Q}^{1} is nef.

Proof: As local coordinates ψ for Q we can take π^{−1} restricted to appropriate open sets such that a coordinate change is described by ψ_{1} ∘ ψ_{0}^{−1} = f, where f ∈ G (cf. [W], Chap. V, Prop. 5.3.). Then we define for q ∈ Q and ξ ∈ T Q_{q}

F (q, ξ) := γ_{X}(ψ(q), Dψ(q).ξ).

Since the Carathéodory-Reiffen metric γ_{X} is invariant under biholomorphic transformations (Proposition 10), this definition does not depend on the choice of the local coordinate and gives us a differential metric F on T Q. Moreover Proposition 11 implies that log F ∈ P SH(T Q\{0}). Now the assertion follows from Proposition 9.

In particular, compact quotients of a bounded domain in ℂ^{n} or in a Stein manifold have nef cotangent bundles.

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Henrik Kratz

Mathematisches Institut Universität Bayreuth 95440 Bayreuth Germany kratz@btm8x1.mat.uni-bayreuth.de 194

Documenta Mathematica 2 (1997)