Compact Complex Manifolds with Numerically Effective Cotangent Bundles

We prove that a projective manifold of dimension n = 2 or 3 and Kodaira dimension 1 has a numerically e ective cotangent bundle if and only if the Iitaka bration is almost smooth, i.e. the only singular bres 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 bred over a rational curve the proof needs an extra assumption concerning the multiplicities of the singular bres. Furthermore, we prove the following theorem: let X be a complex manifold which is hyberbolic with respect to the Carath eodory-Rei en-pseudometric, then any compact quotient of X has a numerically e ective cotangent bundle. 1991 Mathematics Subject Classi cation: 32C10, 32H20


Introduction
It is a natural question in algebraic geometry to classify manifolds by positivity properties of their tangent resp.cotangent bundles.The rst 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 e ectivity.A line bundle L on a projective manifold X is called numerically e ective (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 bres of E, is nef.
Taking the Hartshorne-Frankel conjecture as a guideline, Campana and Peternell considered projective manifolds whose tangent bundles are nef and classi ed 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 1 X are nef.In the rst part we will give a characterization of 2 and 3 dimensional manifolds with Kodaira dimension (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 (X) = 1 and let : X !C be the Iitaka bration of X.Then the following conditions are equivalent: (i) 1 X is nef.(ii) is almost smooth, in the sense that the only singular bres of 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 P mi 1 mi 2, where the m i are the multiplicities of the singular bres.
The equivalence of (i) and (ii) holds also for compact K ahler surfaces.This theorem generalizes a result of Fujiwara Fu] who worked in arbitrary dimension but under the stronger assumption that 1 X is semi-ample, i.e. that some power of O P( 1 X ) ( 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 bration.This follows basically from Gri ths's theory on the variation of the Hodge structure.Then, we will study the base-change which reduces an almost smooth bration 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 bration over a manifold with nef cotangent bundle or (b) it admits an almost smooth Abelian bration over a curve C such that either (i) g(C) 1 or (ii) g(C) = 0 and P mi 1 mi 2.
We remark that the bres F of the Iitaka brations in Theorem 1 are paraAbelian varieties, i.e. there exists an unrami ed 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 bration is almost smooth with para-Abelian bres.
In the second part of this paper we consider complex manifolds X which are hyperbolic with respect to the Carath eodory-Rei en pseudometric.We will show : Theorem 2 Let X be a complex manifold which is hyperbolic with respect to the Carath eodory-Rei en pseudometric and let Q be a compact quotient of X with respect to a subgroup of the automorphism group of X which operates xpointfree and properly discontinuously.Then 1 Q 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 Carath eodory-Rei en pseudometric of X de nes a Finsler structure on the tangent bundle of Q and this gives us a singular hermitian metric on O P( 1 Q ) (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( 1 Q ) (1) is nef.
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 = R 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 !. Then L is said to be nef, if for all > 0 there exists a smooth hermitian metric h on L such that the associated curvature form satis es h (L) !: (iii) Let E be a holomorphic vector bundle on X and P(E) the projective bundle of hyperplanes in the bres 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 bres.
Proposition 2 Let E and F be holomorphic vector bundles.Then

Proposition 1 immediately implies
Proposition 4 Let Y be a nite unrami ed covering of X.Then 1 X is nef if and only if 1 Y is nef.
A bration of X over Y is a surjective holomorphic map : X !Y whose bres are connected.A point x 2 X is said to be critical if the tangent map D (x) has not maximal rank.The images (x) 2 Y of the critical points are the critical values of .They form a proper analytic subset of Y , i.e. in the case, where Y is a curve, a nite subset fa 1 ; : : : ; a l g.Let y 2 Y and let J be the ideal sheaf of y in O Y .Then the bre X y is the complex subspace ( 1 (y); O X = (J ) O X ) of X, and a bre X y is singular if and only if y is a critical value.A bration, for which D has maximal rank everywhere, is called smooth.
When we consider a bration : X !C over a curve C, we will always assume that C is smooth.Such a bration is said to be almost smooth, if the only singular bres of are multiples of smooth irreducible subvarieties.Their multiplicities will be denoted by m i with 1 i l, so that the singular bres are X ai = m i F i , where the F i are smooth irreducible subvarieties.
We will denote the Kodaira dimension of X by (X).Let X be a projective manifold with (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 : X !Z where Z is a projective variety with dim Z = (X).Such a map is called Iitaka bration (cf.Ue]).
2 Manifolds with = 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 (X) = 1 and let : X !C be the Iitaka bration of X.Then the following conditions are equivalent: (i) 1 X is nef.(ii) is almost smooth, in the sense that the only singular bres of 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 P mi 1 mi 2, where the m i are the multiplicities of the singular bres.
The equivalence of (i) and (ii) holds also for compact K ahler surfaces.
Proof: (i) ) (ii) If X is an n-dimensional projective manifold with 1 X nef, it satis es 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 (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 : X !C is the Iitaka bration of X and X ai are the singular bres (1 i l), we calculate e(X) = P e(X ai ) .But now the assertion follows, because e(X ai ) 0 and e(X ai ) = 0 if and only if the bre X ai is a multiple of a smooth elliptic curve (cf.BPV], Chap.III, Prop.11.4).This argument remains true for a compact K ahler 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 bration.

Smooth fibrations
We will consider smooth Abelian brations rst: Proposition 5 Let X and Y be projective manifolds and let : X !Y be a smooth bration, whose bres are Abelian varieties.Then the relative cotangent bundle 1 Proof: (1) We claim that ( 1 X=Y ) = 1 X=Y .For all y 2 Y the cotangent bundle of the bre 1 Xy is trivial, so that 1 X=Y is locally free of rank equal to the dimension Documenta Mathematica 2 (1997) 183{193 of the bres (cf.Ha], Chap.III, Cor.12.9).Moreover for all y 2 Y we have ( 1 X=Y ) y = H 0 (X y ; 1 Xy ) and thus ( ( 1 X=Y )) x = H 0 (X y ; 1 Xy ) for (x) = y .Now, the canonical homomorphism : ( 1 X=Y ) ! 1 X=Y is described stalkwise by x : 7 !(x) with 2 H 0 (X y ; 1 Xy ).Since 1 X=Y j Xy is globally generated, x is surjective and hence bijective.
(2) Any smooth bration : X !Y of projective manifolds gives rise to a variation of the Hodge structure in its bres X y (y 2 Y ).From this Gri ths deduces Gr], Cor.

7.8
Theorem 4 For all n 2 f1; : : :; dim C X y g the bundles R n (O X ) are seminegative in the sense of Gri ths.
Now the bundle E = R n (O X ) is conjugate linear to E = ( n X=Y ) so that the curvature matrices with respect to unitary bases behave as 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 2 f1; : : : ; dim C X y g the bundles ( n X=Y ) are semipositive in the sense of Gri ths.
In particular, since semipositivity implies \nefness", ( n X=Y ) is nef and hence for a smooth Abelian bration 1 X=Y = ( 1 X=Y ) 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 : X !C, because for a smooth one knows from the study of the period map that deg( !X=C ) = 0 (cf.BPV], Chap.III, Thm.18.2).
We have a similar proposition for smooth hyperelliptic brations: Proposition 6 Let X be a projective 3-dimensional manifold and let : X !C be a smooth bration, whose bres are hyperelliptic surfaces.Furthermore, let g(C) 1.
Then 1 X is nef.Proof: We consider the relative Albanese factorization of , i.e. the commutative diagram where A(X=C) is a smooth bration over C whose bres over a 2 C are the Albanese tori Alb(X a ) of the bres X a of .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 j Xa : X a !Alb(X a ) is a surjective submersion with smooth elliptic curves as bres ( DPS], Prop.3.9.).But also A is smooth: let x 2 X; (x) = a and A (x) = y, then both tangent directions of TA(X=C) y lie in the image of DA (x).First, we can nd a tangent v 2 (T A(X=Y ) j Alb(Xa) ) y in the image of DA (x) j Xa (because A j Xa 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 D (x): @ @x1 = @ @z1 .Using the commutativity of the relative Albanese diagram, we get 0 6 = D (x): @ @x 1 = DAlb( )(y) DA (x): @ @x 1 : In particular, w := DA (x): @ @x1 6 = 0; and since DAlb( )(y):v = 0 the vectors v and w have to be linear independent.
We can now apply Proposition 5 twice to conclude that 1 X is nef: Alb( ) : A(X=C) !
C is a smooth bration of projective manifolds whose bres are elliptic curves and by assumption g(C) 1, so that 1 A(X=C) has to be nef.Since A : X !A(X=C) is a smooth elliptic bration too, 1 X is also nef.

Almost smooth fibrations
Let X be a compact complex manifold of dimension n and let : X !C be an almost smooth bration over a smooth curve C. As above we will denote the critical values of by a 1 ; : : : ; a l and their multiplicities bym i where 1 i l, so that the singular bres are X ai = m i F i , where the F i are smooth irreducible subvarieties.
To get rid of the multiple bres 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 nite covering : C 0 !C, which has d mi rami cation 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 bre product X C C 0 gives us a smooth bration ' : X 0 !C 0 and a commutative diagram (cf.Kod], Thm 6.3) Here f is a nite covering which is unrami ed over X 1 (a 0 ), because the multiplicities of and compensate each other over a i (i 1), and f has d m0 rami cation divisors of order m 0 1 over 1 (a 0 ).Assume that we knew 1 X 0 is nef, then we would like to carry this over to 1 X .However, it is not possible to apply Proposition 4 since f is rami ed.But we have the following commutative diagram with exact rows which was already used in Fu] To prove the commutativity of this diagram one uses Documenta Mathematica 2 (1997) 183{193 basically the fact that the restriction of f to a bre of ' is unrami ed.For i 1 we have (a i ) = m i F i .So, de ning A := P l i=1 (mi 1) mi a i we get L = (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 : X !C be an almost smooth bration, whose bres are Abelian varieties.Assume furthermore that (i) g(C) 1 or (ii) g(C) = 0 and deg A 2. Then 1 X is nef.Proof: The process described above allows us to pass to a smooth Abelian bration ', for which 1 X 0 =C 0 is nef by Proposition 5. Moreover the line bundle L = (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 ( 1 X ) are nef (Proposition 3).Since f is a nite surjective map, we nally deduce from Proposition 1 that 1 X 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 (S) = 1 and : S !P 1 is an almost smooth elliptic bration, the condition that deg A 2 (resp.that L is nef) is automatically satis ed.

Similarly we get
Corollary 2 Let X be a projective 3-dimensional manifold with (X) 0 and let : X !C be an almost smooth bration, whose bres are hyperelliptic surfaces.
Assume furthermore that (i) g(C) 1 or (ii) g(C) = 0 and deg A 2. Then 1 X is nef.Proof: To deduce from Proposition 6 that 1 X 0 =C 0 is nef as a quotient of 1 X 0 , we have to assure that g(C 0 ) 1.But g(C 0 ) = 0 leads to 1 = (X 0 ) (X) which contradicts our assumptions.
In particular, these two corollaries yield the direction (ii)) (i) in Theorem 3 which is now completely proved.
The curvature form the singular metric on L is locally given by the closed (1; 1)current c(L) = i @ @' .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 ' are plurisubharmonic.
Remark: We will say that a singular metric is continuous (or simply that it is a continuous metric), if the weight functions ' are continuous on the trivialization sets.The main ingredient for the following arguments will be the next proposition which is independently due to Demailly, Shi man 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 de ne Definition 3 A Finsler structure on E is a continuous function F : E !R 0 , so that for all 2 E: (i) F( ) > 0 for 6 = 0, (ii) F( ) = j jF( ) for all 2 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 bres of E, p : P(E) !X the projection and O P(E) ( 1) the subbundle of p E whose bre over a point in P(E) is given by the complex line represented by that point.Then we have a map p : 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 PSH(Y ).Proposition 8 (a) Any Finsler structure F on E de nes via k k := F p( ); 2 O P(E) ( 1): a continuous metric on O P(E) ( 1).(b) If log F 2 PSH(Enf0g), then ' 2 PSH(U ).Proof: (a) Let : O P(E) ( 1) j U ' !U C be a local trivialization and let s be a local holomorphic section of O P(E) ( 1) j U which describes the trivialization.Then the corresponding weight function is ' (x) = log ks (x)k = log F p(s (x)); x 2 U : The map p s : U ! E is clearly holomorphic.Moreover for x 2 U we have s (x) 6 = 0, so that property (i) in the de nition of Finsler structures leads to F p(s (x)) > 0. From this we conclude ' 2 C 0 (U ).(b) If f : Y !Z is a holomorphic map between complex manifolds and the function u 2 PSH(Z), then u f 2 PSH(Y ) (cf.JP], Appendix, PSH 7).So, since p s is holomorphic, we have ' 2 PSH(U ).
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 2 PSH(Enf0g), 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 ' 2 PSH(U ).For the dual bundle L = O P(E) (1) equipped with the dual metric the weight functions are given by ' = ' , 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 TX is called a di erential (pseudo-) metric.Any such X admits a di erential pseudometric: for p 2 X and 2 TX p we de ne X (p; ) := supfjDg(p): j : g 2 O(X; ); g(p) = 0g; where is the open unit disc in C and O(X; ) the set of all holomorphic maps from X to .Rei en shows in Rei]: Proposition 10 The map X : TX !R 0 is a di erential pseudometric, which has the following invariance property.Let f : X !Y be a holomorphic map of connected complex manifolds, then Y (f(p); Df(p): ) X (p; ); in particular, for a biholomorphic map f the equality holds.The function X is called the Carath eodory-Rei en pseudometric and X is said to be -hyperbolic, if X is a di erential metric.

2.3.2).
Proposition 10 immediately implies: let i : X !Y be a holomorphic immersion and let Y be -hyperbolic, then X is -hyperbolic too.This gives us (ii) Let Y be a Stein manifold and let G be a bounded domain in Y , i.e. there exists an embedding Y , !C N and a bounded domain G C N , such that G = Y \ G is connected.Then G is -hyperbolic.
Proposition 11 Let X be a -hyperbolic manifold.Then the function log X : TXnf0g ! ( 1; +1) is plurisubharmonic.Proof: Since the logarithm is strictly increasing, we have log X (p; ) = supflog jDg(p): j : g 2 O(X; ); g(p) = 0g: The tangent map of a holomorphic map is again holomorphic, so that g(p; ) := log jDg(p): j is in PSH(TX) (see JP], Appendix, PSH 4).Hence log X = sup g fgg is the supremum of plurisubharmonic functions.By assumption X is a di erential metric, i.e.X continuous and X : TXnf0g !R >0 , thus log X : TXnf0g ! ( 1; 1) is also continuous.Now we get our assertion from the following fact ( JP], Appendix, PSH 14).If a family (u ) 2A of plurisubharmonic functions is locally uniformly bounded from above, then the function u 0 := (sup 2A u ) is again plurisubharmonic, where \ " denotes the upper semicontinuous regularization.But we don't need to regularize log X , since it is already continuous and this assures also that the family fgg is locally uniformly bounded from above.Let G be a subgroup of the automorphism group Aut(X), which operates xpointfree and properly discontinuously on X.Then the quotient Q = X=G is a Hausdor space which admits a unique complex structure, such that the projection : X !Q is a holomorphic and locally biholomorphic map.We can now prove Theorem 6 Let X be a -hyperbolic manifold and let Q = X=G be a compact quotient as above.Then the cotangent bundle 1 Q is nef.
Proof: As local coordinates for Q we can take 1 restricted to appropriate open sets such that a coordinate change is described by 1 1 0 = f, where f 2 G (cf. W], Chap.V, Prop.5.3.).Then we de ne for q 2 Q and 2 TQ q F(q; ) := X ( (q); D (q): ): Since the Carath eodory-Rei en metric X is invariant under biholomorphic transformations (Proposition 10), this de nition does not depend on the choice of the local coordinate and gives us a di erential metric F on TQ.Moreover Proposition 11 implies that log F 2 PSH(TQnf0g).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.