RATIONAL CURVES AND AMPLENESS PROPERTIES OF THE TANGENT BUNDLE OF ALGEBRAIC VARIETIES

FRÉDÉRIC CAMPANA AND THOMAS PETERNELL

Abstract. The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold X. Among the results we prove are these:

If X is a projective manifold, and E ⊂ T_{X} is an ample locally free sheaf with n − 2 ≥ rkE ≥ n, then X ≃ ℙ_{n}.

Let X be a projective manifold. If X is rationally connected, then there exists a free T_{X}-ample family of (rational) curves. If X admits a free T_{X}-ample family of curves, then X is rationally generated.

Introduction

The purpose of this paper is, vaguely speaking, to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold X. This study falls into two parts:

(1) a biregular one in which ampleness of some subsheaf of the tangent bundle T_{X} is assumed; one then expects a classification,

(2) a birational one, where the ampleness of T_{X} is assumed only generically; one then expects X to be rationally connected.

In 1979 in his famous solution of the Hartshorne-Frankel conjecture, S. Mori proved that a projective manifold X whose tangent bundle T_{X} is ample, must be the projective space ℙ_{n}. Previously, Mori and Sumihiro showed that ℙ_{n} is the only manifold admitting a vector field vanishing along an ample divisor (and this is an Euler vector field if n ≥ 2). This was generalised by J. Wahl to the extent that if T_{X} ⊗ L^{−1} has a non-zero section with L an ample line bundle (not necessarily effective), then X ≃ ℙ_{n}.

Although these two theorems look at first quite different in nature, they might be only special cases of a more general theorem.

Parts of this paper were written up during a stay of one of the authors at MSRI. He would like to thank the institute for support and the excellent working conditions. Research at MSRI is supported on part by NSF grant DMS-9022140.

Question: Let X be a projective manifold, E an ample locally free sheaf, E ⊂ T_{X}. Is X ≃ ℙ_{n}?

We shall prove

Theorem (1.1) The question has a positive answer if rkE is n − 2, n − 1 or n.

A manifold is said to be rationally connected if two general points can be joined by a chain of rational curves. Examples are Fano manifolds as shown by Campana and Kollár-Miyaoka-Mori. We ask whether it is possible to characterize rational connectedness by a weak positivity property of the tangent bundle. Philosophically speaking we require that T_{X} should be ample on sufficiently many curves; more precisely T_{X} should be ample on the generic curve of a free family (2.5). We shall prove in this spirit (for definitions see (2.1))

Theorem (2.7) Let X be a projective manifold.

(1) If X is rationally connected, then there exists a free T_{X}-ample family of (rational) curves.

(2) If X admits a free T_{X}-ample family of curves, then X is rationally generated.

(1) is actually an easy consequence of [KoMiMo92]. A projective variety X is rationally generated if every variety dominated by X is uniruled, see sect.2. This notion was introduced in [Ca95]); we expect that rationally generated varieties are actually rationally connected; this holds in dimension 3 by [KoMiMo92]. Important special cases of free family are given by complete intersections of hyperplanes. To make this precise we recall the following

Definition (Miyaoka) Let X be a n−dimensional projective manifold, H_{1}, ..., H_{n−1} ample divisors on X. Let E be a vector bundle on X. Then E is said to be generically (H_{1}, ..., H_{n−1}) -ample (nef) if for m_{i} >> 0 and for the general curve C cut out by m_{1} H_{1}, ..., m_{n−1}H_{m−1}, the restriction E|_{C} is ample (nef).

Miyaoka [Mi 87] has shown that X is not uniruled iff Ω_{X}^{1} is generically (H_{1}, ..., H_{n−1})- \underline{nef} for some H_{i}. We obtain as a special case of (2.5):

Corollary If T_{X} is generically (H_{1}, ..., H_{n−1})-ample, then X is rationally generated.

The converse of the corollary can be expected but should be hard (at least with the H_{i} big instead of ample). It is much easier to look only at positivity of large families of curves, not necessarily cut out by divisors. Also a relation to stability is pointed out.

In practice the positivity of T_{X} is difficult to handle. Much easier is the anticanonical bundle −K_{X} = det T_{X}. Therefore we shall investigate positivity properties of −K_{X} in sect. 3 for varieties of negative Kodaira dimension.

0. Notations around rational curves.

Herewe collect for the convenience of the reader various concepts dealing with rational curves. General references are [Ca 92], [KoMiMo 92].

(0.1) A variety X is called rationally connected if and only if two general points on X can be joined by a chain of rational curves.

(0.2) Let (C_{t}) be a covering family of rational curves on X. Introduce the following equivalence relation: x ∼ y ⟺ x, y can be joined by a chain of C′_{t}s. Then there is a variety Y and a rational map f: X ⇀ Y such that for general x, y we have f(x) = f(y) ⟺ x ∼ y. The map f: X ⇀ Y is called the rational quotient of X w.r.t. (C_{t}) (see [Ca94] for a proof, in a more general situation).

(0.3) One can do the construction in (0.2) for all (C_{t}) at the same time and obtains the notion of a ”rational quotient” of X. More precisely define x ≈ y ⟺ x, y can be joined by a chain of rational curves. Then there exists a variety Z and a rational map g: X ⇀ Z such that for very general x, y (i.e. outside a countable union of subvarieties of X) we have g(x) = g(y) ⟺ x ≈ y. An important property of the rational maps f: ⇀ Y and g: ⇀ Z is the almost holomorphicity, i.e. they are holomorphic and proper on a suitable Zariski dense open subset of X.

(0.4) Let X be a variety. Define inductively: X_{o} = X, X_{i} = rational quotient of X_{i−1}. Then X is said to be rationally generated if there exists m such that dim X_{m} = 0. It is conjectured (but known only up to dimension 3, [KoMiMo92]) that X is rationally connected iff X is rationally generated. Rational connectedness and rational generatedness are equivalent provided the following holds: for every holomorphic surjective map f: Xℙ_{1} with X a projective manifold and f having rationally connected fibers, there exists a rational curve C ⊂ X with f(C) = ℙ_{1}. See [Ca95].

1. Ample subsheaves of tangent bundles

In this section we shall prove 1.1 Theorem Let X be a projective manifold of dimension n. Let E ⊂ T_{X} be a locally free sheaf of rank r. If E is ample and rk E ≥ n − 2, then X ≃ ℙ_{n} and E = T_{X} or O(1)^{⊕r}.

Actually (1.1) should hold without any assumption on the rank. If E = T_{X}, then (1.1) is nothing than Mori’s theorem [Mo 79]. Also the other extremal case is known: if rk E = 1, then X ≃ ℙ_{n} by J. Wahl [Wa 83].

The proof of (1.1) proceeds by induction on dim X. In the induction step we will need the following

1.2 Lemma Let X be a n-dimensional projective manifold, φ: X → Y a ℙ_{k}-bundle (k < n) of the form X = ℙ(V) with a vector bundle V of rank k +1. Then the relative tangent sheaf T_{X|Y} does not contain an ample locally free subsheaf.

Remark Lemma 1.2 should only be a special case of a much curve general fact: if φ: X → Y is a fiber space (sufficiently smooth?), then T_{X|Y} = (Ω_{X|Y}^{1} )^{∗} does not contain an ample subsheaf unless Y is a point and X projective space.

Proof. Suppose E ⊂ T_{X|Y} is an ample locally free subsheaf. Let F be a fiber of φ. Then E|_{F} ⊂ T_{X|Y} |F = T_{F} . Hence either E = T_{X|Y} or E = O(1)_{|F}^{⊕r} for all F. In the first case, apply the Euler sequence

(E) 0 → O → φ^{∗}(V^{∗}) ⊗ O_{ℙ(V)}(1) → T_{X|Y} → 0

to derive the ampleness of φ^{∗}(V^{∗}) ⊗ O_{ℙ(V)}(1) ((E) does not split !) which is absurd (or argue as in the second case). So let E|_{F} = O(1)|_{F}^{⊕r} for all F. Then

(*) E ≃ φ^{∗}(E′) ⊗ O_{ℙ(V)}(1)

with a bundle E′ of rank r on Y. Consider again (E). We claim that the inclusion E ↪ T_{X|Y} lifts to an inclusion E ↪ φ^{∗}(V^{∗}) ⊗ O_{ℙ(V)}(1). =46or this we need to show that the canonical map

H^{o}(E^{∗} ⊗ φ^{∗}(V^{∗}) ⊗ O_{ℙ(V)}(1)) → H^{o}(E^{∗} ⊗ T_{X|Y})

is onto, hence that H^{1}(E^{∗}) = 0. By Leray’s spectral sequence,

H^{1}(X, E^{∗}) ≃ H^{1}(Y, E′^{∗} ⊗ φ_{∗}(O_{ℙ(V)}(−1)) = 0.

So we have E = φ^{∗}(E′) ⊗ O_{ℙ(V)}(1) ⊂ φ^{∗}(V^{∗}) ⊗ O_{ℙ(V)}(1), therefore E′ ⊂ V^{∗}. Consider the dual map

α: V → E′^{∗},

which is generically onto. Let S = Im α. Then i : ℙ(S) ↪ ℙ(V) and

i^{∗}(E) = i^{∗}(O_{ℙ(V)}(1) ⊗ φ^{∗}(E′)) = O_{ℙ(S)}(1) ⊗ φ′^{∗}(E′),

where φ′: ℙ(S) → Y is the projection. By construction, O_{ℙ(S)}(1) ⊗ φ′^{∗}(E) is ample, hence, taking det,

O_{ℙ(S)}(r) ⊗ φ′^{∗}(detE′) = O_{ℙ(S⊗ \frac{det E′}{r})}(r)

is ample. Hence S⊗ \frac{det E′}{r} is ample. Now take a general curve C ⊂ Y. Then E′^{∗}⊗ \frac{detE′}{r} |C is ample. But c_{1}(E′^{∗} ⊗ \frac{detE′}{r}) = 0, contradiction.

Now we come to the proof of (1.1). We treat the cases r = n, n−1 and n−2 separately.

(1.3) Case r = n. The inclusion E ⊂ T_{X} yields det E ⊂ −K_{X} . If det E = −K_{X}, then E = T_{X} and the claim is Mori’s theorem. So assume det E ≠ −K_{X}, hence K_{X} + det E ≠ O_{X}, but K_{X} + det E ⊂ O_{X}. So K_{X} + det E is not nef. Now vector bundle adjunction theory [Fu 90, YZ 90] gives X ≃ ℙ_{n} and E = O(1)^{n}.

(1.4) Case r = n − 1. Since all arguments in this case appear also in the more difficult case r = n − 2, we just give a very rough sketch. E ⊂ T_{X} yields det E ⊂ ∧^{n−1}T_{X} = Ω_{X}^{1} ⊗ −K_{X}, hence L = K_{X} + det E ⊂ Ω_{X}^{1}. If L is not nef, then all such pairs (X, E) can be classified ([Fu 90, YZ 90]) and examined to prove our claim. So let L be nef. Hence

H^{0}(X, Ω_{X}^{1} ⊗ L^{∗}) ≠ 0

for some nef line bundle L.

Now H^{o}(Ω_{X}^{1} ⊗ L^{∗}) ≠ 0 implies that X is not rationally connected. However it is uniruled by [Mi 87], so one can consider the rational quotient f: X ⇀ Z w.r.t. some covering family of rational curves. Then the general compact fiber of f is a projective space which leads easily to a contradiction to the nefness of L.

(1.5) Case r = n − 2. Taking det of E ↪ T_{X} gives det E ↪ ∧^{n−2}T_{X} = Ω_{X}^{2} ⊗ −K_{X} hence

L := K_{X} + det E ⊂ Ω_{X}^{2}.

(1.5.1) We first assume that L is not nef. Then by [AM 95], (X, E) is one of the following.

(a) There is a blow-up φ: X → W of a finite set B ⊂ W of smooth points and an ample bundle E′ on W with E = φ^{∗}(E′) ⊗ O_{X}(−A), A = φ^{−1}(B)

(b) X = ℙ_{n}, E splits

(c) X = Q_{n} and E splits or is a twist of the spinor bundle

(d) X = ℙ_{2} × ℙ_{2}, E = O(1, 1) ⊕ O(1, 1)

(e) X is a del Pezzo manifold with b_{2} = 1, E = O(1) ⊕ n − 2

(f) X is a ℙ_{n−1} - bundle or a quadric bundle over a curve Y

(g) X is a ℙ_{n−2}-bundle over a surface. We comment on those cases separately.

(a) We have E = φ^{∗}(E′)⊗O_{X}(−A) ⊂ T_{X} ⊂ φ^{∗}(T_{W}), so that E′⊗I_{B} ⊂ φ_{∗}(T_{X}) ⊂ T_{W} , hence

E′ ⊂ T_{W} .

By induction on b_{2}(X) we may assume W = ℙ_{n} and E′ = O(1)^{⊗(n−2)}. But then E = φ^{∗}(O(1))^{n−2} ⊗ O_{X}(−A) is clearly not ample, at most nef.

(b) Obviously E = O(1)^{⊕(n−2)}.

(c) If E splits, then T_{Q_{n}} would have a section vanishing on ample divisor which is not possible. So E is the twist F (2) of a spinor bundle, see e.g. [PSW 90]. Therefore E has a section vanishing on a hyperplane section, so does T_{Q_{n}} which would imply H^{o}(T_{Q_{n}}(−1)) ≠ 0, contradiction.

(d) Again ℙ_{2} × ℙ_{2} has no vector field vanishing on an ample divisor.

(e) Here K_{X} + det E = O_{X}(−1) and E = O_{X}(1)^{⊕(n−2)}. Now either apply [Wa 83], [MS 78] or apply the Fujita classification of del Pezzo manifolds or argue directly as follows. We have H^{o}(X, T_{X}(−1)) ≠ 0. Choose H ∈ |O_{X}(1)| smooth and consider the sequence

0 → T_{H} → T_{X}|H → N_{H|X} = O(1) → 0.

If the induced map E|_{H} → N_{H} is non-zero, we obtain a map O(1)^{n−3} ↪ T_{H}, otherwise O(1)^{n−2} ⊂ T_{H}. In both cases we get H ≃ ℙ_{n−1} by induction, hence X ≃ ℙ_{n}.

(f) Let F be the general fiber of φ: X → Y. Then E|_{F} ↪ T_{X}|F and since the composition

E|_{F} → N_{F |X} = O_{F}^{a}

is zero, we obtain E|_{F} ↪ T_{F}. Inductively, F ≃ ℙ_{n−1}, hence f cannot be a quadric bundle therefore must be a ℙ_{n−1}-bundle; moreover E ⊂ T_{X|Y} . But this contradicts (1.2).

(g) Finally φ is a ℙ_{n−2} bundle. As in (f), E ⊂ T_{X|Y} , contradicting (1.2). This finishes the case that L is not nef.

(1.5.2) L = K_{X} + det E is nef. So (*) H^{o}(X, Ω_{X}^{2} ⊗ L^{∗}) ≠ 0 with L nef. This implies that X is not rationally connected: otherwise we find by [KoMiMo 92] a family of rational curves C_{t} ⊂ X with T_{X}|C_{t} ample for t general contradicting (*). Hence we have a rational quotient f: X ⇀ Z, [Ca92] and dim Z > 0 since X is not rationally connected. Moreover dim Z < dim X, since X is uniruled by [Mi 87]. Since f is almost holomorphic, i.e. proper on an open set, it has some compact fibers. The general fiber F therefore must be ℙ_{k} (since E|_{F} ↪ T_{F}) with k ≥ n − 2. Therefore L_{F} is never nef, contradiction.

(1.6) We already mentioned Wahl’s theorem several times: if X is a projective manifold, L an ample line bundle, H^{o}(X, T_{X} ⊗L^{∗}) ≠ 0, then X ≃ ℙ_{n}. Previously Mori and Sumihiro [MS 78] proved this under the additional assumption that L is effective. It should actually be possible to deduce Wahl’s theorem from Mori-Sumihiro by covering tricks in the following way. Take d ≥ 2 with H^{o}(X, L^{d}) ≠ 0, and H^{o}(X, T_{X} ⊗ L^{∗}) ≠ O. Suppose s ∈ H^{o}(X, L^{d}), s ≠ 0 with {s = 0} smooth. Let f: Y → X be the associated cyclic cover so that H^{o}(Y, f^{∗}(L)) ≠ 0. Then there is an exact sequence

0 → T_{Y} → f^{∗}T_{X} → N_{R} → 0

where N_{R} is supported on the ramification divisor R ⊂ Y . The map f^{∗}(L) ↪ f^{∗}(T_{X}) yields a map γ: f^{∗}(L) → N_{R}. Now it can be shown that γ = 0, hence we obtain f^{∗}(L) ⊂ T_{Y} . By [MS 78], Y ≃ ℙ_{n}, moreover f^{∗}(L) = O(1), so that deg f = 1 and X ≃ ℙ_{n}. Of course the zero set {s = 0} could be non-smooth. If {s = 0} is at least reduced, then the same arguments basically work, only that Y is merely normal. So one should first prove [MS 78] in case Y is normal. The case where {s = 0} contains a non-reduced component seems more complicated, one should proceed taking roots.

2. The tangent bundle of rationally connected varieties

2.1 Definition Let X be a projective manifold. A covering family (C_{t})_{t∈T} (T compact) of curves is free if

(1) T is irreducible

(2) C_{t} is irreducible for t ∈ T general

(3) for all A ⊂ X at least 2-codimensional there exists t ∈ T with C_{t} ∩ A = ∅.

We shall say that a family (C_{t}) is T_{X}-ample, if T_{X}|C_{t} is ample for general t.

2.2 Theorem Let X be a projective manifold, (C_{t}) a free T_{X}-ample family. Then X is rationally generated. Conversely, a rationally connected manifold has a free T_{X}- ample family of (rational) curves.

Proof : From the definition of a free family it is clear that X is uniruled, applying [MM 86]. Then we consider ”the” rational reduction f: X ⇀ Y and have dimY < dimX, Y smooth. By construction, Y is not uniruled. We need to show that dimY = 0. So assume dimY > 0. By condition (2) of (2.1), f is holomorphic near C_{t} for general t. Since T_{X}|C_{t} is ample, we have dim f(C_{t}) > 0. Now consider the generically surjective map (t general)

α: T_{X}|C_{t} → f^{∗}T_{Y} |C_{t}.

Then Im α is an ample sheaf, hence f^{∗}T_{Y} |C_{t} is ample and so does T_{Y} |f(C_{t}). The family C′_{t} = f(C_{t}) still covers Y with T_{Y} |C′_{t} ample for general t, hence −K_{Y} .C′_{t} > 0 and Y is uniruled by [MM 86], contradiction. The other direction follows from [KoMiMo92] using (2.5(2)).

Next we recall a notion due to Miyaoka.

2.3 Definition Let X be a projective manifold, n = dim X, H_{1}, ⋯, H_{n−1} ample divisors X. A vector bundle E on X is generically (H_{1}, ⋯, H_{n−1})-ample (nef) iff for m_{1} >> 0, ⋯, m_{n−1} >> 0 and for general C cut out by m_{1} H_{1}, ⋯, m_{n−1}H_{n−1} the restriction E|C is ample (nef).

(2.4) The same definition makes sense for H_{1}, ⋯, H_{n−1} big, as long as the general

element cut out by m_{1} H_{1}, ⋯, m_{n−1}H_{n−1} is really a curve, i.e. if dim ⋂_{i=1}^{n−1} Bs(m_{i}H_{i}) ≤ 1.

2.5 Obvious Example (1) If H_{1}, ⋯, H_{n−1} are very ample, then the general C_{t} cut out by H_{1}, ⋯, H_{n−1} deforms to build free families.

(2) If C ⊂ X is a rational curve with T_{X}|C ample, then C moves in a free T_{X}-ample family, provided X is smooth near C.

(2.6) Miyaoka has shown in [Mi 87] that X is not uniruled iff Ω^{1} is generically (H_{1}, ⋯, H_{n−1})-nef for all ample H_{1}, ⋯, H_{n−1} . We are now interested in the ”dual” case.

2.7 Theorem If T_{X} is generically (H_{1}, ⋯, H_{n−1})-ample, then X is rationally generated.

This follows from the more general theorem 2.2.

2.8 Corollary Let X be a smooth projective 3-fold, (C_{t}) a free family of curves such that T_{X}|C_{t} is ample for general t. Then X is rationally connected.

Proof. Rationally generated 3-fold are rationally connected [KoMiMo 92].

2.9 Remark One might expect that (2.2) remains true if one assumes only the ampleness of T_{X}|C_{t}, t general, where (C_{t}) is a family with the following property: for x, y ∈ X general, there is some t such x, y ∈ C_{t} (or joined by a chain of C′_{t}s). However the proof of (2.2) does not give this claim. Probably one should construct from such a family a new, free family of curves.

2.10 Problem Let X be rationally connected. Do there exist H_{1}, ⋯, H_{n−1} ample (or big), such that T_{X} is generically (H_{1}, ⋯, H_{n−1}) - ample? Is there at least a free family (C_{t}) such T_{X}|C_{t} is ample, t general?

(2.11) We discuss (2.10) in several special cases.

(1) If n = dim X = 2, then (2.10) ”clearly” holds.

(2) Assume T_{X} to be (H_{1}, ⋯, H_{n−1}) - semi-stable.

Then T_{X}|C is semi-stable for C cut out by m_{1} H_{1}, ⋯, m_{n−1} H_{n−1} for m_{i} >> 0 (Mehta-Ramanathan). By [Mi 87a], this is equivalent to saying that (T_{X} ⊗ \frac{K_{X}}{n})|C is nef (i.e. S^{n}T_{X} ⊗K_{X}|C is nef) so if we know additionally that −K_{X} ·H_{1} ·...·H_{n−1} > 0, (2.10) holds.

So (2.10) has a positive answer if we can find H_{1}, ⋯, H_{n−1} ample such that

(a) T_{X} is (H_{1}, ⋯, H_{n−1}) semi-stable.

(b) H_{1}, ⋯, H_{n−1} is in the cone generated by classes of covering families of rational curves. (since (b) implies −K_{X}.H_{1}. · · · .H_{n−1} > 0 !)

In sect. 3 we shall discuss the relation of various cones of curves.

(3) If X is Fano, then (b) is always fulfilled. It is expected that for b_{2} = 1, T_{X} is always stable; a lot of cases being checked in [PW 94], so that (2.10) should hold for Fano manifolds with b_{2} = 1 (and probably also for higher b_{2}).

2.12 Remarks (1) If X is rationally generated, does X carry a free, T_{X}-ample family of curves? This could also be considered as a converse of the statement of (2.2).

(2) Is there a direct way to prove that a projective manifold carrying a free T_{X}-ample family of rational curves is rationally connected?

(3) From (2) and (3) it would follow that rationally generated manifolds are rationally connected. While (2) seems accessible, (3) is probably hard.

3. Various cones and the canonical bundle

3.1 Notation K_{nef} denotes the (closed) cone of nef divisors, K_{eff}, that one effective divisors (always modulo numerical equivalence, of course). We let N_{nef} be the dual cone of K_{eff} so that C ∈ N_{1}(X) is in N_{nef} iff D.C ≥ 0 for all D ∈ K_{eff} or all effective divisors D ) and call an element in N_{nef} a nef curve. The dual cone of K_{nef} is of course \overline{NE}(X), the closed cone of curves. Furthermore we let

a) N_{rat} be the closed cone generated by classes [C_{t}], where (C_{t}) is a covering family of rational curves, i.e. C_{t} is a possibly singular irreducible rational curve for general t. Without requiring C_{t} to be irreducible, but connected, with all components of C_{t} being rational, we call the resulting cone Ñ_{rat}.

b) N_{cov} be the analogous cone, omitting ”rational”

c) N_{ci} be the closed cone generated by curves H_{1}, ⋯, H_{n−1}; H_{i} very ample (”ci” stands for complete intersection)

d) N_{sc} be the closed cone generated by curves H^{n−1}; H very ample (”sc” stands for special complete intersection).

We have the obvious inclusions N_{rat} ⊂ Ñ_{rat} ⊂ N_{cov} and N_{sc} ⊂ N_{ci} ⊂ N_{cov}.

3.2 Problems (cp. [DPS 95].

(1) Is N_{nef} = N_{cov}? (Clearly N_{cov} ⊂ N_{nef}).

(2) Assume X Fano. Is N_{rat} = N_{nef} or at least N_{rat} = N_{cov}?

(3) Assume X rationally connected or Fano. Is N_{rat} ∩ N_{ci} ≠ {0}?

(4) Assume X rationally connected or Fano. Is N_{rat} ≠ Ñ_{rat}?

3.3 Remarks

(1) This problem was posed in [DPS 95]. It ”essentially” (up to a limit process) means that given an nef curve C ⊂ X, i.e. D.C ≥ 0 for all effective D, then there should exist a covering family (C′_{t}) such that C′_{t} ≡ α C_{t} for some α ∈ ℚ_{t}. In particular, if every effective divisor is nef, e.g. T_{X} is nef, then every curve C should be ”numerically movable” up to a multiple. For surfaces this is clear. This would give new evidence for the conjecture that Fano manifolds with T_{X} nef are homogeneous ([CP 91, DPS 94]).

RATIONAL CURVES AND AMPLENESS PROPERTIES OF THE TANGENT BUNDLE 11

(2) Assume X Fano. The intention is to investigate ”how much of the geometry is dictated by the moving rational curves.” The cone theorem says that \overline{NE}(X) is already generated by the classes of rational curves. So it seems very natural to suspect that the cone of ”covering curves” is generated by the cone of ”covering rational curves”. As an application, a divisor D would be effective iff D.C ≥ 0 for all rational curves C with T_{X}|C nef. If T_{X} itself is nef, this follows from [CP 91] . Note that for any X the equality N_{rat} = N_{nef} implies in particular that −K_{X} ∈ K_{eff}. So (2) cannot hold for every rationally connected manifold. But it could hold for every rationally connected manifold with −K_{X} ∈ K_{eff}.

(3) This is motivated by stability, see (2.10), (2.11).

3.4 Proposition Let X be a del Pezzo surface. Then N_{rat} = N_{nef}.

Proof. Since dim X = 2, we have N_{amp} = K_{amp}. Now K_{amp} is generated by φ^{∗}H′, where φ: X → X′ is the contraction of a codimension 1-face R in \overline{NE}(X) and H′ is ample on X′. Moreover, X being del Pezzo, X′ = ℙ_{1} or X′ = ℙ_{2} . If X′ = ℙ_{1}, then R is given by φ^{−1} (point p), if X′ = ℙ_{2}, then R is given by φ^{−1} (general line ℓ). Both φ^{−1}(p) and φ^{−1}(ℓ) are nef curves, finishing the proof.

Concerning 3.2.(3) we prove:

3.5 Theorem Let X be a Fano manifold. Then N_{sc} ⊂ Ñ_{rat}.

Proof. By the cone theorem (as in (3.4)) it is sufficient to show the following

Claim (*) Let φ: X → Y be the contraction ofa codimension 1 face R and H ample on Y. Then φ^{∗}(H^{n−1}) ∈ Ñ_{rat}.

a) If dim Y < dim X, we proceed as follows. First note if dim Y ≤ n − 2, then H^{n−1} = 0, so the claim is obvious. So let dim Y = n − 1. Then H^{n−1} can be thought of as a general point of Y (up to a positive rational multiple of H^{n−1}), so that φ^{∗}(H^{n−1}) ≡ general fiber of φ, which is a smooth moving ℙ_{1}, hence in N_{rat}, proving (*) in case dim Y < dim X .

b) φ is birational. Note that ρ(Y) = 1 and that Y is rationally connected (but may be singular). =46irst let us prove the following assertion

(**) Let C ⊂ Y be an irreducible curve not contained in the degeneracy set in Y of φ: X → Y. Then there is a - up to numerical equivalence - unique effective curve C̃ ⊂ X such that

(1) φ_{∗}(C̃) = C

(2) C̃.E_{i} = 0 for all exceptional components of φ.

In fact, let Ĉ ⊂ X be the strict transform of C in X. We make the following ansatz

C̃ = Ĉ + ∑ λ_{i}C_{i},

where C_{i} are the extremal rational curves contracted by φ. We choose the λ_{i} in such a way that C̃.E_{j} = 0 for all divisors E_{j} contracted by φ. Then the λ_{i} are uniquely determined. What remains to be shown is λ_{i} ≥ 0. Take a basis of the φ-ample divisors D, with φ^{∗}(C).D = 0, say D_{1}, ⋯, D_{ρ−1}. Every D_{j} is of the form D_{j} = − ∑ κ_{μ}E_{μ} with κ_{μ} ≥ 0 by Lemma 3.7. Hence we have

D_{j}. ∑ λ_{i}C_{i} = D_{j}.C̃ − D_{j}.Ĉ − D_{j}.Ĉ ∑ κ_{k}E_{k}.Ĉ ≥ 0.

Therefore ∑ λ_{i}C_{i} ∈ \overline{NE}(X|Y), i.e. all λ_{i} ≥ 0. So (**) holds.

Now the verification of (*) in the birational case is easy; choose a covering family (C_{t}) of rational curves in Y. Then, applying (**), we have for t general an effective curve C̃_{t} ⊂ X with φ_{∗}(C̃_{t}) = C_{t} and C̃_{t}.E_{i} = 0 for all φ−exceptional divisors E_{i} ⊂ X. We can choose C̃_{t} rational because φ is the contraction of an extremal face. Now let H be the ample generator of P ic(Y) ≃ ℤ. Take μ ∈ ℕ such that μH is very ample so that we find a smooth curve Γ ≡ (μH)^{n−1}. Then Γ ≡ mC_{t} for some m ∈ ℚ_{+} since ρ(Y) = 1. It follows that Γ ≡ mC̃_{t}, where Γ̃ is again constructed by (**). On the other hand, Γ̃ ≡ φ^{∗}(H)^{n−1}. Hence the (C̃_{t}) yield (taking closure) a covering family of rational curves, numerically of the form φ^{∗}(H^{n−1}), so that φ^{∗}(H^{n−1}) ∈ Ñ_{rat}(X). There are two problems left: first, our C̃ might not be connected and second, the λ_{i} might only be rational and anyway no multiplicities are allowed for the curves in N_{sc}. To get around with these difficulties we first choose C to be a general member of a large family of rational curves in Y, say a free family. Then choose a positive integer k such that kλ_{i} are integers for all i. Now substitute kC by general deformations C_{j} of C. The λ_{j} will then be the same for C and C_{j}. Since all fibers of extremal contractions are uniruled by [Ka92], we can substitute the λ_{i}C̃_{i} by rational curves C_{i,1}, …, C_{i,kλ_{i}} homologous to multiples of C̃_{i} such that the new C̃ = ⋃ Ĉ_{j} ∪ ⋃ C_{i,j} is connected.

3.6 Remark. In order to prove N_{sc} ⊂ N_{rat} in (3.5) we would need the following. Let Y be a singular Fano variety (terminal singularities), ρ(Y) = 1, A ⊂ Y an algebraic subvariety of codimension ≥ 2, then there exists a covering family (C_{t}) of rational curves, C_{t} irreducible for t general, such that A ∩ C_{t} = ∅ for t general.

3.7 Lemma Let X be a projective manifold, f: X → Y be a birational morphism to a normal projective variety Y. Let E_{1}, …, E_{k} be the codimension 1 components of the exceptional set of f. Let D be f-ample. Then

D = − ∑ κ_{μ} E_{μ} + f^{∗}(L)

with κ_{μ} ≥ 0 for some line bundle L on Y.

Proof. Of course an equation as above always exists; we only need to show κ_{μ} ≥ 0. Taking hyperplane sections we reduce to the case dim X = 2. Let B = {μ|κ_{μ} ≤ 0} and A be its complement; assume B ≠ ∅. Since D is f-ample, we have

∑_{μ} κ_{μ} E_{μ} · ∑_{ν∈B}(−κ_{ν})E_{ν} < 0.

Hence

∑_{{μ∈A \atop ν∈B}} κ_{μ}(−κ_{ν})E_{μ} · E_{ν} < ∑_{μ,ν∈B} κ_{μ}κ_{ν}E_{μ} · E_{ν}.

Now the right hand side of the last equation is negative, the intersection matrix (E_{μ} · E_{ν}) being negative definite, while the left hand side is obviously non-negative, contradiction.

We now relate the canonical bundle to various cones. The question we have mind is the following: how negative is the canonical bundle K_{X} of a manifold X with κ(X) = −∞? In dimension > 3 it is even unknown whether K_{X} is not nef, so we will mainly restrict ourselves to dim X = 3. We introduce the following notation:

\overline{NE}_{−}(X) = \{C ∈ \overline{NE}(X)|K_{X} · C < 0\}.

Then vaguely speaking, we ask for the size of \overline{NE}_{−}(X). E.g.: 3.8 Problem Does κ(X) = −∞ imply that \overline{NE}_{−}(X) ∩ N_{clt} ≠ \{0\}? In other words: are there H_{1}, ⋯, H_{n−1} ample such that K_{X} · H_{1}. · · · .H_{n−1} < 0?

A weaker problem would ask for H_{i} only big. Note that this problem is related to the stability considerations in § 2 . We shall now consider the case dimX = 3.

3.9 Proposition Let X be a normal (ℚ-factorial) threefold with at most terminal singularities, φ: X → Y the contraction of an extremal ray with dim Y ≤ 2. Then there exists an ample divisor H on X with K_{X}.H^{2} < 0.

Proof. If ρ(X) = 1, the claim is obvious, so let ρ(X) ≥ 2, i.e. dim Y > 0. Fix H_{o} ∈ P ic(X) ample. Choose any ample divisor L on Y and let

H = H_{o} + φ^{∗}(mL),

m >> 0 to be specified in a moment. H is ample and

K_{X} · H^{2} = K_{X} · H_{o}^{2} + 2K_{X} · H_{o} · φ^{∗}(mL) + K_{X} · φ^{∗}(mL)^{2}.

If dim Y = 2, then K_{X} · φ^{∗}(L)^{2} < 0 since −K_{X} is φ- ample, hence K_{X} · H^{2} < 0 for m >> 0. If dim Y = 1, then L^{2} = 0, but now K_{X} · H_{o} · φ^{∗}(mL) < 0, hence again the claim for m >> 0.

Note that we only used that −K_{X} is ample on the general fiber instead of φ being an extremal contraction.

3.10 Proposition Let X, Y be normal projective (ℚ-factorial) terminal 3-folds, φ: X → Y a divisorial extremal contraction. Assume there exists H′ ample on Y with K_{Y} · H′^{2} < 0. Then there is H ample on X with K_{X} · H^{2} < 0.

Proof. Choose H_{o} ample on X and let H = H_{o} + φ^{∗}(mH). Then H is ample and K_{X} · H_{o}^{2} + 2K_{X} · H_{o} · φ^{∗}(mH) + K_{X} · φ^{∗}(mH)^{2}. Since K_{X} · φ^{∗}(H)^{2} = K_{Y} · H^{2} < 0, we have K_{X} · H^{2} < 0 for m >> 0.

3.11 Proposition Let X be as in (3.10). Let

\begin{aligned}& X & & − − − → & X^{+} \\ φ & & ↘ & ↙ & & φ^{+} \\ & & & Y \end{aligned}  be a flip

so −K_{X} is φ-ample and K_{X}+ is φ^{+}-ample. Let H^{+} be effective on X^{+} whith H^{+2} ∈ NE(X). Let H be its strict transform on X. If K_{X} · H^{2} < 0, then K_{X^{+}} · H^{+2} < 0. Proof. Write

H^{+2} = ∑ λ_{i} C_{i}^{+} + ∑ μ_{j} · A_{j}^{+}, λ_{j} > 0, μ_{j} ≥ 0,

where the A_{j}^{+} are exactly the exceptional curves for φ^{+} .

Analogously: H^{2} = ∑ τ_{i} C_{i} + ∑ ν_{j} A_{j}. (it is clear that H^{2} ∈ NE(X) ! ). We arrange things, such that C_{i} is the strict tansform of C_{i}^{+}. Since K_{X^{+}}.A_{j}^{+} > 0 and K_{X}.A_{j} < 0 it suffices to show (*) K_{X}.C_{i} ≥ K_{X^{+}}.C_{i}^{+}. Let C = C_{i}, C^{+} = C_{i}^{+} for simplicity. Let

X̂
π ↙  ↘ σ
X  − − − →  X^{+}

be a common desingularisation. By [KMM 87, 5-1-11] we have

K_{X} = σ^{∗}(K_{X^{+}}) + ∑ a_{i}^{+} E_{i}

K_{X} = π^{∗}(K_{X}) + ∑ a_{i} E_{i}

and a_{i} ≤ a_{i}^{+} . Therefore (*) follows by intersecting with Ĉ, the strict transform of C_{i} resp. C^{+} in X̂.

3.12 Remarks Of course we would like to show that given a 3-fold X with κ(X) = −∞, then there exists H ample (or at least big) such that K_{X}.H^{2} < 0. The strategy would be to apply the Minimal Model Program (MMP) to finally abtain a contraction X′ → Y with dimY ≤ 2, then apply (3.9) and go back successively. However the flips cause trouble because the inequalities work in the wrong direction. In fact, in the situation of (3.11) we would need H^{+2} ∈ NE(X) and (H^{+})^{2}∩ (the exceptional locus of the flip) = ∅. If say H^{+} is big and globally generated, this could be achieved. However H would in general no longer be globally generated, so we cannot proceed. In special situations however, the problem has a positive solution:

a) there exists φ: X → Y, dim Y ≤ 2, −K_{X} is φ-ample or at least −K_{X} is ample on the general fiber,

b) the Albanese dimension of X is 2 (as a consequence of (a)),

c) | − mK_{X}| ≠ ∅ for some m ∈ ℕ (obvious)

d) there is a divisorial modification φ: X → X′ and X′ is of type a) (e.g. no flips occur in MMP).

It is clear from Prop. 3.9 that in general we have at least:

3.13 Proposition Let X be a normal projective 3-fold, ℚ-factorial, terminal with κ(X) = −∞. Then there exists a covering family (C_{t}) of curves, C_{t} irreducible with \underline{ample} normal bundle for t general, such that K_{X}.C_{t} < 0.

Uniruledness implies only the existence of a family (C_{t}) (however rational) whose normal bundles are nef (t general).

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Frédéric Campana, Université de Nancy I, Département de Mathématiques, B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France

E-mail address: frederic.campana@iecn.u-nancy.fr

Thomas Peternell, Mathematisches Institut, Universität Bayreuth, D - 95440 Bayreuth, Germany

E-mail address: peternel@btm8x1.mat.uni-bayreuth.de