aa r X i v : . [ m a t h . AG ] J a n ON FANO FOLIATIONS
CAROLINA
ARAUJO
AND ST´EPHANE
DRUELAbstract.
In this paper we address Fano foliations on complex projective varieties. These are foliations F whose anti-canonical class − K F is ample. We focus our attention on a special class of Fano foliations,namely del Pezzo foliations on complex projective manifolds. We show that these foliations are algebraicallyintegrable, with one exceptional case when the ambient space is P n . We also provide a classification of delPezzo foliations with mild singularities. Contents
1. Introduction 12. Foliations and Pfaff fields 33. Algebraically integrable foliations 54. Examples 85. The relative anticanonical bundle of a fibration and applications 106. Foliations and rational curves 137. Algebraic integrability of del Pezzo foliations 188. On del Pezzo foliations with mild singularities 279. Del Pezzo foliations on projective space bundles 29References 361.
Introduction
In the last few decades, much progress has been made in the classification of complex projective varieties.The general viewpoint is that complex projective manifolds X should be classified according to the behavior oftheir canonical class K X . As a result of the minimal model program, we know that every complex projectivemanifold can be build up from 3 classes of (possibly singular) projective varieties, namely, varieties X forwhich K X is Q -Cartier, and satisfies K X < K X ≡ K X >
0. Projective manifolds X whose anti-canonical class − K X is ample are called Fano manifolds , and are quite special. For instance, Fano manifoldsare known to be rationally connected (see [Cam92] and [KMM92]).One defines the index ι X of a Fano manifold X to be the largest integer dividing − K X in Pic( X ). Aclassical result of Kobayachi-Ochiai’s asserts that ι X ≤ dim X + 1, and equality holds if and only if X ≃ P n .Moreover, ι X = dim X if and only if X is a quadric hypersurface ([KO73]). Fano manifolds whose indexsatisfies ι X = dim X − del Pezzo manifolds. The philosophy behind these results is that Fano manifolds with high index are the simplestprojective manifolds.Similar ideas can be applied in the context of foliations on complex projective manifolds. If F ( T X is a foliation on a complex projective manifold X , we define its canonical class to be K F = − c ( F ). Inanalogy with the case of projective manifolds, one expects the numerical properties of K F to reflect geometricaspects of F . In fact, ideas from the minimal model program have been successfully applied to the theory offoliations (see for instance [Bru04] and [McQ08]), and led to a birational classification in the case of rank onefoliations on surfaces ([Bru04]). More recently, Loray, Pereira and Touzet have investigated the structure ofcodimension 1 foliations with K F ≡ Mathematics Subject Classification.
ARAUJO
AND ST´EPHANE
DRUEL
In this paper we propose to investigate
Fano foliations on complex projective manifolds. These arefoliations F ( T X whose anti-canonical class − K F is ample (see Section 2 for details). As in the case ofFano manifolds, we expect Fano foliations to present very special behavior. This is the case for instance ifthe rank of F is 1, i.e., F is an ample invertible subsheaf of T X . By Wahl’s Theorem [Wah83], this can onlyhappen if ( X, F ) ≃ (cid:0) P n , O (1) (cid:1) .Guided by the theory of Fano manifolds, we define the index ι F of a foliation F on a complex projectivemanifold X to be the largest integer dividing − K F in Pic( X ). The expected philosophy is that Fano foliationswith high index are the simplest ones. For instance, when X = P n , the index of a foliation F ( T P n of rank r satisfies ι F ≤ r . By [DC05, Th´eor`eme 3.8], equality holds if and only if F is induced by a linear projection P n P n − r , i.e., it comes from the family r -planes in P n containing a fixed ( r − F ( T P n satisfying ι F = r − F is induced by a dominant rational map P n P (1 n − r , n − r linear formsand one quadratic form, or(2) F is the linear pullback of a foliation on P n − r +1 induced by a global holomorphic vector field.In analogy with Kobayachi-Ochiai’s theorem, we have the following result. Theorem ([ADK08, Theorem 1.1]) . Let F ( T X be a Fano foliation of rank r on a complex projectivemanifold X . Then ι F ≤ r , and equality holds only if X ∼ = P n . We say that a Fano foliation F ( T X of rank r on a complex projective manifold X is a del Pezzo foliation if ι F = r −
1. Ultimately we would like to classify del Pezzo foliations. In addition to the above mentionedfoliations on P n , we know examples of del Pezzo foliations of any rank on quadric hypersurfaces, del Pezzofoliations of rank 2 on certain Grassmannians, and del Pezzo foliations of rank 2 and 3 on P m -bundles over P l . These examples are described in Sections 4 and 9.We note that the generic del Pezzo foliation on P n of type (2) above does not have algebraic leaves. Ourfirst main result says that this is the only del Pezzo foliation that is not algebraically integrable. We alsodescribe the geometry of the general leaf in all other cases. Theorem 1.1.
Let F ( T X be a del Pezzo foliation on a complex projective manifold X P n . Then F isalgebraically integrable, and its general leaves are rationally connected. One of the key ingredients in the proof of Theorem 1.1 is the following criterion by Bogomolov andMcQuillan for a foliation to be algebraically integrable with rationally connected general leaf.
Theorem 1.2 ([BM01, Theorem 0.1], [KSCT07, Theorem 1]) . Let X be a normal complex projective variety,and F a foliation on X . Let C ⊂ X be a complete curve disjoint from the singular loci of X and F . Supposethat the restriction F | C is an ample vector bundle on C . Then the leaf of F through any point of C is analgebraic variety, and the leaf of F through a general point of C is moreover rationally connected. Given a del Pezzo foliation F ( T X on a complex projective manifold X , it is not clear a priori how tofind a curve C ⊂ X satisfying the hypothesis of Theorem 1.2. Instead, in order to prove Theorem 1.1, wewill apply Theorem 1.2 in several steps. First we construct suitable subfoliations H ⊂ F for which we canprove algebraic integrability and rationally connectedness of general leaves. Next we consider the the closure W in Chow( X ) of the subvariety parametrizing general leaves of H , as explained in Section 3. We thenapply Theorem 1.2 to the foliation on W induced by F .In the course of our study of Fano foliations, we were led to deal with singularities of foliations. Weintroduce new notions of singularities for foliations, inspired by the theory of singularities of pairs, developedin the context of the minimal model program. In order to explain this, let F ( T X be an algebraicallyintegrable foliation on a complex projective manifold X , and denote by i : ˜ F → X the normalization of theclosure of a general leaf of F . Then there is an effective Weil divisor ˜∆ on ˜ F such that − K ˜ F = i ∗ ( − K F )+ ˜∆.We call the pair ( ˜ F , ˜∆) a general log leaf of F . We say that F has log canonical singularities along a generalleaf if ( ˜ F , ˜∆) is log canonical (see Section 3 for details). Algebraically integrable Fano foliations having logcanonical singularities along a general leaf have a very special property: there is a common point containedin the closure of a general leaf (see Proposition 5.3). This property is useful to derive classification resultsunder some restrictions on the singularities of F , such as the following (see also Theorem 8.1). N FANO FOLIATIONS 3
Theorem 1.3.
Let F ( T X be a del Pezzo foliation of rank r on a complex projective manifold X P n .Suppose that F has log canonical singularities and is locally free along a general leaf. Then either ρ ( X ) = 1 ,or r ≤ , X is a P m -bundle over P l and F T X/ P l . Notice that a del Pezzo foliation F on X P n is algebraically integrable by Theorem 1.1. Hence itmakes sense to ask that F has log canonical singularities along a general leaf in Theorem 1.3 above. Weremark that del Pezzo foliations of codimension 1 on Fano manifolds with Picard number 1 were classifiedin [LPT11a, Proposition 3.7].Theorem 1.3 raises the problem of classifying del Pezzo foliations on P m -bundles π : X → P l . If m = 1,then X ≃ P × P l , and F is the pullback via π of a foliation O (1) ⊕ i ⊂ T P l for some i ∈ { , } (see 9.1). For m ≥
2, we have the following result (see Theorems 9.2 and 9.6 for more details).
Theorem 1.4.
Let F ( T X be a del Pezzo foliation on a P m -bundle π : X → P l , with m ≥ . Supposethat F T X/ P l . Then there is an exact sequence of vector bundles → K → E → Q → on P l suchthat X ≃ P P l ( E ) , and F is the pullback via the relative linear projection X Z = P P l ( K ) of a foliation q ∗ det( Q ) ⊂ T Z . Here q : Z → P l denotes the natural projection. Moreover, one of the following holds. (1) l = 1 , Q ≃ O (1) , K is an ample vector bundle such that K O P ( a ) ⊕ m for any integer a , and E ≃ Q ⊕ K ( r F = 2 ). (2) l = 1 , Q ≃ O (2) , K ≃ O P ( a ) ⊕ m for some integer a > , and E ≃ Q ⊕ K ( r F = 2 ). (3) l = 1 , Q ≃ O (1) ⊕ O (1) , K ≃ O P ( a ) ⊕ m − for some integer a > , and E ≃ Q ⊕ K ( r F = 3 ). (4) l > , Q ≃ O (1) , and K is V -equivariant for some V ∈ H (cid:0) P l , T P l ⊗ O ( − (cid:1) \ { } ( r F = 2 ).Conversely, given K , E and Q satisfying any of the conditions above, there exists a del Pezzo foliation ofthat type. The paper is organized as follows. In Section 2 we introduce the basic notions concerning foliations andPfaff fields on varieties. In Section 3 we focus on algebraically integrable foliations, and develop notions ofsingularities for these foliations. In Section 4 we describe examples of Fano foliations on Fano manifoldswith Picard number 1. In Section 5 we study the relative anti-canonical bundle of a fibration, and provideapplications to the theory of Fano foliations. In Section 6 we recall some results from the theory of rationalcurves on varieties, and explain how they apply to foliations. In Section 7 we prove Theorem 1.1. InSection 8 we address the problem of classifying Fano foliations with mild singularities. In particular weprove Theorem 1.3. In Section 9 we address del Pezzo foliations on projective space bundles.We plan to address Fano foliations on Fano manifolds with Picard number 1 and related questions inforthcoming works.
Notation and conventions.
We always work over the field C of complex numbers. Varieties are alwaysassumed to be irreducible. We denote by Sing( X ) the singular locus of a variety X . Given a sheaf F of O X -modules on a variety X , we denote by F ∗ the sheaf H om O X ( F , O X ). If r is the generic rank of F ,then we denote by det( F ) the sheaf ( ∧ r F ) ∗∗ . If G is another sheaf of O X -modules on X , then we denoteby F [ ⊗ ] G the sheaf ( F ⊗ G ) ∗∗ . If E is a locally free sheaf of O X -modules on a variety X , we denote by P X ( E ) the Grothendieck projectivization Proj X (Sym( E )). If X is a normal variety and X → Y is anymorphism, we denote by T X/Y the sheaf (Ω X/Y ) ∗ . In particular, T X = (Ω X ) ∗ . If X is a smooth variety and D is a reduced divisor on X with simple normal crossings support, we denote by Ω X (log D ) the sheaf ofdifferential 1-forms with logarithmic poles along D , and by T X ( − log D ) its dual sheal Ω X (log D ) ∗ . Noticethat det(Ω X (log D )) ≃ O X ( K X + D ). Acknowledgements.
Much of this work was developed during the authors’ visits to IMPA and InstitutFourier. We would like to thank both institutions for their support and hospitality. We also thank ourcolleagues Julie D´eserti and Jorge Vit´orio Pereira for very helpful discussions.2.
Foliations and Pfaff fields
Definition 2.1.
Let X be normal variety. A foliation on X is a nonzero coherent subsheaf F ( T X satisfying(1) F is closed under the Lie bracket, and(2) F is saturated in T X (i.e., T X / F is torsion free). CAROLINA
ARAUJO
AND ST´EPHANE
DRUEL
The rank r F of F is the generic rank of F .The canonical class K F of F is any Weil divisor on X such that O X ( − K F ) ≃ det( F ).A foliated variety is a pair ( X, F ) consisting of a normal variety X together with a foliation F on X . Definition 2.2.
A foliation F on a normal variety is said to be if its canonical class K F is aCartier divisor. Remark 2.3.
Condition (2) above implies that F is reflexive. Indeed, T X is reflexive by [Har80, Corollary1.2]. Thus, the inclusion F ⊂ T X factors through F ⊂ F ∗∗ . The induced map F ∗∗ → T X / F is genericallyzero. Hence it is identically zero since T X / F is torsion free by (2). Thus F = F ∗∗ . Definition 2.4.
Let X be a variety, and r a positive integer. A Pfaff field of rank r on X is a nonzero map η : Ω rX → L , where L is an invertible sheaf on X (see [EK03]). The singular locus S of η is the closedsubscheme of X whose ideal sheaf I S is the image of the induced map Ω rX ⊗ L ∗ → O X .A closed subscheme Y of X is said to be invariant under η if(1) no irreducible component of Y is contained in the singular locus of η , and(2) the restriction η | Y : Ω rX | Y → L | Y factors through the natural map Ω rX | Y → Ω rY , in other words,there is a commutative diagram Ω rX | Y η | Y / / (cid:15) (cid:15) L | Y , Ω rY ; ; ✈✈✈✈✈✈✈✈✈ where the vertical map is the natural one.Notice that a 1-Gorenstein foliation F of rank r on normal variety X naturaly gives rise to a Pfaff fieldof rank r on X : η : Ω rX = ∧ r (Ω X ) → ∧ r ( T ∗ X ) → ∧ r ( F ∗ ) → det( F ∗ ) ≃ det( F ) ∗ = O X ( K F ) . Definition 2.5.
Let F be a 1-Gorenstein foliation on a normal variety X . The singular locus of F isdefined to be the singular locus S of the associated Pfaff field. We say that F or ( X, F ) is regular at a point x ∈ X if x S . We say that F or ( X, F ) is regular if S = ∅ .Using Frobenius’ theorem, one can prove the following. Lemma 2.6 ([BM01, Lemma 1.3.2]) . Let ( X, F ) be a 1-Gorenstein foliated variety. Suppose that F regularand locally free at a point x ∈ X . Then there exists an analytic open neighborhood U of x , a complex analyticspace W , and a smooth morphism U → W of relative dimension r F such that F U = T U/W . Lemma 2.7.
Let X be a smooth variety, and F a foliation of rank r on X with singular locus S . Let S bethe set of points x ∈ X at which F is not locally free, and S the set of points x ∈ X such that F is locallyfree at x and F ⊗ k ( x ) → T X ⊗ k ( x ) is not injective. (1) Then S ⊂ S ∪ S as sets, and S \ S = S . (2) Let Y ⊂ X be an irreducible subvariety of dimension r F such that Y S ∪ S . Then Y \ S ∪ S isa leaf of F | X \ S ∪ S if and only if Y is invariant under the associated Pfaff field η : Ω rX → O X ( K F ) .Proof. Let x ∈ X be a point at which F is locally free. Then there is an open neighborhood of x wheredet( F ∗ ) is invertible. Thus x ∈ S if and only if x ∈ S , proving (1).Let x ∈ Y \ S ∪ S be a smooth point of Y and let ~v , . . . , ~v r be local vector fields that generate F onan affine neighbourhood U of x . Observe that η | U : Ω rX | U → O X ( K F ) | U is given by H ( U, Ω rX | U ) −→ H ( U, O X ( K F ) | U ) α α ( ~v , . . . , ~v r ) ω where ω ∈ H ( U, O X ( K F ) | U ) is such that ω ( ~v , . . . , ~v r ) = 1. It follows that Y is invariant under η if andonly if, for any local function f on U vanishing along Y ∩ U , and any local ( r − β on U ,we have ( df ∧ β )( ~v , . . . , ~v r ) = 0. This happens if and only if, for any i ∈ { , . . . , r } and any local function f on U vanishing along Y ∩ U , we have df ( ~v i ) = 0 . This is in turn equivalent to requiring that ~v i ( x ) ∈ T Y,x for any i ∈ { , . . . , r } , which is saying precisely that Y \ S ∪ S is a leaf of F | X \ S ∪ S . This proves (2). (cid:3) N FANO FOLIATIONS 5
Next we define Fano foliations and Fano Pfaff fields.
Definition 2.8.
Let X be a normal projective variety.Let F be a 1-Gorenstein foliation on X . We say that F is a Fano foliation if − K F is ample. In thiscase, the index ι F of F is the largest positive integer such that − K F ∼ ι F H for a Cartier divisor H on X .Let L be a line bundle on X , r a positive integer, and η : Ω rX → L a Pfaff field. We say that η is a Fano Pfaff field if L − is ample. In this case, the index ι η of η is the largest positive integer such that L − ∼ A ⊗ ι η for a line bundle A on X . Remark 2.9.
Let X be a smooth complex projective variety. If X admits a Fano foliation or a Fano Pfafffield, then X is uniruled by [Miy87, Corollary 8.6].In analogy with Kobayachi-Ochiai’s theorem, we have the following. Theorem 2.10 ([ADK08, Theorem 1.1]) . Let X be a smooth complex projective variety, L a line bundleon X , r a positive integer, and η : Ω rX → L a Fano Pfaff field. Then: (1) ι η r + 1 ; (2) ι η = r + 1 if and only if r = dim( X ) and ( X, L ) ≃ ( P r , O P r ( − ; (3) ι η = r if and only if either ( X, L ) ≃ ( P n , O P n ( − for some n ≥ r , or r = dim( X ) and ( X, L ) ≃ ( Q r , O Q r ( − , where Q r denotes a smooth quadric hypersurface in P r +1 and O Q r ( − denotes therestriction of O P r +1 ( − to Q r . Definition 2.11.
Let X be a smooth projective variety, and F a Fano foliation on X of rank r F and index ι F . We say that F is a del Pezzo foliation if r F > ι F = r F − Algebraically integrable foliations
Definition 3.1.
Let X be normal variety. A foliation F on X is said to be algebraically integrable if theleaf of F through a general point of X is an algebraic variety. In this situation, by abuse of notation weoften use the word “leaf” to mean the closure in X of a leaf of F . Lemma 3.2.
Let X be normal projective variety, and F an algebraically integrable foliation on X . Thereis a unique irreducible closed subvariety W of Chow( X ) whose general point parametrizes the closure of ageneral leaf of F (viewed as a reduced and irreducible cycle in X ). In other words, if U ⊂ W × X is theuniversal cycle, with universal morphisms π : U → W and e : U → X , then e is birational, and, for a generalpoint w ∈ W , e (cid:0) π − ( w ) (cid:1) ⊂ X is the closure of a leaf of F . Notation 3.3.
We say that the subvariety W provided by Lemma 3.2 is the closure in Chow( X ) of thesubvariety parametrizing general leaves of F . Proof of Lemma 3.2.
First of all, recall that Chow( X ) has countably many irreducible components. On theother hand, since we are working over C , F has uncountably many leaves. Therefore, there is a closedsubvariety W of Chow( X ) such that(1) the universal cycle over W dominates X , and(2) the subset of points in W parametrizing leaves of F (viewed as reduced and irreducible cycles in X )is Zariski dense in W .Let U ⊂ W × X be the universal cycle over W , denote by p : W × X → W and q : W × X → X the naturalprojections, and by π = p | U : U → W and e = q | U : U → X their restrictions to U . We need to show that,for a general point w ∈ W , e (cid:0) π − ( w ) (cid:1) ⊂ X is the closure of a leaf of F .To simplify notation, we suppose that X is smooth. In the general case, in what follows one shouldreplace X with its smooth locus X , W with a dense open subset W ⊂ q ( p − ( X )) and U with U = q − ( X ) ∩ p − ( W ) ∩ U .Let η X : Ω rX → O X ( K F ) be the Pfaff field associated to F . It induces a Pfaff field of rank r on W × X : η W × X : Ω rW × X = ∧ r ( p ∗ Ω W ⊕ q ∗ Ω X ) → ∧ r ( q ∗ Ω X ) ≃ q ∗ Ω rX → q ∗ O X ( K F ) . We claim that U is invariant under η W × X . Indeed, let K be the kernel of the natural morphism Ω rW × X | U ։ Ω rU . The composite map K → Ω rW × X | U → e ∗ O X ( K F ) vanishes on a Zariski dense subset of U by Lemma2.7. Since e ∗ O X ( K F ) is torsion-free, it vanishes identically, and thus the restriction η W × X | U : Ω rW × X | U → e ∗ O X ( K F ) factors through Ω rW × X | U ։ Ω rU . Similarly, the morphism η U : Ω rU → e ∗ O X ( K F ) factors CAROLINA
ARAUJO
AND ST´EPHANE
DRUEL through the natural morphism Ω rU ։ Ω rU/W . Lemma 2.7 then implies that, for a general point w ∈ W , e (cid:0) π − ( w ) (cid:1) ⊂ X is the closure of a leaf of F . (cid:3) Next we come to the definition of a general log leaf of an algebraically integrable foliation.
Definition 3.4.
Let X be normal projective variety, F a 1-Gorenstein algebraically integrable foliation ofrank r on X , and η F : Ω rX → O X ( K F ) the corresponding Pfaff field. Let F be the closure of a general leafof F , and n : ˜ F → F ⊂ X its normalization. By Lemma 2.7, F is invariant under η F , i.e., the restriction η F | F : Ω rX | F → O X ( K F ) | F factors through the natural map Ω rX | F → Ω rF . By Lemma 3.5 below, theinduced map η : Ω rF → O X ( K F ) | F extends uniquely to a generically surjective map ˜ η : Ω r ˜ F → n ∗ O X ( K F ).Hence there is a canonically defined effective Weil divisor ˜∆ on ˜ F such that O ˜ F ( K ˜ F + ˜∆) ≃ n ∗ O X ( K F ).Namely, ˜∆ is the divisor of zeroes of ˜ η .We call the pair ( ˜ F , ˜∆) a general log leaf of F . Lemma 3.5 ([ADK08, Proposition 4.5]) . Let X be a variety and n : e X → X its normalization. Let L bea line bundle on X , r a positive integer, and η : Ω rX → L a Pfaff field. Then η extends uniquely to a Pfafffield ˜ η : Ω r ˜ X → n ∗ L of rank r . Next we define notions of singularity for 1-Gorenstein algebraically integrable foliations according to thesingularity type of their general log leaf. First we recall some definitions of singularities of pairs, developedin the context of the minimal model program. We refer to [KM98, section 2.3] for details. (Singularities of pairs.) . Let X be a normal projective variety, and ∆ = P a i ∆ i an effective Q -divisoron X , i.e., ∆ is a nonnegative Q -linear combination of distinct prime Weil divisors ∆ i ’s on X . Suppose that K X + ∆ is Q -Cartier, i.e., some nonzero multiple of it is a Cartier divisor on X .Let f : ˜ X → X be a log resolution of the pair ( X, ∆). This means that ˜ X is a smooth projective variety, f is a birational projective morphism whose exceptional locus is the union of prime divisors E i ’s, and thedivisor P E i + f − ∗ ∆ has simple normal crossing support. There are uniquely defined rational numbers a ( E i , X, ∆)’s such that K ˜ X + f − ∗ ∆ = f ∗ ( K X + ∆) + X E i a ( E i , X, ∆) E i . The a ( E i , X, ∆)’s do not depend on the log resolution f , but only on the valuations associated to the E i ’s.We say that ( X, ∆) is log terminal (or klt ) if all a i <
1, and, for some log resolution f : ˜ X → X of ( X, ∆), a ( E i , X, ∆) > − f -exceptional prime divisor E i . We say that ( X, ∆) is log canonical if all a i ≤ f : ˜ X → X of ( X, ∆), a ( E i , X, ∆) ≥ − f -exceptional prime divisor E i . If these conditions hold for some log resolution of ( X, ∆), then they hold for every log resolution of( X, ∆). Definition 3.7.
Let X be normal projective variety, F a 1-Gorenstein algebraically integrable foliation on X , and ( ˜ F , ˜∆) its general log leaf. We say that F has log terminal (respectively log canonical) singularitiesalong a general leaf if ( ˜ F , ˜∆) is log terminal (respectively log canonical). In particular, if F has log terminalsingularities along a general leaf, then ˜∆ = 0. Remark 3.8.
Let X be normal projective variety, and F a 1-Gorenstein algebraically integrable foliationof rank r on X . Let W be the closure in Chow( X ) of the subvariety parametrizing general leaves of F ,and U ⊂ W × X the universal cycle. Denote by e : U → X the natural morphism. We saw in the proof ofLemma 3.2 that F induces a Pfaff field η U : Ω rU → e ∗ O X ( K F ), which factors through the natural morphismΩ rU ։ Ω rU/W .Let ˜ W and ˜ U be the normalizations of W and U , respectively. Denote by ˜ π : ˜ U → ˜ W and ˜ e : ˜ U → X the induced morphisms. By Lemma 3.5, η U : Ω rU → e ∗ O X ( K F ) extends uniquely to a Pfaff field η ˜ U :Ω r ˜ U → ˜ e ∗ O X ( K F ). As before, this morphism factors through the natural morphism Ω r ˜ U ։ Ω r ˜ U/ ˜ W , yieldinga generically surjective map Ω r ˜ U/ ˜ W → ˜ e ∗ O X ( K F ) . Thus there is a canonically defined effective Weil divisor ∆ on ˜ U such that det(Ω U/ ˜ W )[ ⊗ ] O ˜ U (∆) ≃ ˜ e ∗ O X ( K F ). N FANO FOLIATIONS 7
Let w be a general point of ˜ W , set ˜ U w := ˜ π − ( w ) and ∆ w := ∆ | ˜ U w . Then ( ˜ U w , ∆ w ) coincides withthe general log leaf ( ˜ F , ˜∆) defined above. In particular, by [BCHM10, Corollary 1.4.5], F has log terminal(respectively log canonical) singularities along a general leaf if and only if ( ˜ U , ∆) has log terminal (respectivelylog canonical) singularities over the generic point of ˜ W .The same construction can be carried out by replacing W with a general closed subvariety of it.Next we compare the notions of singularities for algebraically integrable foliations introduced in Defini-tion 3.7 with those introduced earlier in [McQ08]. We recall McQuillan’s definitions, which do not requirealgebraic integrability. ([McQ08, Definition I.1.2]) . Let ( X, F ) be a foliated variety. Given a birational morphism ϕ : ˜ X → X ,there is a unique foliation ˜ F on ˜ X that agrees with ϕ ∗ F on the open subset of ˜ X where ϕ is an isomorphism.We say that ϕ : ( ˜ X, ˜ F ) → ( X, F ) is a birational morphism of foliated varieties.From now on assume moreover that K F is Q -Cartier and ϕ is projective. Then there are uniquely definedrational numbers a ( E, X, F )’s such that K ˜ F = ϕ ∗ K F + X E a ( E, X, F ) E, where E runs through all exceptional prime divisors for ϕ . The a ( E, X, ∆)’s do not depend on the birationalmorphism ϕ , but only on the valuations associated to the E ’s.For an exceptional prime divisor E over X , define ǫ ( E ) := (cid:26) E is invariant by the foliation,1 if E is not invariant by the foliation.The foliated variety ( X, F ) is said to be terminalcanonicallog terminallog canonical in the sense of McQuillan if, for all E exceptional over X , a ( E, X, F ) > , > ,> − ǫ ( E ) , ≥ − ǫ ( E ) . Lemma 3.10.
Let ( X, F ) be a -Gorenstein foliated variety. If F is regular, then ( X, F ) is canonical inthe sense of McQuillan.Proof. Let ϕ : ( ˜ X, ˜ F ) → ( X, F ) be a birational projective morphism of foliated varieties with ˜ X smooth.Let η F : Ω r F X → O X ( K F ) and η ˜ F : Ω r F ˜ X → O ˜ X ( K ˜ F ) be the associated Pfaff fields. Since F is regular, ϕ ∗ η F : ϕ ∗ Ω r F X → ϕ ∗ O X ( K F ) is a surjective morphism.We claim that the the composite map ϕ ∗ Ω r F X → Ω r F ˜ X → O ˜ X ( K ˜ F ) factors through ϕ ∗ η F : ϕ ∗ Ω r F X → ϕ ∗ O X ( K F ). Indeed, denote by K be the kernel of ϕ ∗ η F : ϕ ∗ Ω r F X → ϕ ∗ O X ( K F ). The composite map K → ϕ ∗ Ω r F X → Ω r F ˜ X → O ˜ X ( K ˜ F ) vanishes over a dense subset of ˜ X . Since O ˜ X ( K ˜ F ) is torsion-free, itvanishes identically on ˜ X . This proves the claim. So we obtain a nonzero map ϕ ∗ O X ( K F ) → O ˜ X ( K ˜ F ).Thus there is an effective divisor E on ˜ X such that K ˜ F = ϕ ∗ K F + E . (cid:3) Proposition 3.11.
Let X be a normal projective variety, and F a -Gorenstein algebraically integrablefoliation on X . Let W be the closure in Chow( X ) of the subvariety parametrizing general leaves of F .If ( X, F ) is log terminal (respectively log canonical) in the sense of McQuillan, then F has log terminal(respectively log canonical) singularities along a general leaf.Proof. We follow the notation in Remark 3.8.Let w ∈ ˜ W be a general point and let ( ˜ U w , ∆ w ) be the corresponding log leaf. We denote by ˜ e w : ˜ U w → X the natural morphism. Recall that(3.1) K ˜ U w + ∆ w = ˜ e ∗ w ( K F ) . Suppose that ( X, F ) is log terminal (respectively log canonical) in the sense of McQuillan. We have to showthat the pair ( ˜ U w , ∆ w ) is log terminal (respectively log canonical).Let d : Y → ˜ U be a log resolution of singularities, and consider the commutative diagram CAROLINA
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AND ST´EPHANE
DRUEL Y d / / g $ $ ˜ U ˜ e / / ˜ π (cid:15) (cid:15) X. ˜ W Denote by F Y the foliation induced by F on Y , and notice that F Y = T Y/ ˜ W . Write K F Y = g ∗ K F + X a ( E, X, F ) E, where E runs through all exceptional prime divisors for g . Note that the support of the divisor ∆ on ˜ U defined in Remark 3.8 is exceptional over X , so the strict transforms of its components in Y appear amongthe E ’s.Set Y w := d − ( ˜ U w ), d w := d | Y w : Y w → ˜ U w , and E w := E | ˜ U w . Since w ∈ ˜ W is general, K F Y | Y w = K Y w .Thus(3.2) K Y w = d ∗ w ˜ e ∗ w ( K F ) + X a ( E, X, F ) E w . Notice that d w : Y w → ˜ U w is a log resolution of singularities. From (3.2) and (3.1) we deduce that K Y w = d ∗ ( K ˜ U w + ∆ w ) + X a ( E, X, F ) E w . This proves the result. (cid:3)
Remark 3.12.
The notions of singularities of foliations discussed above do not say anything about thesingularities of the ambient space. For instance, let Y be a smooth variety, T any normal variety, andset X := Y × T , with natural projection p : X → Y . Set F := p ∗ T Y ⊂ T X . Then ( X, F ) is a regular1-Gorenstein foliated variety, canonical in the sense of McQuillan, while X may be very singular.4. Examples (Foliations of rank r and index r on P n ) . Let F ( T P n be a Fano foliation of rank r and index ι F = r on P n . These are classically known as degree 0 foliations on P n . By [DC05, Th´eor`eme 3.8], F is defined by alinear projection P n P n − r . The singular locus of F is a linear subspace S of dimension r −
1. The closureof the leaf through a point p S is the r -dimensional linear subspace L of P n containing both p and S . Let p , . . . , p r ∈ S be r linearly independent points in S , and v i ∈ H ( P n , T P n ( − p i . Then the v i ’s define an injective map O P n (1) ⊕ r → T P n whose image is F . Thus the restricted map F | L → T L is induced by the sections v i | L ∈ H ( L, T L ( − ⊂ H ( L, T P n ( − | L ). In particular, the zerolocus of the map det( F ) | L → det( T L ) is the codimension one linear subspace S ∩ L ⊂ L . Thus the log leaf( ˜ F , ˜∆) = (
L, S ∩ L ) is log canonical, and F has log canonical singularities along a general leaf. (Foliations of rank r and index r − P n ) . Let F ( T P n be a Fano foliation of rank r and index ι F = r − P n . By [LPT11a, Theorem 6.2], • either F is defined by a rational dominant map P n P (1 n − r , n − r linear forms andone quadric form, where P (1 n − r ,
2) denotes the weighted projective space of type (1 , . . . , | {z } r times , • or F is the linear pullback of a foliation on P n − r +1 induced by a global holomorphic vector field.Note that a foliation on P n − r +1 induced by a global holomorphic vector field may or may not havealgebraic leaves. Moreover, algebraically integrable foliations of rank r and index r − P n may or maynot have log canonical singularities along a general leaf. (Fano foliations on Grassmannians) . Let m and n be nonnegative integers, and V a complex vector spaceof dimension n + 1. Let G = G ( m + 1 , V ) be the Grassmannian of ( m + 1)-dimensional linear subspaces of V , with tautological exact sequence 0 → K → V ⊗ O G → Q → . Let k be an integer such that 0 k n − m −
1, and W a ( k + 1)-dimensional linear subspace of V . Set F := W ⊗ K ∗ ⊂ V ⊗ K ∗ . N FANO FOLIATIONS 9
The map V ⊗ K ∗ → Q ⊗ K ∗ induced by V ⊗ O G → Q yields a map F → Q ⊗ K ∗ ≃ T G . For a generalpoint [ L ] ∈ G , L ∩ W = { } since k + m n −
1. Thus the map F → T G is injective at [ L ]. Since F is locallyfree, F ֒ → T G is injective. Let P be the linear span of L and W in V . It has dimension m + k + 2 n + 1.Notice that the Grassmannian G ( m + 1 , P ) ⊂ G is tangent to F at a general point of G ( m + 1 , P ).Suppose that k n − m − P ) < dim( V )). Then F is a subbundle of T G incodimension one, and thus saturated in T G by lemma 9.7. In particular F is a Fano foliation on G of rank r = ( m + 1)( k + 1). Its singular locus S is the set of points [ L ] ∈ G such that dim( L ∩ W ) > G ) = Z [ O G (1)] where O G (1) ≃ det( Q ) is the pullback of O P ( ∧ m +1 V ) (1) under the Pl¨uckerembedding. It follows that F has index ι F = k + 1. In particular, ι F = r − m = 1 and k = 0.In this case, G = G (2 , V ) and F is the rank 2 foliation on G whose general leaf is the P of 2-dimensionallinear subspaces of a general 3-plane containing the line W .Finally, observe that S ∩ G ( m + 1 , P ) is irreducible and has codimension one in G ( m + 1 , P ). Moreover,det( T G ( m +1 ,P ) ) ≃ O G ( m +1 ,P ) ( m + k + 2), and det( F ) | G ( m +1 ,P ) ≃ O G ( m +1 ,P ) ( k + 1). It follows that the mapdet( F ) | G ( m +1 ,P ) → det( T G ( m +1 ,P ) ) vanishes at order m + 1 along S ∩ G ( m + 1 , P ). So the general log leafof F is ( ˜ F , ˜∆) = (cid:16) G ( m + 1 , P ) , ( m + 1) · (cid:0) S ∩ G ( m + 1 , P ) (cid:1)(cid:17) . In particular, F has log canonical singularities along a general leaf if and only if m = 0, i.e., G = P n , and F is the foliation described in 4.1 above. In all other cases, the closures of the leaves of F do not have acommon point in G .When m = 1 and k = 0, we obtain a rank 2 del Pezzo foliation on G = G (2 , V ) with general log leaf( ˜ F , ˜∆) ≃ ( P , H ), where H is a line in P .Next we want to discuss Fano foliations on hypersurfaces of projective spaces. In order to do so, it willbe convenient to view foliations as given by differential forms. (Foliations as q -forms) . Let X be a smooth variety of dimension n >
2, and F ( T X a foliation of rank r on X . Set N ∗ F := ( T X / F ) ∗ , and N F := ( N ∗ F ) ∗ . These are called the conormal and normal sheaves ofthe foliation F , respectively. The conormal sheaf N ∗ F is a saturated subsheaf of Ω X of rank q := n − r .The q -th wedge product of the inclusion N ∗ F ⊂ Ω X gives rise to a nonzero twisted differential q -form ω with coefficients in the line bundle L := det( N F ), which is locally decomposable and integrable . To saythat ω ∈ H ( X, Ω qX ⊗ L ) is locally decomposable means that, in a neighborhood of a general point of X , ω decomposes as the wedge product of q local 1-forms ω = ω ∧ · · · ∧ ω q . To say that it is integrable meansthat for this local decomposition one has dω i ∧ ω = 0 for i ∈ { , . . . , q } . Conversely, given a twisted q -form ω ∈ H ( X, Ω qX ⊗ L ) \ { } which is locally decomposable and integrable, we define a foliation of rank r on X as the kernel of the morphism T X → Ω q − X ⊗ L given by the contraction with ω . Lemma 4.5.
Fix n > , and let X ⊂ P n +1 be a smooth hypersurface of degree d > . Let k and q be integerssuch that k q n − and q > . Then h ( X, Ω qX ( k )) = 0 . Before we prove the lemma, we recall Bott’s formulae. (Bott’s Formulae) . Let n, p, q and k be integers, with n positive and p and q nonnegative. Then h q ( P n , Ω p P n ( k )) = (cid:0) k + n − pk (cid:1)(cid:0) k − p (cid:1) for q = 0 , p n and k > p, k = 0 and 0 p = q n, (cid:0) − k + p − k (cid:1)(cid:0) − k − n − p (cid:1) for q = n, p n and k < p − n, r, s and t be integers, with r and s nonnegative. Observe that the natural pairing Ω p P n ⊗ Ω n − p P n → Ω n P n isperfect. It induces an isomorphism ∧ r T P n ( t ) ≃ Ω n − r P n ( t + n + 1). So the formulae above become h s ( P n , ∧ r T P n ( t )) = (cid:0) t + n +1+ rt + n +1 (cid:1)(cid:0) t + nn − r (cid:1) for s = 0 , r n and t + r > , t = − n − n − r = s n, (cid:0) − t − r − t − n − (cid:1)(cid:0) − t − n − r (cid:1) for s = n, r n and t + n + r + 2 , Proof of Lemma 4.5.
By [Fle81, Satz 8.11],
ARAUJO
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DRUEL (1) h ( X, Ω rX ( s )) = 0 for s < r n − h ( X, Ω rX ( s )) = 0 for 0 r n − s r − h ( X, Ω qX ( q )) = 0 for 1 q n −
2. Let q ∈ { , . . . , n − } . By Bott’sformulae,(1) h ( P n +1 , Ω r P n +1 ( r )) = 0 for r > h ( P n +1 , Ω r P n +1 ( s )) = 0 for s < r − P n +1 → Ω q P n +1 ( q − d ) → Ω q P n +1 ( q ) → Ω q P n +1 ( q ) | X → , and the vanishing of H ( P n +1 , Ω q P n +1 ( q )) and H ( P n +1 , Ω q P n +1 ( q − d )) imply the vanishing of H ( X, Ω q P n +1 ( q ) | X ).The cohomology of the exact sequence of sheaves on X → Ω q − X ( q − d ) → Ω q P n +1 ( q ) | X → Ω qX ( q ) → , and the vanishing of H ( X, Ω q P n +1 ( q ) | X ) and H ( X, Ω q − X ( q − d )) yield the result. (cid:3) Proposition 4.7 (Fano foliations on hypersurfaces) . Fix n > , and let X ⊂ P n +1 be a smooth hypersurfaceof degree d > . Let r ∈ { , · · · , n − } , and ι be a positive integer. Then there exists a Fano foliation ofrank r and index ι on X if and only if d + ι r + 1 .Proof. Let F be a Fano foliation on X of rank r and index ι defined by a twisted ( n − r )-form ω ∈ H ( X, Ω n − rX ( n + 2 − d − ι )). Notice that 1 n − r n −
2. By lemma 4.5, we must have n − r < n + 2 − d − ι, or, equivalently, d + ι r + 1 . Conversely, let r ∈ { , · · · , n − } and ι be such that d + ι r + 1. Let ω ∈ H ( P n +1 , Ω n − r P n +1 ( n + 2 − d − ι ))be a general twisted ( n − r )-form defining a Fano foliation of rank r + 1 and index d + ι ≤ r + 1 on P n +1 .Then ω | X ∈ H ( X, Ω n − rX ( n + 2 − d − ι )) defines a foliation on X of rank r and index ι . (cid:3) Corollary 4.8.
Fix n > , and let X ⊂ P n +1 be a smooth hypersurface of degree d > . Then there existsa Fano foliation on X of rank r ∈ { , · · · , n − } and index ι = r − if and only if d = 2 .Proof. Suppose there exists a Fano foliation on X of rank r ∈ { , · · · , n − } and index ι = r − X . ByProposition 4.7, we must have d
2. Conversely, a foliation of rank r + 1 and and index ι = r + 1 on P n +1 induces a foliation of rank r and index ι = r − X . (cid:3) Question 4.9.
Let X ⊂ P n +1 be a smooth hypersurface of degree d > and dimension n > . Let F ( T X be a Fano foliation of rank r and index ι on X , with d + ι = r + 1 . Is F induced by a Fano foliation of rank r + 1 and index r + 1 on P n +1 ? In Section 9, we provide several examples of del Pezzo foliations on projective space bundles.5.
The relative anticanonical bundle of a fibration and applications
In [Miy93, Theorem 2], Miyaoka proved that the anticanonical bundle of a smooth projective morphism f : X → C onto a smooth proper curve cannot be ample. In [ADK08, Theorem 3.1], this result wasgeneralized by dropping the smoothness assumption, and replacing − K X/C with − ( K X/C + ∆), where ∆ isan effective Weil divisor on X such that ( X, ∆) is log canonical over the generic point of C . In this sectionwe give a further generalization of this result and provide applications to the theory of Fano foliations. Theorem 5.1.
Let X be a normal projective variety, and f : X → C a surjective morphism with connectedfibers onto a smooth curve. Let ∆ + ⊆ X and ∆ − ⊆ X be effective Weil Q -divisors with no commoncomponents such that f ∗ O X ( k ∆ − ) = O C for every non negative integer k . Set ∆ := ∆ + − ∆ − , and assumethat K X + ∆ is Q -Cartier. (1) If ( X, ∆) is log canonical over the generic point of C , then − ( K X/C + ∆) is not ample. (2) If ( X, ∆) is klt over the generic point of C , then − ( K X/C + ∆) is not nef and big.
N FANO FOLIATIONS 11
Proof.
To prove (1), we assume to the contrary that ( X, ∆) is log canonical over the generic point of C , and − ( K X/C + ∆) is ample. Let π : ˜ X → X be a log resolution of singularities of ( X, ∆), A an ample divisor on C , and m ≫ D = − m ( K X/C + ∆) − f ∗ A is very ample. Then K ˜ X + π − ∗ ∆ + − π − ∗ ∆ − = π ∗ ( K X + ∆ + − ∆ − ) + E + − E − , where E + and E − are effective π -exceptional divisors with no common components and the support of π − ∗ ∆ + E + + E − is a snc divisor.Set ˜ f := f ◦ π and let ˜ D ∈ | π ∗ D | be a general member. Setting ˜∆ + = π − ∗ ∆ + + m ˜ D + E − , we obtainthat ( ˜ X, ˜∆ + ) is log canonical over the generic point of C and that K ˜ X + ˜∆ + ∼ Q ˜ f ∗ K C + E + + π − ∗ ∆ − − m ˜ f ∗ A. Furthermore, since E + is effective and π -exceptional, π ∗ O ˜ X ( lE + ) = O X for any l ∈ N . Then for any l ∈ N ,˜ f ∗ O ˜ X ( lm ( K ˜ X/C + ˜∆ + )) ≃ ˜ f ∗ O ˜ X ( l ( mE + + mπ − ∗ ∆ − − ˜ f ∗ A )) ≃ ˜ f ∗ O ˜ X ( l ( mE + + mπ − ∗ ∆ − )) ⊗ O C ( − lA ) . Observe that ˜ f ∗ O ˜ X ( lm ( E + + π − ∗ ∆ − )) = O C . Indeed, let U ⊆ C be a non empty open subset and let˜ λ ∈ H ( ˜ f − ( U ) , O ˜ X ( lm ( E + + π − ∗ ∆ − ))) that is, ˜ λ is a rational function on ˜ X such that div(˜ λ ) + lm ( E + + π − ∗ ∆ − ) > f − ( U ). Let λ be the unique rational function on X such that ˜ λ = π ◦ λ . Thendiv( λ ) + lm ∆ − > f − ( U ) since E + is π -exceptional. Since f ∗ O X ( lm ∆ − ) = O C by assumption,there exists a regular function µ on U such that λ = f ◦ µ over f − ( U ). Thus the natural map O C ֒ → ˜ f ∗ O ˜ X ( lm ( E + + π − ∗ ∆ − )) is an isomorphism as claimed.Finally, observe that ˜ f ∗ O ˜ X ( lm ( K ˜ X/C + ˜∆ + )) is semi-positive by [Cam04, Theorem 4.13], but that con-tradicts the fact that A is ample. This proves (1).To prove (2), we assume to the contrary that ( X, ∆) is klt over the generic point of C , and − ( K X/C +∆) isnef and big. There exists an effective Q -Cartier Q -divisor N on X such that − ( K X/C + ∆) − εN is ample for0 < ε ≪
1. Let 0 < ε ≪ X, ∆ + εN ) is klt over the generic point of C . Set ∆ ′ + := ∆ + + εN ,∆ ′− := ∆ − , and ∆ ′ := ∆ + εN . Then − ( K X/C + ∆ ′ ) = − ( K X/C + ∆) − εN is ample, contradicting part (1) above. This proves (2). (cid:3) Remark 5.2.
Examples arising from Fano foliations show that Theorem 5.1 is sharp. To fix notation, let F ( T X be an algebraically integrable Fano foliation on a smooth projective variety X , with general logleaf ( ˜ F , ˜∆). We let C ⊂ Chow( X ) be a general complete curve contained in the closure of the subvarietyparametrizing general leaves of F . We denote by U the normalization of the universal cycle over C , withuniversal morphism e : U → X . Since C is general, e : U → X is birational onto its image. By Remark 3.8,there is a canonically defined Weil divisor ∆ on U such that − ( K U/C + ∆) = e ∗ ( − K F ). In particular, since − K F is ample, − ( K U/C + ∆) is always nef and big. It is ample if and only if the leaves parametrized by C have no common point. Moreover, ( U, ∆) is log canonical over the generic point of C if and only if ( ˜ F , ˜∆)is log canonical.By choosing F to have log canonical singularities along a general leaf, we see that we cannot strengthenthe conclusion of Theorem 5.1 by replacing “ample” with “nef and big”.On the other hand, consider the rank 2 del Pezzo foliation F on X = G (2 , V ) defined in 4.3. Then( ˜ F , ˜∆) ≃ ( P , H ), where H is a line in P . So it is not log canonical, while in this case − ( K U/C + ∆)is ample. So we cannot relax the assumption that ( U, ∆) is log canonical over the generic point of C inTheorem 5.1.As a first application of Theorem 5.1, we derive a special property of Fano foliations with mild singularities.This property will play a key role in our study of Fano foliations. ARAUJO
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Proposition 5.3.
Let X be a normal projective variety, and F ( T X an algebraically integrable Fanofoliation on X . If F has log canonical singularities along a general leaf, then there is a common point inthe closure of a general leaf of F .Proof. Let W be the normalization of the closure in Chow( X ) of the subvariety parametrizing general leavesof F , and U the normalization of the universal cycle over W , with universal family morphisms: U e / / π (cid:15) (cid:15) X .W
Denote by U w the fiber of π over a point w ∈ W .For every x ∈ X , π | e − ( x ) : e − ( x ) → W is finite. If we show that dim( e − ( x )) > dim( W ) for some x ∈ X ,then we conclude that π (cid:0) e − ( x ) (cid:1) = W , and thus x ∈ e (cid:0) U w (cid:1) for every w ∈ W , i.e., x is contained in theclosure of a general leaf of F .Suppose to the contrary that dim (cid:0) e − ( x ) (cid:1) < dim( W ) for every x ∈ X . Let C ⊂ W be a general completeintersection curve, and let U C be the normalization of π − ( C ), with natural morphisms π C : U C → C and e C : U C → X . Since C is general, C is not contained in π (cid:0) e − ( x ) (cid:1) for any x ∈ X , and thus the morphism e C : U C → X is finite onto its image. In particular, e ∗ C ( − K F ) is ample.By Remark 3.8, F induces a generically surjective morphism Ω r F U C /C → e ∗ C det( F ) ∗ . By Lemma 5.4 belowfollowed by Lemma 3.5, after replacing C with a finite cover if necessary, we may assume that π C has reducedfibers. This implies that det(Ω U C /C ) ≃ O U C ( K U C /C ). Thus there exists an effective integral divisor ∆ C on U C such that − ( K U C /C + ∆ C ) = e ∗ C ( − K F ) . Since F has log canonical singularities along a general leaf, the pair ( U C , ∆ C ) is log canonical over thegeneric point of C . But this contradicts Theorem 5.1, and the result follows. (cid:3) Lemma 5.4 ([BLR95, Theorem 2.1’]) . Let X be a quasi-projective variety, and f : X → C a flat surjectivemorphism onto a smooth curve with reduced general fiber. Then there exists a finite morphism C ′ → C suchthat f ′ : X ′ → C ′ is flat with reduced fibers. Here X ′ denotes the normalization of C ′ × C X and f ′ : X ′ → C ′ is the morphism induced by the projection C ′ × C X → C ′ . Using these ideas, next we give elementary proof of the Lipman-Zariski conjecture for klt spaces (see[GKKP10, Theorem 6.1]).
Proposition 5.5.
Let F be an algebraically integrable foliation on a normal projective variety X . Supposethat F has log terminal singularities and is locally free along a general leaf. Then the leaf through a generalpoint of X is proper and smooth and there exists an almost proper map X Y whose general fibers areleaves of F .Proof. Let W be the normalization of the closure in Chow( X ) of the subvariety parametrizing general leavesof F , and U the normalization of the universal cycle over W . By [Kol07, Theorem 3.35, 3.45] (see also[GKK10, Corollary 4.7] and Theorem 8.2), there is a log resolution of singularities d : Y → U such that d ∗ T Y ( − log Σ) = T U where Σ ⊂ Y is the largest reduced divisor contained in d − (Sing( U )). Consider thecommutative diagram: Y d / / ' ' g U e / / π (cid:15) (cid:15) XW By assumption, there is a dense open subset W ⊂ W such that e ∗ F is locally free along U := π − ( W ).We set Y := d − ( U ) and Σ := Σ | Y . By Remark 3.8, since e ∗ F is locally free along U , F induces afoliation ( e ∗ F ) | U ⊂ T U . Thus there is an injection g ∗ F | Y ֒ → T Y ( − log Σ ) ⊂ T Y . N FANO FOLIATIONS 13
Hence there exists an effective integral divisor Σ ′ on Y such that K Y + Σ + Σ ′ = ( g ∗ K F ) | Y . Let w ∈ W be a general point, and ( U w , ∆ w ) the corresponding log leaf. Set Y w := d − ( U w ), d w := d | Y w : Y w → U w , Σ w := Σ | Y w , and Σ ′ w := Σ ′ | Y w . By assumption, ( U w , ∆ w ) is log terminal. Thus ∆ w = 0. Hence K U w = ( e ∗ K F ) | U w , and we get K Y w + Σ w + Σ ′ w = ( g ∗ K F ) | Y w = d ∗ w K U w . Notice that d w : Y w → U w is a log resolution of singularities, and recall that U w is log terminal. On theother hand, Σ w and Σ ′ w are effective integral divisors on Y w . So we must have Σ w = Σ ′ w = 0. This impliesthat U w is smooth, and by Lemma 5.6 below, F is regular along the image of U w . (cid:3) Lemma 5.6.
Let Y be a variety with normalization morphism n : ˜ Y → Y . Let G be a locally free sheaf ofrank r G on Y and η : Ω Y → G be any morphism. Let ˜ η : Ω Y → n ∗ G be the extension given by [Sei66] . Let r bea positive integer and let S r ( η ) (resp. S r (˜ η ) ) be the locus where ∧ r η : Ω rY → ∧ r G (resp. ∧ r ˜ η : Ω r ˜ Y → n ∗ ∧ r G )is not surjective. Then S r (˜ η ) = n − ( S r ( η )) .Proof. If η has rank > r at a some point y in Y then n ∗ η has rank > r at every point in n − ( y ) and thus ˜ η has rank > r at every point in n − ( y ). Therefore S r (˜ η ) ⊂ n − ( S r ( η )).Let us assume that η has rank r at some point y in Y . By shrinking Y if necessary, we may decompose G as O ⊕ r G − rY ⊕ G in such a way that the induced morphism η : Ω Y → O ⊕ r G − rY is zero at y . Write η = ( η , η )and let ˜ η : Ω Y → n ∗ O ⊕ r G − rY (respectively ˜ η : Ω Y → n ∗ G ) be the extension of η : Ω Y → O ⊕ r G − rY (respectively η : Ω Y → G ) given by [Sei66]. Then ˜ η = (˜ η , ˜ η ) and the claim then follows from [Dru04,Lemme 1.2]. (cid:3) Corollary 5.7 (Lipman-Zariski conjecture for klt spaces. See also [GKKP10, Theorem 6.1]) . Let X be a kltspace such that the tangent space T X is locally free. Then X is smooth.Proof. The result follows from proposition 5.5 applied to the foliation induced by the projection morphism X × C → C , where C is a smooth complete curve. (cid:3) Proposition 5.8.
Let F be a -Gorenstein algebraically integrable foliation on a normal projective variety X . Suppose that F has log terminal singularities along a general leaf. Then det( F ) is not nef and big.Proof. We let C ⊂ Chow( X ) be a general complete curve contained in the closure of the subvarietyparametrizing general leaves of F . We denote by U the normalization of the universal cycle over C , withnatural morphisms π : U → C and e : U → X . Since C is general, e : U → X is birational onto its image.Thus if − K F is nef and big, then so is e ∗ ( − K F ).By Remark 3.8, F induces a Pfaff field Ω rU/C → e ∗ O C ( − K F ), where r denotes the rank of F . By Lemma5.4 followed by Lemma 3.5, after replacing C with a finite cover if necessary, we may assume that π hasreduced fibers. This implies that det(Ω rU/C ) ≃ O U ( K U/C ). Thus there exists a canonically defined effectivedivisor ∆ on U such that − ( K U/C + ∆) = e ∗ ( − K F ) . By assumption, ( U, ∆) is log terminal over the generic point of C . So, by Theorem 5.1, e ∗ ( − K F ) cannotbe nef and big. (cid:3) Foliations and rational curves
If a smooth projective variety X admits a Fano foliation F , then it is uniruled, as we have observed inRemark 2.9. In order to study the pair ( X, F ), it is useful to understand the behavior of F with respectto families of rational curves on X . This is the theme of this section. We start by recalling some definitionsand results from the theory of rational curves on smooth projective varieties. We refer to [Kol96] for moredetails.Let X be a smooth projective variety, and H a family of rational curves on X , i.e., an irreduciblecomponent of RatCurves n ( X ). If C is a curve from the family H , with normalization morphism f : P → C ⊂ X , then we denote by [ C ] or [ f ] any point of H corresponding to C . We denote by Locus ( H ) the locusof X swept out by curves from H . We say that H is unsplit if it is proper, and minimal if, for a generalpoint x ∈ Locus ( H ), the closed subset H x of H parametrizing curves through x is proper. We say that H is ARAUJO
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DRUEL dominating if Locus ( H ) = X . In this case we say that a curve C parametrized by H is a moving curve on X , and that any curve from H is a deformation of C . (Minimal dominating families of rational curves) . Let X be a smooth projective uniruled variety. Then X always carries a minimal dominating family of rational curves. Fix one such family H , and let [ f ] ∈ H bea general point. By [Kol96, IV.2.9], f ∗ T X ≃ O P (2) ⊕ O P (1) ⊕ d ⊕ O ⊕ ( n − d − P , where d = deg( f ∗ T X ) − ≥ x ∈ X , let ˜ H x be the normalization of H x . By [Kol96, II.1.7, II.2.16], ˜ H x isa finite union of smooth projective varieties of dimension d = deg( f ∗ T X ) −
2. Define the tangent map τ x : ˜ H x P ( T x X ∗ ) by sending a curve that is smooth at x to its tangent direction at x . Define C x to bethe image of τ x in P ( T x X ∗ ). This is called the variety of minimal rational tangents at x associated to theminimal family H . The map τ x : ˜ H x → C x is in fact the normalization morphism by [Keb02] and [HM04]. Definition 6.2.
Let π : X → Y be a proper morphism defined on a dense open subset of X . A family ofrational curves H on X is said to be horizontal (with respect to π ) if the general member of H meets X and is not contracted by π . If moreover Locus ( H ) dominates Y , then we say that H is h-dominating (withrespect to π ) .Notice that if X admits a horizontal family of rational curves, then it admits a minimal horizontal familyof rational curves. Indeed, it is enough to take a horizontal family having minimal degree with respect tosome fixed ample line bundle on X . Similarly for h-dominating families. Lemma 6.3.
Let X be a smooth projective variety, and π : X → Y a surjective proper morphism definedon a dense open subset of X . Suppose that H is a minimal horizontal family of rational curves with respectto π . Then − K X · H ≤ dim Y + 1 , where − K X · H denotes the intersection number of − K X with any curvefrom the family H . Moreover • If − K X · H = dim Y + 1 , then H is dominating. • If − K X · H = dim Y , then Locus ( H ) has codimension at most 1 in X .Proof. Let x be a general point in Locus ( H ), and denote by Locus ( H x ) the locus of X swept out by curvesfrom H x . By assumption, any irreducible component Z of Locus ( H x ) is proper. Moreover, by [Kol96,IV.3.13.3], any curve in Z is numerically proportional in X to a curve from the family H . In particular Z cannot contain any curve contracted by π . Therefore dim (cid:0) Locus ( H x ) (cid:1) ≤ dim( Y ).On the other hand, by [Kol96, IV.2.6.1], dim( X ) + ( − K X · H ) ≤ dim (cid:0) Locus ( H ) (cid:1) + dim (cid:0) Locus ( H x ) (cid:1) + 1.Thus − K X · H ≤ dim( Y ) + 1 − (cid:16) dim( X ) − dim (cid:0) Locus ( H ) (cid:1)(cid:17) ≤ dim Y + 1 . If − K X · H = dim Y + 1, then we must have dim (cid:0) Locus ( H ) (cid:1) = dim( X ), i.e., H is dominating. If − K X · H =dim Y , then we must have dim( X ) − dim (cid:0) Locus ( H ) (cid:1) ≤ (cid:3) (Rationally connected quotients) . Let H , . . . , H k be families of rational curves on X . For each i , let H i denote the closure of H i in Chow( X ). Two points x, y ∈ X are said to be ( H , . . . , H k )-equivalent if theycan be connected by a chain of 1-cycles from H ∪ · · · ∪ H k . This defines an equivalence relation on X . By[Cam92] (see also [Kol96, IV.4.16]), there exists a proper surjective equidimensional morphism π : X → T from a dense open subset of X onto a normal variety whose fibers are ( H , . . . , H k )-equivalence classes. Wecall this map the ( H , . . . , H k ) -rationally connected quotient of X . When T is a point we say that X is( H , . . . , H k )-rationally connected.From now on we investigate the behavior of foliations on a smooth projective variety X with respect tofamilies of rational curves on X . We start with a simple but useful observation. Lemma 6.5.
Let X be a smooth projective variety, H a family of rational curves on X , and F an alge-braically integrable foliation on X . Suppose that ℓ is contained in a leaf of F and avoids the singular locusof F for some [ ℓ ] ∈ H . Then the same holds for general [ ℓ ] ∈ H .Proof. Let W be the closure in Chow( X ) of the subvariety parametrizing general leaves of F , with universalfamily morphisms: U p (cid:15) (cid:15) q / / X.W
N FANO FOLIATIONS 15
Let A W be a general very ample effective divisor on W , and set A = q ∗ ( p ∗ ( A W )).The condition that ℓ is contained in a leaf of F and avoids the singular locus S of F is equivalent to thecondition that ℓ ∩ S = ∅ and A · ℓ = 0. Hence, if this condition holds for some [ ℓ ] ∈ H , then it holds forgeneral [ ℓ ] ∈ H . (cid:3) Lemma 6.6.
Let X be a smooth projective uniruled variety, H , · · · , H k unsplit families of rational curveson X , and F an algebraically integrable foliation on X . Denote by π : X → T the ( H , · · · , H k ) -rationallyconnected quotient of X . Suppose that a general curve from each of the families H i ’s is contained in a leafof F and avoids the singular locus of F . Then there is an inclusion T X /T ⊂ F | X .Proof. Let W be the closure in Chow( X ) of the subvariety parametrizing general leaves of F , with universalfamily morphisms: U p (cid:15) (cid:15) q / / X.W
Let A W be a general very ample effective divisor on W , and set A = q ∗ ( p ∗ ( A W )). By assumption, a generalcurve ℓ ⊂ X parametrized by each H i is contained in a leaf of F , and avoids the singular locus of F . Thus A · ℓ = 0.Let X t = ( π ) − ( t ) be a general fiber of π . By [Kol96, IV.3.13.3], every proper curve C ⊂ X t isnumerically equivalent in X to a linear combination of curves from the families H i ’s , and so A · C = 0. Thisshows that A | X t ≡
0, and thus X t ⊂ q ( p − ( w )) for some w ∈ W , i.e., X t is contained in a leaf of F . Weconclude that T X /T ⊂ F | X by Lemma 6.7 below. (cid:3) Lemma 6.7.
Let F be a foliation of rank r F on a normal variety X , and π : X → Y an equidimensionalmorphism with connected fibers onto a normal variety. Suppose that the general fiber of π is contained in aleaf of F . Then F induces a foliation G of rank r G = r F − (cid:0) dim( X ) − dim( Y ) (cid:1) on Y , together with anexact sequence → T X/Y → F → ( π ∗ G ) ∗∗ . Definition 6.8.
Under the hypothesis of Lemma 6.7, we say that F is the pullback via π of the foliation G . Proof of Lemma 6.7.
Notice that the induced map T X/Y → T X / F is generically zero by assumption. Since T X / F is torsion free, it must be identically zero, hence we have an inclusion T X/Y ⊂ F .First we define the foliation G ⊂ T Y induced by F analytically. Let y ∈ Y be a general point. Choosean analytic open neighborhood V ⊂ Y of y , and a local holomorphic section s : V → X of π . There existsan analytic open neighborhood U ⊂ X of x = s ( y ), and a complex analytic space W such that the leaves of F | U are the fibers of a holomorphic map p : U → W . After shrinking V if necessary, we get a holomorphicmap p ◦ s : V → W , which defines a foliation G s on V . Notice that(6.1) T y G s = dπ x (cid:0) T x F (cid:1) , where dπ x : T x X → T y Y denotes the tangent map of π at x , T x F and T y G denote the fibers of F ⊂ T X and G s ⊂ T Y at x and y , respectively. Notice that G s does not depend on the choice of local section s : V → X ,since X is normal, T X/Y ⊂ F , and π has connected fibers. Moreover, these foliations defined locally glueand extend to a foliation G of rank r G = r F − (cid:0) dim( X ) − dim( Y ) (cid:1) on Y .Next we give an algebraic description of G . Since π is equidimensional, there are dense open subsets X ⊂ X and Y ⊂ Y such that codim X (cid:0) X \ X ) ≥
2, codim Y (cid:0) Y \ Y ) ≥ π := π | X maps X into Y , F := F | X is a subbundle of T X , and G := G | Y is a subbundle of T Y . Consider the tangent map dπ : T X → ( π ) ∗ T Y . By (6.1), G coincides with the saturation of the subsheaf ( π ) ∗ (cid:0) dπ ( F ) (cid:1) in T Y ,and the induced map α : dπ ( F ) → ( π ) ∗ T Y (cid:14) ( π ) ∗ G is generically zero. Since G is a subbundle of T Y ,( π ) ∗ T Y (cid:14) ( π ) ∗ G is torsion free, and hence the map α must be identically zero. So we have an exact sequence0 → T X /Y → F → ( π ) ∗ G . Note that the sheaves T X/Y , F and ( π ∗ G ) ∗∗ are reflexive. Since reflexivesheaves on a normal variety are normal sheaves ([Har80, Proposition 1.6]), and codim X (cid:0) X \ X ) ≥
2, weobtain an exact sequence 0 → T X/Y → F → ( π ∗ G ) ∗∗ . (cid:3) In the setting of Lemma 6.6, if moreover the families H i ’s are dominating, then we may drop the assump-tion that F is algebraically integrable. This is the content of the next lemma. ARAUJO
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Lemma 6.9.
Let X be a smooth projective variety, H , · · · , H k unsplit dominating families of rational curveson X , and F a foliation on X . Denote by π : X → T the ( H , · · · , H k ) -rationally connected quotient of X . If T P ⊂ f ∗ F for general [ f ] ∈ H i , ≤ i ≤ k , then there is an inclusion T X /T ⊂ F | X .Proof. Recall that for general [ f ] ∈ H i one has f ∗ T X ≃ O P (2) ⊕ O P (1) ⊕ d i ⊕ O ⊕ ( n − d i − P , where n = dim X and d i = deg( f ∗ T X ) −
2. Hence, the assumption O P (2) ≃ T P ⊂ f ∗ F implies that the natural inclusion T P ⊂ f ∗ T X factors through f ∗ F ֒ → f ∗ T X . Therefore a general curve from each of the families H i ’s iscontained in a leaf of F .Let x ∈ X be a general point. We define inductively a sequence of (irreducible) subvarieties of X asfollows. Set V ( x ) := { x } , and let V j +1 ( x ) be the closure of the union of curves from the families H i ,0 ≤ i ≤ k , that pass through a general point of V j ( x ).Then dim V j +1 ( x ) ≥ dim V j ( x ), and equality holds if and only if V j +1 ( x ) = V j ( x ). In particular, thereexists j such that V j ( x ) = V j ( x ) for every j ≥ j . We set V ( x ) = V j ( x ). Since x is general, V ( x ) issmooth at x . Notice also that V ( x ) is irreducible, and that V ( x ) is contained in the leaf of F through x byconstruction.We define the subfoliation V ⊂ F by setting V x = T x V ( x ) for general x ∈ X . The leaf of V through x is precisely V ( x ). In particular V is an algebraically integrable foliation of X . Moreover, by construction, ageneral curve from each of the families H i ’s is contained in a leaf of V , and avoids the singular locus of V by [Kol96, II.3.7]. The result then follows from Lemma 6.6. (cid:3) Next we apply the results from the previous section to characterize pairs ( X, F ) when F is a Fanofoliation that is ample when restricted to a general member of a minimal covering family of rational curveson X . Lemma 6.10.
Let X be an n -dimensional smooth projective variety admitting a minimal dominating familyof rational curves H . Let F ( T X be a Fano foliation of rank r on X . If f ∗ F is an ample vector bundlefor general [ f ] ∈ H , then ( X, F ) ≃ (cid:0) P n , O (1) ⊕ r (cid:1) .Proof. Denote by π : X → T the H -rationally connected quotient of X . By [ADK08, Proposition2.7], after shrinking X and T if necessary, we may assume that π is a P k -bundle, and the inclusion F | X ֒ → T X factors through the natural inclusion T X /T ֒ → T X . Recall that f ∗ T X /T ∼ = O (2) ⊕O (1) ⊕ k − .If O (2) ⊂ f ∗ F for general [ f ] ∈ H , then the general curve from H is tangent to the foliation F . Hencethe general fiber of π is contained in a leaf of F . Since F | X ⊂ T X /T , we must have F | X = T X /T .If O (2) f ∗ F for general [ f ] ∈ H , then f ∗ F ∼ = O P k (1) ⊕ r . Since F | X is saturated in T X /T , we musthave r < k . Then [DC05, Th´eor`eme 3.8] implies that the foliation induced by F on a general fiber of π is O P k (1) ⊕ r ֒ → T P k . In either case, we conclude that F is algebraically integrable and has log canonicalsingularities along a general leaf. Proposition 5.3 then implies that there is a point x ∈ X contained in theclosure of a general leaf of F . This is only possible if T is a point, and ( X, F ) ∼ = (cid:0) P n , O (1) ⊕ r (cid:1) . (cid:3) Lemma 6.11.
Let X be a smooth projective variety, X ⊂ X a dense open subset, T a positive dimen-sional normal variety, and π : X → T a proper surjective equidimensional morphism with and rationallyconnected general fiber. Let T be the normalization of the closure of T in Chow( X ) , and U the universalcycle over T .Through a general point of T there exists a curve C ⊂ T such that the following holds. Let C → T bethe normalization of the closure of C in T , U C the normalization of U × T C , and π C : U C → C the inducedmorphism. Then • all irreducible fibers of π C : U C → C are reduced, • U C → X is finite, and • no fiber of π C is entirely mapped into the exceptional locus of the universal morphism U → X .Moreover, given any subset Z ⊂ T such that codim T ( Z ) ≥ , C can be chosen so that the image of C in T avoids Z .Proof. Consider the universal morphisms U π (cid:15) (cid:15) e / / X.T
N FANO FOLIATIONS 17
Given t ∈ T , we write U t = π − ( t ).Since X is smooth, the exceptional locus E of e has pure codimension one in U . Let F be an irreduciblecomponent of E . Since X ∩ e ( E ) = ∅ and π is equidimensional, π ( F ) has codimension one in T . Moreover, ∀ t ∈ T , either F ∩ U t = ∅ , or F ∩ U t is a union of irreducible components of U t . So we may assumethat X = X \ e (cid:0) π − ( π ( E )) (cid:1) . Let E ′ ⊂ E be the union of irreducible components F of E such that π − ( π ( F )) ⊂ E . If t ∈ π ( E ) \ π ( E ′ ), then U t has at least two irreductible components, at least one of whichis not contained in E .Let S ⊂ T be the locus over which the fibers of π are multiple, and let S be its closure in T . By[GHS03], codim T ( S ) ≥
2. Let C ⊂ X \ e (cid:0) E ∪ π − ( S ∪ Z ) (cid:1) be a general complete intersection curve, andset C := π ( C ∩ X ) ⊂ T \ S . Let U C be the normalization of U × T C , and π C : U C → C the inducedmorphism. By construction, the irreducible fibers of π C over C are reduced. Moreover, the image of C in T does not meet π ( E ′ ). Hence the fibers of π C over C \ C have at least two irreductible components, atleast one of which is not mapped into E . (cid:3) Lemma 6.12.
Let X be a smooth projective variety, X ⊂ X a dense open subset, T a positive dimensionalnormal variety, and π : X → T a proper surjective equidimensional morphism of relative dimension r − .Let F be a rank r Fano foliation on X , and assume that there is an exact sequence → T X /T → F | X → ( π ∗ G ) , where G is an invertible subsheaf of T T . Suppose that the general fiber F of π is rationally connected, andsatisfies c ( A ) r − · F ≤ for some ample line bundle A on X .Then G defines a foliation by rational curves on T .Proof. Let T be the normalization of the closure of T in Chow( X ), and U the normalization of the universalcycle over T . We denote by π : U → T and e : U → X the universal morphisms, and by E the exceptionallocus of e . Let F U be the foliation induced by F on U . Notice that F U is regular along the general fiberof π . Moreover F U and e ∗ F agree on U \ E .By Lemma 6.7, there exists a smooth open subset T ⊂ T with codim T ( T \ T ) ≥
2, a rank 1 subbundle G ⊂ T T , and an exact sequence 0 → T U /T → F U → ( π ∗ G ) , where U = π − ( T ), F U = F U | U , and π = π | U : U → T . In particular, there is a canonically definedeffective divisor D on U such that(6.2) O U ( − K F U ) ≃ π ∗ G ⊗ (cid:16) O U ( − D ) [ ⊗ ] det (cid:0) T U /T (cid:1)(cid:17) . Moreover, since F U is regular along the general fiber of π , the divisor D does not dominate T .Let C → T be the curve provided by Lemma 6.11, and n : C → T the normalization of its closure in T .We also require that n ( C ) ⊂ T . Let U C be the normalization of U × T C , and denote by π C : U C → C , q : U C → U , and e C : U C → X the natural morphisms: U Cπ C (cid:15) (cid:15) q / / e C U π (cid:15) (cid:15) e / / X.C n / / T By Lemma 6.11,(a) all irreducible fibers of π C : U C → C are reduced,(b) e C : U C → X is finite, and(c) no fiber of π C is entirely mapped into E by q .Since c ( A ) r − · F ≤
2, condition (a) above implies that every fiber of π C is reduced at all of its genericpoints. Thus π C is smooth at the generic points of every fiber (see [Gro66, Chap. IV Corollaires 15.2.3 and14.4.2]). By shrinking T if necessary, we may assume that the same holds for π . By [Dru99, Lemme 4.4],det( T U C /C ) ≃ O U C ( − K U C /C ), and det( T U /T ) ≃ O U ( − K U /T ). Thus(6.3) O U C ( − K U C /C ) ≃ q ∗ det( T U /T ) . ARAUJO
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Since F U and e ∗ F agree on U \ E , there are effective divisors ∆ + and ∆ − on U C , both supported oncomponents of fibers of π C that are mapped into E by q , such that(6.4) q ∗ ( − K F U ) = e ∗ C ( − K F ) + ∆ + − ∆ − . Condition (c) above implies that ∆ + and ∆ − are supported on reducible fibers of π C , and no fiber of π C isentirely contained in their supports. In particular, ( π C ) ∗ O U C ( k ∆ − ) = O C ∀ k ≥ U C , and combining it with (6.2) and (6.3), we get that O U C (cid:0) − ( K U C /C + D + ∆ + − ∆ − ) (cid:1) ≃ O U C (cid:0) e ∗ C ( − K F ) (cid:1) ⊗ π ∗ C ( n ∗ G ) ∗ , where D is an effective divisor on U C that does not dominate C . By Theorem 5.1, − ( K U C /C + D + ∆ + − ∆ − )is not ample. On the other hand, condition (b) above implies that e ∗ C ( − K F ) is ample. So we must havedeg C ( n ∗ G ) >
0, and thus the general leaf of the foliation on T defined by G is a rational curve byTheorem 1.2. (cid:3) Algebraic integrability of del Pezzo foliations
In this section we prove Theorem 1.1. Our argument involves constructing subfoliations of del Pezzofoliations which inherit some of their positivity properties. One way to construct such subfoliations is viaHarder-Narasimhan filtrations, as we now explain. (Harder-Narasimhan filtration) . Let X be an n -dimensional projective variety, and A an ample linebundle on X . Let F be a torsion-free sheaf of rank r on X . We define the slope of F with respect to A to be µ A ( F ) = c ( F ) · A n − r . We say that F is µ A -semistable if for any subsheaf E of F we have µ A ( E ) ≤ µ A ( F ).Given a torsion-free sheaf F on X , there exists a filtration of F by subsheaves0 = E ( E ( . . . ( E k = F , with µ A -semistable quotients Q i = E i / E i − , and such that µ A ( Q ) > µ A ( Q ) > . . . > µ A ( Q k ). This iscalled the Harder-Narasimhan filtration of F (see [HN75], [HL97, 1.3.4]). Lemma 7.2.
Let X be a normal projective variety, A an ample line bundle on X , and F ( T X a foliationon X . Let F ⊂ F ⊂ · · · ⊂ F k = F be the Harder-Narasimhan filtration of F with respect to A .Then F i ( T X defines a foliation on X for every i such that µ A ( F i ) > .Proof. This is well known. See for instance [She92, Lemma 9.1.3.1]. (cid:3)
Notation 7.3.
Let X be a normal projective variety, A an ample line bundle on X , and F a coherenttorsion free sheaf of O X -modules. Let m i ∈ N , 1 ≤ i ≤ dim( X ) −
1, be large enough integers, H i ∈ | m i A | be general members, and set C := H ∩ · · · ∩ H dim( X ) − . By the Mehta-Ramanathan Theorem (see [MR82,6.1] or [HL97, 7.2.1]), the Harder-Narasimhan filtration of F with respect to A commutes with restrictionto C . In this case we say that C is a general complete intersection curve for F and A in the sense ofMehta-Ramanathan . If F and A are clear from the context, we simply say that C is a general completeintersection curve . Lemma 7.4.
Let X and Y be smooth complex projective varieties with dim( Y ) > , X an open subset of X with codim X ( X \ X ) ≥ , Y a dense open subset of Y and π : X → Y a proper surjective equidimensionalmorphism. Let G be a coherent torsion-free sheaf of O Y -modules on Y , and F a coherent torsion-free sheafof O X -modules such that F | X ≃ π ∗ G | Y . Let A be an ample line bundle on X , and C ⊂ X a generalcomplete intersection curve for F and A in the sense of Mehta-Ramanathan. Suppose that F | C is notsemistable.Then there exists a subsheaf H ( G such that µ (cid:16)(cid:0) π ∗ ( H | Y ) (cid:1) | C (cid:17) > µ ( F | C ) .Proof. Consider general elements H i ∈ | m i A | , for i ∈ { , . . . , dim( X ) − } , where the m i ∈ N are largeenough so that the Harder-Narasimhan filtration of F commutes with restriction to the complete intersectioncurve C = H ∩ · · · ∩ H dim( X ) − . Set Z := H ∩ · · · ∩ H dim( X ) − dim( Y ) , and Z := Z ∩ X . Then Z is a smoothvariety of dimension equal to dim( Y ), and the restriction ϕ := π | Z : Z → Y is a finite morphism. Denoteby j : Z ֒ → Z the inclusion. N FANO FOLIATIONS 19
Let K be a splitting field of the function field K ( Z ) over K ( Y ), and let ψ : Z ′ → Z ⊂ X be thenormalization of Z in K . Set Z ′ := ψ − ( Z ), and let j ′ : Z ′ ֒ → Z ′ be the inclusion. Let ψ be the restrictionof ψ to Z ′ , and set F ′ := ( ψ ∗ F ) ∗∗ = j ′∗ (cid:0) ψ ∗ ϕ ∗ ( G | Y ) (cid:1) . Let G be the Galois group of K ( Z ′ ) over K ( Y ).By [HL97, Lemma 3.2.2], F ′ is not semistable with respect to ψ ∗ H | Z . Let E ′ be the maximally destabi-lizing subsheaf of F ′ . Then µ ( ψ ∗ A ) ( E ′ ) > µ ( ψ ∗ A ) ( F ′ ). This implies µ ( ψ ∗ A ) ( E ′ | C ′ ) > µ ( ψ ∗ A ) ( F ′ | C ′ ), where C ′ = C × Z Z ′ .Because of its uniqueness, the maximally destabilizing subsheaf E ′ of F ′ is invariant under the actionof G on F ′ . Thus, up to shrinking Y if necessary, we may assume that there is a subsheaf H ⊂ G suchthat E ′ ≃ (cid:0) j ∗ ψ ∗ ϕ ∗ ( H | Y ) (cid:1) . By [HL97, Lemma 3.2.1], µ ( ψ ∗ A ) ( E ′ ) = µ ( ψ ∗ A ) ( E ′ | C ′ ) = d · µ A ( H | C ), and µ ( ψ ∗ A ) ( F ′ ) = µ ( ψ ∗ A ) ( F ′ | C ′ ) = d · µ A ( G | C ), where d denotes the degree of C ′ → C . (cid:3) Proposition 7.5.
Let X be a normal projective variety, A an ample line bundle on X , and F ( T X afoliation on X . Suppose that µ A ( F ) > , and let C ⊂ X be a general complete intersection curve. Theneither • F is algebraically integrable with rationally connected general leaves, or • there exits an algebraically integrable subfoliation G ( F with rationally connected general leaf suchthat det( G ) · C > det( F ) · C . Remark 7.6.
Let F be coherent torsion-free sheaf of O X -modules on X , and C ⊂ X a general completeintersection curve. Then F is locally free along C , so that det( F ) · C is well-defined. Proof of Proposition 7.5.
Suppose that F is semistable with respect to A . Then F | C is semistable withslope µ ( F | C ) >
0, and F | C is an ample vector bundle by [Har71, Theorem 2.4]. Thus F is algebraicallyintegrable with rationally connected general leaf by Theorem 1.2.Suppose that F is not semistable. Let 0 = F ⊂ F ⊂ · · · ⊂ F k = F be the Harder-Narasimhanfiltration of F with respect to A . Note that k >
2. Set i := max { i > | µ A ( F i / F i − ) > } , and G := F i . Since µ A ( G ) > G ( T X defines a foliation on X by Lemma 7.2 above. Since µ A ( F ) >µ A ( F / F ) > · · · > µ A ( F k / F k − ), we must have • µ A ( F i / F i − ) > i i , and • µ A ( F i / F i − ) i > i + 1.Thus(1) ( F i / F i − ) | C is an ample vector bundle on C for any i i by [Har71, Theorem 2.4], and(2) det( F i / F i − ) · C i > i + 1.From (1) it follows that G | C is an ample vector bundle on C . Thus G is algebraically integrable withrationally connected general leaf by Theorem 1.2. From (2) it follows thatdet( G ) · C = P i i det( F i / F i − )) · C = det( F ) · C − P i +1 i k det( F i / F i − )) · C > det( F ) · C. (cid:3) These results allow us to prove Theorem 1.1 in the special case when X has Picard number 1. Proposition 7.7.
Let F be a del Pezzo foliation on a smooth projective variety X P n with ρ ( X ) = 1 .Then F is algebraically integrable with rationally connected general leaf.Proof. By Remark 2.9, X is uniruled. Since ρ ( X ) = 1, X is in fact a Fano manifold. Let A be an ampleline bundle on X such that Pic( X ) = Z [ A ]. By assumption, det( F ) ≃ A ⊗ r − , where r = r F > F is not algebraically integrable with rationally connected general leaf. By Proposition 7.5,there exits a subfoliation G ( F such that det( G ) ≃ A ⊗ k for some k > r − > r G . By [ADK08, Theorem1], ( X, A ) ≃ ( P n , O P n (1)). (cid:3) In the next two propositions we address algebraically integrability of del Pezzo foliations on projectivespace bundles. Recall from 4.1 the description of degree 0 foliations on P n , H = O P n (1) ⊕ r ( T P n . Theleaves of H are fibers of a linear projection P n P n − r from an ( r − P n .We want to describe families of such foliations. ARAUJO
AND ST´EPHANE
DRUEL (Families of degree 0 foliations on P m ) . Let T be a smooth positive dimensional variety, E a locally freesheaf of rank m + 1 ≥ T , and set X := P T ( E ). Denote by O X (1) the tautological line bundle on X , by π : X → T the natural projection, and by Y ≃ P m a general fiber of π . Let H ( T X/T be a foliation of rank s ≥ X , and suppose that H | Y ≃ O P m (1) ⊕ s ( T P m . We first observe that there is an open subset T ⊂ T ,with codim T ( T \ T ) ≥
2, such that, for any t ∈ T , H | X t ≃ O P n (1) ⊕ s ( T P m , where X t = π − ( t ) ≃ P m .Indeed, there exists an open subset T ⊂ T , with codim T ( T \ T ) ≥ (cid:0) T X/T (cid:1) / H is flat over T .Then, for any t ∈ T , the inclusion H ( T X/T restricts to an inclusion H | X t ( T P m . In particular, H | X t is torsion free for any t ∈ T . By removing a subset of codimension ≥ T if necessary, we may assumethat O P m ( s ) ≃ (cid:0) det( H ) (cid:1) | X t ≃ det (cid:0) H | X t (cid:1) ⊂ ∧ s T P m for any t ∈ T . By Bott’s formulae, H | X t is saturatedin T P m . So H | X t is a degree 0 foliations of rank s on P m , i.e., H | X t ≃ O P m (1) ⊕ s .Set V ′ := π ∗ ( H ( − ⊂ π ∗ ( T X/T ( − ≃ E ∗ , and denote by V the saturation of V ′ in E ∗ . Note that π ∗ V is a reflexive sheaf by [Har80, Proposition 1.9]. The above observations imply that over T we have V ′ = V , and π ∗ V ≃ H ( − H = ( π ∗ V )(1). In particular,det( H ) ≃ π ∗ (det V ) ⊗ O X ( s ) . Let K be the kernel of the dual map E → V ∗ . By removing a subset of codimension ≥ T ifnecessary, we may assume that there is an exact sequence of vector bundles on T :0 → K | T → E | T → V ∗ | T → . Consider the P m − s -bundle Z := P T (cid:0) K | T (cid:1) , with natural projection q : Z → T . The above exact sequenceinduces a rational map p : π − ( T ) Z over T , which restricts to a surjective morphism p : X → Z ,where X is the complement in π − ( T ) of the P s − -subbundle P T (cid:0) V ∗ | T (cid:1) ⊂ P ( E | T ). By construction, H | X = T X /Z . Note also that codim X ( X \ X ) ≥ Proposition 7.9.
Let C be a smooth complete curve, E an ample locally free sheaf of rank m + 1 ≥ on C , and set X := P C ( E ) . Denote by O X (1) the tautological line bundle on X , and by π : X → C the naturalprojection. Let F ⊂ T X/C be a foliation of rank r ≥ on X such that det( F ) ≃ O X ( r − ⊗ π ∗ L for somenef line bundle L on C . Then F is algebraically integrable with rationally connected general leaf.Proof. Denote by Y ≃ P m the general fiber of π . We may assume that F ( T X/C . So the restriction of F to Y is a Fano foliation of rank r and index r − P m . Recall from 4.2 the classification of such foliationsestablished in [LPT11a]. The foliation F is algebraically integrable if and only if so is F | Y ( T Y . So we mayassume that F | Y is the pullback via a linear projection P m P m − r +1 of a foliation on P m − r +1 inducedby a global holomorphic vector field, i.e., F | Y ≃ O P m (1) r − ⊕ O P m .Set V ′ := π ∗ ( F ( − ⊂ π ∗ ( T X/C ( − ≃ E ∗ , and denote by V the saturation of V ′ in E ∗ . Notice thatthe inclusion ( π ∗ V ′ )(1) ⊂ F extends to an inclusion H := ( π ∗ V )(1) ⊂ F ⊂ T X/C . So H is a subfoliationof F of rank r − H | Y ≃ O P n (1) r − .Let the notation be as in 7.8, with T = T = C and s = r −
1. There is a surjective morphism p : X → Z such that H | X = T X /Z , and det( H ) ≃ π ∗ (cid:0) det( V ) (cid:1) ⊗ O X ( r − F induces a rank 1 foliation G ⊂ T Z on Z such that det( F | X ) ≃ det (cid:0) T X /Z (cid:1) ⊗ p ∗ G .(Notice that G is an invertible sheaf by [Har80, Proposition 1.9].) Recall that codim X ( X \ X ) ≥
2. So G ≃ q ∗ (cid:0) det( V ∗ ) ⊗ L (cid:1) , and det( V ∗ ) is ample since so is E . Let B ⊂ Z be a general complete intersectioncurve. Then G | B is an ample line bundle. Thus G is a foliation by rational curves by Theorem 1.2. Thegeneral leaf of F is the closure of the inverse image by p of a general leaf of G . Hence it is algebraic andrationally connected. (cid:3) Proposition 7.10.
Let E be an ample locally free sheaf of rank m + 1 ≥ on P l , and set X := P P l ( E ) .Denote by O X (1) the tautological line bundle on X , π : X → P l the natural projection, and Y ≃ P m thegeneral fiber of π . Let F ( T X be a foliation of rank r ≥ on X such that det( F ) ≃ O X ( r − ⊗ π ∗ L forsome nef line bundle L on P l . Suppose that F * T X/ P l , and set H := F ∩ T X/ P l . Then (1) F is algebraically integrable with rationally connected general leaf; (2) r ∈ { , } , and r = 3 implies l = 1 ; (3) L ≃ O P l unless r = 2 , l = 1 , and L ≃ O P (1) ; (4) if m ≥ , then H | Y ≃ O P m (1) ⊕ r − ( T P m , and there is an exact sequence → H → F → π ∗ R , N FANO FOLIATIONS 21 where R ⊂ T P l is an ample invertible subsheaf. (5) if m = 1 , then l ≥ r = 3 , X ≃ P × P l , H = T X/ P l , and F is the pullback via the natural projection P × P l → P l of a degree zero foliation O P l (1) ⊕ O P l (1) ( T P l on P l .Proof. Denote by ℓ ⊂ Y ≃ P m a general line. Set Q := F / H ⊆ π ∗ T P l . Notice that H is saturated in T X , and stable under the Lie bracket. So it defines a foliation of rank r H < r on X . Note also that Q is torsion-free. The sheaves F , H and Q are locally free in a neighborhood of ℓ , and we have an exactsequence of vector bundles 0 → H | ℓ → F | ℓ → Q | ℓ → . Notice that Q | ℓ ⊂ ( π ∗ T P l ) | ℓ ≃ O ⊕ l P , and det( H | ℓ ) ≃ O P ( r − ⊗ det( Q | ℓ ) ∗ ⊂ ∧ r H ( T P m | ℓ ). Sodeg (cid:0) det( Q ) | ℓ (cid:1) ∈ {− , } . We claim that det( Q ) | ℓ ≃ O P . Suppose to the contrary that det( Q ) | ℓ ≃ O P ( − r H = r − m , H = T X/ P l , and F | ℓ ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O P ( − F induces arank 1 foliation on P l . Let C ⊂ P l be the germ of a (complex analytic) leaf of this foliation. Then π − ( C )is a germ of a leaf of F , and thus f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O P , a contradiction. This proves thatdet( Q ) | ℓ ≃ O P . Since det( H | ℓ ) ≃ O P ( r − ⊂ ∧ r H ( T P m | ℓ ) and r H < r , one of the following occurs.(a) m ≥ r , H | ℓ ≃ O P (1) ⊕ r − , and Q | ℓ ≃ O P ; or(b) m = r − H = T X/ P l , and Q | ℓ ≃ O P ⊕ O P .First we treat case (a). By generic flatness, we have H | Y ( T P m . Notice that H | Y is closed under theLie bracket, and det (cid:0) H | Y (cid:1) ≃ O P m ( r − ⊂ ∧ r − T P m . By Bott’s formulae, H | Y is saturated in T P m . So H | Y is a degree 0 foliations of rank r − P m , i.e., H | Y ≃ O P m (1) ⊕ r − ( T P m .By [Har80, Proposition 1.9], Q ∗∗ is locally free, and thus Q ∗∗ ≃ π ∗ R for some invertible subsheaf R ⊂ T P l .We have O X ( r − ⊗ π ∗ L ≃ det( F ) ≃ det( H ) ⊗ π ∗ R .Let the notation be as in 7.8, with T = P l and s = r −
1. So we have a a surjective morphism p : X → Z such that H | X = T X /Z , and det( H ) ≃ π ∗ (cid:0) det( V ) (cid:1) ⊗ O X ( r − R ≃ det( V ∗ ) ⊗ L ⊂ T P l . Recallfrom 7.8 the exact sequence of vector bundles on T :0 → K | T → E | T → V ∗ | T → , where T ⊂ P l is an open subset such that codim P l ( P l \ T ) ≥
2. Since E is ample, and V has rank r −
1, wemust have det( V ∗ ) ≃ O P l ( k ) for some k ≥ r −
1. By Bott’s formulae, r
3. Moreover, if l ≥
2, then r = 2and L ≃ O P l . If r = 3, then l = 1 and L ≃ O P . If r = 2 and l = 1, then L ≃ O P ( k ) for some k ∈ { , } .By Lemma 6.7, F induces a rank 1 foliation G ⊂ T Z on Z such that det( F | X ) ≃ det (cid:0) T X /Z (cid:1) ⊗ p ∗ G .Notice that G is an invertible sheaf by [Har80, Proposition 1.9]. Recall that codim X ( X \ X ) ≥
2. So we have G ≃ q ∗ (cid:0) det( V ∗ ) ⊗ L (cid:1) , and det( V ∗ ) is ample since so is E . Recall also that q : Z → T is a P m − r +1 -bundle.So through a general point of Z there exists a complete curve B not contracted by q , and avoiding thesingular locus of the foliation G ⊂ T Z . The restriction G | B is an ample line bundle. Thus G is a foliation byrational curves by Theorem 1.2. The general leaf of F is the closure of the inverse image by p of a generalleaf of G . Hence it is algebraic and rationally connected.Next we consider case (b): H = T X/ P l , l ≥
3, and Q | ℓ ≃ O P ⊕ O P . By Lemma 6.7, F is thepullback via π of a rank 2 foliation G ⊂ T P l . In particular, det( F ) ≃ det( T X/ P l ) ⊗ π ∗ det( G ). Hencedet( G ) ≃ det( E ) ⊗ L ⊂ ∧ T P l . By assumption, L is nef and E is ample of rank ≥
2. So, by Bott’sformulae, we must have det( Q ) ≃ O P l (2), rank( E ) = 2 and L ≃ O P l . Since E is an ample vector bundle, E | P ≃ O P (1) ⊕ for any line P ⊂ P l . By [OSS80, Theorem 3.2.1], E ≃ O P l (1) ⊕ O P l (1). Thus X ≃ P × P l ,and G is a degree zero foliation on P l . The leaves of G are 2-planes containing a fixed line in P l . Hence theleaves of F are algebraic and rationally connected. (cid:3) Remark 7.11.
Let the notation and assumptions be as in Proposition 7.10, and denote by ( ˜
F , ˜∆) thegeneral log leaf of F . If m ≥
2, then π induces a P r − -bundle structure π ˜ F : ˜ F → P . If l = 1, then then˜∆ is a prime divisor of π ˜ F -relative degree 1. If l > r = 2), then ˜∆ is the union of a primedivisor of π ˜ F -relative degree 1 and a fiber of π ˜ F . If m = 1, π induces a P -bundle structure π ˜ F : ˜ F → P ,and ˜∆ is a fiber of π ˜ F . Notice that in all cases ( ˜ F , ˜∆) is log canonical.
Remark 7.12.
In Section 9 we will classify locally free sheaves E on P l for which X = P ( E ) admits a delPezzo foliation F * T X/ P l . Moreover, we will give a precise geometric description of such foliations. ARAUJO
AND ST´EPHANE
DRUEL
Now we consider another special case of Theorem 1.1.
Proposition 7.13.
Let F be a del Pezzo foliation of rank r ≥ on a smooth projective variety X . Let H be minimal dominating family of rational curves on X , with associated rationally connected quotient π : X → T . Suppose that F | X T X /T , and f ∗ F ≃ O P (1) ⊕ r − ⊕ O P for a general member [ f ] ∈ H .Then there are integers l ≥ , m ≥ , and an ample locally free sheaf E on P l such that X ≃ P P l ( E ) .Moreover, under this isomorphism, π becomes the natural projection P P l ( E ) → P l , and det( F ) ≃ O X ( r − ,where O X (1) is the tautological line bundle on P P l ( E ) .In particular, F is algebraically integrable with rationally connected general leaf by Proposition 7.10.Proof. Write det( F ) = A ⊗ r − for an ample line bundle A on X . Then f ∗ A ≃ O P (1) for any [ f ] ∈ H ,which implies that H is unsplit. By [ADK08, Lemma 2.2], we may assume that codim X ( X \ X ) ≥ T issmooth, and π is proper, surjective, equidimensional, and has irreducible and reduced fibers. We denote by m the relative dimension of π , and set l := dim T .First we show that π is a P m -bundle. Set H := F | X ∩ T X /T = ker( F | X → π ∗ T T ). Let [ f ] ∈ H be a general member. By assumption, f ∗ F ≃ O P (1) ⊕ r − ⊕ O P and F | X T X /T . Moreover f ∗ T T ≃ O ⊕ dim( T ) P . So we conclude that H has rank equal to r − < m , and f ∗ H ≃ O P (1) ⊕ r − . Thus π is a P m -bundle by [ADK08, Proposition 2.7]. Denote by Y ≃ P m a general fiber of π .Let H ⊂ F be a saturated subsheaf extending H ⊂ F | X , and set Q := ( F / H ) ∗∗ . Then H isreflexive, Q is locally free of rank one by [Har80, Proposition 1.9], and det( F ) ≃ det( H ) ⊗ Q . Moreover, Q | Y ≃ O P m . Therefore there exists an invertible subsheaf G ⊂ T T such that Q | X = π ∗ G , and an inclusion(7.1) ( A | X ) ⊗ r − ≃ det( F | X ) ֒ → ∧ r − T X /T ⊗ π ∗ G . The next step is to show that G induces a foliation by rational curves on T . For this purpose, let B ⊂ X be a general smooth complete curve, set G B := ( π | B ) ∗ G , X B := X × T B , and consider theinduced P m -bundle π B : X B → B . Denote by A X B the ample line bundle on X B obtained by pulling back A from X . Then (7.1) yields an inclusion A ⊗ r − ⊗ π ∗ B ( G ∗ B ) ⊂ ∧ r − T X B /B . It follows from [ADK08, Lemma 5.2] that deg B ( G B ) >
0. By Theorem 1.2, this implies that the general leafof the foliation induced by G ⊂ T T is a rational curve.Let C ≃ P be a smooth compactification of a general leaf C of the foliation induced by G ⊂ T T , andlet X C be the normalization of the closure of π − ( C ) in X , with induced morphism π C : X C → C . Denoteby A X C the pullback of A to X C . Every fiber of π C is generically reduced and irreducible since it has degreeone with respect to the ample line bundle A X C . Since C is smooth, π C is flat, and X C is normal, every fibersatisfy Serre’s condition S , and hence it is integral. Therefore π C : X C → C ≃ P is a P m -bundle by [Fuj75,Corollary 5.4]. Notice that the image of X C in X is invariant under F . Thus, by Lemma 3.5, F inducesa foliation F X C of rank r on X C such that det( F X C ) ≃ ( A X C ) ⊗ r − ⊗ π ∗ C L for some nef line bundle L on C . It follows from Proposition 7.10 that r ∈ { , } , and F X C is algebraically integrable with rationallyconnected general leaf. Hence the same holds for F .Let ( ˜ F , ˜∆) be a general log leaf of F . Denote by ˜ e : ˜ F → X the natural morphism, and by A ˜ F thepullback of A to ˜ F . Recall the formula:( A ⊕ r − F ) ⊗ O ˜ F ( ˜∆) ≃ O ˜ F (cid:0) ˜ e ∗ ( − K F ) + ˜∆ (cid:1) = O ˜ F ( − K ˜ F ) . Notice that ˜ F is also the normalization of a general leaf of F X C for some C as above. By Remark 7.11, π C induces a P r − -bundle structure π ˜ F : ˜ F → P . So we can write ˜ F ≃ P P (cid:0) ( π ˜ F ) ∗ A ˜ F (cid:1) , and ( π ˜ F ) ∗ A ˜ F ≃ O P ( a ) ⊕ · · · ⊕ O P ( a r ), with 1 ≤ a ≤ · · · ≤ a r . Then O ˜ F ( − K ˜ F ) ≃ π ∗ ˜ F O P ( − a − · · · − a r + 2) ⊗ A ⊗ r ˜ F .Substituting this in the formula above, we get:˜∆ ∈ (cid:12)(cid:12) π ∗ ˜ F O P ( − a − · · · − a r + 2) ⊗ A ˜ F (cid:12)(cid:12) . In particular, we see that ˜∆ contains a unique irreducible component that dominates P under π ˜ F , whichwe denote by ˜ σ . Moreover, the restriction of π ˜ F to ˜ σ makes it a P r − -bundle over P .Let σ ′ : P → ˜ F be the section of π ˜ F corresponding to a general surjection E ։ O P ( a r ), and set C ′ = σ ′ ( P ) ⊂ ˜ F . Then C ′ is a moving curve on ˜ F , and thus0 ≤ ˜∆ · C ′ = − a − · · · − a r − + 2 ≤ − r. N FANO FOLIATIONS 23 If r = 3, then ˜∆ · C ′ = 0, which implies that ˜∆ does not contain any fiber of π ˜ F as irreducible component,i.e., ˜∆ = ˜ σ . Similarly, if r = 2, then either ˜∆ = ˜ σ , or ˜∆ = ˜ σ + ˜ f , where ˜ f is a fiber of π ˜ F . In any case,we see that ( ˜ F , ˜∆) is log canonical. Therefore, by Proposition 5.3, there is a point x ∈ X contained in theclosure of a general leaf of F .Suppose that ˜∆ = ˜ σ . We will show that l = 1. Let T be the normalization of the closure of T inChow( X ), with universal family morphisms π : U → T and e : U → X . We denote by F U the foliation on U induced by F , and by G T the foliation on T induced by G . Consider the commutative diagram:˜ F π ˜ F ❆❆❆❆❆❆❆❆ / / ˜ e $ $ X Cπ C (cid:15) (cid:15) / / U π (cid:15) (cid:15) e / / X.C / / T Let E ⊂ U be the exceptional locus of e . Since X is smooth, E has pure codimension one in U . Since c ( A ) m · Y = 1, the fibers of π are irreducible, and thus E is a union of fibers of π .We claim that the image of ˜ F in U does not meet E . This implies that e is an isomorphism over aneighborhood of x . Suppose otherwise that there is a point c ∈ C that is mapped into π ( E ) ⊂ T . Set˜ f := π − F ( c ) ⊂ ˜ F , and denote by x ∈ e ( E ) ⊂ X the image of a general point of ˜ f . Note that e − ( x ) ispositive dimensional, while its intersection with the image of ˜ F in U is zero-dimensional. Hence there is apositive dimensional family of general leaves of F U meeting e − ( x ), yielding a positive dimensional familyof general leaves of F passing through x . This shows that ˜ f ⊂ ˜∆, contradicting the assumption that ˜∆ = ˜ σ ,and proving the claim.Since e is an isomorphism over a neighborhood of x , and the general leaf of F contains x , we concludethat the general leaf of G T contains the point t = π (cid:0) e − ( x ) (cid:1) ∈ T . Let u ∈ π − ( t ) be a general point, andlet c ∈ C be a point mapped to t . Then there is a leaf of F X C whose image in U contains the point u .Thus, if l = dim T >
1, then we can find a positive dimensional family of general leaves of F U containing u .Since e is birational at u , we conclude that u lies in the singular locus of F . Thus e (cid:0) π − ( t ) (cid:1) is contained inthe singular locus of F , and ˜∆ contains a fiber of π ˜ F , contradicting our assumptions. We have just provedthat if ˜∆ = ˜ σ , then l = 1, and thus X = X C ≃ P P (cid:0) π ∗ A (cid:1) .From now on suppose that r = 2 and ˜∆ = ˜ σ + ˜ f . In particular we must have l >
1. We must show thatin this case π extends to a P m -bundle π : X → P l . Let H ′ ⊂ RatCurves( X ) be a family that parametrizes(among possibly other curves) the image of ˜ σ in X . Since A ˜ F · ˜ σ = 1, H ′ is unsplit. Notice that H and H ′ are numerically independent in N ( X ), and X is ( H, H ′ )-rationally connected. The latter is because˜ F is itself rationally connected with respect to families obtained from restriction of H and H ′ , and thereis a point x ∈ X contained in the closure of a general leaf of F . It follows from [Kol96, IV.3.13.3] that ρ ( X ) = 2.Next we show that H generates an extremal ray of the Mori cone NE( X ) (see [KM98] for the definitionand properties of the Mori cone). First we claim that the common point x lies in the image of ˜ σ in X .In any case, x ∈ ˜ e ( ˜∆) by Lemma 5.6. Notice that F X C is regular at the generic point of any fiber of π C .Thus the image of the singular locus of F X C in ˜ F is ˜ σ . Hence, given any point in the image of ˜ F \ ˜ σ in X C , there is a leaf of F X C that does not pass through this point. So we must have x ∈ ℓ ′ := ˜ e (˜ σ ). Nowlet Z ⊂ X be the closure of the union of the curves ℓ ′ when F runs through general leaves of F . Then Z is irreducible, it dominates T , and dim Z = dim T . By [Kol96, IV.3.13.3], N ( Z ) is generated by [ ℓ ′ ]. Byconstruction, a general point of X can be connected to Z by a curve from H . Since H is unsplit, this is aclosed condition, and it holds for every point of X . It follows from [BSW92, (Proof of) Lemma 1.4.5] (seealso [Occ06, Remark 3.3]) that any curve on X is numerically equivalent to a linear combination λℓ ′ + µℓ ,where λ ≥ ℓ is a curve parametrized by H . This implies that [ ℓ ] generates an extremal ray of NE( X ).Indeed, suppose [ ℓ ] = α + α , with α , α ∈ NE( X ) \ { } . Write α i = λ i [ ℓ ′ ] + µ i [ ℓ ], with λ i ≥
0. Then(1 − µ − µ ) ℓ ≡ ( λ + λ ) ℓ ′ . Since ℓ and ℓ ′ are numerically independent, we must have λ = λ = 0. Thus[ ℓ ] generates an extremal ray of NE( X ), and π extends to a morphism ¯ π : X → W , namely the contractionof the extremal ray generated by H . ARAUJO
AND ST´EPHANE
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We claim that the morphism ¯ π : X → W is equidimensional. Let X t be a component of a fiber of ¯ π .Since ¯ π is the contraction of the extremal rays generated by H , N ( X t ) is generated by classes of curvesfrom the family H . Moreover, since any point of X can be connected to Z by a curve from H , X t meets Z . On the other hand, N ( Z ) is generated by [ ℓ ′ ]. Thus Z ∩ X t must be 0-dimensional. We conclude fromthese observations that dim X t + dim Z = dim X , and ¯ π is equidimensional.By [Fuj87, Lemma 2.12], W is smooth and π is a P m -bundle. Recall from the beginning of the proof thatthere exists a line bundle G ⊂ T W on W such that G · B > B ⊂ W . Since ρ ( W ) = 1, it followsthat G is ample. By [Wah83], since l >
1, we must have ( W, G ) ≃ ( P l , O P l (1)). (cid:3) We end this section by proving Theorem 1.1.
Proof of Theorem 1.1.
Write det( F ) = A ⊗ r − for an ample line bundle A on X . Set r := r F > n := dim( X ) ≥
3. The proof is by induction on n .By Remark 2.9, we know that X is uniruled. Fix a minimal dominating family H of rational curves on X , and let π : X → T be the H -rationally connected quotient of X . Step . Let [ f ] ∈ H be a general member. We show that one of the following holds.(1) f ∗ F ≃ O P (1) ⊕ r − ⊕ O P , and H is unsplit, or(2) f ∗ F ≃ O P (2) ⊕ O P ( r = 2), or(3) f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O ⊕ P ( r > H is unsplit.Write f ∗ F ≃ O P ( a ) ⊕ · · · ⊕ O P ( a r ), where a ≤ · · · ≤ a r and a + · · · + a r = ( r − A · ℓ . By[Kol96, IV.2.9], f ∗ T X ≃ O P (2) ⊕ O P (1) ⊕ d ⊕ O ⊕ ( n − d − P where d = deg( f ∗ T X ) − ≥
0. By Lemma 6.10, f ∗ F cannot be ample. Therefore, either the a i ’s are as in one of the three cases listed above, or f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O P ( − A · ℓ = 1,and hence H is unsplit.We claim that f ∗ F O P (2) ⊕ O P (1) ⊕ r − ⊕ O P ( − T P ⊂ f ∗ F forgeneral [ f ] ∈ H , and Lemma 6.9 implies that T X /T ( F | X . Hence f ∗ T X /Y ≃ O P (2) ⊕ O P (1) ⊕ r − .By [Ara06], π is a projective space bundle, and we may assume that codim X ( X \ X ) ≥
2. By lemma6.7, F induces a foliation by curves on T . For a general point t in T , let C ⊂ T be the germ of theleaf (in the complex analytic topology) through t . Then π − ( C ) is the germ of a leaf of F , and thus f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O P , a contradiction. Step . We treat case (1): f ∗ F ≃ O P (1) ⊕ r − ⊕ O P and H is unsplit.By [ADK08, Lemma 2.2], we may assume that codim X ( X \ X ) ≥ T is smooth, and π is proper,surjective, equidimensional, and has irreducible and reduced fibers.If dim( T ) = 0, then, since the family H is unsplit, ρ ( X ) = 1, and the result follows from Proposition 7.7.So we may assume that dim( T ) >
1. By Proposition 7.13, we may assume also that F | X ( T X /T . Denoteby Y a general fiber of π , and set m = dim Y . Then F | Y ⊂ T Y is a Del Pezzo foliation on the smoothprojective variety Y . If Y P m , then the result follows by induction on the dimension. So we may assumethat Y ≃ P m . Since A | Y ≃ O P m (1), π is a P m -bundle by [Fuj75, Corollary 5.4]. By removing a subset ofcodimension ≥ T if necessary, we may assume that F | X is a subbundle of T X /T .Let B ⊂ X be a general smooth complete curve, set X B := X × T B , and consider the induced P m -bundle π B : X B → B . Denote by A X B the pullback from X of the line bundle A , and by F X B ( T X B /B the pullback of F . Then det( F X B ) = A ⊗ r − X B . Thus F X B is algebraically integrable and has rationallyconnected general leaf by Proposition 7.9. Since B is general, the same holds for F . Step . We treat case (2): r = 2 and f ∗ F ≃ O P (2) ⊕ O P . We will show in particular that if F | X T X /T ,then π is a P -bundle, and F is the pullback via π of a foliation by rational curves on T .Let x ∈ X be a general point, and C x ⊂ P ( T x X ∗ ) the variety of minimal rational tangents at x associatedto H . Since T P ⊂ f ∗ F for general [ f ] ∈ H , we have C x ⊂ P ( F ∗ x ) ⊂ P ( T x X ∗ ).We claim that dim( H x ) = 0. Indeed, suppose dim( H x ) >
0. Then C x = P ( F ∗ x ) ≃ P . By [Ara06],after shrinking X and T if necessary, π : X → T becomes a P -bundle over a smooth base. Hence F | X = T X /T ֒ → T X , and thus f ∗ F ≃ O P (2) ⊕ O P (1), a contradiction. This proves the claim. N FANO FOLIATIONS 25
Suppose ♯ ( H x ) >
2, and fix [ ℓ ] ∈ H x . Then the surface obtained as the union of curves from H meeting ℓ at general points is invariant under F , and thus it is the leaf of F through x . This shows that the generalleaf of F is algebraic and rationally connected.From now on we assume that ♯ ( H x ) = 1. After shrinking X and T if necessary, we may assume that π is a P -bundle over a smooth base, and there is an exact sequence0 → T X /T → F | X → ( π ∗ G ) , where G is an invertible subsheaf of T T . By Lemma 6.12, G defines a foliation by rational curves on T .The general leaf of F is the closure of the inverse image by π of a general leaf of G . Hence it is algebraicand rationally connected. Step . Finally we treat case (3): r > f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O ⊕ P and H is unsplit. In particular,we show that one of the following holds: • π is a quadric bundle of relative dimension r −
1, and F is the pullback via π of a foliation byrational curves on T . • π is a P r − -bundle, and F is the pullback by π of a foliation by rationally connected surfaces on T .Since O P (2) ⊂ f ∗ F , we must have T X /T ⊂ F | X by lemma 6.9. Thus f ∗ T X /T ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O ⊕ k P with k ∈ { , , } , and dim T > k = 2. Then F | X = T X /T . In particular, F has log canonical singularities along ageneral leaf. But this contradicts Proposition 5.3, which asserts that there is a common point through ageneral leaf of F .Next suppose that k = 1, and denote by Y the general fiber of π . Then dim( T ) >
2, dim Y = r −
1, anddet( T Y ) ≃ ( A | Y ) ⊗ r − . Thus Y ≃ Q r − by [KO73]. By Lemmas 6.7 and 6.12, F induces a foliation G byrational curves on T . The general leaf of F is the closure of the inverse image under π of a general leaf of G . Hence it is algebraic and rationally connected.Finally we suppose that k = 0, and denote by Y the general fiber of π . Then dim( T ) >
3, dim Y = r − T Y ) ≃ ( A | Y ) ⊗ r − . Thus Y ≃ P r − by [KO73], and, since H is unsplit, π can be extended incodimension 1 to a P r − -bundle over a smooth base. We still denote this extension by π : X → T . Bylemma 6.7, F induces a rank 2 foliation G on T . By removing a subset of codimension ≥ T ifnecessary, we may assume that G is a subbundle of T T . Since π is smooth, F | X = ( dπ ) − ( π ∗ G ), andthere exists an exact sequence of vector bundles0 → T X /Y → F | X → π ∗ G → . Let Q be a coherent sheaf of O X -modules extending π ∗ G . Let B ⊂ X be a general complete intersectioncurve for Q and A in the sense of Mehta-Ramanathan. Consider the induced P r − -bundle π B : X B = B × T X → B , with natural morphisms: X Bπ B (cid:15) (cid:15) / / q X π (cid:15) (cid:15) / / X.B n / / T Then O X B (cid:0) q ∗ ( − K F ) (cid:1) = O X B ( − K X B /B ) ⊗ π ∗ B (cid:0) n ∗ (det( G )) (cid:1) . We know that − K X B /B is not ample by [Miy93,Theorem 2]. Hence deg B (cid:0) n ∗ (det( G )) (cid:1) > n ∗ G is ample, then the general leaf of G is algebraic and rationally connected by Theorem 1.2. Sincethe general leaf of F is the closure of the inverse image under π of a general leaf of G , it follows that thegeneral leaf of F is algebraic and rationally connected. So we may assume that n ∗ G is not ample, andhence not semistable.By Lemma 7.4, there is a saturated rank 1 subsheaf M ⊂ G such that deg B ( n ∗ M ) > deg B ( n ∗ G ) > M ⊂ T T is a rational curve. Let C ≃ P be a smoothcompactification of a general leaf C of M , and let X C be the normalization of the closure of π − ( C ) in X , with induced morphism π C : X C → C . Denote by A X C the pullback of A to X C . Every fiber of π C is ARAUJO
AND ST´EPHANE
DRUEL generically reduced and irreducible since it has degree one with respect to the ample line bundle A X C . Since C is smooth, π C is flat, and X C is normal, every fiber satisfy Serre’s condition S , and hence it is integral.Therefore π C : X C → C ≃ P is a P m -bundle by [Fuj75, Corollary 5.4]. In particular, X C is smooth.By removing a subset of codimension ≥ T if necessary, we may assume that M is a line bundle on T , and that there is a line bundle M ′ on T fitting into an exact sequence0 → M → G → M ′ → . Let L and L ′ be the unique line bundles on X extending π ∗ M and π ∗ M ′ , respectively. Let K bethe saturated subsheaf of T X extending ( dπ ) − ( π ∗ M ). Then K is a foliation of rank r − X , anddet( K ) ≃ A ⊗ r − ⊗ L ′∗ . Notice also that X C is the normalization of a general leaf of K , and that K isregular along a general fiber of π .Denote by ( X C , D ) the general log leaf of K , and by q : X C → X the natural morphism. Since K is regular along a general fiber of π , D is supported on a finite union of fibers of π C . Recall that O X C ( − K X C ) ≃ q ∗ det( K ) ⊗ O X C ( D ). So we have q ∗ L ′ ≃ O X C ( K X C ) ⊗ A ⊗ r − X C ⊗ O X C ( D ) . On the other hand, since dim X C = r − A X C is ample, Fujita’s theorem ([, ]) implies that O X C ( K X C ) ⊗ A ⊗ r − X C is nef. We remark that exactly one of the following holds.(a) For any moving section σ ⊂ X C of π C , we have q ∗ L ′ · σ > O X C ( − K X C ) ≃ A ⊗ r − X C and D = 0. In this case X C ≃ P × P , A X C ≃ O P × P (1 ,
1) and q ∗ L ′ ≃ O X C .Let W be the normalization of the closure in Chow( X ) of the subvariety parametrizing general leaves of K , and U the normalization of the universal cycle over W , with universal family morphisms: U e / / π (cid:15) (cid:15) X .W
There are open subsets X ⊂ X and W ⊂ W , with codim X ( X \ X ) ≥
2, such that e is an isomorphismover X , and π induces an equidimensional morphism π : X → W with connected fibers. Notice that K | X = T X /W . By Lemma 6.7, F induces a rank 1 foliation N ⊂ T W , and an exact sequence0 → K | X → F | X → ( π ∗ N ) ∗∗ . Thus, there is a canonically defined effective divisor D on X such that L ′ | X ⊗ O X ( D ) ≃ ( π ∗ N ) ∗∗ . By removing a subset of codimension ≥ W if necessary, we may assume that W is smooth and N islocally free. Our aim is to show that the general leaf of N is a rational curve. Since the general leaf of F is the closure of the inverse image under π of a general leaf of N , it will follow that the general leaf of F is algebraic and rationally connected.Let σ ⊂ X C be a moving section. Then its image in X is a moving curve. Let ℓ ′ be a general deformationof q ( σ ). In particular, we may assume that ℓ ′ ⊂ X ∩ X , and that both F and K are regular along ℓ ′ by[Kol96, II.3.7].Suppose that ℓ ′ is not tangent to K . By Lemma 6.5, we must have D = 0. So we are in case (a) above,and thus L ′ · ℓ ′ = q ∗ L ′ · σ >
0. Now consider the curve π ( ℓ ′ ) in W . It is a moving curve, the foliation N ⊂ T W is regular along π ( ℓ ′ ), and N · π ( ℓ ′ ) >
0. By Theorem 1.2, this implies that the leaves of N are rational curves.Next we suppose that ℓ ′ is tangent to K . Set C ′ = π ( ℓ ′ ) ⊂ T . Then C ′ is a leaf of M ⊂ T , and M isregular along C ′ , i.e., M | C ′ ≃ T C ′ . Moreover, the same holds for a general deformation of C ′ by Lemma 6.5.Thus T T | C ′ ≃ O P (2) ⊕ O ⊕ dim T − P . As in Step 1, we see that G | C ′ ≃ O P (2) ⊕ O P . This implies that L ′ · ℓ ′ = M · C ′ = 0. So we are in case (b) above: X C ≃ P × P , A X C ≃ O P × P (1 ,
1) and D = 0. Wemay assume that σ is the ruling of X C dominating C . Thus ℓ ′ determines an unsplit dominating family H ′ of rational curves on X , and π : X → W is nothing but the ( H, H ′ )-rationally connected quotient of X .By Lemma 6.12, N is a foliation by rational curves on W . (cid:3) N FANO FOLIATIONS 27 On del Pezzo foliations with mild singularities
In this section we prove Theorem 1.3. In fact, Theorem 1.3 follows immediately from Theorem 8.1 belowand Proposition 5.3.
Theorem 8.1.
Let F ( T X be a del Pezzo foliation of rank r on a smooth projective n -dimensional variety X . Suppose that F is locally free along the closure of a general leaf. Then (1) either X a P m -bundle over P n − m , with ≤ m ≤ n − , or (2) for any minimal dominating family of rational curves on X , with associated rationally connectedquotient π : X → T , we have F | X ⊂ T X /T . When X P n , we know from Theorem 1.1 that a del Pezzo foliation F on X is algebraically integrable.In the proof of Theorem 8.1 in this case, we will consider a suitable resolution of singularities of the generalleaf of F , whose existence is guaranteed by the following theorem. Theorem 8.2 ([Kol07, Theorem 3.35, 3.45] and [GKK10, Corollary 4.7]) . Let X be a normal variety. Thenthere exists a resolution of singularities d : Y → X such that • d is an isomorphism over X \ Sing( X ) , and • d ∗ T Y ( − log D ) ≃ T X where D is the largest reduced divisor contained in d − (Sing( X )) . We call a resolution d as in Theorem 8.2 a canonical desingularization of X . Let X be a normal projective variety, and F an algebraically integrable 1-Gorenstein foliation on X .We denote by F the closure of a general leaf of F , ˜ F its normalization, ( ˜ F , ˜∆) the corresponding log leaf,and Y a canonical desingularization of ˜ F : Y d / / ¯ e $ $ ˜ F / / ˜ e @ @ F e / / X. Suppose that F is locally free along F . Let D ⊂ Y be the largest reduced divisor contained in d − (Sing( ˜ F )). Then ¯ e ∗ F ⊂ T Y ( − log D ) ⊂ T Y . Therefore there exists an effective divisor ∆ on Y suchthat ¯ e ∗ ( − K F ) + ∆ + D = − K Y . Recall from Definition 3.4 that K ˜ F + ˜∆ = ˜ e ∗ K F . Hence we have ∆ + D = d ∗ ˜∆ − K Y/ ˜ F . MoreoverSing( ˜ F ) ⊂ Supp( ˜∆) by Lemma 5.6. Thus Supp(∆ + D ) ⊂ d − (Supp( ˜∆)). Remark 8.4.
Let the notation and assumptions be as in 8.3 above, and suppose moreover that F is a Fanofoliation. Then ∆ + D = 0. Indeed, if ∆ + D = 0, it follows from the above discussion that ˜ F is smooth and˜∆ = 0. Therefore, F is induced by an almost proper map X T , contradicting Proposition 5.3. Proof of Theorem 8.1.
Write det( F ) = A ⊗ r − , with A an ample line bundle on X , and denote by S thesingular locus of F . We follow the notation introduced in 8.3 above.Let H be a minimal dominating family of rational curves on X , and π : X → T the associated rationallyconnected quotient. Let [ f ] ∈ H be a general member. Suppose that F | X T X /T . Recall from the proofof Theorem 1.1 that one of the following holds.(1) Either f ∗ F ≃ O P (1) ⊕ r − ⊕ O P , or(2) r = 2, f ∗ F ≃ O P (2) ⊕ O P , π is a P -bundle, and F is the pullback via π of a foliation by rationalcurves on T , or(3) r ≥ H is unsplit, f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O ⊕ P , and one of the following holds.(a) Either π is a quadric bundle of relative dimension r −
1, and F is the pullback via π of afoliation G by rational curves on T , or(b) π is a P r − -bundle, and F is the pullback by π of a foliation G by rationally connectedsurfaces on T .If we are in case (1), then π makes X a P m -bundle over P n − m , with 1 ≤ m ≤ n −
1, by Proposition 7.13.Suppose we are in case (2). Notice that F is regular along a general curve from H . ARAUJO
AND ST´EPHANE
DRUEL
The restriction of π to F ∩ X induces a surjective morphism with connected fibers ϕ : Y → P . Let f ⊂ Y be a general fiber of ϕ , and set ℓ := ¯ e ( f ) ⊂ F . Then ℓ ∩ S = ∅ , and F is smooth along ℓ . Thus f ∩ Supp(∆ + D ) = ∅ . Hence Supp(∆ + D ) is a union of irreducible components of fibers of ϕ .We claim that Y = ˜ F ∼ = P × P . Suppose otherwise. Then there exists a section C of ϕ such that C <
0. Since C Supp(∆ + D ), we have:1 ≥ C + 2 = − K Y · C = − K F · ¯ e ∗ ( C ) + (∆ + D ) · C ≥ . Hence we must have (∆+ D ) · C = 0, and thus the inclusion ¯ e ∗ F ⊂ T Y is an isomorphism in a neighborhoodof C . In particular, no smooth fiber of ϕ is contained in Supp(∆ + D ). Suppose ϕ has reducible fibers,and let C be an irreducible component of a reducible fiber such that C ∩ C = ∅ . Then C < C Supp(∆ + D ). As before, we get that (∆ + D ) · C = 0, and thus the inclusion ¯ e ∗ F ⊂ T Y is anisomorphism in a neighborhood of C . Proceeding by induction, we conclude that no irreducible componentof a reducible fiber of ϕ is contained in Supp(∆+ D ). Thus ∆+ D = 0. But this is impossible by Remark 8.4.Therefore we must have Y = ˜ F ∼ = P × P , as claimed. In particular, D = 0.Let C be a section of ϕ such that C = 0, and denote by f a fiber of ϕ . By Remark 8.4, ∆ = 0. Thus∆ ≡ mf for some integer m ≥
1. We have:2 = − K Y · C = − K F · ¯ e ∗ ( C ) + ∆ · C ≥ m ≥ . Hence we must have − K F · ¯ e ∗ ( C ) = 1 and ∆ = f . Set ℓ ′ = ¯ e ( C ), and let H ′ be the family of rationalcurves on X containing [ ℓ ′ ]. Then − K F · ℓ ′ = 1, and thus H ′ is unsplit. Let [ f ′ ] ∈ H ′ be a general member.As in Step 1 of the proof of Theorem 1.1, we see that ( f ′ ) ∗ F ≃ O P (1) ⊕ O P . Let π ′ : X ′ → T ′ be the H ′ -rationally connected quotient of X . Notice that ℓ and ℓ ′ are numerically independent in X . Therefore ℓ is not contracted by π ′ . On the other hand, ℓ is contained in a leaf of F . So we must have F | X ′ T X ′ /T ′ .From the analysis of case (1) above and Proposition 7.13, we conclude that π ′ makes X a P n − -bundle over P .Next we show that case (3a) does not occur.Suppose to the contrary that H is unsplit, π : X → T is a quadric bundle of relative dimension r − F is the pullback via π of a foliation G by rational curves on T . By [ADK08, Lemma 2.2], π canbe extended in codimension 1 in X to a proper surjective equidimensional morphism with irreducible andreduced fibers. We still denote this extension by π : X → T .The morphism π induces a morphism π B : Y → P , where P B is a smooth compactification of ageneral leaf B of the foliation G on T . Since F is regular along a general fiber of π , Supp (cid:0) ∆ + D (cid:1) is aunion of irreducible components of fibers of π B .Let ℓ ′ ⊂ Y be a general curve from a minimal horizontal family of rational curves with respect to π B .Then ℓ ′ Supp(∆ + D ), and − K Y · ℓ ′ = − K F · ¯ e ∗ ℓ ′ + (∆ + D ) · ℓ ′ = ( r − A · ¯ e ∗ ℓ ′ + (∆ + D ) · ℓ ′ ≥ r − ≥ . On the other hand, by Lemma 6.3, − K Y · ℓ ′ ≤
2. So we must have r = 3, A · ℓ ′ = 1, and (∆ + D ) · ℓ ′ = 0.The latter implies that ¯ e ( ℓ ′ ) ∩ S = ∅ . Thus ¯ e ( ℓ ′ ) ∩ Sing( F ) = ∅ , and all fibers of π B meet the regular locusof F . By Remark 8.4, ∆ + D = 0. Thus π B has at least one reducible fiber, and at least one irreduciblecomponent of such reducible fiber meets the regular locus of F . By letting B run through general leaves ofthe foliation G , the images in X of reducible fibers of π B sweep out a divisor on X . But this contradictsthe fact that π has irreducible fibers in codimension 1.Finally we show that case (3b) can only occur if X is a P m -bundle over P n − m .So suppose H is unsplit, r ≥ f ∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O ⊕ P , π is a P r − -bundle, and F is thepullback via π of a foliation G by rationally connected surfaces on T . Recall that π can be extended toa P r − -bundle in codimension 1 in X . We still denote this extension by π : X → T .Let Z be the normalization of a general leaf of G . Then π induces a P r − -bundle π Z : Y → Z , where Y and Z are dense open subsets of Y and Z , respectively. Let H ′ Y be a minimal h-dominating family ofrational curves with respect to π Z , and [ ℓ ′ ] ∈ H ′ Y a general member. Fix a family H ′ of rational curves on X containing a point of RatCurves n ( X ) corresponding to ¯ e ( ℓ ′ ).Since F is regular along a general fiber of π , ℓ ′ Supp(∆ + D ), and − K Y · ℓ ′ = − K F · ¯ e ∗ ℓ ′ + (∆ + D ) · ℓ ′ = ( r − A · ¯ e ∗ ℓ ′ + (∆ + D ) · ℓ ′ ≥ r − ≥ . N FANO FOLIATIONS 29
On the other hand, by Lemma 6.3, − K Y · ℓ ′ ≤
3. So − K Y · ℓ ′ ∈ { , } . Moreover A · ¯ e ∗ ℓ ′ = 1, and thus H ′ is unsplit.First let us assume that − K Y · ℓ ′ = 3. Then r = 4 − (∆ + D ) · ℓ ′ ∈ { , } . By Lemma 6.3, H ′ Y is adominating family of rational curves on Y . Thus H ′ is an unsplit dominating family of rational curves on X . Let [ f ′ ] ∈ H ′ be a general member, and π ′ : X ′ → T ′ the H ′ -rationally connected quotient of X . Noticethat H and H ′ are numerically independent in X . Therefore the general curve from H is not contractedby π ′ , while it is contained in a leaf of F . So we must have F | X ′ T X ′ /T ′ . From the analysis of theprevious cases, we conclude that either f ′ falls under case (1) above, and so X is a P m -bundle over P n − m , or f ′∗ F ≃ O P (2) ⊕ O P (1) ⊕ r − ⊕ O ⊕ P , π ′ is a P r − -bundle, and T X ′ /T ′ ⊂ F | X ′ . In the latter case, F is regularalong a general fiber of π ′ . Thus (∆ + D ) · ℓ ′ = 0 and r = 4. Let π ′′ : X ′′ → T ′′ be the ( H, H ′ )-rationallyconnected quotient of X . By Lemma 6.9, T X ′′ /T ′′ ⊂ F | X ′′ , and thus rank ( T X ′′ /T ′′ ) ≤ rank ( F ) = 4. Onthe other hand, since H and H ′ are numerically independent in X , the fibers of the P -bundles π and π ′ cannot meet along a positive dimensional variety. Therefore the fibers of π ′′ have dimension at least 4. Weconclude that F | X ′′ = T X ′′ /T ′′ . But this is impossible by Proposition 5.3.From now on we assume that − K Y · ℓ ′ = 2. Then (∆ + D ) · ℓ ′ = 0 and r = 3. By Lemma 6.3, either H ′ Y is a dominating family of rational curves on Y , or Locus ( H ′ Y ) has codimension 1 in Y .If H ′ Y dominating, then H ′ is an unsplit dominating family of rational curves on X . Let π ′ : X ′ → T ′ bethe H ′ -rationally connected quotient of X . As before, we conclude that T X ′ /T ′ ⊂ F | X ′ . Let π ′′ : X ′′ → T ′′ be the ( H, H ′ )-rationally connected quotient of X . By Lemma 6.9, T X ′′ /T ′′ ⊂ F | X ′′ . By Proposition 5.3, T X ′′ /T ′′ = F | X ′′ . Thus rank ( T X ′′ /T ′′ ) = 2, and F is the pullback via π ′′ of a foliation by rational curveson T ′′ . The same argument used in case (3a) above shows that this is impossible.Finally, we assume that Locus ( H ′ Y ) has codimension 1 in Y . Since (∆ + D ) · ℓ ′ = 0, we have ¯ e ( ℓ ′ ) ∩ S = ∅ .Therefore the general member of H ′ avoids S and is tangent to F by Lemma 6.5. Let π ′′ : X ′′ → T ′′ bethe ( H, H ′ )-rationally connected quotient of X , and denote by F ′′ a general fiber of π ′′ . By Lemma 6.6, T X ′′ /T ′′ ⊂ F | X ′′ . In particular, dim F ′′ ≤ rank ( F ). We will show that dim F ′′ = 3. From this it followsthat F | X ′′ = T X ′′ /T ′′ , contradicting Proposition 5.3, and finishing the proof of Theorem 8.1.Let y ∈ Locus ( H ′ Y ) be a general point. By [Kol96, IV.2.6.1],3 + 2 = dim Y + ( − K Y · ℓ ′ ) ≤ dim (cid:0) Locus ( H ′ Y ) (cid:1) + dim (cid:16) Locus (cid:0) ( H ′ Y ) y (cid:1)(cid:17) + 1 ≤ (cid:16) Locus (cid:0) ( H ′ Y ) y (cid:1)(cid:17) + 1 . Thus dim (cid:16)
Locus (cid:0) ( H ′ Y ) y (cid:1)(cid:17) = 2. Since Locus (cid:0) ( H ′ Y ) y (cid:1) ⊂ Locus ( H ′ Y ), and the latter is irreducible and 2-dimensional, we conclude that (cid:16) Locus (cid:0) ( H ′ Y ) y (cid:1)(cid:17) = Locus ( H ′ Y ). Then the image of Locus ( H ′ Y ) in X iscontained in a general fiber of π ′′ . Moreover, it does not contain any curve from the family H ′ , since H and H ′ are numerically independent in X . Thus dim F ′′ = 3. (cid:3) Del Pezzo foliations on projective space bundles
Our first aim in this section is to give a precise geometric description of del Pezzo foliations F on projectivespace bundles X → P l such that F * T X/ P l . (Two special cases) . Let E be an ample locally free sheaf of rank m + 1 ≥ P l , and set X := P P l ( E ).Denote by O X (1) the tautological line bundle on X , and by π : X → P l the natural projection. Let F * T X/ P l be a del Pezzo foliation on X , and write det( F ) ≃ A ⊗ r F − for an ample line bundle A on X .By Proposition 7.10, r F ∈ { , } . We first determine the restriction of A to a general line ℓ on a fiber of π .As in Step 1 of the proof of Theorem 1.1, we verify that A · ℓ = 1 unless m = 1, r F = 2 ≤ l , F | ℓ ≃ O (2) ⊕ O ,and T X/ P l ( F .Suppose we are in the latter case. We claim that X ≃ P × P l , and F is the pullback via π of a degree 0foliation of rank 1 on P l . Indeed, by Lemma 6.7, F is the pullback by π of a rank 1 foliation G ⊂ T P l . Sodet( F ) ≃ det( T X/ P l ) ⊗ π ∗ det( G ), and A ≃ O X (2) ⊗ π ∗ (cid:0) det( E ∗ ) ⊗ G (cid:1) . Write G ≃ O P l ( k ) for some integer k . If k
0, then det( T X/ P l ) ≃ A ⊗ π ∗ O P m ( − k ) is ample, contradicting [Miy93, Theorem 2]. By Bott’sformulae, k = 1. Let P ⊂ P l be a line, and write E | P ≃ O P ( a ) ⊕ O P ( b ) with a b . Let σ : P → P P ( E | P )be the section corresponding to the projection O P ( a ) ⊕ O P ( b ) ։ O P ( a ), and set C := σ ( P ). Then1 A · C = (cid:0) O X (2) ⊗ π ∗ (det( E ∗ ) ⊗ G ) (cid:1) · C = 2 a − ( a + b ) + 1 . Thus a > b , and so a = b . By [OSS80, Theorem 3.2.1], E ≃ O P m ( a ) ⊕ . This proves the claim. ARAUJO
AND ST´EPHANE
DRUEL
So we may restrict ourselves to the case when A restricts to O (1) on the fibers of π . Then, by replacing E with π ∗ A if necessary, we may assume that det( F ) ≃ O X ( r F − m = 1, then l ≥ r = 3, X ≃ P × P l , H = T X/ P l , and F is the pullback via thenatural projection P × P l → P l of a degree zero foliation O P l (1) ⊕ O P l (1) ( T P l on P l . So we may assumethat m ≥ Theorem 9.2.
Let E be an ample locally free sheaf of rank m + 1 ≥ on P l , and set X := P P l ( E ) . Denoteby O X (1) the tautological line bundle on X , and by π : X → P l the natural projection. Let F * T X/ P l be afoliation of rank r ≥ on X such that det( F ) ≃ O X ( r − . (1) The possible values for the pair ( l, r ) are (1 , , (1 , and ( l, , with l ≥ . (2) There exists a subbundle V ⊂ E ∗ such that F ∩ T X/ P l ≃ ( π ∗ V )(1) , an inclusion j : det( V ∗ ) ֒ → T P l ,and a commutative diagram of exact sequences / / F ∩ T X/ P l ≃ ( π ∗ V )(1) / / (cid:15) (cid:15) F / / (cid:15) (cid:15) I W ⊗ π ∗ det( V ∗ ) π ∗ j (cid:15) (cid:15) / / / / T X/ P l / / T X / / π ∗ T P l / / where W ⊂ X is a closed subscheme with codim X W ≥ . (3) If l ≥ , then V ≃ O P l ( − . If l = 1 , then either V ≃ O P ( − , or V ≃ O P ( − , or V ≃ O P ( − ⊕ . (4) Let K be the kernel of the dual map E ։ V ∗ , and consider the P m − r +1 -bundle Z := P P l ( K ) , withnatural projection q : Z → P l . Then F is the pullback by the linear projection X/ P l Z/ P l of afoliation on Z induced by a nonzero global section of T Z ⊗ q ∗ det( V ) . (5) If l = 1 and m ≥ r + 1 , then F is locally free and W = ∅ . If moreover V ≃ O P ( − , then F ≃ ( π ∗ V )(1) ⊕ π ∗ det( V ∗ ) .Proof. Item (1) was proved in Proposition 7.10Set H := F ∩ T X/ P l , and recall from the proof of Proposition 7.10 that H ≃ ( π ∗ V )(1), where V ⊂ E ∗ is a saturated subsheaf of rank r − P l ). Moreover, there isan inclusion det( V ∗ ) ⊂ T P l , and an isomorphism (cid:0) F / H (cid:1) ∗∗ ≃ π ∗ det( V ∗ ). We claim that V is in fact asubbundle of E ∗ . If l = 1, then this is clear. Moreover, in this case either V ≃ O P ( − V ≃ O P ( − V ≃ O P ( − ⊕ . If l ≥
2, then r = 2, and V is locally free of rank 1. The condition det( V ∗ ) ⊂ T P l impliesthat V ≃ O P l ( − E is ample, V must be a subbundle of E ∗ . This proves (2) and (3).Let K be the kernel of the dual map E ։ V ∗ . Consider the P m − r +1 -bundle Z := P P l ( K ), with naturalprojection q : Z → P l . The surjection E ։ V ∗ induces a rational map p : X Z over P l , which restricts toa surjective morphism p : X → Z , where X is the complement in X of the P r − -subbundle P ( V ∗ ) ⊂ P ( E ).As in 7.8, we have H | X = T X /Z . By Lemma 6.7, F | X is the pullback via p of a rank 1 foliation G ( T Z .One checks easily that G ≃ q ∗ det( V ∗ ). This proves (4).In order to prove (5), recall from 7.8 that, since l = 1, H | F is a degree 0 foliation of rank r − F ≃ P m for any fiber F of π . Thus the map det( H ) ֒ → ∧ r − T X vanishes along a closed subset of dimension equalto 1 + ( r −
2) = r − dim( X ) −
3. I.e., H is a subbundle of F in codimension
2. By lemma 9.9, F islocally free and W = ∅ . If moreover V ≃ O P ( − H (cid:0) X, H ⊗ π ∗ det( V ) (cid:1) ≃ H (cid:0) X, π ∗ V (1) ⊗ π ∗ O P ( − (cid:1) ≃ H (cid:0) P , O P ( − ⊕ ⊗ E (cid:1) by Leray’s spectral sequence= 0 since E is ample.Hence F ≃ H ⊕ π ∗ det( V ∗ ). (cid:3) Our next goal is to classify locally free sheaves E on P l for which X = P P l ( E ) admits a del Pezzo foliation F * T X/ P l . For that purpose, we first recall the definition and basic properties of the Atiyah class of locallyfree sheaves on smooth varieties. (The Atiyah class of a locally free sheaf) . Let T be a smooth variety, and E a locally free sheaf ofrank m + 1 ≥ T . Let J T ( E ) be the sheaf of 1-jets of E . I.e., as a sheaf of abelian groups on T , J T ( E ) ≃ E ⊕ (Ω T ⊗ E ), and the O T -module structure is given by f ( e, α ) = ( f e, f α − df ⊗ e ), where f , e and α are local sections of O T , E and Ω T ⊗ E , respectively. The Atiyah class of E is defined to be the element N FANO FOLIATIONS 31 at ( E ) ∈ H ( T, E nd ( E ) ⊗ Ω T ) corresponding to the Atiyah extension0 → Ω T ⊗ E → J T ( E ) → E → . It can be explicitly described as follows. Choose an affine open cover ( U i ) i ∈ I of T such that E admits aframe f i : O m +1 U i ∼ → E | U i for each U i . For i, j ∈ I , define f ij := f − j | U ij ◦ f i | U ij . Then at ( E ) = (cid:2) ( − f j | U ij ◦ df ij | U ij ◦ f − i | U ij ) i,j (cid:3) ∈ H ( T, E nd ( E ) ⊗ Ω T ) . (See [Ati57, Proof of Theorem 5].).Set X := P T ( E ), and denote by π : X → T the natural projection. The push-forwarded Euler sequence0 → O T → E nd ( E ) → π ∗ T X/T → H ( T, E nd ( E ) ⊗ Ω T ) → H ( T, π ∗ T X/T ⊗ Ω T ) ≃ H ( X, T
X/T ⊗ π ∗ Ω T ) , where the last isomorphism is given by Leray’s spectral sequence. We denote by ¯ at ( E ) ∈ H ( X, T
X/T ⊗ π ∗ Ω T )the image of at ( E ) under this map.We claim that ¯ at ( E ) is the class in H ( X, T
X/T ⊗ π ∗ Ω T ) of the exact sequence(9.1) 0 → T X/T → T X → π ∗ T T → . To show this, we compute a cocycle that represents the extension class of (9.1). Let ( U i ) i ∈ I be the affineopen cover of T chosen above. By shrinking U i if necessary, we may assume that T T admits a frame t i : O lU i ∼ → T T | U i for each U i , where l = dim( T ). Let t ∨ i be dual frame of Ω T , and set π i := π | U i . The frame f i induces an isomorphism U i × P m ≃ P U i ( E | U i ) over U i , and a splitting s i : π ∗ i ( T T | U i ) → T X | V i of (9.1) over V i := π − i ( U i ). For i, j ∈ I , define a i,j := ( s j | V ij ⊗ id π ∗ Ω T | V ij − s i | V ij ⊗ id π ∗ Ω T | V ij )( π ∗ i t i | V ij ⊗ π ∗ i t ∨ i | V ij ) , where t i is viewed as a line vector whose entries are local sections of T T , and t ∨ i as a column vector. Then (cid:2) ( a i,j ) i,j (cid:3) ∈ H ( X, T
X/T ⊗ π ∗ Ω T ) is a cocycle representing the class of (9.1). Write df ij = ( α ijkn ) k,n ∈{ ,...,m } with α ijkn ∈ H ( U ij , Ω T | U ij ) for 0 k, n m , and set df ij := (cid:16) α ijkn y n ∂∂y n (cid:17) k,n ∈{ ,...,m } , where ( y : · · · : y m )are homogeneous coordinates on P m associated to the frame f i , and α ijkn y n ∂∂y n ∈ H (cid:0) V ij , T X/T ⊗ π ∗ Ω T | V ij (cid:1) .We get a i,j = (cid:0) s j | V ij ⊗ id π ∗ Ω T | V ij − s i | V ij ⊗ id π ∗ Ω T | V ij (cid:1)(cid:0) π ∗ i t i | V ij ⊗ π ∗ i t ∨ i | V ij (cid:1) = (cid:0) s j | V ij ( π ∗ i t i | V ij ) − s i | V ij ( π ∗ i t i | V ij ) (cid:1) ⊗ π ∗ i t ∨ i | V ij = (cid:0) − f j | U ij · df ij | U ij (cid:1) ⊗ π ∗ i t ∨ i | V ij . This proves our claim. (Equivariance for locally free sheaves) . Let the notation be as in 9.3. Let W be an invertible sheaf on T , and V ∈ H ( T, T T ⊗ W ) a twisted vector field on T . We say that E is V -equivariant if there exists a C -linear map ˜ V : E → W ⊗ E lifting the derivation V : O T → W (see [CL77]). By [CL77, Proposition 1.1], E is V -equivariant if and only if V ∗ at ( E ) ∈ H ( T, E nd ( E ) ⊗ W ) vanishes. Lemma 9.5.
Let T be a smooth variety, and E a locally free sheaf of rank m + 1 ≥ on T . Set X := P T ( E ) ,denote by π : X → T the natural projection, and by O X (1) the tautological line bundle on X . Suppose thatthere are locally free subsheaves i : H ֒ → T X/T and j : Q ֒ → T T fitting into an exact sequence → H → F → π ∗ Q → . Denote by e ∈ H ( X, H ⊗ π ∗ Q ∗ ) the class of this extension. (1) There exists a morphism of O X -modules F → T X extending i : H → T X/Y and π ∗ j : π ∗ Q → π ∗ T T if and only if i ∗ e = j ∗ ¯ at ( E ) in H ( X, T
X/T ⊗ π ∗ Q ∗ ) ≃ H ( T, π ∗ T X/T ⊗ Q ∗ ) . (2) The set of morphisms of O X -modules F → T X extending i and π ∗ j is either empty, or it is a torsorunder Hom O X ( π ∗ Q , T X/T ) ≃ Hom O T ( Q , E nd ( E )) . (3) Suppose that H ( −
1) = π ∗ V for some locally free sheaf V on T . Notice that i : H → T X/T induces a map V → π ∗ ( T X/Y ( − ≃ E ∗ . Denote by ¯ e the image of e under the composite map H ( X, H ⊗ π ∗ Q ∗ ) ≃ H ( T, V ⊗ E ⊗ Q ∗ ) → H ( T, E nd ( E ) ⊗ Q ∗ ) . Then there exists a morphismof O X -modules F → T X extending i and π ∗ j if and only if ¯ e − j ∗ at ( E ) ∈ H ( T, E nd ( E ) ⊗ Q ∗ ) is inthe image of the natural map H ( T, Q ∗ ) → H ( T, E nd ( E ) ⊗ Q ∗ ) . ARAUJO
AND ST´EPHANE
DRUEL
Proof.
Notice that i ∗ e is the extension class of the lower exact sequence in the commutative diagram0 / / H / / i (cid:15) (cid:15) F / / (cid:15) (cid:15) π ∗ Q / / / / T X/T / / T X/T ⊔ H F / / π ∗ Q / / j ∗ ( ¯ at ( E )) is the extension class of the upper exactsequence in the commutative diagram0 / / T X/T / / T X × π ∗ T T π ∗ Q / / (cid:15) (cid:15) π ∗ Q π ∗ j (cid:15) (cid:15) / / / / T X/T / / T X / / π ∗ T T / / O X -modules k : F → T X thatfits into a commutative diagram 0 / / H / / i (cid:15) (cid:15) F / / k (cid:15) (cid:15) π ∗ Q π ∗ j (cid:15) (cid:15) / / / / T X/T / / T X / / π ∗ T T / / i ∗ e = j ∗ ( ¯ at ( E )) in H ( X, T
X/T ⊗ π ∗ Q ∗ ) ≃ H ( T, π ∗ T X/T ⊗ Q ∗ ). This proves (1).Let k , k : F → T X be morphisms of O X -modules extending i and π ∗ j . Then their difference k − k lies in Hom O X ( F , T X/T ) ⊂ Hom O X ( F , T X ), and ( k − k ) | H ≡
0. Conversely, if ϕ ∈ Hom O X ( π ∗ Q , T X/T ),then k + ϕ ◦ p : F → T X extends i and π ∗ j , where p : F → π ∗ Q is the surjective map from above. Thisproves (2).For statement (3), observe that j ∗ ( ¯ at ( E )) is the image of at ( E ) under the composite map H ( T, E nd ( E ) ⊗ Ω T ) → H ( T, E nd ( E ) ⊗ Q ∗ ) → H ( T, π ∗ T X/T ⊗ Q ∗ ) . Moreover, i : H → T X/T induces a map V → π ∗ ( T X/Y ( − ≃ E ∗ . The class i ∗ e is the image of e ∈ H ( H ⊗ π ∗ Q ∗ ) ≃ H ( T, V ⊗ E ⊗ Q ∗ ) under the composite map H ( T, V ⊗ E ⊗ Q ∗ ) → H ( T, E nd ( E ) ⊗ Q ∗ ) → H ( T, π ∗ T X/T ⊗ Q ∗ )since the map π ∗ E nd ( E ) → T X/T factors through the natural map π ∗ E nd ( E ) → π ∗ E ∗ (1). The cohomologyof the exact sequence 0 → O T → E nd ( E ) → π ∗ T X/T → Q ∗ yields the exact sequence H ( T, Q ∗ ) → H ( T, E nd ( E ) ⊗ Q ∗ ) → H ( T, π ∗ T X/T ⊗ Q ∗ ) . These observations put together prove (3). (cid:3)
We return to the problem of classifying locally free sheaves E on P l for which X = P P l ( E ) admits a delPezzo foliation F * T X/ P l . Theorem 9.6.
Let E be an ample locally free sheaf of rank m + 1 ≥ on P l , and set X := P P l ( E ) . Denote by O X (1) the tautological line bundle on X , and by π : X → P l the natural projection. Let r be an integer suchthat ≤ r ≤ m + l − . Then there exists a foliation F * T X/ P l of rank r on X such that det( F ) ≃ O X ( r − if and only if one of the following holds. (1) l = 1 , r = 2 , and E ≃ O P (1) ⊕ K for some ample vector bundle K on P such that K O P ( a ) ⊕ m for any integer a . (2) l = 1 , r = 2 , and E ≃ O P (2) ⊕ O P ( a ) ⊕ m for some integer a > . (3) l = 1 , r = 3 , and E ≃ O P (1) ⊕ ⊕ O P ( a ) ⊕ m − for some integer a > . (4) l > , r = 2 , and there exists a V -equivariant vector bundle K on P l for some V ∈ H ( P l , T P l ⊗ O P l ( − \ { } and an exact sequence → K → E → O P l (1) → .Proof. First we show that these are necessary conditions. Suppose there exists a foliation F * T X/ P l ofrank r on X such that det( F ) ≃ O X ( r − l, r ) are (1 , ,
3) and ( l, l ≥
2. Set H := F ∩ T X/ P l , and recall from Theorem 9.2 that H ≃ ( π ∗ V )(1), where V ⊂ E ∗ is a subbundle of rank r −
1. Moreover if l ≥
2, then V ≃ O P l ( − l = 1,then either V ≃ O P ( − V ≃ O P ( − V ≃ O P ( − ⊕ . There is an inclusion j : det( V ∗ ) ֒ → T P l . N FANO FOLIATIONS 33
Finally, let K be the kernel of the dual map E ։ V ∗ , and consider the P m − r +1 -bundle Z := P P l ( K ), withnatural projection q : Z → P l . By Theorem 9.2, F is the pullback by the linear projection X/ P l Z/ P l of a foliation on Z induced by an inclusion q ∗ det( V ∗ ) ⊂ T Z that lifts q ∗ j : q ∗ det( V ∗ ) ֒ → q ∗ T P l .Let at ( K ) ∈ H ( P l , E nd ( K ) ⊗ Ω P l ) be the Atiyah class of K . Let V ∈ H ( P l , T P l ⊗ det( V )) be thesection associated to j : det( V ∗ ) ֒ → T P l . By Lemma 9.5, there exists a map q ∗ det( V ∗ ) → T Z lifting q ∗ j if and only if j ∗ at ( K ) ∈ H ( P l , E nd ( K ) ⊗ det( V )) is in the image of the natural map H ( P l , det( V )) → H ( P l , E nd ( K ) ⊗ det( V )).If V ≃ O P l ( − H ( P l , det( V )) = 0. Thus there exists q ∗ det( V ∗ ) → T Z lifting q ∗ j : q ∗ det( V ∗ ) ֒ → q ∗ T P l if and only if K is V -equivariant. If l ≥
2, this is case (4) above.From now on suppose l = 1, and write K ≃ O P ( a ) ⊕ · · · ⊕ O P ( a k ), with k = m − r + 1. In terms ofthis decomposition we have: at ( O P ( a )) 0 · · ·
00 . . . . . . ...... . . . 00 · · · at ( O P ( a k )) = at ( K ) ∈ H ( P , E nd ( K ) ⊗ Ω P ) = H ( P , O P ( a − a ) ⊗ Ω P ) · · · H ( P , O P ( a − a k ) ⊗ Ω P )... ... H ( P , O P ( a k − a ) ⊗ Ω P ) · · · H ( P , O P ( a k − a k ) ⊗ Ω P ) If det( V ) ≃ O P ( − ≃ Ω P , then there exists a map q ∗ det( V ∗ ) → T Z lifting q ∗ j if and only if a = · · · = a k . Since E is an ample vector bundle, this implies that E ≃ V ∗ ⊕ K . This is case (2) or (3) above.Finally, suppose that V ≃ O P ( − E is ample, we must have E ≃ O P (1) ⊕ K , and K must beample. Suppose that K ≃ O P ( a ) ⊕ n for some integer a . Then Z ≃ P × P m − . Denote by g : Z → P m − the second projection. Then T Z ⊗ q ∗ det( V ) ≃ q ∗ O P ( − ⊕ (cid:0) q ∗ O P ( − ⊗ g ∗ T P m − (cid:1) . Thus any nonzeroglobal section of T Z ⊗ q ∗ det( V ) vanishes along an hypersurface in Z , contradicting the fact that q ∗ det( V )is saturated in T Z .Conversely, let us show that these are sufficient conditions. Given l , E and r satisfying one of the conditionsabove, we will construct a foliation F * T X/ P l of rank r on X such that det( F ) ≃ O X ( r −
1) in steps. Firstwe will find a vector bundle V of rank r − P l fitting into an exact sequence of vector bundles0 → K → E → V ∗ → , and such that there is an inclusion j : det( V ∗ ) ֒ → T P l . We then set Z := P P l ( K ), with natural projection q : Z → P l . The exact sequence above induces a rational map p : X Z over P l , which restricts to asurjective morphism p : X → Z , where X is the complement in X of the P r − -subbundle P ( V ∗ ) ⊂ P ( E ).Note that codim X ( X \ X ) ≥
2. The next step consists of lifting the inclusion j : det( V ∗ ) ֒ → T P l to aninclusion q ∗ det( V ) ⊂ T Z . We then check that q ∗ det( V ) is saturated in T Z , and let F be the unique saturatedsubsheaf of T X extending dp − (cid:0) q ∗ det( V ∗ ) (cid:1) ⊂ T X . It is a foliation on X satisfying det( F ) ≃ O X ( r − l = 1, r = 2, and E ≃ O P (1) ⊕ K for some ample vector bundle K on P such that K O P ( a ) ⊕ m for any integer a . We set V := O P ( −
1) and let j : V ∗ ≃ O P (1) ֒ → T P bethe inclusion associated to some V ∈ H (cid:0) P , T P ⊗ O P ( − (cid:1) \ { } . Then K is V -equivariant, and so thereexists a map q ∗ O P (1) ֒ → T Z lifting q ∗ j . It remains to show that q ∗ O P (1) is saturated in T Z . To provethis, by Lemma 9.7, it is enough to show that q ∗ O P (1) is a subbundle of T Z in codimension 1. Suppose tothe contrary that the map q ∗ O P (1) → T Z vanishes along an hypersurface Σ in Z . Then the composed map q ∗ O P (1) → T Z → q ∗ T P vanishes along Σ, and Σ must be a fiber of q . By Lemma 9.5, K ≃ O P ( a ) ⊕ n forsome a >
1, contradicting our assumptions.Case (2). Suppose that l = 1, r = 2, and E ≃ O P (2) ⊕ O P ( a ) ⊕ m for some integer a >
1. We set V := O P ( −
2) and fix an isomorphism j : V ∗ ≃ T P . Then Z ≃ P × P m − , and q ∗ j lifts to a foliation q ∗ T P ⊂ T Z .Case (3). Suppose that l = 1, r = 3, and E ≃ O P (1) ⊕ ⊕ O P ( a ) ⊕ m − for some integer a >
1. We set V := O P ( − ⊕ and fix an isomorphism j : det( V ∗ ) ≃ T P . Then Z ≃ P × P m − , and q ∗ j lifts to a foliation q ∗ T P ⊂ T Z . ARAUJO
AND ST´EPHANE
DRUEL
Case (4). Suppose that l > r = 2, and there exists a V -equivariant vector bundle K on P l for some V ∈ H ( P l , T P l ⊗ O P l ( − \ { } and an exact sequence 0 → K → E → O P l (1) →
0. We set V := O P l ( − j : V ∗ ≃ O P l (1) ֒ → T P l be the inclusion associated to V . Since K is V -equivariant, there existsa map q ∗ O P l (1) ֒ → T Z lifting q ∗ j . It remains to show that q ∗ O P l (1) is saturated in T Z . To prove this,by Lemma 9.7, it is enough to show that q ∗ O P l (1) is a subbundle of T Z in codimension 1. Suppose tothe contrary that the map q ∗ O P l (1) → T Z vanishes along an hypersurface Σ in Z . Then the composedmap q ∗ O P l (1) → T Z → q ∗ T P l vanishes along Σ, and q (Σ) has codimension 1 in P l . This is saying that O P l (1) ֒ → T P l vanishes in codimension 1 in P l , which is impossible since l > (cid:3) Lemma 9.7.
Let X be a normal variety, and E ⊂ F coherent sheaves of O X -modules, with E locally freeand F torsion-free. Then E is saturated in F if and only if E is a subbundle of F in codimension .Proof. To say that E is saturated in F is equivalent to saying that F / E is torsion-free. To say that E is asubbundle of F in codimension 1 is equivalent to saying that F / E is locally free in codimension 1. Since X is normal, if F / E is torsion-free, then it is locally free in codimension 1. Conversely, suppose that F / E is locally free in codimension 1, and let us show that F / E is torsion-free. Let f be a nonzero local sectionof O X and s a local section of F such that f s is a local section of E . Since F is torsion-free, s is a rationalsection of E . By assumption, s is regular in codimension 1. Since X is normal and E is locally free, it followsthat s is a regular local section of E . (cid:3) Remark 9.8.
Lemma 9.7 fails to be true if F is not torsion-free. Let F := E ⊕ T where T is a torsionsheaf whose support has codimension > X . Then E is a subbundle of F in codimension 1 but F / E = T is a torsion sheaf. Lemma 9.9.
Let → H → F → Q → be an exact sequence of coherent sheaves on a noetherian integralCohen-Macaulay scheme X . Suppose that F is reflexive, H and Q ∗∗ are locally free, and Q is locally freein codimension . Then F and Q are locally free.Proof. By hypothesis, there is an open subset U ⊂ X , with codim X ( X \ U ) ≥
3, such that H | U and Q | U are locally free. Moreover H ⊗ Q ∗ is locally free, and hence a Cohen-Macaulay sheaf. So we have H ( X, H ⊗ Q ∗ ) ≃ H (cid:0) U, ( H ⊗ Q ∗ ) | U (cid:1) . Therefore, the extension class of the exact sequence 0 → H | U → F | U → Q | U → U yields an exact sequence 0 → H → F ′ → Q ∗∗ → X such that F ′ | U ≃ F | U .Since X is reduced and both H and Q ∗∗ are locally free, so is F ′ . Since F is reflexive, F ≃ F ′ by [Har80,Proposition 1.6]. (cid:3) Our next goal is to construct del Pezzo foliations F on projective space bundles X → T such that F ( T X/T . These can be viewed as families of del Pezzo foliations on projective spaces.
Construction 9.10.
Let T be a positive dimensional smooth projective variety, E an ample locally freesheaf of rank m + 1 ≥ T , and set X := P T ( E ). Denote by O X (1) the tautological line bundle on X ,and by π : X → T the natural projection. Let r be an integer such that 2 r m − F of rank r on X such that F ( T X/T .Suppose that there are locally free sheaves Q and K on T , of rank r − > m − r + 2 > → K → E → Q → . In particular, Q is ample. Denote by e ∈ H ( T, K ⊗ Q ∗ ) the class of this extension.Set Z := P T ( K ) → T , denote by O Z (1) the tautological line bundle on Z , and by q : Z → T the naturalprojection. Recall the pushed-forwarded Euler’s sequence:(9.2) 0 → O T → E nd O T ( K ) → q ∗ T Z/T → . Let c : Y → X be the blow up of X along L := P T ( Q ) where the inclusion L ⊂ X is induced by thesurjection E ։ Q . Let E ⊂ Y be the exceptional divisor of c . By Leray’s spectral sequence, there is anatural isomorphism H ( Z, O Z (1) ⊗ q ∗ Q ∗ ) ≃ H ( T, K ⊗ Q ∗ ). Let H be the vector bundle on Z associatedto the image of e in H ( Z, O Z (1) ⊗ q ∗ Q ∗ ). Then H has rank r , q ∗ H = E , and there is a commutativediagram of exact sequences: N FANO FOLIATIONS 35 / / q ∗ K / / (cid:15) (cid:15) q ∗ E / / (cid:15) (cid:15) q ∗ Q / / / / O Z (1) / / H / / q ∗ Q / / . Denote by g : P Z ( H ) → Z the natural projection, and by O P Z ( H ) (1) the the tautological line bundle.Observe that there is an isomorphism Y ≃ P Z ( H ) that fits into the commutative diagram Y ≃ P Z ( H ) c w w ♦♦♦♦♦♦♦♦♦♦♦♦ g ' ' PPPPPPPPPPPP (cid:15) (cid:15) X = P T ( E ) π ( ( PPPPPPPPPPPPP Z = P T ( K ) , q v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ T where c is induced by the surjection c ∗ E = g ∗ ( q ∗ E ) ։ g ∗ H ։ O P Z ( H ) (1) . In order to construct a del Pezzo foliation F on X we make the following assumptions.(1) There is an injective map ϕ : det( Q ) ֒ → E nd O T ( K ) (equivalently, h (cid:0) T, det( Q ) ∗ ⊗ E nd O T ( K ) (cid:1) = 0).(2) The inclusion q ∗ det( Q ) ⊂ T Z/T ⊂ T Z induced by ϕ via (9.2) defines a foliation on Z . By Lemma 9.7this is equivalent to requiring that the map q ∗ det( Q ) ֒ → T Z/T is nonzero in codimension 1.We then set F Y := dg − ( q ∗ det( Q )) ⊂ T Y , and F := df ( F Y ) ⊂ T X/T ⊂ T X .For any t ∈ T , let v t ∈ End C ( K t ) be an endomorphism induced by ϕ at t ∈ T . Then the foliation on X t ≃ P m induced by F is the linear pullback of a foliation G t on Z t ≃ P m − r +1 induced by the globalholomorphic vector field ~v t associated to v t . One can prove that the closure of a general leaf of G t is arational curve C meeting the singular locus of G t at a single point. Moreover, either this point is a cusp on C , or ~v t viewed as a local vector field on C vanishes with multiplicity at least 2.Once assumption (1) above is fulfilled, we investigate when assumption (2) holds.First we claim that v t is a nilpotent endomorphism for any t ∈ T . Indeed, the composite mapdet( Q ) ⊗ k −→ E nd O T ( K ) ⊗ k −→ O T α ⊗ · · · ⊗ α k ϕ ( α ) ⊗ · · · ⊗ ϕ ( α k ) Tr( ϕ ( α ) ◦ · · · ◦ ϕ ( α k ))is zero since det( Q ) is ample. Thus Tr( v t ◦ · · · ◦ v t | {z } k times ) = 0 for any k >
1, showing that v t is nilpotent.Notice that q ∗ det( Q ) is saturated in T Z/T if and only if the following holds. For a general point t ∈ T , v t has rank >
2, and there exists an open subset T ⊂ T , with codim T ( T \ T ) >
2, such that v t has rank > t ∈ T .Finally, observe that the assumptions are fulfilled if K contains det( Q ) ⊕ det( Q ) ⊗ ⊕ det( Q ) ⊗ as a directsummand, and det( Q ) ֒ → E nd O T (cid:0) det( Q ) ⊕ det( Q ) ⊗ ⊕ det( Q ) ⊗ (cid:1) is associated to . We end this section by addressing Fano Pfaff fields on projective space bundles.
Proposition 9.11.
Let T be a smooth projective variety, E a locally free sheaf of rank m + 1 ≥ on T ,and set X := P T ( E ) . Denote by O X (1) the tautological line bundle on X , and by π : X → T the naturalprojection. Let r > be an integer. (1) If r > m + 3 then h (cid:0) X, ∧ r T X ( − r + 1) (cid:1) = 0 . (2) If h (cid:0) X, ∧ r T X ( − r + 1) (cid:1) = 0 , then h (cid:0) X, ∧ r − s T X/T ( − r + 1) ⊗ π ∗ ( ∧ s T T ) (cid:1) = 0 for some s ∈ { , , } .If h (cid:0) X, ∧ r − T X/T ( − r + 1) ⊗ π ∗ ( ∧ T T ) (cid:1) = 0 then r = m + 2 > . (3) If E is an ample vector bundle and h (cid:0) X, ∧ r − T X/T ( − r +1) ⊗ π ∗ ( ∧ T T ) (cid:1) = 0 , then either T ≃ P × P and r − m = 1 , or T ≃ P and r − m = 2 , or T ≃ P l ( l > ) and r − m = 1 . ARAUJO
AND ST´EPHANE
DRUEL (4) If E is an ample vector bundle, l = dim( T ) > , h (cid:0) X, ∧ r − T X/T ( − r + 1) ⊗ π ∗ T T (cid:1) = 0 and ρ ( T ) = 1 ,then T ≃ P l and r .Proof. The short exact sequence 0 → T X/T → T X → π ∗ T T → ∧ r T X = F ⊃ F ⊃ · · · ⊃ F r +1 = 0such that F i /F i +1 ≃ ∧ i T X/T ⊗ π ∗ ( ∧ r − i T T ) . By Bott’s formulae, h (cid:0) P m , ∧ i T P m ( − r + 1) (cid:1) = 0 if either 0 i r − r − m , or i = r − r − < m . So in these cases we have h (cid:0) X, ( F i /F i +1 )( − r + 1) (cid:1) = 0, proving (1) and (2).From now on suppose that E is an ample vector bundle. If h (cid:0) X, ∧ r − T X/T ( − r + 1) ⊗ π ∗ ( ∧ T T ) (cid:1) = 0,then r − m > r >
3, and ∧ r − T X/T ( − r + 1) ≃ π ∗ det( E ∗ ). Hence h (cid:0) T, ∧ T T ⊗ det( E ∗ ) (cid:1) = 0. By[DP10], either T ≃ P × P and r − m = 1, or T ≃ P and r − m = 2, or T ≃ P l ( l >
2) and r − m = 1, proving (3).Now suppose that ρ ( T ) = 1, and h (cid:0) X, ∧ r − T X/T ( − r + 1) ⊗ π ∗ T T (cid:1) = 0. Euler’s sequence0 → O X → π ∗ E ∗ (1) → T X/T → → ∧ r − T X/T ( − r + 1) → ∧ r − π ∗ ( E ∗ ) → ∧ r − T X/T ( − r + 1) → . By Bott’s formulae, h (cid:0) P m , ∧ r − T P m ( − r + 1) (cid:1) = 0 and h (cid:0) P m , ∧ r − T P m ( − r + 1) (cid:1) = 0. Hence π ∗ (cid:0) ∧ r − T X/T ( − r + 1) (cid:1) = 0 and R π ∗ (cid:0) ∧ r − T X/T ( − r + 1) (cid:1) = 0. Thus, by pushing forward by π the above exactsequence, we conclude that ∧ r − E ∗ ≃ π ∗ (cid:0) ∧ r − T X/T ( − r + 1) (cid:1) . The projection formula then yields anisomorphism H ( T, ∧ r − E ∗ ⊗ T T ) ≃ H (cid:0) X, ∧ r − T X/T ( − r + 1) ⊗ p ∗ T T (cid:1) = { } . By [AKP08, Corollary 4.3], we must have T ≃ P l .A nonzero section of ∧ r − E ∗ ⊗ T P l yields a nonzero map α : ∧ r − E → T P l . Let ℓ ⊂ P l be a general line,and write E | ℓ ≃ O P ( a ) ⊕ · · · ⊕ O P ( a m ), with 1 a · · · a m . Then α induces a nonzero map ∧ r − (cid:0) O P ( a ) ⊕ · · · ⊕ O P ( a m ) (cid:1) → O P (2) ⊕ O P (1) ⊕ l − . Thus r − a + · · · + a r −
2, proving (4). (cid:3)
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E-mail address : [email protected] St´ephane Druel: Institut Fourier, UMR 5582 du CNRS, Universit´e Grenoble 1, BP 74, 38402 Saint Martind’H`eres, France
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