CAROLINA ARAUJO AND STEPHANE DRUEL

Del Pezzo

Description

Niveau: Supérieur, Doctorat, Bac+8
ON FANO FOLIATIONS CAROLINA ARAUJO AND STEPHANE DRUEL Abstract. In this paper we address Fano foliations on complex projective varieties. These are foliations F whose anti-canonical class ?KF 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 algebraically integrable, with one exceptional case when the ambient space is Pn. We also provide a classification of del Pezzo foliations with mild singularities. Contents 1. Introduction 1 2. Foliations and Pfaff fields 3 3. Algebraically integrable foliations 5 4. Examples 8 5. The relative anticanonical bundle of a fibration and applications 10 6. Foliations and rational curves 13 7. Algebraic integrability of del Pezzo foliations 18 8. On del Pezzo foliations with mild singularities 27 9. Del Pezzo foliations on projective space bundles 29 References 36 1. 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 manifoldsX should be classified according to the behavior of their canonical class KX . As a result of the minimal model program, we know that every complex projective manifold can be build up from 3 classes of (possibly singular) projective varieties, namely, varieties X for which KX is Q-Cartier, and satisfies KX < 0, KX ? 0 or KX > 0.

• open neighborhood


• then either


• index ?f


• normal variety


• normal complex projective


• anti- canonical class


• cycle over


• zariski dense


• projective manifold

Sujets

• Neighbourhood (mathematics)
• Normal scheme
• Zariski topology

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 22
• Langue: English

Extrait

ON FANO FOLIATIONS

´ CAROLINAARAUJOAND STEPHANEDRUEL

Abstract. These are foliationsIn this paper we address Fano foliations on complex projective varieties. Fwhose anti-canonical class−K F is ample. focus our attention on a special class of Fano foliations, We namelydel Pezzo Wefoliations on complex projective manifolds. show that these foliations are algebraically integrable, with one exceptional case when the ambient space isP n . We also provide a classification of del Pezzo foliations with mild singularities.

Contents

1. Introduction 2. Foliations and Pfaff fields 3. Algebraically integrable foliations 4. Examples 5. The relative anticanonical bundle of a fibration and applications 6. Foliations and rational curves 7. Algebraic integrability of del Pezzo foliations 8. On del Pezzo foliations with mild singularities 9. Del Pezzo foliations on projective space bundles References

1 3 5 8 10 13 18 27 29 36

1.itcudortInno In the last few decades, much progress has been made in the classification of complex projective varieties. The general viewpoint is that complex projective manifoldsXshould be classified according to the behavior of their canonical classK X  a result of the minimal model program, we know that every complex projective. As manifold can be build up from 3 classes of (possibly singular) projective varieties, namely, varietiesXfor whichK X isQ-Cartier, and satisfiesK X <0,K X ≡0 orK X >0. Projective manifoldsXwhose anti- canonical class−K X is ample are calledFano manifolds, and are quite special. For instance, Fano manifolds are known to be rationally connected (see [Cam92] and [KMM92]). One defines the indexι X of a Fano manifoldXto be the largest integer dividing−K X in Pic(X). A classical result of Kobayachi-Ochiai’s asserts thatι X ≤dimX and equality holds if and only if+ 1,X'P n . Moreover,ι X = dimXif and only ifX manifolds whose index Fanois a quadric hypersurface ([KO73]). satisfiesι X = dimX− are called These1 were classified by Fujita in [Fuj82a] and [Fuj82b].del Pezzo manifolds. The philosophy behind these results is that Fano manifolds with high index are the simplest projective manifolds. Similar ideas can be applied in the context offoliationson complex projective manifolds. IfF(T X is a foliation on a complex projective manifoldX, we define its canonical class to beK F =−c 1 (F). In analogy with the case of projective manifolds, one expects the numerical properties ofK F to reflect geometric aspects ofF fact, ideas from the minimal model program have been successfully applied to the theory of. In foliations (see for instance [Bru04] and [McQ08]), and led to a birational classification in the case of rank one foliations on surfaces ([Bru04]). More recently, Loray, Pereira and Touzet have investigated the structure of codimension 1 foliations withK F ≡0 in [LPT11b]. 2010Mathematics Subject Classification.14M22, 37F75. The first named author was partially supported by CNPq and Faperj Research Fellowships. The second named author was partially supported by the CLASS project of the A.N.R. 1

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´ CAROLINAARAUJOAND STEPHANEDRUEL

In this paper we propose to investigateFano foliations are Theseon complex projective manifolds. foliationsF(T X whose anti-canonical class−K F  Asis ample (see Section 2 for details). in the case of Fano manifolds, we expect Fano foliations to present very special behavior. This is the case for instance if the rank ofFis 1, i.e.,Fis an ample invertible subsheaf ofT X  Wahl’s Theorem [Wah83], this can only. By happen if (XF)'  P n O(1)  . Guided by the theory of Fano manifolds, we define the indexι F of a foliationFon a complex projective manifoldXto be the largest integer dividing−K F in Pic(X expected philosophy is that Fano foliations). The with high index are the simplest ones. For instance, whenX=P n , the index of a foliationF(T P n of rank rsatisfiesι F ≤rdsifyholalit,equflniynaodC0[DBy.]8.3eme`roe´hT,5Fis induced by a linear projection P n 99KP n−r , i.e., it comes from the familyr-planes inP n containing a fixed (r−1)-plane. Fano foliations in F(T Pn satisfyingι F =r− fall into one of the following1 were classified in [LPT11a, Theorem 6.2]. They two classes. (1) EitherFis induced by a dominant rational mapP n 99KP(1 n−r 2), defined byn−rlinear forms and one quadratic form, or (2)Fis the linear pullback of a foliation onP 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]).LetF(T X be a Fano foliation of rankron a complex projective manifoldX. Thenι F ≤r, and equality holds only ifX ∼ = P n . We say that a Fano foliationF(T X of rankron a complex projective manifoldXis adel Pezzo foliation ifι F =r− addition to the above mentioned In we would like to classify del Pezzo foliations.1. Ultimately foliations onP n any rank on quadric hypersurfaces, del Pezzo, we know examples of del Pezzo foliations of foliations of rank 2 on certain Grassmannians, and del Pezzo foliations of rank 2 and 3 onP m -bundles over P l . These examples are described in Sections 4 and 9. We note that the generic del Pezzo foliation onP n of type (2) above does not have algebraic leaves. Our first main result says that this is the only del Pezzo foliation that is not algebraically integrable. We also describe the geometry of the general leaf in all other cases.

Theorem 1.1.LetF(T X be a del Pezzo foliation on a complex projective manifoldX6'P n . ThenFis algebraically 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 and McQuillan for a foliation to be algebraically integrable with rationally connected general leaf.

Theorem 1.2([BM01, Theorem 0.1], [KSCT07, Theorem 1]).LetXbe a normal complex projective variety, andFa foliation onX. LetC⊂Xbe a complete curve disjoint from the singular loci ofXandF. Suppose that the restrictionF| C is an ample vector bundle onC. Then the leaf ofFthrough any point ofCis an algebraic variety, and the leaf ofFthrough a general point ofCis moreover rationally connected.

Given a del Pezzo foliationF(T X on a complex projective manifoldX, it is not clear a priori how to find a curveC⊂X in order to prove Theorem 1.1, wesatisfying the hypothesis of Theorem 1.2. Instead, will apply Theorem 1.2 in several steps. First we construct suitable subfoliationsH⊂Ffor which we can prove algebraic integrability and rationally connectedness of general leaves. Next we consider the the closure Win Chow(X) of the subvariety parametrizing general leaves ofH We, as explained in Section 3. then apply Theorem 1.2 to the foliation onWinduced byF. In the course of our study of Fano foliations, we were led to deal with singularities of foliations. We introduce new notions of singularities for foliations, inspired by the theory of singularities of pairs, developed in the context of the minimal model program. In order to explain this, letF(T X be an algebraically ˜ integrable foliation on a complex projective manifoldX, and denote byi:F→Xthe normalization of the ˜ ˜ ˜ closure of a general leaf ofF. Then there is an effective Weil divisor Δ onFsuch that−K F ˜ =i ∗ (−K F ) + Δ. ˜ ˜ We call the pair (F Δ) a generallog leafofF. We say thatFhaslog canonical singularities along a general ˜ ˜ leafif (F  AlgebraicallyΔ) is log canonical (see Section 3 for details). integrable Fano foliations having log canonical singularities along a general leaf have a very special property: there is a common point contained in the closure of a general leaf (see Proposition 5.3). This property is useful to derive classification results under some restrictions on the singularities ofF, such as the following (see also Theorem 8.1).

ON FANO FOLIATIONS 3 Theorem 1.3.LetF(T X be a del Pezzo foliation of rankron a complex projective manifoldX6'P n . Suppose thatFhas log canonical singularities and is locally free along a general leaf. Then eitherρ(X) = 1, orr≤3,Xis aP m -bundle overP l andF6⊂T X/Pl . Notice that a del Pezzo foliationFonX6'P n  it Henceis algebraically integrable by Theorem 1.1. makes sense to ask thatFhas log canonical singularities along a general leaf in Theorem 1.3 above. We remark that del Pezzo foliations of codimension 1 on Fano manifolds with Picard number 1 were classified in [LPT11a, Proposition 3.7]. Theorem 1.3 raises the problem of classifying del Pezzo foliations onP m -bundlesπ:X→P l . Ifm= 1, thenX'P 1 ×P l , andFis the pullback viaπof a foliationO(1) ⊕i ⊂T P l for somei∈ {12}(see 9.1). For m≥2, we have the following result (see Theorems 9.2 and 9.6 for more details). Theorem 1.4.LetF(T X be a del Pezzo foliation on aP m -bundleπ:X→P l , withm≥2. Suppose thatF6⊂T X/Pl . Then there is an exact sequence of vector bundles0→K→E→Q→0onP l such thatX'P Pl (E), andFis the pullback via the relative linear projectionX99KZ=P Pl (K)of a foliation q ∗ det(Q)⊂T Z . Hereq:Z→P l denotes the natural projection. Moreover, one of the following holds. (1)l= 1,Q'O(1),Kis an ample vector bundle such thatK6'O P 1 (a) ⊕m for any integera, and E'Q⊕K(r F = 2). (2)l= 1,Q'O(2),K'O P 1 (a) ⊕m for some integera>1, andE'Q⊕K(r F = 2). (3)l= 1,Q'O(1)⊕O(1),K'O P1 (a) ⊕m−1 for some integera>1, andE'Q⊕K(r F = 3). (4)l>2,Q'O(1), andKisV-equivariant for someV∈H 0  P l  T P l ⊗O(−1)  \ {0}(r F = 2). Conversely, givenK,EandQthe conditions above, there exists a del Pezzo foliation ofsatisfying any of that type.

The paper is organized as follows. In Section 2 we introduce the basic notions concerning foliations and Pfaff fields on varieties. In Section 3 we focus on algebraically integrable foliations, and develop notions of singularities for these foliations. In Section 4 we describe examples of Fano foliations on Fano manifolds with Picard number 1. In Section 5 we study the relative anti-canonical bundle of a fibration, and provide applications to the theory of Fano foliations. In Section 6 we recall some results from the theory of rational curves on varieties, and explain how they apply to foliations. In Section 7 we prove Theorem 1.1. In Section 8 we address the problem of classifying Fano foliations with mild singularities. In particular we prove 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 in forthcoming works.

Notation and conventions.We always work over the fieldC are always Varietiesof complex numbers. assumed to be irreducible. We denote by Sing(X) the singular locus of a varietyX a sheaf. GivenFof O X -modules on a varietyX, we denote byF ∗ the sheafHom O X (FO X ). Ifris the generic rank ofF, then we denote by det(F) the sheaf (∧ r F) ∗∗ . IfGis another sheaf ofO X -modules onX, then we denote byF[⊗]Gthe sheaf (F⊗G) ∗∗ . IfEis a locally free sheaf ofO X -modules on a varietyX, we denote by P X (E) the Grothendieck projectivization Proj X (Sym(E)). IfXis a normal variety andX→Yis any morphism, we denote byT X/Y the sheaf (Ω 1 X/Y ) ∗  particular,. InT X = (Ω 1 X ) ∗ . IfXis a smooth variety and Dis a reduced divisor onXcrossings support, we denote by Ωwith simple normal 1 X (logD) the sheaf of ∗ differential 1-forms with logarithmic poles alongD, and byT X (−logD) its dual sheal Ω 1 X (logD) . Notice that det(Ω 1 X (logD))'O X (K X +D). Acknowledgements.Much of this work was developed during the authors’ visits to IMPA and Institut Fourier. We would like to thank both institutions for their support and hospitality. We also thank our colleaguesJulieD´esertiandJorgeVito´rioPereiraforveryhelpfuldiscussions.

2.Foliations and Pfaff fields

Definition 2.1.LetXbe normal variety. AfoliationonXis a nonzero coherent subsheafF(T X satisfying (1)Fis closed under the Lie bracket, and (2)Fis saturated inT X (i.e.,T X /Fis torsion free).