aa r X i v : . [ m a t h . AG ] N ov ABOUT THE HYPERBOLICITY OF COMPLETEINTERSECTIONS
SIMONE DIVERIO
Abstract.
This note is an extended version of a thirty minutes talk given atthe “XIX Congresso dell’Unione Matematica Italiana”, held in Bologna fromSeptember 12th to September 17th, 2011. This was essentially a survey talkabout connections between Kobayashi hyperbolicity properties and positivityproperties of the canonical bundle of projective algebraic varieties. Notations and preliminary material
Let X be a compact complex space. Definition 1.1. An entire curve in X is a non constant holomorphic map f : C → X .We shall give a definition of Kobayashi hyperbolicity which is often referred toas Brody hyperbolicity. Nevertheless, by Brody’s lemma, the two concepts coincidein the compact case: in the sequel, we shall freely speak of “hyperbolicity” refer-ring to these equivalent concepts. For more details on Brody’s lemma and Brodyhyperbolicity, see for instance [15]. Definition 1.2.
A compact complex space X is hyperbolic if and only if there areno entire curves in X . Remark . Fix any hermitian metric ω on X . By Brody’s lemma, if there existson X an entire curve, then there exists on X also a so-called Brody curve, i.e. anentire curve f with bounded derivative || f ′ || ω ≤ C .Let us give some basic examples of hyperbolic compact manifolds. Example 1.4.
Let C be a compact Riemann surface. Of course if C is the Riemannsphere or an elliptic curve, then C is not hyperbolic. On the other hand, by theUniformization Theorem, if the geometric genus of C is greater than or equal totwo, then the universal cover of C is the complex unit disc. Thus, since everyholomorphic map f : C → C lifts to a holomorphic map to the universal cover, byLiouville’s theorem it must be constant. Therefore, C is hyperbolic if and only if g ( C ) ≥ Example 1.5 (Kobayashi) . Let X be a compact complex manifold with amplecotangent bundle T ∗ X , i.e. such that there exists a positive integer k such that allevaluation maps H ( X, S k T ∗ X ) → J S k T ∗ X,x , H ( X, S k T ∗ X ) → S k T ∗ X,x ⊕ S k T ∗ X,y , Mathematics Subject Classification.
Primary 32Q45; Secondary 14J70, 14M10, 14J32.
Key words and phrases.
Kobayashi hyperbolic, variety of general type, Lang’s conjectures,rational curve, Calabi-Yau threefold, jet differentials, projective complete intersection.The author is partially supported by the ANR project “POSITIVE”, ANR-2010-BLAN-0119-01. for x, y ∈ X , x = y , are surjective for all k ≥ k . Here, S k T ∗ X is the k -th symmetricpower of the cotangent bundle and J S k T ∗ X is the bundle of 1-jets of sections of S k T ∗ X .We claim that X is hyperbolic. To see this, fix an integer k ≥ k as above anda basis σ , . . . , σ N of H ( X, S k T ∗ X ). Then, the function η : T X → R v (cid:18) N X j =0 | σ j ( x ) · v ⊗ k | (cid:19) / k , v ∈ T X,x , is easily seen to be plurisubharmonic, and strictly plurisubharmonic outside thezero section, thanks to the ampleness assumption. Now, suppose that X is nothyperbolic. Then, there exists on X a non constant Brody curve f : C → X andtherefore η ◦ f : C → R is a bounded subharmonic function defined on the entirecomplex plane which is strictly subharmonic outside { f ′ = 0 } . Hence, f is constantand we get a contradiction.Examples of compact complex manifolds with ample cotangent bundle are givenfor instance by the intersection of at least n/ n , and by general linear section of smalldimension in a product of sufficiently many smooth projective varieties with bigcotangent bundle (see [3] for more details). Other examples are given by compacthermitian manifolds with negative holomorphic bisectional curvature.It is conjectured in [3] that the intersection, in P N , of at least N/ Example 1.6.
More generally, a compact hermitian manifold (
X, ω ) such that itsholomorphic sectional curvature is negative is hyperbolic. Any compact quotient∆ n / Γ, n ≥
2, of a polydisc by a group Γ ⊂ Aut(∆) n acting freely and properlydiscontinuously on ∆ n gives an example of compact hermitian manifold with neg-ative holomorphic sectional curvature but with no hermitian metrics of negativeholomorphic bisectional curvature.In the case of complex projective curves, the hyperbolicity can be characterizedin a completely algebraic way: a complex projective curve is hyperbolic if and onlyif its geometric genus is greater than or equal to two, and this happens if and only ifits cotangent bundle, which in complex dimension one coincides with its canonicalbundle, is ample. By the Riemann-Roch formula, this is equivalent to being ofgeneral type. Let us give the precise definition. Definition 1.7.
Let X be a compact complex manifold and L → X a holomorphicline bundle. The Kodaira-Iitaka dimension κ ( L ) of L is defined as follows: if H ( X, L ⊗ m ) = { } for each m ≥ κ ( L ) = −∞ , otherwise 0 ≤ κ ( L ) ≤ dim X is the unique integer such that C − m κ ( L ) ≤ dim H ( X, L ⊗ m ) ≤ C m κ ( L ) , for some positive constant C and for all sufficiently large and divisible integers m . If κ ( L ) = dim X , L is said to be big . We denote by κ ( X ) the Kodaira-Iitaka dimensionof the canonical bundle K X = det T ∗ X of X and call it the Kodaira dimension of X .The manifold X is said to be of general type if it has maximal Kodaira dimension. Remark . By Hirzebruch-Riemann-Roch and Kodaira’s vanishing, if L → X isample then L is big. Thus, manifolds with ample canonical bundle are of generaltype. Beside projective curve of genus ≥
2, a typical example of manifold of generaltype is given by smooth projective hypersurfaces X ⊂ P n +1 such that deg X ≥ n +3 (they have in fact ample canonical bundle, as it can be straightforwardly checkedby adjunction). 2. Guiding conjectures
Let us begin with the following proposition.
Proposition 2.1 (Demailly [4]) . Let X be a compact complex manifold and ω ahermitian form on X . Consider the following statements:(i) X is Kobayashi hyperbolic.(ii) There exists ε > such that for every curve C ⊂ X − χ ( b C ) = 2 g ( b C ) − ≥ ε deg ω C, where b C → C is the normalization of C and deg ω C := R C ω .(iii) Every holomorphic map Z → X , where Z is a complex torus, is constant.Then, ( i ) = ⇒ ( ii ) = ⇒ ( iii ) . Property ( ii ) above is often referred to as algebraic hyperbolicity . Now, if X issupposed to be projective algebraic, the three properties above are conjectured tobe equivalent. Conjecture 2.2 (Lang [17], Demailly [4]) . In the proposition above, if X is more-over projective algebraic, then ( iii ) = ⇒ ( i ). Remark . If X is supposed to be merely K¨ahler, then the conjecture is false.One can show, for instance, that a K K X will be always projective algebraic. In this case, it is verytempting to believe that the analytic property of being hyperbolic should translateto completely algebraic properties of the manifold (generalizing then the case ofcomplex dimension one). This belief is formalized in the following. Conjecture 2.4 (Lang [17]) . Let X be a complex projective manifold. Then, X is Kobayashi hyperbolic if and only if X as well as all its subvarieties are of generaltype.The sufficiency of the condition in the conjecture above is the object of this otherone. Conjecture 2.5 (Green-Griffiths) . Let X be a projective manifold of general type.Then, there should exists a proper algebraic subvariety Y ( X such that all entirecurves in X are actually contained in Y .The necessity of the condition in Conjecture 2.4, since a projective manifold ofgeneral type without rational curves has ample canonical bundle, would imply thefollowing. Conjecture 2.6 (Kobayashi) . Let X be a projective hyperbolic manifold. Then,the canonical bundle K X should be ample.Observe that all these conjectures are obvious in dimension one, but already nontrivial (and, apart from the last one, not known) in dimension two.To end this section, let us mention that there is also an arithmetic counterpartof the story, which is moreover already highly non trivial in dimension one. Innumber theory, the Mordell conjecture states that a curve of genus greater than1 defined over a number field K has only finitely many rational points over anyfinite field extension L of K : it was proved by G. Faltings in 1983. Remind that SIMONE DIVERIO until Faltings’ theorem, there was not known a single example of a curve which wasproved to have only a finite number of rational points in every finitely generatedfield over Q . “Accidentally”, the curves satisfying Faltings’ theorem are exactly thehyperbolic ones. According to Lang, this should be the right framework in orderto have a higher dimensional analogous of Faltings’ theorem. Conjecture 2.7 (Lang [16]) . Let X be a projective manifold defined over a numberfield K . Then, X is hyperbolic if and only if it contains only finitely many rationalpoints over any finite field extension L of K .At present, almost nothing is known about this very fascinating theme.3. Overview of old and recent results
We now survey some results relative to the conjectures mentioned above.3.1.
The Kobayashi conjecture.
As mentioned in the previous sections, for thecase of curves this conjecture is a trivial consequence of the Uniformization Theoremand Liouville’s theorem.For surfaces, the Enriques-Kodaira birational classification implies that, in orderto prove the conjecture, it suffices to show that K3 surfaces are not hyperbolic.Since Kummer surfaces (which are obviously non hyperbolic) are dense in the mod-uli space of K3’s and since hyperbolicity is an open property (with respect toanalytic topology), the conjecture follows. In fact, much more is true: by [21] everyprojective K3 surface does contain at least one rational curve.Unfortunately, we do not dispose of an analogous result in higher dimension.It is likely that every compact projective manifold with vanishing first real Chernclass is not hyperbolic and even more: it should admit a non trivial holomorphicmap from a complex torus (or, more optimistically, a rational curve).Now, we concentrate on dimension three. To begin with, observe that severalpowerful machineries from birational geometry —such as the characterization ofuniruledness in terms of negativity of the Kodaira dimension (which holds true upto dimension three and is conjectural in general), the Iitaka fibration, the abundanceconjecture (which is actually a theorem up to dimension three)— permit to reducethis conjecture to the following statement: a projective threefold X of Kodairadimension κ ( X ) = 0 cannot be hyperbolic. This is the analogous of the reductionto K3 surfaces in dimension two. By abundance, the Beauville-Bogomolov decom-position theorem and elementary properties of hyperbolic manifolds, in order toprove the Kobayashi conjecture in dimension three it suffices to show that a Calabi-Yau threefold is not hyperbolic . Here, by a Calabi-Yau threefold we mean a simplyconnected compact projective threefold with trivial canonical class K X ≃ O X and H i ( X, O X ) = 0, i = 1 , • The article [14] is the culmination of a series of papers by Wilson in which hestudies in a systematic way the geometry of Calabi-Yau threefolds; amongmany other things, it is shown there that if the Picard number ρ ( X ) > X . • Following somehow the same circle of ideas, it was proven in [24] (see also[22]) that a Calabi-Yau threefold X has a rational curve provided thereexists on X a non-zero effective non-ample line bundle on X .By the Cone Theorem, if there exists on a Calabi-Yau manifold X a non-zeroeffective non-nef line bundle, then there exists on X a rational curve (generating an extremal ray). Therefore, we can always suppose that such an effective line bundle isnef. Remark, on the other hand, that in Peternell’s result, the effectivity hypothesisis crucial (regarding it in a more modern way) in order to make the machinery ofthe logMMP work. In this spirit, Oguiso asked in [22] the following question: isit true that if a Calabi-Yau threefold X possesses a non-zero nef non-ample linebundle, then there exists a rational curve on X ? Here is a positive answer, under a mild condition on the Picard number of X . Theorem 3.1 (Diverio-Ferretti [9]) . Let X be a Calabi-Yau threefold and L → X a non-zero nef non-ample line bundle. Then, X has a rational curve provided ρ ( X ) > . The proof of this theorem is of arithmetic nature and relies upon a careful studyof the diophantine properties of the cubic intersection form defining the nef bound-ary in the N´eron-Severi space of X . In particular, it is exploited the link betweenthe boundedness and convexity properties of the nef cone and the arithmetic of itsboundary.The results stated above thus permit, as in [24], to exclude a certain number ofcases to be checked in order to prove such a non-hyperbolicity statement. Finally,as far as we know, there is no known example of a Calabi-Yau threefold withoutnon-constant holomorphic images of complex tori.3.2. The Green-Griffiths-Lang conjecture.
This conjecture in full generalityis still unknown even in dimension two, so let us start by discussing this case.To begin with, observe that if the Zariski closure of all entire curves in a surface S is positive dimensional (otherwise the surface is hyperbolic) but a proper subvariety,then by uniformization it must be a finite union of rational and elliptic curves. Ifthe second Segre number c − c of a minimal surface of general type S is positive,then it is know, since the work of Bogomolov, that there is abundance of symmetricdifferentials on S . Thanks to this, Bogomolov himself was able to show that thereis only a finite number of rational and elliptic curves on such a surface S , see [7].Several years later, McQuillan [18], proving deep results on holomorphic (multi)-foliations on surfaces, was able to extend the work of Bogomolov to the transcen-dental setting, thus showing that the Green-Griffiths conjecture is true for surfacesof general type with positive second Segre number. More precisely, using Miyaoka’ssemi-positivity result for cotangent bundles of nonuniruled projective varieties anda dynamic diophantine approximation, he derived strong Nevanlinna Second MainTheorems for entire curves tangent to the leaves of a holomorphic foliation. Inparticular, he obtains that every parabolic leaf of an algebraic (multi)foliation ona surface S of general type is algebraically degenerate. The assumption c > c guarantees the existence of an algebraic multi-foliation such that every entire curveis contained in one of its leaves.This last sentence can be rephrased by saying that under the assumption on thesecond Segre class, every entire curve must satisfy an algebraic differential equationof order one. This can be put in perspective as follows (see [4] for all the details).Let X be a complex manifold, p k : J k X → X the holomorphic fiber bundle of k -jetsof germs of holomorphic curves γ : ( C , → X and J k X reg its subset of regularones, i.e. such that γ ′ (0) = 0. There is a natural action of the group G k of k -jetsof biholomorphisms of ( C ,
0) on J k X , and the quotient J k X reg / G k admits a nicegeometric relative compactification J k X reg / G k ֒ → X k . Here, π k : X k → X is atower of projective bundles over X . In particular, it is naturally endowed with atautological line bundle O X k ( − π k ) ∗ O X k ( m ), m ≥
1, is the sheaf of holomor-phic sections of a holomorphic vector bundle E k,m T ∗ X → X on X . For instance, SIMONE DIVERIO when k = 1, we have that E ,m T ∗ X ≃ S m T ∗ X is just the m -th symmetric power ofthe cotangent bundle. Definition 3.2.
The holomorphic sections of the vector bundle E k,m T ∗ X are called invariant jet differentials of order k and (weighted) degree m .Invariant jet differentials of order k and (weighted) degree m act on ( k -jets of)holomorphic curves traced in X as polynomial differential operators. More precisely,let P ∈ O X ( E k,m T ∗ X )( U ) be an invariant jet differentials over an open set U ⊂ X .Then, P is nothing but a holomorphic function P : p − k ( U ) ⊂ J k X → C , such that P ( γ ◦ ϕ ) = ( ϕ ′ ) m P ( γ ), for γ : ( C , → U a k -jet of germs of holomorphic curveand ϕ : ( C , → ( C ,
0) a k -jet of biholomorphism of the origin. This immediatelyexplains why E ,m T ∗ X ≃ S m T ∗ X and why, in general, invariant jet differentials canbe considered as algebraic differential operators acting on holomorphic curves. Theorem 3.3 (Green-Griffiths [13], Siu-Yeung [28], Demailly [4]) . Let X be pro-jective algebraic and A → X an ample line bundle. Then, for all invariant jetdifferentials P ∈ H ( X, E k,m T ∗ X ⊗ A − ) ≃ H ( X k , O X k ( m ) ⊗ π ∗ k A − ) with valuesin the anti-ample divisor A − and all entire curves f : C → X , we have P ( f ) ≡ . Therefore, abundance of invariant jet differentials is translated by this theoremin abundance of constraints for entire curves. From this point of view, the firststep to prove hyperbolicity type (or, more generally, algebraic degeneracy type)statements is to produce some invariant jet differential.For instance, it is conjectured (and proved for surfaces) by Green and Griffithsin [13] that if X is a projective algebraic manifold of general type then there shouldexist a k ≫ O X k (1) is big (or rather, another less refined version ofthis line bundle is big, when X k is obtained by modding out just by the groupof homotheties instead of the full group of reparametrization of the origin). Thisconjecture has been proved in full generality only very recently by J.-P. Demaillyin [5], by means of an astonishing combination of his holomorphic Morse inequalitiesand a probabilistic interpretation of higher order jets.3.2.1. Projective hypersurfaces and complete intersections.
Since in full generalitythe Green-Griffiths-Lang conjecture seems out of reach for the moment, severalefforts have been made in some particular cases. Let us concentrate on projectivehypersurfaces and, later on, more generally, on complete intersections.Let X ⊂ P n +1 , n ≥
2, be a projective hypersurface of degree deg X = d . If X issmooth, then adjunction formula together with the Euler exact sequence show that K X ≃ O X ( d − n − K X is ample if (and only if) d ≥ n + 3; in particular,if deg X ≥ n + 3, then X is of general type. Therefore, if X is a smooth projectivehypersurface of degree deg X ≥ n + 3, the Green-Griffiths conjecture predicts theexistence of a proper subvariety Y ( X containing all entire curve in X .If, moreover, deg X ≥ n +1 and X is (very) generic, then Voisin proved in [29,30](among other things) that all its subvarieties are of general type (this was firstproven in [12]) and, as a consequence of her proof, that X is algebraically hyperbolic.Here, by very generic we mean a hypersurface whose modulus is outside a countableunion of proper subvarieties of the parameter space. Then, in view of Conjectures2.2 and 2.4, in this case X should be hyperbolic: this was in fact the content of thefollowing conjecture made by Kobayashi in 1970. Conjecture 3.4 (Kobayashi) . Let X ⊂ P n +1 , n ≥
2, be a smooth (very) genericprojective hypersurface of degree deg X ≥ n + 1. Then, X is hyperbolic.Observe, first of all, that the genericity assumption is necessary since, for ex-ample, Fermat’s hypersurfaces always contain lines (which shows also that ample canonical bundle does not imply hyperbolic). Second, the conjecture implies inparticular that every generic projective complete intersection of high (multi)degreeshould be hyperbolic.For surfaces in projective 3-space, the conjecture is known without optimality ofthe degree d , which should be starting from five: it has been proven independentlyby McQuillan [19] for d ≥
36 and Demailly-El Goul [6] for d ≥
21. The lower boundon the degree has been further refined by P˘aun [23] to d ≥ X ≥
593 and X is generic, then there does not exist any Zariski dense entirecurve in X (but observe that the locus covered by the image of entire curve is notexcluded to be Zariski dense).Next, let us give a brief account about more recent and somehow general results. Theorem 3.5 (Diverio-Merker-Rousseau [10], Diverio-Trapani [11]) . Let X ⊂ P n +1 be a generic projective hypersurface of degree deg X ≥ n . Then, there exists aproper algebraic subvariety Y ( X which contains the image of all entire curves in X . If dim X ≥ or dim X = 2 and X is very generic, then the codimension of Y in X is at least two. The information about the codimension of Y is the contribution of [11]: in orderto obtain it, it is crucial that every effective divisor is ample (this explains the verygenericity assumption for surfaces, in view of the Noether-Lefschetz theorem).Before discussing the key ingredients in the proof of the theorem above, let usmake some comments and state a couple of corollaries. As far as we know, this isthe first result in all dimensions about algebraic degeneracy of entire curves in man-ifolds of general type. On the other hand, unfortunately, projective hypersurfacesform a too small class to see this result really as an evidence for the Green-Griffithsconjecture. Nevertheless, this theorem gives also an effective estimate on the degreeof the hypersurfaces only in terms of their dimension (even if this degree is veryfar from being optimal...). It would be desirable to drop the genericity assump-tion (which is intrinsic in the methods used in the proof) as far as only algebraicdegeneracy (and not the full hyperbolicity) is concerned. Corollary 3.6 (Diverio-Trapani [11]) . A very generic threefold in P of degree atleast is Kobayashi hyperbolic. This follows from the fact that the degeneracy locus is of dimension one in thiscase and thus it should consist of rational or elliptic curves, which are absent inthese hypersurfaces.
Corollary 3.7 (Brotbek [1]) . Let X be a generic projective complete intersection of(multi)degree greater than or equal to (2 n , . . . , n ) . Then, X is hyperbolic provided dim X ≤ X . This corollary is obtained by one further genericity argument combined with theaction of the group of projective automorphisms. Observe that this last result canbe seen as a partial confirmation of Debarre’s conjecture stated in the introduction(for more developments on Debarre’s conjecture, especially for generic completeintersection surfaces of high degree in P which are shown to have ample cotangentbundle, see [1]). Heuristic idea of the proof of Theorem 3.5.
Thanks to the strategy proposed in thework [26] of Siu and inherited somehow by Voisin’s work [29, 30], it was quite clearthat in order to prove such an algebraic degeneracy statement one had, above all,to overcome two main difficulties:
SIMONE DIVERIO (i) Produce some invariant jet differential vanishing on an ample divisor onevery smooth projective hypersurface of large degree.(ii) Produce low pole order meromorphic vector fields on the universal hyper-surface, in order to produce by differentiation new jet differentials, enoughto control the geometry of entire curves.Step (i) has been done in [8], as an application of algebraic holomorphic Morseinequalities: the method is intrinsically effective and permits to estimate explicitly,in terms of the dimension only, how large the degree should be. Step (ii) has beenachieved in [20].Once we have at our disposal a “first” jet differentials, thanks to a semi-continuityargument we can extend it holomorphically to all projective hypersurfaces parame-trized by a Zariski open set of their moduli space (this semi-continuity argumentmakes the result “only” generic). Next, the extended jet differential can be seen asa jet differential on an open set of the universal hypersurface. Therefore, one cannow use low pole order meromorphic tangent vector fields to derive this algebraicoperator in order to obtain new ones, algebraically independent from the first one.These new operators, once restricted to every projective hypersurface parametrizedby this Zariski open set, give enough informations to conclude that their base locusprojects onto a proper subvariety. The low pole order property enables us to be surethat after differentiation the jet differential remains holomorphic and with valuesin an anti-ample divisor. (cid:3)
Very recently, Siu, in [27], was able to push forward his own strategy in order toobtain Kobayashi hyperbolicity of generic projective hypersurfaces of high degree.The details of his beautiful and delicate proof are underway of validation by theexperts.
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