Unbendable rational curves of Goursat type and Cartan type
aa r X i v : . [ m a t h . AG ] F e b UNBENDABLE RATIONAL CURVES OF GOURSAT TYPEAND CARTAN TYPE
JUN-MUK HWANG AND QIFENG LI
Abstract.
We study unbendable rational curves, i.e., nonsingular ra-tional curves in a complex manifold of dimension n with normal bundlesisomorphic to O P (1) ⊕ p ⊕ O ⊕ ( n − − p ) P for some nonnegative integer p . Well-known examples arise from alge-braic geometry as general minimal rational curves of uniruled projectivemanifolds. After describing the relations between the differential geo-metric properties of the natural distributions on the deformation spacesof unbendable rational curves and the projective geometric properties oftheir varieties of minimal rational tangents, we concentrate on the caseof p = 1 and n ≤
5, which is the simplest nontrivial situation. In thiscase, the families of unbendable rational curves fall essentially into twoclasses: Goursat type or Cartan type. Those of Goursat type arise fromordinary differential equations and those of Cartan type have specialfeatures related to contact geometry. We show that the family of lineson any nonsingular cubic 4-fold is of Goursat type, whereas the familyof lines on a general quartic 5-fold is of Cartan type, in the proof ofwhich the projective geometry of varieties of minimal rational tangentsplays a key role.
MSC2010: 58A30, 32C25, 14J701.
Introduction
A nonsingular rational curve in a complex manifold of dimension n is saidto be unbendable if its normal bundle is isomorphic to O P (1) ⊕ p ⊕ O ⊕ ( n − − p ) P for some nonnegative integer p . There are many examples of unbendable ra-tional curves arising from algebraic geometry. Any uniruled projective man-ifold has minimal rational curves and general minimal rational curves areunbendable. In this case, unbendable rational curves are sometimes calledstandard rational curves, indicating that they are general among minimalrational curves (see Section 1.1 of [HM] or Section 1.2 of [Hw01]). Under-standing the germ of an unbendable rational curve in a complex manifold,i.e. the biholomorphic structure of neighborhoods of such a curve, is impor-tant especially when they are minimal rational curves in a Fano manifold of This work was supported by the Institute for Basic Science (IBS-R032-D1).
Picard number 1, because quite often the germ of the curve determines thebiregular type of the ambient Fano manifold by Cartan-Fubini type exten-sion theorem (see Section 3 of [HM] or Section 3 of [Hw01]).One approach to study the germ of an unbendable rational curve is bylooking at certain distributions (i.e. Pfaffian systems) on the deformationspace of the unbendable rational curves in the ambient complex manifold(i.e. the corresponding open subset in the Douady space of the ambientmanifold). In many cases, the germ of an unbendable rational curve isdetermined by the germ of these distributions in a neighborhood of thecorresponding point in the Douady space. In this paper, we employ thiscorrespondence to study the germ of unbendable rational curves.The relation between the geometry of rational curves on a complex man-ifold and certain natural differential systems on the corresponding Douadyspace has been studied much in twistor theory (e.g. [Hi]). What is new inour approach is the role of the varieties of minimal rational tangents (VMRTin abbreviation), the complex submanifold in the projectivized tangent bun-dle of the ambient complex manifold traced by tangents to the deformationsof the unbendable rational curves. The original notion of VMRT (e.g. in[HM]) is for minimal rational curves on uniruled projective manifolds andthey are defined as certain projective subvarieties in the projectivized tan-gent spaces of the uniruled projective manifolds. In this paper, we considerVMRT for unbendable rational curves and they are defined (see Definition3.15) as certain complex submanifolds in the projectivized tangent spaces.In fact, for our purpose, it is sufficient to consider the germs of VMRT.We show that the projective geometry of VMRT is intricately related tothe properties of the distributions on the deformation space, as explained inSection 3.To explain our main results, let us recall briefly the notion of the growthvector of a distribution D on a complex manifold M (see Section 2.1 fora precise definition). Associated to D ⊂ T M is a sequence of saturatedsubsheaves D = ∂ D ⊂ ∂ D ⊂ ∂ D ⊂ · · · ⊂ ∂ k D = ∂ k +1 D of T M for some nonnegative integer k , generated by the successive bracketsof local vector fields belonging to D . The growth vector of D is the strictlyincreasing sequence of positive integers(rank( D ) , rank( ∂ D ) , . . . , rank( ∂ k D )) . It is the most basic invariant of a distribution. We say that D is bracket-generating if ∂ k D = T M , i.e., the last entry of its growth vector is dim M .In the current article, we concentrate on the case p = 1, i.e. when there isexactly one O (1)-factor in the normal bundle of unbendable rational curves.This is the case when the anti-canonical degree of the rational curves is 3.In this case, the deformation space has a natural rank 2 distribution andthe germ of unbendable rational curves can be recovered from some extra NBENDABLE RATIONAL CURVES 3 structure on the germ of this rank 2 distribution (Proposition 3.20). Thereare two particularly well-understood classes of rank 2 distributions: rank 2distributions D ⊂ T M satisfyingrank( ∂ i D ) = rank( ∂ i − D ) + 1 for all 1 ≤ i ≤ k and ∂ k D = T M, which are called Goursat distributions (see [MZ]) and distributions with thegrowth vector (2 , , , which ´E. Cartan studied extensively in [Ca]. Weinvestigate the geometry of unbendable rational curves whose associateddistributions belong to one of these two classes of distributions. Let uscall them unbendable rational curves of Goursat type and of Cartan type,respectively.Unbendable rational curves of Goursat type arise from ordinary differen-tial equations and we describe precisely which ordinary differential equationsgive rise to unbendable rational curves (see Theorem 4.7 for a precise ver-sion): Theorem 1.1.
Unbendable rational curves of Goursat type arise from or-dinary differential equations of the type d n u d t n = a ( d n − u d t n − ) + a ( d n − u d t n − ) + a d n − u d t n − + a where a , a , a , a are local holomorphic functions of the n variables t , u , d u d t , . . . , d n − u d t n − . When n = 2, Theorem 1.1 is precisely Theorem 3.1 of [Hi].To study unbendable rational curves of Cartan type, we use some struc-ture theory of (2 , , Theorem 1.2.
There is a natural 1-1 correspondence between germs ofa (2 , , -distribution at general points and germs of general members ofa bracket-generating family of unbendable rational curves of anti-canonicaldegree 3 in a complex manifold of dimension equipped with a contact struc-ture such that the rational curves are tangent to the contact structure. Here, a family of unbendable rational curves is bracket-generating if theassociated distribution in the deformation space of the curves is a bracket-generating distribution. There is a well-known geometric interpretation of(2 , , , , JUN-MUK HWANG AND QIFENG LI compute. The simplest nontrivial question in this direction is to determinewhether a given family of unbendable rational curves with p = 1 in a com-plex manifold of dimension 5 is of Goursat type or of Cartan type. Our keyresult is the following criterion (see Theorem 4.13 and Theorem 5.3): Theorem 1.3.
A bracket-generating family of unbendable rational curveswith p = 1 in a 5-dimensional manifold is of Goursat type if and only if thethird fundamental forms of the VMRT along a general member is zero atsome point. If it is not of Goursat type, then it is of Cartan type. Using this criterion, we prove the following (see Theorem 6.1).
Theorem 1.4.
The family of lines on a general hypersurface of degree 4 in P is a family of unbendable rational curves of Cartan type. This illustrates the advantage of using VMRT in our approach. We shouldmention, however, that the computation can be still tricky even after usingTheorem 1.3. For the proof of Theorem 1.4, we need to select a hypersurfacewith a special involution to simplify the computation.In this context, Theorem 1.4 leads to the following two questions whichwe leave for future studies.
Question 1.5.
Is the family of lines on any nonsingular hypersurface ofdegree 4 in P of Cartan type? Question 1.6.
What is the growth vector of the rank 2 distribution D inDefinition 3.4 for the family of lines on a general hypersurface of degree n − P n +1 ?This paper is organized as follows. In Section 2, we collect standard factson some differential geometric notions to fix our terminology and notation.In Section 3, we present a general theory of the natural distribution on thedeformation space of unbendable rational curves and its relation to VMRT.Section 4 studies unbendable rational curves of Goursat type and Section 5studies unbendable rational curves of Cartan type. Finally, in Section 6, weprove Theorem 1.4. 2. Preliminaries
We work in the complex analytic category: all manifolds, bundles andmaps are holomorphic. Open subsets are taken in Euclidean topology, unlessstated otherwise. The tangent bundle of a complex manifold M is denoted by T M . For a vector bundle W of rank r on M , its projectivization P W → M is the P r − -bundle whose fibers consist of 1-dimensional subspaces in fibersof W → M . The tautological line bundle on P W is denoted by O P W ( − O P W (1) . NBENDABLE RATIONAL CURVES 5
Distributions. A distribution D on a complex manifold M is a vectorsubbundle D ⊂ T M of the tangent bundle. By abuse of notation, we denoteby D ⊂ T M also the locally free subsheaf of local sections of D , a subsheafof the sheaf of vector fields on M . A meromorphic distribution D of rank r on a complex manifold M is a coherent subsheaf D of rank r in the sheafof vector fields on M which is saturated in the sense that there exists nosubsheaf D ′ of the sheaf of vector fields satisfying D ⊂ D ′ , D = D ′ andrank( D ) = rank( D ′ ). By abuse of notation, we write D ⊂ T M to denote ameromorphic distribution. When the restriction of D to a neighborhood of apoint x ∈ M is a distribution, we write D x ⊂ T x M to denote the fiber of thecorresponding vector subbundle at x . The saturation condition implies thata meromorphic distribution induces a distribution on a Zariski open subsetof M which is the complement of a closed analytic subset of codimensionat least 2. It implies also that D is determined by the fibers D x ⊂ T x M atgeneral points x ∈ M . Given a distribution D ⊂ T M , Lie brackets of localsections [ , ] : ∧ D → T M induces a homomorphism of vector bundlesLevi D : ∧ D → T M/D, called the
Levi tensor of D .The Cauchy characteristic of a meromorphic distribution D on M is thesubsheaf Ch( D ) ⊂ D defined as follows: a local section v of D belongs toCh( D ) if and only if [ v, w ] belongs to D for any local section w of D . TheCauchy characteristic is involutive, defining a meromorphic foliation.Associated to a meromorphic distribution D ⊂ T M are the derived mero-morphic distributions ∂ i D and ∂ ( i ) D defined inductively by ∂ D = ∂ (1) D = [ D, D ] +
D,∂ i +1 D = [ ∂ i D, ∂ i D ] + ∂ i D, and ∂ ( i +1) D = [ D, ∂ ( i ) D ] + ∂ ( i ) D. Writing ∂ − D = 0 and ∂ D = D , we have an integer d ≥ ∂ d − D ) = rank( ∂ d D ) = rank( ∂ d + i D )for any i >
0. The strictly increasing sequence of integers(rank( D ) , rank( ∂D ) , rank( ∂ D ) , . . . , rank( ∂ d D ))is called the growth vector of D . If rank( ∂ d D ) = dim M , we say that D is bracket-generating .We say that a meromorphic distribution D ⊂ T M is regular at a point x ∈ M (equivalently, x is a regular point of D ) if Ch( D ) , ∂ i D and ∂ ( i ) D forall i ≥ x ∈ M . Regular pointsof a meromorphic distribution form a Zariski-open subset in M called the regular locus of D . We say that D is a regular distribution if every point of M is a regular point of D .A vector field v on M is an infinitesimal automorphism of a meromorphicdistribution D , if the 1-parameter germs of local biholomorphisms of M JUN-MUK HWANG AND QIFENG LI (and consequently
T M ) generated by v preserve D. The Lie algebra of allinfinitesimal automorphisms of D is denoted by aut ( D ).When π : Y → M is a submersion of complex manifolds, denote by T π ⊂ T Y the distribution given by Ker(d π ). For a meromorphic distribution D on M , the inverse image π − D is a meromorphic distribution on Y whosefiber at a general point y ∈ Y is( π − D ) y = (d y π ) − ( D π ( y ) )where d y π : T y Y → T π ( y ) M is the differential of π at y . It is easy to see that(2.1) ∂ k ( π − D ) = π − ( ∂ k D ) , ∂ ( k ) ( π − D ) = π − ( ∂ ( k ) D )for any k ≥ π − D ) = π − Ch( D ).For a distribution D ⊂ T M , let π : P D → M be the projectivizationof the vector bundle D . Define the distribution pr( D ) on P D , called the prolongation of D , whose fiber at [ v ] ∈ P D corresponding to 0 = v ∈ D ispr( D ) [ v ] := (d [ v ] π ) − ( C v ) . Then(2.2) ∂ (pr( D )) = π − D from Proposition 5.1 of [MZ] (this is also a special case of Proposition 1 in[HM04]).2.2. Fundamental forms of projective submanifolds.
A good refer-ence for fundamental forms is Section 12.1 of [IL], where they are formu-lated in terms of differential forms. For our purpose, it is convenient toreformulate them in terms of vector fields as follows.For a vector space V , its projectivization P V is the set of 1-dimensionalsubspaces of V . For a (not necessarily closed) complex submanifold Z ⊂ P V ,denote by ˆ Z ⊂ V be the union of 1-dimensional subspaces corresponding to Z . The affine tangent space ˆ T z Z ⊂ V at a point z ∈ Z is the tangent spaceof ˆ Z at a point of ˆ z \
0. There is a natural identification T z Z = Hom(ˆ z, ˆ T z Z/ ˆ z ) , (equivalently, ˆ T z Z/ ˆ z = ˆ z ⊗ T z Z )) . Let ξ : V \ → P V be the natural C ∗ -bundle. Fix a linear coordinate system x , . . . , x n on V where n = dim V . Let v , . . . , v k be local vector fields on Z near a point z ∈ Z. Given any point z ′ ∈ V \ ξ ( z ′ ) = z , we can chooseholomorphic functions v i,j in a neighborhood of z ′ in V such thatˆ v i = n X j =1 v i,j ∂∂x j are local vector fields tangent to ˆ Z near z ′ satisfying d ξ (ˆ v i ) | ˆ Z = v i . Writeˆ v (ˆ v ) = n X j =1 ˆ v ( v ,j ) ∂∂x j NBENDABLE RATIONAL CURVES 7 and inductivelyˆ v k (ˆ v k − · · · ˆ v ( v )) = n X j =1 ˆ v k (ˆ v k − · · · ˆ v ( v ,j )) ∂∂x j . Using this notation, fundamental forms of Z ⊂ P V can be described asfollows.(1) Consider the map sending v ,z ⊗ v ,z ∈ T z Z ⊗ T z Z to d z ′ ξ (ˆ v (ˆ v ))modulo T z Z where ˆ v (resp. ˆ v ) is a local vector field on ˆ Z whosevalue at z ′ ∈ ˆ z \ v ,z (resp. v ,z ) by d z ′ ξ . This determinesa homomorphismFF z,Z : Sym ( T z Z ) −→ T z P V /T z Z called the second fundamental form of Z at z , independent of thechoice of z ′ and ˆ v i . There exists a Zariski open subset Dom(FF Z ) ⊂ Z such that the images of FF y,Z , y ∈ Dom(FF Z ) , form a vectorsubbundle T (2) Z ⊂ ( T P V ) | Dom(FF Z ) . Thus we have a surjective vector bundle homomorphismFF Z : Sym ( T Z ) → T (2) Z/T Z on Dom(FF Z ) ⊂ Z. (2) For z ∈ Dom(FF Z ) , consider the map sending v ,z ⊗ v ,z ⊗ v ,z ∈ T z Z ⊗ T z Z ⊗ T z Z to d z ′ ξ (ˆ v (ˆ v (ˆ v ))) modulo T (2) z Z , where ˆ v i is alocal vector field on Z whose value at z ′ ∈ ˆ z \ v i,z by d z ′ ξ for each i = 1 , ,
3. This determines a homomorphismFF z,Z : Sym ( T z Z ) −→ T z P V /T (2) z ( Z ) , called the third fundamental form of Z at z , independent of thechoices of z ′ and ˆ v i . There exists a Zariski open subset Dom(FF Z ) ⊂ Dom(FF Z ) such that the images of FF y,Z , y ∈ Dom(FF Z ) form avector subbundle T (3) Z ⊂ ( T P V ) | Dom(FF Z ) . We obtain a surjective vector bundle homomorphismFF Z : Sym ( T Z ) → T (3) Z/T (2) Z on Dom(FF Z ) ⊂ Z. (3) Repeating the above construction inductively, we have a homomor-phism FF kz,Z : Sym k ( T z Z ) −→ T z P V /T ( k − z ( Z ) , called the k -th fundamental form of Z at z , where Dom(FF kZ ) ⊂ Dom(FF k − Z ) is a Zariski open subset such that the images of FF k − y,Z , y ∈ Dom(FF kZ ) , form a vector subbundle T ( k − Z ⊂ ( T P V ) | Dom(FF kZ ) . JUN-MUK HWANG AND QIFENG LI
The quotient bundle T ( k ) Z/T ( k − Z on Dom(FF k +1 Z ) ⊂ Dom(FF kz )is denoted by N ( k ) Z and called the k -th normal space of Z with thefiber N ( k ) Z,z called the k -th normal space of Z at z . Thus we have asurjective vector bundle homomorphismFF kZ : Sym k ( T Z ) → T ( k ) Z/T ( k − Z = N ( k ) Z on Dom(FF k +1 Z ) ⊂ Z. We say that Z ⊂ P V is linearly nondegenerate if it is not contained inany hyperplane in P V . It is well-known that there exists k ≥ T ( k ) z Z = T z P V for any z ∈ Dom(FF k +1 Z ) if and only if Z ⊂ P V is linearlynondegenerate.2.3. Jet spaces.
We recall the construction of jet spaces of functions for oneindependent and one dependent variables. In comparison with the referencessuch as Section I.3 of [BCG] or Chapter 4 of [Ol], we emphasize the projectivebundle structure of the prolongation for the application in Section 4.Let ∆ be a 1-dimensional complex manifold. Let J = ∆ × C be thetrivial line bundle on ∆ with the natural projection p − : J → ∆ and set J = T J to be the tangent bundle.Let p : P J → J be the projectivization of J and let J ⊂ T P J be the prolongation of J . Note that J is a contact structure on the 3-dimensional manifold P J in the sense of Definition 5.4. In fact, the contactform is ϑ J of Example 5.5 under the natural isomorphism P T J = P T ∗ J .Let p : P J → P J be the projectivization of J and let J ⊂ T P J bethe prolongation of J . Assuming we have defined a distribution J k of rank2 on a manifold P J k − , we define a distribution J k +1 of rank 2 on P J k asthe prolongation of J k .The space J k of k -jets of functions on ∆ is an open subset of P J k − defined as follows. Assume that ∆ has a coordinate function t and let u be a linear coordinate function on C such that ( t, u ) is a coordinate systemfor J . Let J be the open subset P J \ P T p − . Regarding d t and d u asholomorphic functions on J = T J , we can view the meromorphic function u (1) := d u d t on P J as a holomorphic function on J defining affine coordinates alongthe fibers of J → J . Let J be the open subset P J | J \ P T p . Then themeromorphic function u (2) := d u (1) d t on P J can be viewed as a holomorphic function on J defining affine co-ordinates along the fibers of J → J . Inductively define J k +1 as the open NBENDABLE RATIONAL CURVES 9 subset P J k | J k \ P T p kk − . Then the meromorphic function u ( k +1) := d u ( k ) d t on P J k − can be viewed as a holomorphic function on J k defining affinecoordinates along the fibers of J k +1 → J k . A section of the fibration J k +1 → J k over an open subset U ⊂ J k is givenby u ( k +1) = F ( t, u, u (1) , . . . , u ( k ) )for a holomorphic function F on U . This is equivalent to an ordinary differ-ential equation of order k +1 in the independent variable t and the dependentvariable u with initial conditions in U .For k > i ≥
1, denote by p ki : J k → J i the composition p i +1 i ◦ p i +2 i +1 ◦ · · · ◦ p k − k − ◦ p kk − . It is easy to check from (2.2) (see also Theorem 6.5 of [Mo]) that the sequence J k ⊂ ( p kk − ) − J k − ⊂ · · · ⊂ ( p k ) − J ⊂ T J k coincides with the sequence of derived distributions on J k , both J k ⊂ ∂ J k ⊂ · · · ⊂ ∂ k − J k ⊂ ∂ k J k and J k ⊂ ∂ (1) J k ⊂ · · · ⊂ ∂ ( k − J k ⊂ ∂ ( k ) J k . Distributions on the deformation spaces of unbendablerational curves
Definition 3.1.
Let C ⊂ M be a nonsingular projective rational curve on acomplex manifold of dimension n . We say that C is an unbendable rationalcurve if its normal bundle N C is isomorphic to N C ∼ = O P (1) ⊕ p ⊕ O ⊕ ( n − − p ) P (3.1)for some nonnegative integer p . The number p + 2 is the anti-canonicaldegree of C , i.e., the degree of the line bundle ∧ n T M | C .By the deformation theory of rational curves, the family of nontrivialdeformations of C in M has dimension n − p = dim H ( C, N C ) and allgeneral deformations of C in M are also unbendable. It is convenient toformulate this in terms of the Douady space (or Hilbert scheme when M isalgebraic) of M as follows. A good reference for Douady spaces is SectionVIII.1 of [GPR]. Definition 3.2. A family of unbendable rational curves on a complex man-ifold M means a connected open subset K in the Douady space Douady( M ) such that each member of K is an unbendable rational curve. The anti-canonical degree of K is the anti-canonical degree of a member of K , namely,the integer p + 2 in Definition 3.1. We have dim K = n − p . Denote by K ρ ←− U µ −→ M the associated universal family morphisms. The morphism ρ is a P -bundleand the morphism µ is a submersion with the fiber dimension p . A point y ∈ K corresponds to the unbendable rational curve C y := µ ( ρ − ( y )) in M .The following lemma is well-known from the deformation theory of ratio-nal curves (see page 58 of [HM04]). Lemma 3.3.
In Definition 3.2, for a point α ∈ U with y = ρ ( α ) and x = µ ( α ) ∈ C y , we have the following natural identifications. (i) T y K = H ( C y , N C y ) . (ii) T α U = H ( C y , T M | C y ) /H ( C y , T C y ⊗ m x ) . (iii) Under the identification in (ii), the subspace T ρα ⊂ T α U tangent tothe fiber of ρ is identified with the subspace H ( C y , T C y ) /H ( C y , T C y ⊗ m x ) ⊂ H ( C y , T M | C y ) /H ( C y , T C y ⊗ m x ) induced by T C y ⊂ T M | C y . (iv) Under the identification in (ii), the subspace T µα tangent to the fiberof µ is identified with H ( C y , T M | C y ⊗ m x ) /H ( C y , T C y ⊗ m x ) = H ( C y , N C y ⊗ m x ) . In particular, we have T µα ∩ T ρα = 0 for any α ∈ U from T M | C y ∼ = T C y ⊕ N C y . The space K of unbendable rational curves carries a natural distribution: Definition 3.4.
For each member C of K , let N + C be the subbundle of thenormal bundle N C corresponding to the O P (1) ⊕ p factor of (3.1), which isindependent of the choice of the isomorphism (3.1). Define a distribution D ⊂ T K of rank 2 p such that its fiber at [ C ] ∈ K is D [ C ] := H ( C, N + C ) ⊂ H ( C, N C ) = T [ C ] K via the natural identification in Lemma 3.3 (i).This distribution D on K is our main object of study. We relate it tosome natural distributions on U defined as follows. Definition 3.5.
In Lemma 3.3, write V := T µ and F := T ρ on U . Set T = V ⊕ F , which is a distribution on U by Lemma 3.3. Define meromorphicdistributions T k , k ≥ U inductively by T := T and T k +1 := [ V , T k ] + T k for each k ≥ . Write T k +1 := T k +1 / T k for each k ≥ T := F .The followings are straight-forward from the definition of T k . NBENDABLE RATIONAL CURVES 11
Lemma 3.6.
In Definition 3.5, there are homomorphisms of sheaves ψ k : V ⊗ T k → T k +1 , k ≥ on a Zariski open subset on U such that ψ ( v ⊗ f ) = [ v, f ] modulo T and ψ k ( v k +1 ⊗ [ v k , [ v k − , · · · , [ v , f ]]]) = [ v k +1 , [ v k , · · · , [ v , f ]]] for each k ≥ , where f is a local section of F , and v, v , . . . , v k +1 arelocal sections of V . In the above equation, the repeated brackets representingelements of T k and T k +1 are used to denote the corresponding elements of T k and T k +1 . A key fact is the following result from Proposition 1 and Proposition 8 of[HM04].
Proposition 3.7.
In Definitions 3.4 and 3.5, the meromorphic distribution T is a distribution on U and satisfies T = ρ − D . Proposition 3.8.
In the setting of Definition 3.5, (i) [ F , T ] ⊂ T ; (ii) [ F , T ] ⊂ T ; and (iii) If [ F , T k ] ⊂ T k for ≤ k ≤ m − for some integer m ≥ , then [ F , T m ] ⊂ T m .Proof. (i) is a direct consequence of Proposition 3.7. (ii) follows from (i)and (iii).To prove (iii), we introduce the following notation. Fix local sections f of F and v , . . . , v m of V . Write[ v , . . . , v k , f ] := [ v , [ v , [ · · · [ v k , f ] · · · ]]]which is a local section of T k , respectively. The following lemma is imme-diate from Jacobi identity and [ V , V ] ⊂ V . Lemma 3.9.
For any k ≥ and any permutation σ of the set { , , . . . , k } , [ v , . . . , v k , f ] − [ v σ (1) , . . . , v σ ( k ) , f ] is a local section of T k − . Assuming that [ F , T k ] ⊂ T k for all 1 ≤ k ≤ m − , we have the followingthree lemmata. Lemma 3.10.
For any ≤ i ≤ j ≤ m − satisfying i + j ≤ m − , ( † ) i,j [[ v , . . . , v i , f ] , [ v i +1 , . . . , v i + j , f ]] is a section of T i + j . Here [ v , . . . , v i , f ] means f if i = 0 .Proof. It is sufficient to prove( ♮ ) i : the statement ( † ) i,j is true for all i ≤ j ≤ m − − i for all 0 ≤ i < m . The statement ( † ) , is immediate from [ F , F ] ⊂ F . When i = 0 and 1 ≤ j ≤ m −
1, the statement ( † ) ,j follows from the assumption [ F , T j ] ⊂ T j for j ≤ m −
1. This proves ( ♮ ) .Now fix 1 ≤ i < m and assume that ( ♮ ) k is true for all 0 ≤ k < i . Takeany i ≤ j ≤ m − − i . Then[[ v , . . . , v i , f ] , [ v i +1 , . . . , v i + j , f ]]is equal to[[ v , v i +1 , . . . , v i + j , f ] , [ v , . . . , v i , f ]] + [ v , [[ v , . . . , v i , f ] , [ v i +1 , . . . , v i + j , f ]]] . The first term belongs to T i + j by the induction hypothesis ( † ) i − ,j +1 andthe second term belongs to [ V , T i + j − ] ⊂ T i + j by the induction hypothesis( † ) i − ,j . This proves ( ♮ ) i and the lemma. (cid:3) Lemma 3.11.
For any ≤ k ≤ m, [[ v , . . . , v k − , f ] , [ v k , . . . , v m , f ]] ≡ − [[ v , . . . , v k , f ] , [ v k +1 , . . . , v m , f ]] modulo T m . Here [ v , . . . , v k − , f ] means f if k = 1 and [ v k +1 , . . . , v m , f ] means f if k = 2 m .Proof. By Jacobi identity,[[ v , . . . , v k − , f ] , [ v k , . . . , v m , f ]]= [[ v , . . . , v k − , f ] , [ v k , [ v k +1 , . . . , v m , f ]]]= − [[ v k , [ v , . . . , v k − , f ]] , [ v k +1 , . . . , v m , f ]]+[ v k , [[ v , . . . , v k − , f ] , [ v k +1 , . . . , v m , f ]]] . The last term is a local section of T m by Lemma 3.10. The second lastterm is − [[ v k , v , . . . , v k − , f ] , [ v k +1 , . . . , v m , f ]]which is equal to − [[ v , . . . , v k , f ] , [ v k +1 , . . . , v m , f ]] modulo T m by Lemma3.9. (cid:3) Lemma 3.12.
For any ≤ k ≤ m , we have [ f, [ v , . . . , v m , f ]] ≡ ( − k [[ v , . . . , v k , f ] , [ v k +1 , . . . , v m , f ]] modulo T m . Here [ v k +1 , . . . , v m , f ] means f if k = 2 m .Proof. This follows from Lemma 3.11 by induction on k . (cid:3) Lemma 3.12 with k = 2 m says[ f, [ v , . . . , v m , f ]] ≡ ( − m [[ v , . . . , v m , f ] , f ]modulo T m . Thus [ f, [ v , . . . , v m , f ]] is a local section of T m . Since ele-ments of the form [ v , . . . , v m , f ] span T m locally, we obtain Proposition3.8 (iii). (cid:3) Remark 3.13.
Proposition 3.8 (i) and (ii) cannot be extended further. Infact, there are cases where [ F , T ]
6⊂ T , as explained in Sections 5 and 6below. NBENDABLE RATIONAL CURVES 13
Corollary 3.14.
In setting of Definition 3.5, we have ∂ (2) T = ∂ T = T = ρ − ( ∂ D ) . Proof.
By definition, we have T = [ V , T ] + T ⊂ ∂ (2) T ⊂ ∂ T = ∂ T . For a local section f of F and local sections v , v of V , we have[[ v , f ] , [ v , f ]] = [[ v , [ v , f ]] , f ] + [ v , [ f, [ v , f ]]] . As [[ v , [ v , f ]] , f ] and [ v , [ f, [ v , f ]]] are both local sections of T by Propo-sition 3.8 (i) and (ii), so is [[ v , f ] , [ v , f ]]. Since elements of the form [ v i , f ]span T locally, we have ∂ T ⊂ T and consequently T = ∂ (2) T = ∂ T . Using T = ρ − D from Proposition 3.7, we obtain ∂ T = ∂ T = ρ − ( ∂ D )by (2.1). (cid:3) We recall the definition of varieties of minimal rational tangents. Origi-nally (as in [HM], [Hw01]), they are defined for minimal rational curves onuniruled projective manifolds. But essentially the same definition works forany unbendable rational curve in complex manifolds.
Definition 3.15.
In the terminology of Lemma 3.3, define the tangent mor-phism τ : U → P T M by setting τ ( α ) := [ T x C y ] ∈ P T x M. Both τ and the restriction τ x : µ − ( x ) → P T x M for each x ∈ M are immersions (e.g. by Proposition 1.4 of [Hw01]). Theimage C x of τ x is called the variety of minimal rational tangents (VMRT inabbreviation) of K at x ∈ M . For α ∈ µ − ( x ), the τ x -image of the germ of µ − ( x ) at α is denoted by C α ⊂ P T x M . Since τ x is immersive, the germ C α is a submanifold and an irreducible component of the germ of C x at τ x ( α ).The next proposition is a generalization of Proposition 2 of [HM04]. Proposition 3.16.
In the setting of Lemma 3.6, the following holds. (i)
The homomorphism ψ is an isomorphism between V ⊗ F and T . (ii) Pick α ∈ U such that z := τ ( α ) is in Dom(FF k C α ) for some k ≥ .Then for a local section f of F and local sections v , . . . , v k of V near α , the value of the vector field [ v k , [ v k − , · · · , [ v , f ]]] at α satisfies d µ ([ v k , [ v k − , · · · , [ v , f ]]] α ) ≡ d µ ( f α ) ⊗ FF kz, C α ( v , . . . , v k ) modulo T ( k − z C α . (iii) In the setting of (ii), we have a natural identification T k,α = ˆ z ⊗ N ( k ) C α ,z where N ( k ) C α ,z is the k -th normal space of C α ⊂ P T x M at the point z ∈C α . In particular, the homomorphism ψ j of Lemma 3.6 is surjectiveon Dom(FF k C α ) for all ≤ j ≤ k . (iv) If C α is linearly nondegenerate in P T x M, x = µ ( α ) for a general α ∈ U , then T U = T k on a Zariski open subset of U for a sufficientlylarge integer k . (v) In the setting of (iv), the distribution D on K is bracket-generating.Proof. (i) is from Section 2 of [HM04].To prove (ii), we modify the computation in the proof of Proposition 2 in[HM04] as follows. Fix a local coordinate system x , · · · , x n in a neighbor-hood of x ∈ M . Then λ := dx , . . . , λ n := dx n give linear coordinates inthe vertical directions of T M → M .Via the immersion τ , let us identify a neighborhood U of α in U with asubmanifold of P T M such that α is identified with z and µ agrees with therestriction of the natural projection P T M → M . Let ξ : T M \ (0-section) → P T M be the natural C ∗ -bundle. We can choose a point z ′ ∈ ξ − ( z ), aneighborhood O of z ′ in T M and local holomorphic functions f i , v i,j on O with 1 ≤ i ≤ n and 1 ≤ j ≤ k such that the vector fields on O ˆ f = n X j =1 f j ∂∂λ j + n X j =1 λ j ∂∂x j , ˆ v i = n X j =1 v i,j ∂∂λ j are tangent to ξ − ( U ) and their images under d ξ induce f and v i on U .Direct calculations show that[ˆ v , ˆ f ] = n X j =1 v ,j ∂∂x j modulo ∂∂λ , . . . , ∂∂λ n and[ˆ v m , [ˆ v m − , · · · , [ˆ v , ˆ f ]]] = n X j =1 ˆ v m (ˆ v m − · · · ˆ v ( v ,j )) ∂∂x j modulo ∂∂λ , . . . , ∂∂λ n for each 2 ≤ m ≤ k . This proves (ii).(iii) and (iv) follow from (ii). Since T = ρ − D by Proposition 3.7, wehave T k +1 ⊂ ∂ k T = ρ − ∂ k D for each k ≥
1. Thus (v) follows from (iv). (cid:3)
The next proposition is well-known. (i) is from Proposition 2.3 of [Hw01]and the rest follows easily from (i) (see Proposition 3.1 of [HH] or the proofof Propositions 2.2 and 3.1 of [Mk]).
NBENDABLE RATIONAL CURVES 15
Proposition 3.17.
In the setting of Lemma 3.6 and Definition 3.15, for apoint y ∈ K , define C y := ∪ α ∈ ρ − ( y ) C α ⊂ P T M | C y . It is a complex manifold with a submersion π y : C y → C y . (i) The relative tangent bundle of π y : C y → C y restricted to ρ − ( y ) ∼ = P is isomorphic to O P ( − ⊕ p . (ii) The relative normal bundle of C y ⊂ P T M | C y restricted to ρ − ( y ) isisomorphic to O P ( − ⊕ ( n − − p ) and the relative second fundamentalforms of C y ⊂ P T M | C y along ρ − ( y ) are given by a section of avector bundle on ρ − ( y ) isomorphic to Hom(Sym O P ( − ⊕ p , O P ( − ⊕ ( n − − p ) ) = O ⊕ ( p + p )( n − − p ) / P . (iii) If a point α ∈ ρ − ( y ) is in Dom(FF C α ) , then any point β ∈ ρ − ( y ) is in Dom(FF C β ) . In this case, the relative second normal spacesof C y ⊂ P T M | C y along ρ − ( y ) determine a vector bundle on ρ − ( y ) isomorphic to O P ( − ⊕ q for some nonnegative integer q ≤ n − − p. (iv) When a point α ∈ ρ − ( y ) is in Dom(FF C α ) , the relative third funda-mental forms of C y ⊂ P T M | C y along ρ − ( y ) are given by a sectionof a vector bundle on ρ − ( y ) isomorphic to Hom(Sym O P ( − ⊕ p , O P ( − ⊕ ( n − − p − q ) ) = O P (1) ⊕ r for r = ( p + 3 p + 2 p )( n − − p − q ) . The above results say that the properties of D are intimately relatedto the projective geometry of VMRT, especially their fundamental forms.Many different types of fundamental forms of VMRT can arise dependingon the geometry of the unbendable rational curves and it is difficult to obtainuniform results. We have to restrict to specific classes of unbendable rationalcurves to proceed further. Definition 3.18.
A family K of unbendable rational curves on a complexmanifold is said to be bracket-generating if the distribution T is a bracket-generating distribution on U . By Proposition 3.7, this is equivalent to sayingthat D in Definition 3.4 is a bracket-generating distribution on K .The next proposition shows that to study germs of unbendable rationalcurves, we may concentrate on bracket-generating families. Proposition 3.19.
Suppose K is a family of unbendable rational curveson a complex manifold M , which is not bracket-generating. Then a generalmember C ⊂ M of K has a neighborhood C ⊂ U ⊂ M equipped with asubmersion η : U → B to a complex manifold B such that η ( C ) is a point b ∈ B and the family K b , which consists of members of K representingrational curves contained in η − ( b ) , is bracket-generating. In particular, thecurve C is an unbendable rational curve in the lower-dimensional complexmanifold η − ( b ) . Proof.
To prove Proposition 3.19, we may replace K by a neighborhood ofa general member of K . In particular, we may assume that µ has connectedfibers and C x is irreducible for a general point x ∈ M . We define a meromor-phic distribution E ⊂
T M such that its fiber E x ⊂ T x M at a general point x ∈ M is the linear span of C x in T x M . Then T k = µ − E on an open subsetof U for sufficiently large k . It follows that E is not bracket-generating on M because T is not bracket-generating on U . Thus ∂ m E becomes involutivefor some m ≥ M. We have a closedanalytic subset Z ⊂ M of codimension ≥ ∂ m E is an involutivedistribution on M \ Z . Then a general member C of K is disjoint from Z by Lemma 2.1 of [Hw01]. Then the leaves of the foliation defined by ∂ m E induces a submersion η : U → B in a neighborhood U of C such that C iscontained in a fiber of η . It is clear that C is an unbendable rational curveon the fiber and D| K b ⊂ T K b is bracket-generating. (cid:3) For the rest of the paper, we concentrate on unbendable rational curveswith p = 1 , which is exactly when D is a distribution of rank 2. In this case,we have the following general result. Proposition 3.20.
For a family of unbendable rational curves of anti-canonical degree on M , let π : P D → K be the projectivization of thevector bundle D of rank 2 on K . (i) Let ι : U → P D be the map sending α ∈ U with y = ρ ( α ) and x = µ ( α ) to the 1-dimensional subspace of H ( C y , N + C y ) = D y givenby H ( C y , N + C y ⊗ m x ) . Then ι is a biholomorphic map satisfying ρ = π ◦ ι. (ii) The image d ι ( T µ ) is a line subbundle of pr( D ) ⊂ π − D satisfying d ι ( T µ ) ∩ T π = 0 , i.e., it splits the exact sequence → T π → pr( D ) → O P D ( − → . Conversely, given a distribution D of rank on a complex manifold Y , let π : P D → Y be its projectivization and let V ⊂ pr( D ) be a line subbundle ona neighborhood U of a fiber of π that splits → T π → pr( D ) → O P D ( − → on U . Then after shrinking U if necessary, we obtain a submersion µ : U → M to a complex manifold M whose fibers are the leaves of V such thatthe images of the fibers of π under µ define a family of unbendable rationalcurves of anti-canonical degree 3 on M and D can be identified with thedistribution D associated with the family.Proof. Let n be the dimension of M . (i) follows easily from N C y ∼ = O P (1) ⊕O ⊕ ( n − P . (ii) follows from Lemma 3.3 and Proposition 3.17 (i). To see theconverse statement, the curve C y = µ ( π − ( y )) corresponding to a point y ∈ Y has the normal bundle N C y of degree 1 which is generated by globalsections. Thus it is isomorphic to O P (1) ⊕ O ⊕ ( n − P , implying that C y is NBENDABLE RATIONAL CURVES 17 unbendable. This proves that a neighborhood of y in Y can be regardedas the family of unbendable rational curves in an n -dimensional complexmanifold M . (cid:3) Unbendable rational curves of Goursat type
Definition 4.1.
A distribution D of rank 2 on an n -dimensional complexmanifold Y is a Goursat distribution if ∂ k D is a distribution of rank k + 2 foreach 1 ≤ k ≤ n − . A family K of unbendable rational curves of anticanonicaldegree 3 in an n -dimensional complex manifold is said to be of Goursat type ,if the distribution D on K in Definition 3.4 is a Goursat distribution on itsregular locus. By Proposition 3.7, this is equivalent to saying that thedistribution T on U is a Goursat distribution on its regular locus.An example of Goursat distribution is the distribution J n on the jetspace J n of Subsection 2.3. The following result of ´E. Cartan (Theorem 6.5of [Mo]) says that any Goursat distribution is isomorphic to J n at generalpoints. Theorem 4.2.
Let D ⊂ T Y be a Goursat distribution on a manifold Y ofdimension n ≥ . Then at a general point y ∈ Y , there exist a neighborhood y ∈ U ⊂ Y and an open subset O ⊂ J n − in the jet space of dimension n (see Subsection 2.3) with a biholomorphic map ϕ : U → O such that d ϕ ( D | U ) = J n − | O . A direct consequence of Proposition 3.20 and Theorem 4.2 is the following.
Theorem 4.3.
Let K be a family of unbendable rational curves of Goursattype on a complex manifold M of dimension n with the universal morphisms K ρ ← U µ → M . By Theorem 4.2, a general point of K has a neighborhood U biholomorphic to an open subset O ⊂ J n − such that D ⊂ T K corresponds to J n − on O . Then there is a biholomorphism φ with a commutative diagram ρ − ( U ) φ −→ P J n − | O ↓ ↓ U ϕ −→ O such that d φ sends T µ | ρ − ( U ) to a line subbundle of J n − on P J n − | O thatsplits the exact sequence → T p n − n − → J n − → O P J n − ( − → over the open subset ( p n − n − ) − ( O ) ⊂ P J n − , where p n − n − : P J n − | J n − → J n − is the natural projection. Conversely, given a line subbundle V of J n − over an open subset O ⊂ J n − that splits the above exact sequence,let M be the space of leaves of V in a neighborhood of a fiber of p n − n − . Thenthe images of the fibers of p n − n − in M give a family of unbendable rationalcurves of Goursat type on M . The following proposition gives an interesting class of examples of un-bendable rational curves of Goursat type.
Proposition 4.4.
Let Z ⊂ P n − be a submanifold of dimension 1. Regard P n − as a hyperplane in P n and let β : M → P n be the blowup of P n along Z . Let K be the set of lines on P n which intersect Z and are not containedin P n − . Regard K as the parameter space of the strict transforms to M of such lines. Then K is a family of unbendable rational curves of anti-canonical degree on M . If furthermore Z is linearly nondegenerate in P n − , then K is of Goursat type.Proof. A line in P n belonging to K intersects Z transversally and its normalbundle in P n is isomorphic to O (1) ⊕ ( n − . Thus its strict transform in M isa rational curve with normal bundle O (1) ⊕ O ⊕ ( n − , namely, an unbendablerational curve of anti-canonical degree 3.Now assume that Z is linearly nondegenerate in P n − and let us showthat K is of Goursat type. The linear nondegeneracy of Z ⊂ P n − impliesrank( T i ) = i + 2 for 1 ≤ i ≤ n − M o ⊂ M be the Zariski open subset given by β − ( P n \ P n − ) andfix affine coordinates ( x , . . . , x n ) on M o induced by an inhomogeneous co-ordinate system on P n \ P n − . Let λ = d x , . . . , λ n = d x n be the fibercoordinates on T M o . The members of K intersect M o on affine lines in thedirection of Z ⊂ P n − via the induced trivialization P T M o ∼ = P n − × M o .In other words, the submanifold C := τ ( µ − ( M o )) ⊂ P T M o is isomorphic tothe product Z × M o ⊂ P n − × M o = P T M o . In terms of the coordinates( λ , . . . , λ n , x , . . . , x n )on T M o , the vector field ˆ f := λ ∂∂x + · · · + λ n ∂∂x n on T M o restricted to the cone ˆ C ⊂
T M o \ (0-section) gives a section of ξ − F where ξ : ˆ C → C is the natural C ∗ -bundle. Local vector fields on Z can beextended to local vector fields on C ∼ = Z × M o . Thus fibers of ξ − V on ˆ C arespanned by restrictions to ˆ C of local vector fields of the form a ∂∂λ + · · · + a n ∂∂λ n where a , . . . , a n are suitable holomorphic functions in λ , . . . , λ n defined onsome open subset in C n such that they represent local vector fields tangentto ˆ Z ⊂ C n . It follows that T = V + F + [ V , F ] is generated by V , F andlocal vector fields of the form[ a ∂∂λ + · · · + a n ∂∂λ n , ˆ f ] = a ∂∂x + · · · + a n ∂∂x n NBENDABLE RATIONAL CURVES 19 where a , . . . , a n are suitable local holomorphic functions in λ , . . . , λ n . Byinduction, we see that T k for any k is generated by V , F and local vectorfields of the form b ∂∂x + · · · + b n ∂∂x n where b , . . . , b n are suitable local holomorphic functions in λ , . . . , λ n and P ni =1 b i ∂∂λ i is a local section of T ( k ) ˆ Z. This implies that[ F , T i ] ⊂ T i and [ T i , T j ] ⊂ T j +1 ⊂ T i + j for any 1 ≤ i ≤ j. From T = ρ − D , we have ρ − ( ∂ ( i ) D ) = T i +1 = ρ − ( ∂ i D ) . Combined with (4 . D is (2 , , . . . , n ) and D is a Goursat distribution on its regular locus. (cid:3) Definition 4.5.
Let O ⊂ J n − be a connected open subset and let V ⊂ J n − | ( p n − n − ) − ( O ) be a line subbundle splitting the exact sequence in Theorem 4.3 on ( p n − n − ) − ( O ) ⊂ P J n − . The line bundle V determines a section of p nn − : P J n − → P J n − on ( p n − n − ) − ( O ). This section restricted to the open subset J n − ∩ ( p n − n − ) − ( O )determines an ordinary differential equation u ( n ) = F ( t, u, u (1) , . . . , u ( n − )(4.2)for some holomorphic function F on J n − ∩ ( p n − n − ) − ( O ) as explained inSubsection 2.3. If O and V arise from a neighborhood of a general point y ∈ K of a family of unbendable rational curves of Goursat type as describedin Theorem 4.3, we say that (4.3) is an ODE associated with K . It is notunique, depending on the choice of the biholomorphism ϕ .What kind of ordinary differential equations are associated with unbend-able rational curves in Definition 4.5? To answer it, we look at one specialexample first. Proposition 4.6.
Let Z ⊂ P n − be the rational normal curve of degree n − ≥ , in other words, the set of pure symmetric tensors in P Sym n − ( C ) ,and let K be the family of unbendable rational curve of Goursat type deter-mined by Z ⊂ P n − in Proposition 4.4. Then we can choose ϕ in Theorem4.3 such that u ( n ) = 0 is an ODE associated with K .Proof. The semi-direct product (Sym n − C ) × GL(2) = C n × GL(2) acts onthe configuration Z ⊂ P n − ⊂ P n = P ( C ⊕ Sym n − C )such that the kernel of the action is the finite subgroup { (0 , a ) ∈ C n × GL(2) | a ∈ C ∗ Id C and a n − = Id C } . This action can be lifted to the blowup M → P n along Z with an inducedaction on K . It follows that dim aut ( D ) ≥ n + 4. It is a classical result (forexample, see Theorem 6.44 of [Ol]) that any ordinary differential equationof order n ≥ n + 4)-dimensional Lie algebra of infinitesimal auto-morphisms is equivalent (up to a choice of ϕ ) to u ( n ) = 0 . This completesthe proof when n ≥ n = 3, the curve Z ⊂ P is a conic and the strict transform of P ⊂ P under the blowup M → P can be contracted to a point yielding abirational morphism from M to the nonsingular quadric hypersurface Q ⊂ P (e.g. see Lecture 22 of [Ha], page 288). Then K can be viewed as a familyof unbendable rational curves on Q and the infinitesimal automorphisms of Q induce infinitesimal automorphisms of D . It follows thatdim aut ( D ) ≥ dim so ( C ) = 10 . We have the classical result (see Theorem 6.44 of [Ol] again) that any thirdorder ordinary differential equation with 10-dimensional Lie algebra of in-finitesimal (contact) automorphisms is equivalent (up to a choice of ϕ ) to u (3) = 0 . This completes the proof when n = 3. (cid:3) We have the following general result. This can be viewed as a generaliza-tion of Theorem 3.1 in [Hi] which corresponds to the case n = 2. Theorem 4.7.
Let K be a family of unbendable rational curves of Goursattype in a complex manifold of dimension n ≥ . Then there exists an ODEassociated with K of the form u ( n ) = a ( u ( n − ) + a ( u ( n − ) + a u ( n − + a (4.3) where a , . . . , a are holomorphic functions of t, u, u (1) , . . . , u ( n − on a do-main in J n − . Conversely, any ODE of this type gives rise to a familyof unbendable rational curves of Goursat type in some complex manifold ofdimension n .Proof. By Theorem 4.3, a family of unbendable rational curves of Goursattype gives a splitting of the exact sequence0 → T p n − n − → J n − → O P J n − ( − → O in J n − . The unbendable curves in Proposition 4.6give another splitting of this exact sequence. Their difference is a sectionof the line bundle T p n − n − ⊗ O P J n − (1) on ( p n − n − ) − ( O ). Along the P -fibersof p n − n − , this line bundle is isomorphic to O P (3). Thus the difference fromthe equation u ( n ) = 0 of Proposition 4.6 is given by a cubic polynomial in u ( n − . Thus it is of the form (4.3).Conversely, the ODE (4.3) gives rise to a section of the line bundle T p n − n − ⊗O P J n − (1) on ( p n − n − ) − ( O ) for a suitable open subset O ⊂ J n − , inducinga splitting of the exact sequence. Thus it is associated with a family ofunbendable rational curves via Theorem 4.3. (cid:3) NBENDABLE RATIONAL CURVES 21
When dim M ≤
4, unbendable rational curves of anti-canonical degree 3are essentially of Goursat type:
Proposition 4.8.
Let M be a complex manifold of dimension ≤ . If K isa bracket-generating family of unbendable rational curves of anti-canonicaldegree , then it is of Goursat type.Proof. Under the assumption, the dimension of K is equal to dim M ≤ D is bracket-generating. It is easy to see that a bracket-generating distribution of rank 2 on a manifold of dimension ≤ K is of Goursat type. (cid:3) Example 4.9.
Let X ⊂ P be a nonsingular cubic hypersurface. The family K of lines on X is a family of unbendable rational curves. The VMRT C x ata general point x ∈ X is irreducible and linearly nondegenerate in P T x X (seeSection 1.4.2 of [Hw01]). The family K is bracket-generating by Proposition3.16. Thus it is of Goursat type by Proposition 4.8 and there is an ODE ofthe type given in Theorem 4.7 associated with it. However, it seems hardto find holomorphic functions a , . . . , a in Theorem 4.7 explicitly from thecubic equation defining X .Now we discuss important features of the VMRT of unbendable rationalcurves of Goursat type. Proposition 4.10.
Use the notation of Definition 3.15 for a family of un-bendable rational curves of Goursat type on a manifold M . Then the VMRT C α ⊂ P T x M for a general point α ∈ U is linearly nondegenerate and themeromorphic distributions T k satisfy [ F , T k ] ⊂ T k for all k ≥ .Proof. We claim that rank( T i ) < rank( T i +1 ) for i ≤ n − n is thedimension of M . Since ∂ k D has rank k + 2 and by (2.1) ∂ k T = ∂ k ( ρ − D ) = ρ − ( ∂ k D )has rank k + 3 for each 0 ≤ k ≤ n −
2, the claim implies that T k +1 = ρ − ( ∂ k D ) for all k ≥
0, which proves the proposition.To prove the claim, we may replace K by any open subset in K . Thus byTheorem 4.3, we may assume that the distribution D on K is biholomorphicto the distribution J n − on an open subset O ⊂ J n − . Since the deriveddistributions of J n − are (see the last paragraph of Subsection 2.3) J n − ⊂ ( p n − n − ) − J n − ⊂ · · · ⊂ ( p n − ) − J ⊂ T J n − , we see that for a general element v ∈ J n − y ,Levi ( p n − i ) − J i ( v, ( p n − i ) − J i ) = 0for each 1 ≤ i ≤ n − . By choosing a general α ∈ ρ − ( y ) such that d ρ ( V α )corresponds to such a general element v ∈ J n − y , we see that rank( T i ) < rank( T i +1 ) for 1 ≤ i ≤ n − (cid:3) Proposition 4.11.
Let K be a family of unbendable rational curves of Gour-sat type on a complex manifold M of dimension n ≥ . Then for a general y ∈ K , the third fundamental form of C α ⊂ P T µ ( α ) M vanishes for some α ∈ ρ − ( y ) .Proof. Since [ F , T ] ⊂ T and [ F , T ] ⊂ T by Proposition 4.10, the linebundle T / T is trivial along ρ − ( y ) for a general y ∈ K . Thus Proposition3.16 (iii) implies that the relative normal spaces { N (3) C α , α ∈ ρ − ( y ) } define avector bundle on ρ − ( y ) isomorphic to T ⊗ T ∗ ( ρ − ( y )) ∼ = O P ( − . Then as in Proposition 3.17 (iv), the third fundamental forms of {C α , α ∈ ρ − ( y ) } is isomorphic to a section of the line bundleHom(Sym O P ( − , O P ( − O P (1) . It follows that the third fundamental form vanishes at some point of ρ − ( y ). (cid:3) Lemma 4.12.
Let D be a bracket-generating distribution of rank 2 ona -dimensional manifold Y . Then rank( ∂D ) = 3 and rank( ∂ (2) D ) =rank( ∂ D ) . Proof.
We may assume that D is regular at every point of Y . Since rank( D ) =2 , the image of Levi D : ∧ D → T Y /D has rank 1 and ∂D has rank 3. If ∂ (2) D has rank 5, then ∂ D must have rank 5. It remains to exclude the casewhen rank( ∂ (2) D ) = 4 and rank( ∂ D ) = 5. In this case, the homomorphismof vector bundles Levi ∂D : ∧ ( ∂D ) → T Y /∂D is surjective on the regular locus of D . For a general point y ∈ Y , choosea basis u, v, w ∈ ( ∂D ) y such that u, v ∈ D y . Then Levi ∂D ( u, v ) = 0, whileLevi ∂D ( u, w ) and Levi ∂D ( v, w ) have values in ∂ (2) D/∂D . Thus Levi ∂D can-not be surjective, a contradiction. (cid:3) Now we have the following characterization of unbendable rational curvesof Goursat type in dimension 5.
Theorem 4.13.
Let M be a 5-dimensional complex manifold and let K bea bracket-generating family of unbendable rational curves on M with anti-canonical degree . Then the following three statements are equivalent. (1) The growth vector of the distribution D on K is (2 , , , (i.e., it isof Goursat type); (2) [ F , T ] ⊂ T ; (3) for a general y ∈ K , the third fundamental form of C α at τ ( α ) van-ishes for some α ∈ ρ − ( y ) .Proof. The implications (1) ⇒ (2) ⇒ (3) are by Propositions 4.10 and 4.11. NBENDABLE RATIONAL CURVES 23
By Corollary 3.14, T = ∂ (1) T = ρ − ( ∂ (1) D ) . Then (2) implies T = [ T , T ] + T = ∂ (2) T = ρ − ( ∂ (2) D ) , where the first equality follows from Lemma 3.10 and the third equalityfollows from (2.1). Thus ∂ (2) D has rank 4, which implies (1) by Lemma4.12.By Proposition 3.17 (iii), (3) implies that the image of the relative thirdfundamental forms of {C α , α ∈ ρ − ( y ) } span a line subbundle of N C y ∼ = O P (1) ⊕ O ⊕ P corresponding to one of the O P -factors of O ⊕ P . Thus we havea distinguished vector subbundle N ♯C y ⊂ N C y isomorphic to O P (1) ⊕ O ⊕ P spanned by O P (1) and the images of the second and the third fundamentalforms of C α , α ∈ ρ − ( y ). Define D ♯y := H ( C y , N ♯C y ) ⊂ D y for general y ∈ K . This gives a meromorphic distribution D ♯ on K of rank4 containing ∂ D such that ρ − D ♯ = T at general points of U . This implies(2). (cid:3) Unbendable rational curves of Cartan type
We introduce the following terminology for convenience.
Definition 5.1.
Let Y be a complex manifold of dimension 5. A regulardistribution D ⊂ T Y of rank 2 with the growth vector (2 , ,
5) is called a
Cartan distribution . For a Cartan distribution D ⊂ T Y , the vector bundlehomomorphisms given by Lie brackets ∧ D → ( ∂D ) /D and D ⊗ ( ∂D ) /D → T Y /∂D are surjective. A family K of unbendable rational curves in a 5-dimensionalcomplex manifold is said to be of Cartan type if the distribution D on K inDefinition 3.4 is a Cartan distribution on its regular locus.The following is straightforward. Lemma 5.2.
Let D ⊂ T Y be a bracket-generating distribution of rank 2on a 5-dimensional complex manifold Y . Then the restriction of D on itsregular locus is either a Cartan distribution or a Goursat distribution. Theorem 5.3.
Let K be a bracket-generating family of unbendable rationalcurves of anti-canonical degree 3 on a complex manifold of dimension 5.Then it is of Cartan type if and only if for a general y ∈ K , the thirdfundamental form of C α is nonzero at τ ( α ) for each α ∈ ρ − ( y ) . Proof.
If the third fundamental form of C α is nonzero at τ ( α ) for each α ∈ ρ − ( y ), then K is not of Goursat type by Theorem 4.13. Thus it is of Cartantype by Lemma 5.2.Assuming that K is of Cartan type, the distribution D is a Cartan distri-bution in a neighborhood of a general point y ∈ K . ThenLevi ∂ D ( v, ( ∂ D ) y ) = 0for any nonzero element v ∈ D y from Lemma 4.12. Since T = ρ − ∂ D byCorollary 3.14, this implies that the third fundamental form of C α at τ ( α )is nonzero for each α ∈ ρ − ( y ) by Proposition 3.16 (ii) and Proposition 3.20(i). (cid:3) There are many examples of Cartan distributions. In fact, a generic dis-tribution of rank 2 on a 5-dimensional manifold is a Cartan distribution onits regular locus. Applying Proposition 3.20, we obtain many examples ofunbendable rational curves of Cartan type. There is a particularly interest-ing class of examples related to contact structures. To describe them, weneed to recall some basics of contact structures.
Definition 5.4.
For a complex manifold M , a regular distribution H ⊂ T M of corank 1 is called a contact structure if Ch( H ) = 0. The line bundle T M/H is called the contact line bundle and the line bundle-valued 1-form θ : T M → T M/H is called the contact form . Then d θ | H agrees with − Levi H .When n = dim M , there is a natural isomorphism ∧ n T M ∼ = ( T M/H ) ( n +1) / . In particular, the dimension of M should be an odd number. Example 5.5.
Let Y be a complex manifold and let φ : T ∗ Y → Y be theprojection of the cotangent bundle. The natural 1-form θ Y on T ∗ Y sends avector v ∈ T ζ ( T ∗ Y ) at ζ ∈ T ∗ y Y to θ Y ( v ) = ζ (d φ ( v )) ∈ C . Then d θ Y is the natural symplectic form on T ∗ Y . Let ϕ : P T ∗ Y → Y bethe projective bundle and let ξ : T ∗ Y \ (0-section) → P T ∗ Y be the C ∗ -bundle. For a nonzero element ζ ∈ T ∗ y Y , denote by [ ζ ] ∈ P T ∗ y Y the image ξ ( ζ ) and by ζ ⊥ ⊂ T y Y the hyperplane annihilated by ζ. Define ahyperplane H Y [ ζ ] ⊂ T [ ζ ] ( P T ∗ Y ) by H Y [ ζ ] = (d [ ζ ] ϕ ) − ( ζ ⊥ )where d [ ζ ] ϕ : T [ ζ ] ( P T ∗ Y ) → T y Y is the differential of ϕ at the point [ ζ ] ∈ P T ∗ y Y . The resulting distribution H Y ⊂ T ( P T ∗ Y ) is a contact structure on P T ∗ Y satisfying Ker( θ Y ) = ξ − H Y on T ∗ Y \ (0-section). We denote thecorresponding contact form by ϑ Y : T ( P T ∗ Y ) → T ( P T ∗ Y ) /H Y ∼ = O P T ∗ Y (1) . We omit the proof of the following elementary lemma.
NBENDABLE RATIONAL CURVES 25
Lemma 5.6.
Let S ⊂ X be a compact complex submanifold of a complexmanifold X and let ϑ : T X → L be a surjective homomorphism to a linebundle L on X defining a distribution H = Ker( ϑ ) ⊂ T X of corank 1.Assume that (1) Null(Levi H x ) := { v ∈ H x , Levi H ( v, u ) = 0 for all u ∈ H x } has thesame dimension for all x ∈ X and (2) Null(Levi H x ) ∩ T x S = 0 for all x ∈ S .Then there exists a neighborhood U ⊂ X of S and a submersion ν : U → M to a complex manifold with a contact structure H ⊂ T M such that H| U = ν − H and T νx = Null(Levi H x ) for all x ∈ U . Proposition 5.7.
Let H ⊂ T M be a contact structure on a complex man-ifold of dimension 5. Let K be a bracket-generating family of unbendablerational curve of anti-canonical degree 3 whose members are tangent to H .Then K is of Cartan type.Proof. The varieties of minimal rational tangents of K at general points of M are contained in P H ⊂ P T M . Thus Proposition 4.10 implies that K cannot be of Goursat type. Then it must be of Cartan type by Lemma5.2. (cid:3) Definition 5.8.
A family of contact unbendable rational curves of Cartantype means a family of unbendable rational curves of Cartan type describedin Proposition 5.7.There are many examples of contact unbendable rational curves of Cartantype. To see this, we need the following properties of Cartan distributionsproved by Zelenko in [Z99] and [Z06]. They were discussed also in Section4.3 of [BH] in a less precise form.
Lemma 5.9.
Let D ⊂ T Y be a Cartan distribution on a 5-dimensionalcomplex manifold Y and let W ⊂ T ∗ Y be the subbundle of rank annihilat-ing ∂D . Using the notation of Example 5.5, let σ := d θ Y | W be the 2-formon W given by the restriction of d θ Y . Then the null-space Null( σ ) w := { u ∈ T w W, σ ( u, v ) = 0 for all v ∈ T w W } is 1-dimensional for each nonzero w ∈ W, defining a line subbundle V ♯ ⊂ T W outside the -section which satisfies the following. (i) θ Y ( V ♯ ) = 0 . (ii) V ♯ ∩ T η = 0 where η : W → Y is the natural projection. (iii) V ♯ ⊂ η − D. Proof.
All the statements are proved in [Z06] (Zelenko worked in the settingof real differentiable manifolds, but the arguments are valid also in complexanalytic setting). That Null( σ ) w is a 1-dimensional subspace in Ker( θ Y )is the equation (3.13) of [Z06] which was proved in Proposition 2.2 andCorollary 2.1 of [Z99]. (ii) and (iii) follow from the equations (3.13) and(3.16) in the proof of Proposition 3.1 in [Z06]. (cid:3) The following theorem says that each germ of a Cartan distribution de-termines in a natural way a germ of a family of contact unbendable rationalcurves of Cartan type.
Theorem 5.10.
Let D ⊂ T Y be a Cartan distribution. Let ϑ Y be thecanonical contact form on P T ∗ Y from Example 5.5. Let ̺ : P W → Y bethe projectivization of the vector bundle W ⊂ T ∗ Y in Lemma 5.9 and let ϑ : T P W → O P W (1) be the line-bundle valued 1-form on P W ⊂ P T ∗ Y obtained from the restriction of ϑ Y . Then the following holds. (1) Ker( ϑ ) ⊂ T P W is a distribution of corank 1 on P W . (2) In the terminology of Lemma 5.6, define V ♭x := Null(Levi Ker( ϑ ) x ) for x ∈ P W. Then dim V ♭x = 1 for all x ∈ P W defining a line subbundle V ♭ ⊂ T P W , which is transversal to F ♭ := T ̺ and satisfies V ♭ ⊂ ̺ − D . (3) For any point y o ∈ Y , there exists a neighborhood K ♭ ⊂ Y of y o suchthat the leaves of V ♭ define a submersion ν : U ♭ := ̺ − ( K ♭ ) → M ♭ to a 5-dimensional complex manifold M ♭ equipped with a contactstructure H ♭ satisfying ν − H ♭ = Ker( ϑ ) | U ♭ . (4) In (3), for any y ∈ K ♭ , the morphism ν sends ̺ − ( y ) to an unbend-able rational curve C ♭y ⊂ M ♭ of anti-canonical degree satisfying T C ♭y ⊂ H ♭ | C ♭y . (5) Regarding K ♭ ̺ ←− U ♭ ν −→ M ♭ in (3) as a family of unbendable rational curves on M ♭ via (4), wehave D | K ♭ = D where D is the distribution determined by the familyof unbendable rational curves on M ♭ as in Definition 3.4.Proof. (1) is immediate from the definition of θ Y and the submersion ̺ : P W → Y .(2) is a reformulation of Lemma 5.9: the line bundle V ♭ is just the imageof V ♯ of Lemma 5.9 under the projection λ : W \ (0-section) → P W . To seethis, let θ be the restriction of θ Y to W such that Ker( θ ) = λ − (Ker( ϑ ))from Ker( θ Y ) = ξ − Ker( ϑ Y ) in Example 5.5. For a nonzero w ∈ W , it iseasy to see that Null(Levi Ker( θ ) w ) = Null(d θ | Ker( θ ) w ) . Thus it suffices to showNull(d θ | Ker( θ ) w ) ⊂ Null(d θ ) w + T λw . (5.1)Fix e ∈ T λw and u ∈ T w W satisfying d θ ( e, u ) = 1 and θ ( u ) = 0 . For any v ∈ Null(d θ | Ker( θ ) w ), we have(5.2) d θ ( v − d θ ( v, u ) e, u ) = 0 . NBENDABLE RATIONAL CURVES 27
Note that v − d θ ( v, u ) e ∈ Null(Levi
Ker( θ ) w ) from T λw ⊂ Null(Levi
Ker( θ ) w ) . Thus(5.2) implies that v − d θ ( v, u ) e ∈ Null(d θ ) w . This proves (5.1) and (2).(3) is a consequence of (2) and Lemma 5.6.In (4), the inclusion T C ♭y ⊂ H | C ♭y is immediate from T ̺ ⊂ Ker( ϑ ). Thenormal bundle N C ♭y of C ♭y in M ♭ is a quotient bundle of the normal bundle N ̺ − ( y ) of ̺ − ( y ) in U ♭ which is a trivial vector bundle. Thus N C ♭y is semi-positive. From ∧ T M ♭ ∼ = ( T M/H ) ⊗ in Definition 5.4 and ( T M/H ) | C ♭y ∼ = O P (1)in Example 5.5, we have ∧ T M ♭ | C ♭y ∼ = O P (3), which implies N C ♭y ∼ = O P (1) ⊕ O ⊕ P . This shows that C ♭y is an unbendable rational curve.To see (5), note that V ♭ + F ♭ ⊂ ̺ − D from (2). Combining it withProposition 3.7, we have V ♭ + F ♭ ⊂ ̺ − ( D ∩ D ). This inclusion must be anidentity if rank( D ∩ D ) ≤
1. But then F ♭ must be the Cauchy characteristicof V ♭ + F ♭ , a contradiction to Proposition 3.16 (i). It follows that rank( D ∩D ) = 2 , implying D = D . (cid:3) Remark 5.11.
The construction in Theorem 5.10 is inspired by the idea ofJacobi curves in [Z99] and [Z06]. Our argument is a translation of Zelenko’ssymplectic approach into contact geometry. Zelenko’s Jacobi curves can beinterpreted as the images of VMRT under the Gauss map.The next theorem says that the unbendable rational curves of Cartantype arising from the construction of Theorem 5.10 cover all examples ofcontact unbendable rational curves of Cartan type.
Theorem 5.12.
In the setting of Proposition 5.7, let Y ⊂ K be the regularlocus of D and let D ⊂ T Y be the Cartan distribution given by the restrictionof D to Y . We can apply Theorem 5.10 to D to obtain K ♭ , U ♭ , M ♭ and H ♭ .Then there exists an open embedding χ : U ♭ ⊂ U satisfying ̺ = ρ ◦ χ and d χ (Ker( ϑ )) = µ − H | χ ( U ♭ ) . In particular, there is an open embedding M ♭ ⊂ M satisfying H ♭ = H | M ♭ and a commutative diagram K ♭ ̺ ← U ♭ ν → M ♭ ∩ ↓ χ ∩K ρ ← U µ → M. Proof.
To prove the theorem, we may replace K by Y and assume that D = D is a Cartan distribution. Then T is a distribution on U satisfying T = ρ − ∂D by Corollary 3.14. Since elements of K are tangent to H , the germ C α is contained in P H x forall x ∈ M and α ∈ µ − ( x ). Then T m ⊂ µ − H for all m ≥ µ − H define a section σ : U → P ( T U / T ) ∗ of the projection P ( T U / T ) ∗ → U . Let Σ ⊂ P ( T U / T ) ∗ be theimage of σ . It is easy to check from the definition of ϑ U thatKer( ϑ U | Σ ) ∼ = µ − H (5.3)under the biholomorphic map Σ ∼ = U .Let W ⊂ T ∗ Y be the annihilator of ∂D and define a holomorphic map η : U → P W by sending α ∈ U to the annihilator of the hyperplaned ρ ( µ − H ) α ⊂ T y Y containing ( ∂D ) y with y = ρ ( α ). From the commu-tative diagram of P -bundles U η → P W ↓ ρ ↓ Y = Y, if η is not biholomorphic, it contracts each fiber of ρ to a single point. Inthe latter case, we obtain a distribution H ′ on Y such that µ − H = ρ − H ′ ,which implies F ⊂
Ch( µ − H ), a contradiction toCh( µ − H ) = µ − Ch( H ) = T µ . It follows that η is a biholomorphic map.The homomorphism of vector bundles ρ ∗ ( T ∗ Y ) → T ∗ U induces an iso-morphism of vector bundles ρ ∗ W ∼ = ( T U / T ) ∗ because T = ρ − ∂D . Let ξ : P ( T U / T ) ∗ → P W be the composition of the holomorphic maps P ( T U / T ) ∗ ∼ = → ρ ∗ P W → P W. It is easy to see that ξ − Ker( ϑ Y | P W ) = Ker( ϑ U | P ( T U / T ) ∗ )(5.4)and ξ ◦ σ = η. Equations (5.3) and (5.4) say that the biholomorphism η sends µ − H to Ker( ϑ Y | P W ). By the definitions of H ♭ and M ♭ in Theorem5.10, this implies that the inclusions U ♭ ⊂ U and M ♭ ⊂ M obtained fromthe inverse of η satisfy the required properties. (cid:3) Lines on quartic 5-folds
Main result and the strategy of its proof.
The main result of thissection is the following.
Theorem 6.1.
A general hypersurface X ⊂ P of degree and a generalprojective line ℓ on X have the following properties. (1) X is smooth along ℓ . (2) The normal bundle N ℓ of ℓ ⊂ X is isomorphic to O (1) ⊕ O ⊕ . (3) The second fundamental form FF ℓ x , C x is nonzero for all x ∈ ℓ . (4) The third fundamental form FF ℓ x , C x is nonzero for all x ∈ ℓ . NBENDABLE RATIONAL CURVES 29 (5)
The fourth fundamental form FF ℓ P , C P is nonzero for some P ∈ ℓ and consequently, the VMRT C P at P is linearly nondegenerate in P T P X. In particular, general lines on X are unbendable rational curves of Cartantype by Proposition 3.16 and Theorem 5.3. To prove Theorem 6.1, it suffices to prove the next theorem because theproperties (1)–(5) are open conditions in the space of pairs (
X, ℓ ) consistingof a quartic hypersurface X ⊂ P and an unbendable line ℓ contained in thesmooth locus of X. Theorem 6.2.
Let P be the projective space with homogeneous coordinates x , . . . , x . Define f := X i =2 x i + x x + x x + x x x + x x x + x x + x x +( x + x x + x ) x + x x + x x + ( x + x ) x . Then ℓ := ∩ i =2 Zero(x i ) is a projective line on the hypersurface X := Zero(f) of degree with the following properties. (1) X is smooth along ℓ . (2) The normal bundle N ℓ of ℓ ⊂ X is isomorphic to O (1) ⊕ O ⊕ . (3) The second fundamental form FF ℓ x , C x is nonzero for all x ∈ ℓ . (4) The third fundamental form FF ℓ x , C x is nonzero for all x ∈ ℓ . (5) The fourth fundamental form FF ℓ P , C P is nonzero for some P ∈ ℓ and consequently, the VMRT C P at P is linearly nondegenerate in P T P X. Lemma 6.3.
The hypersurface X is smooth along the projective line ℓ .Proof. Take any point x = [ t : t : 0 : · · · : 0] ∈ ℓ . Then the tangent spaceof the affine cone ˆ X ⊂ C of X at x is { t x + t x + t t x + t t x = 0 } , (6.1)which is properly contained in C . Thus X is smooth at x . (cid:3) Notation 6.4.
We can identify the space of lines through a point x ∈ P n with P T x P n in a natural way. Using this identification and abusing thenotation, let us denote by C x ⊂ P T x X ⊂ P T x P n the subscheme of lines lyingon X passing through x . If ℓ is an unbendable rational curve, the germ of C x at ℓ x agrees with the germ C α of the VMRT at α = P T x ℓ as defined inDefinition 3.15.The next lemma is easy to check. Lemma 6.5.
Define an involution γ of P by [ λ : λ : λ : λ : λ : λ : λ ] [ λ : λ : λ : λ : λ : λ : λ ] . Then we have γ · [ C f ] = [ C f ] , γ · X = X , γ · ℓ = ℓ , and the points of ℓ fixedby γ are P := [1 : 1 : 0 : · · · : 0] and Q := [ − · · · : 0] . In the terminology of Notation 6.4 and Lemma 6.5, we state the followingtwo propositions, whose proofs are to be given later.
Proposition 6.6.
The subscheme C P ⊂ P T P X is smooth at ℓ P , the germof C P at ℓ P is of dimension 1 and its fundamental forms at ℓ P , FF ℓ P , C P , FF ℓ P , C P , and FF ℓ P , C P are nonzero. Proposition 6.7.
The subscheme C Q ⊂ P T Q X is smooth at ℓ Q , the germof C Q at ℓ Q is of dimension 1 and its fundamental forms at ℓ Q , FF ℓ Q , C Q and FF ℓ Q , C Q are nonzero. Theorem 6.2 can be proved from these two propositions as follows.
Proof of Theorem 6.2. (1) is Lemma 6.3.Since the normal bundle of ℓ in P is isomorphic to O ℓ (1) ⊕ , its normalbundle N ℓ in X is isomorphic to ⊕ i =1 O ( d i ) with each d i ≤
1. By the adjunc-tion formula, the anti-canonical degree of ℓ in X is deg T P | ℓ − deg X = 3which implies deg N ℓ = P i =1 d i = 1. By the deformation theory of rationalcurves, there is a natural identification T ℓ P C P = H ( ℓ, N ℓ ⊗ m P )) = H ( P , ⊕ i =1 O ( d i − . By Proposition 6.6, we have dim T ℓ P C P = 1. By reordering d i , we deducefrom the above formula that d = 1 and d i ≤ i = 1. Since P i =1 d i = 1,we have N ℓ ∼ = O (1) ⊕ O ⊕ . This proves (2).(3) follows from Proposition 6.6 because the second fundamental form ofthe VMRT is unchanged along an unbendable rational curve by Proposition3.17 (ii).To check (4), we use Proposition 3.17 (iii) to obtain a holomorphic linesubbundle L in the O (1) ⊕ -factor of N ℓ in (3) such that the third funda-mental form FF ℓ x , C x , x ∈ ℓ is an element of H ( ℓ, L ). In particular, eitherFF ℓ x , C x is zero for all x ∈ ℓ or there is at most one point z ∈ ℓ such thatFF ℓ z , C z = 0. By Propositions 6.6 and 6.7, we have FF ℓ z , C z = 0 for at mostone z ∈ ℓ \{ P, Q } . But if such a point z exists, as the involution γ in Lemma6.5 induces an isomorphism between FF ℓ x , C x with FF ℓ γ ( x ) , C γ ( x ) for all x ∈ ℓ ,we have γ ( z ) = z and FF ℓ γ ( z ) , C γ ( z ) = 0 , a contradiction. This proves (4).Finally, (5) is from Proposition 6.6. (cid:3) NBENDABLE RATIONAL CURVES 31
Proof of Proposition 6.6.Lemma 6.8.
Let P ⊂ P be the hyperplane Zero(x ) with homogenouscoordinates [ y : · · · : y ] given by y i = x i , ≤ i ≤ . Let ψ : P P be theprojection with vertex P which induces an isomorphism [ ψ ] : P ( T P P ) → P .Then [ ψ ] identifies C P ⊂ P T P X ⊂ P ( T P P ) with the subscheme Z ⊂ P defined by the system of homogeneous equations h = h = h = h = 0 , where h := y + y + y + y ,h := 3 y y + y y + 2 y y + y + y + 3 y ,h := 3 y y + y y + 2 y y + 3 y y + y + y + 2 y ,h := X i =2 y i + y y + y y + y y + y y + y y . Under this identification, the point ℓ P ∈ C P corresponds to α := [1 : 0 : · · · :0] ∈ Z. Proof.
The projective line joining P and [ y : · · · : y ] ∈ P intersects P \ P along { [1 : λy + 1 : λy : · · · : λy ] , λ ∈ C } . This line lies on X , i.e., belongs to C P if and only if f (1 , λy + 1 , λy , . . . , λy ) = X k =1 h k λ k = 0for all λ ∈ C . It follows that h = h = h = h = 0 describes the image ofthe subscheme C P under the identification [ ψ ]. It is clear that ℓ P is sent to α . (cid:3) The following translation of Lemma 6.8 into affine coordinates is imme-diate.
Lemma 6.9.
Let A ⊂ P be the complement of Zero(y ) with the affinecoordinates z i := y i y , i = 2 , . . . , . Then α ∈ Z is defined by z = · · · = z =0 and the subscheme Z ∩ A is defined by g = g = g = g = 0 , where g := z + z + z + z ,g := 3 z + z + 2 z + z + z + 3 z ,g := 3 z + z + 2 z + 3 z + z + z + 2 z ,g := X i =2 z i + z + z + z + z + z . Lemma 6.10.
In Lemma 6.9, the Zariski tangent space T α Z ∼ = T ℓ P C P cor-responds to the 1-dimensional subspace z = z = z = z = 0 . In particular,the subscheme C P is of dimension 1 and smooth at ℓ P . Proof.
This is immediate from T α (Zero(g )) = { z + z + z + z = 0 } ,T α (Zero(g )) = { + z + 2z = 0 } ,T α (Zero(g )) = { + z = 0 } ,T α (Zero(z )) = { z = 0 } . (cid:3) Lemma 6.11.
By Lemma 6.10, the function t := z | Z ∩ A gives a localcoordinate on a neighborhood U ⊂ Z of α . Write z i = a i t + b i t + c i t + d i t + O ( t )(6.2) on U for all i = 2 , . . . , . Then a = ( a , a , a , a , a ) , b = ( b , b , b , b , b ) , c = ( c , c , c , c , c ) , d = ( d , d , d , d , d ) . are given by a = (0 , , , , , b = (1 , − , , , , (6.3) c = ( − , − , , , , d = (2 , − , − , , . In particular, the vectors a , b , c and d are linearly independent.Proof. From t = z | U , we already have a = 1 and b = c = d = 0. Recallthat g k ( z , . . . , z ) = 0 on Z ∩ A for k = 1 , . . . ,
4. Putting (6.2) in theseequations, we obtain g k ( z , . . . , z ) = P i =1 G k,i t i + O ( t ) on U , where each G k,i is an explicit polynomials of a , b , c , d , . . . , a , b , c , d . Since thesecoefficients G k,i of t i vanish, we can determine the vectors a , b , c and d oneby one to obtain (6.3). (cid:3) Lemma 6.12.
The second, the third and the fourth fundamental forms of Z at α are nonzero.Proof. By the definition of vectors a , b , c and d , we have T α ( Z ∩ A ) = C a ,T (2) α ( Z ∩ A ) = C a + C b ,T (3) α ( Z ∩ A ) = C a + C b + C c ,T (4) α ( Z ∩ A ) = C a + C b + C c + C d , where T ( k ) α ( Z ∩ A ) is the k -th osculating space of Z ∩ A at α . By Lemma6.11, we obtain the result. (cid:3) Lemma 6.10 and Lemma 6.12 complete the proof of Proposition 6.6.
NBENDABLE RATIONAL CURVES 33
Proof of Proposition 6.7.
The proof of Proposition 6.7 is completelyparallel to the proof of Proposition 6.6. It consists of the following lemmata.We skip the proofs which are direct translations of those in the proof ofProposition 6.6.
Lemma 6.13.
Let P ⊂ P be the hyperplane Zero(x ) with homogenouscoordinates [ y : · · · : y ] given by y i = x i , ≤ i ≤ . Let ψ : P P be theprojection with vertex Q which induces an isomorphism [ ψ ] : P ( T Q P ) → P .Then [ ψ ] identifies C Q ⊂ P T Q X ⊂ P ( T Q P ) with the subscheme Z ⊂ P defined by the system of homogeneous equations h = h = h = h = 0 , where h := − y + y + y − y ,h := 3 y y + y y − y y + y + y + y ,h := 3 y y − y y + 2 y y + y y − y + y ,h := X i =2 y i + y y + y y + y y + y y + y y . Under this identification, the line ℓ Q ∈ C Q corresponds to α := [1 : 0 : · · · :0] ∈ Z. Lemma 6.14.
Let A ⊂ P be the complement of Zero(y ) with the affinecoordinates z i := y i y , i = 2 , . . . , . Then α ∈ Z is defined by z = · · · = z =0 and the subscheme Z ∩ A is defined by g = g = g = g = 0 , where g := − z + z + z − z ,g := 3 z + z − z + z + z + z ,g := 3 z − z + 2 z + z − z + z ,g := X i =2 z i + z + z + z + z + z . Lemma 6.15.
In Lemma 6.14, the Zariski tangent space T α Z ∼ = T ℓ Q C Q corresponds to the 1-dimensional subspace z = z = z = z = 0 . Inparticular, the subscheme C Q is of dimension 1 and smooth at ℓ Q . Lemma 6.16.
By Lemma 6.15, the function t := z | Z ∩ A gives a localcoordinate on a neighborhood U ⊂ Z of α . Write z i = a i t + b i t + c i t + d i t + O ( t ) on U for all i = 2 , . . . , . Then a = ( a , a , a , a , a ) , b = ( b , b , b , b , b ) , c = ( c , c , c , c , c ) , d = ( d , d , d , d , d ) . are given by a = (0 , , , , , b = ( − , − , − , − , , c = ( − , − , − , − , , d = ( − , − , − , − , . In particular, the vectors a , b and c are linearly independent, and we have d = 2 b . Lemma 6.17.
The second and the third fundamental forms of Z at α arenonzero. Lemma 6.15 and Lemma 6.17 complete the proof of Proposition 6.7.
References [BCG] Bryant, R., Chern, S.-S., Gardner, R. B., Goldschmidt, H. and Griffiths, P.:
Exte-rior differential systems . MSRI Publ. , Springer, New York, 1991.[BH] Bryant, R. L., Hsu, L.: Rigidity of integral curves of rank 2 distributions. Invent.math. (1993) 435-461.[Ca] Cartan, ´E.: Les syst`emes de Pfaff `a cinq variables et les ´equations aux d´eriv´eespartielles du second ordre. Ann. Ec. Norm. Sup. (1910) 109-192.[GPR] Grauert, H., Peternell, T., Remmert, R.: Several Complex Variables VII . Encycl.Math. Sci. , Springer-Verlag, Berlin-Heidelberg, 1994.[Ha] Harris, J.: Algebraic geometry, a first course . Grad. Texts Math. , Springer-Verlag, New York, 1992.[Hi] Hitchin, N.: Complex manifolds and Einstein’s equations. in
Twistor geometry andnonlinear systems . Lect. Notes Math. , 73-99, Springer-Verlag, Berlin-New York,1982.[HH] Hong, J., Hwang, J.-M.: Characterization of the rational homogeneous space asso-ciated to a long simple root by its variety of minimal rational tangents. in
AlgebraicGeometry in East Asia-Hanoi 2005 , Advanced Stud. Pure Math. (2008) 217-236.[Hw01] Hwang, J.-M.: Geometry of minimal rational curves on Fano manifolds. in Schoolon Vanishing Theorems and Effective Results in Algebraic Geometry. Trieste, 2000 ,ICTP Lect. Notes, , 335-393, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.[HM] Hwang, J.-M., Mok, N.: Varieties of minimal rational tangents on uniruled projec-tive manifolds. in Several complex variables (Berkeley, CA, 1995–1996) , MSRI Publ. , 351-389, Cambridge Univ. Press, Cambridge, 1999.[HM04] Hwang, J.-M., Mok, N.: Birationality of the tangent map for minimal rationalcurves. Asian J. Math. (2004) 51-63.[IL] Ivey, T. A., Landsberg, J. M.: Cartan for beginners: Differential geometry via movingframes and exterior differenital systems.
Second Ed., Amer. Math. Soc., Providence,2016.[Mk] N. Mok: Recognizing certain rational homogeneous manifolds of Picard number1 from their varieties of minimal rational tangents.
Third International Congress ofChinese Mathematicians , AMS/IP Stud. Adv. Math., , 41-61, Amer. Math. Soc.,Providence, 2008.[Mo] Montgomery, R.: A tour of subriemannian geoemtries, their geodesics and applica-tions.
Math. Surv. Monographs , Amer. Math. Soc., Providence, 2002.[MZ] Montgomery, R., Zhitomirskii, M.: Geometric approach to Goursat flags. Ann. I. H.Poincar´e (2001) 459-493.[Ol] Olver, P.: Equivalence, invariants and symmetry.
Cambridge Univ. Press, Cambridge,1995.[Z99] Zelenko, I.: Nonregular abnormal extremals of 2-distribution: existence, secondvariation, and rigidity. J. Dynam. Control Syst. (1999) 347-383. NBENDABLE RATIONAL CURVES 35 [Z06] Zelenko, I.: On variational approach to differential invariants of rank two distribu-tions. Diff. Geom. Appl. (2006) 235-259.(2006) 235-259.