aa r X i v : . [ m a t h . DG ] J u l PROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES
JOE S. WANG
Abstract.
Consider an immersed Legendrian surface in the five dimensional complex projective spaceequipped with the standard homogeneous contact structure. We introduce a class of fourth order projectiveLegendrian deformation called Ψ -deformation , and give a differential geometric characterization of surfacesadmitting maximum three parameter family of such deformations. Two explicit examples of maximallyΨ-deformable surfaces are constructed; the first one is given by a Legendrian map from P blown up atthree distinct collinear points, which is an embedding away from the -2-curve and degenerates to a pointalong the -2-curve. The second one is a Legendrian embedding of the degree 6 del Pezzo surface, P blownup at three non-collinear points. In both cases, the Legendrian map is given by a system of cubics throughthe three points, which is a subsystem of the anti-canonical system. Contents
1. Introduction 12. Legendrian surface 32.1. Structure equation 42.2. Flat asymptotic 3-web 92.3. Vanishing cubic differential Ψ 102.4. Isothermally asymptotic 122.5. Asymptotically ruled 133. Second order deformation 163.1. Structure equation 164. Ψ-deformation 204.1. Structure equation 204.2. Surfaces with maximum ∞ Ψ-deformations 224.3. (Ψ , χ )-deformations 255. Examples 275.1. Flat surface 275.2. Tri-ruled surface 29References 321.
Introduction
Let Z m +1 be a complex manifold of odd dimension 2 m + 1 ≥
3. A contact structure on Z is bydefinition a hyperplane field H ⊂
T Z such that it is locally defined by H = h α i ⊥ for a 1-form α thatsatisfies the nondegeneracy condition α ∧ ( dα ) m = 0. A manifold with a contact structure is called a contact manifold . Typical examples of contact manifolds are the homogeneous adjoint varieties of simpleLie algebras including the odd dimensional projective spaces P m +1 , [LM]. Mathematics Subject Classification.
Key words and phrases.
Legendrian surface, projective deformation, del Pezzo surface.
There exist a distinguished class of subvarieties in a contact manifold. A
Legendrian subvariety M m ⊂ Z m +1 is a H -horizontal subvariety of maximum dimension m . Typical examples of Legen-drian subvarieties are the homogeneous sub-adjoint varieties, [LM]. In contrast to the case of real andsmooth category, where both contact structures and Legendrian subvarieties are flexible and admit un-obstructed local deformation, there are relatively small set of known Legendrian subvarieties in complexcontact manifolds, even in the simplest odd dimensional projective spaces.Bryant showed that every compact Riemann surface can be embedded in P as a Legendrian curvein relation to the twistorial description of minimal surfaces in 4-sphere, [Br1]. Landsberg and Maniveladopted the idea from [Br1] and showed that a K3 surface blown up at certain twelve points can beembedded in P as a Legendrian surface via an explicit birational contactomorphism from the projectivecotangent bundle P ( T ∗ P ) to P . Buczynski established a general principle of hyperplane sections forLegendrian subvarieties in projective spaces, [Bu2]. Successive hyperplane sections of the known exam-ples then gave many new smooth Legendrian subvarieties. Buczynski also showed that the algebraiccompletion of the special linear group SL C ⊂ P is a smooth, Fano Legendrian subvariety with Picardnumber 1, [Bu1].The purpose of the present paper is to propose a differential geometric study of Legendrian surfaces inthe five dimensional projective space P from the perspective of projective Legendrian deformation. Letus explain the motive. In his general investigation of the deformation of a submanifold in a homogeneousspace during the period 1916 -1920, Cartan considered the following problem of third order projectivedeformation of a surface in the projective space P , which built upon the earlier work of Fubini, [Ca] andthe reference therein; Let x: Σ ֒ → P be a surface. Let x ′ : Σ ֒ → P be a deformation of x . x ′ is a thirdorder deformation of x when there exists an application map g : Σ → SL C such that for each p ∈ Σ , x and g ( p ) ◦ x ′ agree up to order three at p . Which surfaces in P admit a nontrivial(application map g isnonconstant) third order deformation? Cartan showed that a generic surface is rigid and does not admit such deformations, but that thereexist two special sets of surfaces with nondegenerate second fundamental form that admit maximumthree parameter family of third order deformations.The present work was inspired especially by the observation that a quartic Kummer surface is one ofthe maximally deformable surfaces, [Fe]. This led us to consider the analogous problem for Legendriansurfaces in P , which may allow one to obtain the detailed structure equations for a set of Legendriansurfaces with special properties. The integration of the structure equation so obtained may suggest amethod of construction for new examples. In particular, one may hope to find the Legendrian analogueof Kummer surfaces. Main results.
1. Let x:
M ֒ → P be a Legendrian surface in P viewed as a homogeneous space of the symplecticgroup Sp C . Assuming the second fundamental form of x is nondegenerate, the moving frame methodis employed to determine the basic local invariants of a Legendrian surface as a set of three symmetricdifferentials ( Φ , Ψ , χ ) of degree (3 , ,
2) and order (2 , , -deformation is introduced. Given a Legendrian surfacex: M ֒ → P , a deformation x ′ : M ֒ → P is a Ψ-deformation when the pair ( Φ ′ , Ψ ′ ) for x ′ is isomorphic tothat of x at each point of M up to motion by Sp C . This choice of deformation is justified later by theanalogy with the aforementioned Cartan’s result that there exist two special sets of Legendrian surfacescalled D -surfaces and D -surfaces that admit maximum three parameter family of Ψ-deformations,Proposition 4.9.3. The structure equations for the maximally Ψ-deformable surfaces are determined. It turns out thatthe local moduli space of D -surfaces depends on 1 arbitrary function of one variable, whereas the localmoduli space of D -surfaces is finite dimensional, Theorem 4.21. ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 3
4. Two global examples of D -surfaces are constructed explicitly. The first one is given by a Legendrianmap from P blown up at three distinct collinear points, which is an embedding away from the -2-curve(the proper transform of the line through the three points) and degenerates to a point along the-2-curve, Theorem 5.1. The second one is given by a Legendrian embedding of the degree 6 del Pezzosurface, P blown up at three non-collinear points, Theorem 5.7.The paper is organized as follows. In Section 2, the basic local invariants of a Legendrian surface aredefined as a set of three symmetric differentials ( Φ , Ψ , χ ). By imposing natural geometric conditionsin terms of these invariants, we identify four distinguished classes of Legendrian surfaces, and determinetheir structure equations, Section 2.2 through 2.5. In Section 3, as a preparatory step for the analysisof deformation with geometric constraints in Section 4, we determine the structure equations for thesecond order Legendrian deformation, or equivalently the Legendrian deformation preserving the secondorder cubic differential Φ. The analysis shows that there is no local obstruction for the second orderdeformation of a Legendrian surface with nondegenerate cubic Φ. In Section 4, we introduce the Ψ-deformation, which is the main object of study in this paper. It is the second order deformation whichalso preserves the fourth order cubic differential Ψ. The analysis shows that a generic Legendrian surfacedoes not admit any nontrivial Ψ-deformations, but that the two special sets of Legendrian surfaces called D -surfaces and D -surfaces admit maximum three parameter family of Ψ-deformations. Moreover, asubset of these maximally deformable surfaces admit Ψ-deformations that also preserve the fifth orderquadratic differential χ , Theorem 4.34. In Section 5, we choose and integrate two simple examples ofstructure equations for D -surfaces. In Section 5.1, the flat case is examined, where all the structurecoefficients vanish. The structure equation is integrated, and one gets a rational Legendrian variety witha single, second order branch type isolated singularity. Its smooth resolution is P blown up at threedistinct collinear points, with the exceptional divisor being the -2-curve. In Section 5.2, the case called tri-ruled (Section 2.5) is examined, where each leaf of the three Φ-asymptotic foliations lies in a linearLegendrian P . The structure equation is integrated, and one gets a smooth Legendrian embedding of P blown up at three non-collinear points. The embedding is given by a subsystem of the anti-canonicalsystem and each of the six -1-curves is mapped to a line.Throughout the paper, we freely apply the methods and results of exterior differential systems. Werefer the reader to [BCG3] for the standard reference on the subject.2. Legendrian surface
The method of moving frames is a process of equivariant frame adaptation for a submanifold ina homogeneous space. The algorithmic operation of successive normalizations reveals the basic localinvariants of the submanifold as the coefficients of the structure equation and their derivatives. Themethod was developed by E. Cartan, and Cartan himself applied it extensively to a variety of problems.In this section, the method of moving frames is applied to immersed Legendrian surfaces in the fivedimensional complex projective space P . We establish the fundamental structure equation which de-pends on the sixth order jet of the Legendrian immersion, and identify the basic local invariants as theset of three symmetric differentials Φ , Ψ, and χ of order 2 ,
4, and 5 respectively, Proposition 2.22. Φand Ψ are cubic, and χ is quadratic.In order to understand the geometric implication of these invariants, we impose a set of conditionsin terms of Φ , Ψ, and χ , and give an analysis for the Legendrian surfaces that satisfy these conditions.This in turn leads to four classes of Legendrian surfaces with interesting geometric properties, Section2.2, 2.3, 2.4, and 2.5 respectively.Let us give an outline of the analysis. • Φ, cubic differential of order 2: Φ represents the second fundamental form of the Legendrian surface.Assuming Φ is nondegenerate, the base locus of Φ defines a 3-web called asymptotic web . It is the lowestorder local invariant of a Legendrian surface.
JOE S. WANG
We give an analysis for surfaces with flat asymptotic web, Section 2.2. The condition for the asymptoticweb to be flat is expressed by a single fourth order equation for the Legendrian immersion, (2.26). Adifferential analysis shows that this PDE becomes involutive after a partial prolongation with the generalsolution depending on five arbitrary functions of 1 variable, Proposition 2.28. It turns out that all of thesurfaces that are of interest to us necessarily have flat asymptotic web, e.g., surfaces admitting maximumfamily of nontrivial Legendrian deformation, Section 4.The moving frame computation associates to each asymptotic foliation a unique Legendrian P -fieldthat has second order contact with the given foliation, (2.40). An asymptotic foliation is called ruled when the associated Legendrian P -field is leafwise constant. We give an analysis for surfaces with ruledasymptotic foliations, Section 2.5. • Ψ, cubic differential of order 4: The moving frame computations show that there is no third orderlocal invariants for a Legendrian surface. The pencil of cubics (Ψ; Φ) based at Φ accounts for roughlyone-half of the fourth order invariants of a Legendrian surface.We give an analysis for surfaces with vanishing Ψ, called Ψ -null surfaces, which can be considered asthe Legendrian analogue of quadrics in P , Section 2.3. The condition for Ψ to vanish is expressed bya pair of fourth order equations for the Legendrian immersion, (2.30). A differential analysis shows thatthe structure equation for Ψ-null surfaces closes up with the general solution depending on one constant,Proposition 2.34.More generally, we give an analysis for the class of surfaces called isothermally asymptotic surfaces,which is the case when Ψ is proportional to Φ and the pencil (Ψ; Φ) degenerates, Section 2.4. It will beshown that this class of surfaces are examples of surfaces admitting maximum three parameter family ofΨ-deformations, Section 4. • χ -quadratic differential of order 5: The geometry of χ is examined in Section 4. It will be shown thatthere exist Legendrian surfaces which admit maximum one parameter family of deformations preservingthe triple (Φ , Ψ , χ ).For a modern exposition of Cartan’s equivalence method, we refer to [Ga][IL].2.1. Structure equation.
Let V = C be the six dimensional complex vector space. Let ̟ be thestandard symplectic 2-form on V . Let P = P ( V ) be the projectivization equipped with the inducedcontact structure. The contact hyperplane field H on P is defined by H x = [(ˆx y ̟ ) ⊥ ] , for x ∈ P , where ˆx ∈ V is any de-projectivization of x. (ˆx y ̟ ) ⊥ ⊂ V is a codimension one subspace containing ˆx,and its projectivization [(ˆx y ̟ ) ⊥ ] ⊂ P is a hyperplane at x. The symplectic group Sp C acts transitivelyon P as a group of contact transformation. H inherits a conformal class of nondegenerate symplectic 2-form determined by the restriction of ̟ onthe quotient space (ˆx y ̟ ) ⊥ / h ˆx i . A two dimensional Lagrangian subspace of H x is called Legendrian . LetΛ → P be the bundle of Legendrian 2-planes. Let Lag ( V ) be the set of three dimensional Lagrangiansubspaces of V . The symplectic group Sp C acts transitively on both Λ and Lag ( V ), and there existsthe incidence double fibration; Sp ( C ) π ↓ Λ = Sp ( C ) /P ցւ Lag ( V ) P π π Figure 2.1. Double fibrationThe fiber of π is isomorphic to Lag (2 , C ), and the fiber of π is P . ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 5
To fix the notation once and for all, let us define the projection maps π, π , and π explicitly. Let( e, f ) = ( e , e , e , f , f , f ) denote the Sp C ⊂ SL C frame of V such that the 2-vector ̟ ♭ = e ∧ f + e ∧ f + e ∧ f is dual to the symplectic form ̟ . Define π ( e, f ) = ([ e ] , [ e ∧ e ∧ e ]) , (2.1) π ([ e ] , [ e ∧ e ∧ e ]) = [ e ] ,π ([ e ] , [ e ∧ e ∧ e ]) = [ e ∧ e ∧ e ] . In this formulation, the stabilizer subgroup P in Figure 2.1 is of the form(2.2) P = { (cid:18) A B · ( A t ) − (cid:19) } , where ( A − B ) t = A − B , and A = { ∗ ∗ ∗· ∗ ∗· ∗ ∗ } . Here ’ · ’ denotes 0 and ’ ∗ ’ is arbitrary.The Sp C -frame ( e, f ) satisfies the structure equation(2.3) d ( e, f ) = ( e, f ) φ for the Maurer-Cartan form φ of Sp C . The components of φ are denoted by φ = (cid:18) ω ηθ − ω t (cid:19) , where { ω, θ, η } are 3-by-3 matrix 1-forms such that θ t = θ, η t = η . φ satisfies the structure equation(2.4) dφ + φ ∧ φ = 0 . Let x :
M ֒ → P be a H -horizontal, immersed Legendrian surface. We employ the method of movingframes to normalize the Sp C -frame along x. Our argument is local, and the action of certain finitepermutation group that occurs in the course of normalization shall be ignored. This does not affectthe analysis nor the result of moving frame computation for our purpose. The process of equivariantreduction terminates at the sixth order jet of the immersion x. . By definition, there exists a unique lift ˜x : M ֒ → Λ. Let ˜x ∗ Sp C → M be thepulled back P -bundle. We continue to use φ to denote the pulled back Maurer-Cartan form on ˜x ∗ Sp C .From (2.1), (2.3), the initial state of φ on ˜x ∗ Sp C takes the form φ = ω ω ω η η η ω ω ω η η η ω ω ω η η η · · · − ω − ω − ω · θ θ − ω − ω − ω · θ θ − ω − ω − ω , where θ ij = θ ji , η ij = η ji , and we denote ω i = ω i , i = 1 ,
2. For any section s : M → ˜x ∗ Sp ( C ), { s ∗ ω , s ∗ ω } is a local coframe of M . . Differentiating θ = 0 , θ = 0, one gets (cid:18) θ θ θ θ (cid:19) ∧ (cid:18) ω ω (cid:19) = 0 . By Cartan’s lemma, there exist coefficients t ijk ; i, j, k = 1 ,
2, fully symmetric in indices such that θ ij = t ijk ω k . JOE S. WANG
The structure equation shows that the cubic differential(2.5) Φ = θ ij ω i ω j = t ijk ω i ω j ω k is well defined on M up to scale. Φ represents the second fundamental form of the Legendrian immersion. Definition 2.6.
Let x :
M ֒ → P be an immersed Legendrian surface. Let Φ be the cubic differential (2.5) which represents the second fundamental form of the immersion x . The Legendrian surface is nondegenerate if the cubic differential Φ is equivalent to an element in the unique open orbit of thegeneral linear group GL C action on cubic polynomials in two variables, [Mc] . Remark 2.7.
The Segre embedding P × Q ⊂ P is ruled by lines, and it has a degenerate secondfundamental cubic. We assume the Legendrian surface is nondegenerate from now on. By a frame adaptation, one maynormalize t ijk such that Φ = 3 ω ω ( ω + ω ) , (2.8) = − ω ω ω , where ω = − ( ω + ω ). This is equivalent to(2.9) (cid:18) θ θ θ θ (cid:19) = (cid:18) ω ω + ω ω + ω ω (cid:19) . The structure group P , (2.2), for the 2-adapted frame is reduced such that A = { (cid:18) a ∗· a A ′ (cid:19) | a = 0 } , where A ′ is the finite subgroup of GL C whose induced action leaves Φ invariant.Three asymptotic line fields are determined by { ( ω ) ⊥ , ( ω ) ⊥ , ( ω ) ⊥ } . The set of respective foliationsdefines a 3-web called asymptotic web on the Legendrian surface. Since a planar 3-web has local invariants,e.g., web curvature, asymptotic web is the lowest order invariant of a nondegenerate Legendrian surface. . On the 2-adapted frame satisfying (2.9), set(2.10) ω ij = δ ij ω + s ijk ω k , for i, j = 1 , , for coefficients s ijk . Differentiating (2.9), one gets − s + s + 2 s + 2 s = 0 , (2.11) − s + 32 s + 12 s − s + s − s = 0 , − s + s + 2 s + 2 s = 0 . Exterior derivatives of (2.10) show that ds ≡ − η − η + 13 ω ,ds ≡ − η ,ds ≡ − η − η + ω ,ds ≡ − η − η + ω ,ds ≡ − η ,ds ≡ − η − η + 13 ω , mod ω , ω ; ω . ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 7
By a frame adaptation, one may translate the coefficients { s = 0 , s = 0 , s − s = 0 } , whichforces 3 s − s = 0 by (2.11). This set of normalizations is equivalent to adapting the Sp C -frame sothat de ≡ f ω mod e , e ; ω ,de ≡ f ω mod e , e ; ω ,d ( e − e ) ≡ ( f + f ) ω mod e , ( e − e ); ω . By a further frame adaptation, one may translate s ijk = 0(we omit the details), and we have(2.12) ω ij = δ ij ω . For this 3-adapted frame, the structure equation shows that a triple of Legendrian P -fields is well de-fined along the Legendrian surface. Let ( L , L , L ) be the triple of Legendrian P -fields, or equivalentlythe triple of Lagrangian 3-plane fields, defined by( L , L , L ) : M → Lag ( V ) × Lag ( V ) × Lag ( V ) , (2.13) L = [ e ∧ e ∧ f ] ,L = [ e ∧ e ∧ f ] ,L = [ e ∧ ( e − e ) ∧ ( f + f )] . Each L i is the unique Legendrian P -field that has second order contact with the asymptotic foliationdefined by h ω i i ⊥ . . On the 3-adapted frame satisfying (2.12), set ω = h k ω k , (2.14) ω = h k ω k ,η ij = η ji = h ijk ω k , for i, j = 1 , . The structure equation shows that the cubic differential Ψ = η ij ω i ω j is well defined up to scale, and upto translation byΨ → Ψ + ( s ω + s ω )(( ω ) + ( ω ) ) , for arbitrary coefficients s , s . By a frame adaptation, one may translate h = 0 , h = 0 so that(2.15) Ψ = ω ω (( h + 2 h ) ω + (2 h + h ) ω ) . Ψ is now well defined up to scale. It is a fourth order invariant of the nondegenerate Legendrian surface.For the problem of projective deformation of Legendrian surfaces, Ψ will play the role of the thirdfundamental form for surfaces in P .Note that the derivative of (2.12) with the relation h = 0 , h = 0 gives the compatibility equations h = h + h , (2.16) h = h + h ,h = 35 ( − h + h ) ,h = 35 ( h − h ) . . On the 4-adapted frame satisfying (2.15), (2.16), set η = h k ω k , (2.17) η = h k ω k . JOE S. WANG
The structure equation shows that the quadratic differential χ = η ω + η ω is well defined up toscale, and up to translation by χ → χ + s (( ω ) + ( ω ) ) , for arbitrary coefficient s . By a frame adaptation, one may translate h + h = 0(we omit the details) so that(2.18) χ = h ( ω ) + ( h + h ) ω ω − h ( ω ) .χ is now well defined up to scale. It is a fifth order invariant of the nondegenerate Legendrian surface.Restricting to the sub-bundle defined by the equation h + h = 0, we set(2.19) η = h ω + h ω . At this stage, no more frame adaptation is available, and the components of the induced Maurer-Cartanform φ is uniquely determined, modulo at most a finite group action (this finite group does not enterinto our analysis, and we shall not pursue the exact expression for the representation of this group). Thereduction process of moving frame method stops here.For a notational purpose, let us make a change of variables; h = a , h = b , h = c , (2.20) h = a , h = b , h = c .h = a , h = b ,h = a , The covariant derivatives are denoted by da i = − a i ω + a ik ω k ,db i = − b i ω + b ik ω k ,dc i = − c i ω + c ik ω k . Differentiating (2.14), (2.17), (2.19), one gets a set of compatibility equations among the covariant deriva-tives. a = − b , (2.21) a = − a + 152 b + 8 b ,a = − a + 152 b + 10 b ,a = a + 5 b − b − b ,a = − b ,b = b − a + c + a a − a a ,b = − b − c + 35 a + 25 a a − a a ,c = c − a b − a b + 2 a b + 2 a b . The structure equation (2.4) for φ is now an identity with these relations. One may check that thestructure equation with the coefficients { a i , b j , c k } becomes involutive after one prolongation with thegeneral solution depending on one arbitrary function of 2 variables in the sense of Cartan, [BCG3]. Proposition 2.22.
Let x :
M ֒ → P be a nondegenerate immersed Legendrian surface. Let ˜x : M ֒ → Λ be the associated lift to the bundle of Legendrian 2-planes. Let ˜x ∗ Sp ( C ) → M be the pulled back bundle,Figure 2.1. ˜x ∗ Sp ( C ) admits a reduction to a sub-bundle with 1-dimensional fibers such that the induced ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 9
Maurer-Cartan form φ satisfies the structure equations (2.9) , (2.12) , (2.14) , (2.15) , (2.16) , (2.17) , (2.19) , (2.20) , and (2.21) . We shall work with the 5-adapted frame for the rest of the paper. Unless stated otherwise, ’the structureequation’ would mean the structure equation for the 5-adapted frame.Note that under the notations we chose, the invariant differentials Φ, Ψ, and χ , (2.8), (2.15), (2.18),are expressed by Φ = 3 ( ω ) ω + 3 ω ( ω ) , (2.23) Ψ = ω ω (( a + 2 a ) ω + (2 a + a ) ω ) , = ( a + 2 a ) ( ω ) ω + (2 a + a ) ω ( ω ) ,χ = b ( ω ) + ( b + b ) ω ω − b ( ω ) . In the next two sections, Section 3 and Section 4, we shall examine the deformability, or the rigidity, ofLegendrian surfaces preserving these invariant differentials.Before we proceed to the problem of deformation, let us examine four classes of Legendrian surfaceswith special geometric properties. There exist a number of surfaces in P with notable characteristics,which have been the subject of extensive study, [Fe]. Some of the surfaces described below can beconsidered as the Legendrian analogues of these classical surfaces.2.2. Flat asymptotic 3-web.
In this sub-section, we consider the class of Legendrian surfaces with flatasymptotic 3-web. For a comprehensive introduction to web geometry, we refer to [PP].
Definition 2.24.
Let
M ֒ → P be a nondegenerate Legendrian surface. The asymptotic 3-web is the setof three foliations defined by { ( ω ) ⊥ , ( ω ) ⊥ , ( ω ) ⊥ } at 2-adapted frame, where ω + ω + ω = 0 . The following analysis shows that the differential equation describing the Legendrian surfaces with flatasymptotic web is in good form(involutive) and admits arbitrary function worth solutions locally.The web curvature of the asymptotic 3-web can be expressed in terms of the structure coefficients ofthe Legendrian surface. From the structure equation(for 5-adapted frame), dω i = 23 ω ∧ ω i , i = 1 , , . The web curvature K of the 3-web is given by23 dω = K ω ∧ ω , (2.25) = −
45 ( a − a ) ω ∧ ω . The asymptotic web is flat when(2.26) a − a = 0 . We wish to give an analysis of the compatibility equations derived from this vanishing condition.Differentiating a − a = 0, one gets a = − b + 3 b ,b = − b + b . Differentiating the second equation for b , one gets b = 2 b − b − a + a a − c + 2 c ,b = − b + b . Exterior derivative d ( d ( a )) = 0 with these relations then gives b = − b + a − a a + 3 c − c . The identities from d ( d ( b )) = 0 , d ( d ( b )) = 0 determine the derivative of b by db + 83 b ω = ( − a b − a a − b a − b a + b a + 14 a a + c + 2 c − c ) ω (2.27) + ( a b + 14 a a + 2 b a + 2 b a − b a − a a − c + c ) ω . At this step, we interrupt the differential analysis and invoke a version of Cartan-K¨ahler theorem, ageneral existence theorem for analytic differential systems, [BCG3].
Proposition 2.28.
The structure equation for the nondegenerate Legendrian surfaces with flat asymptoticweb is in involution with the general solution depending on five arbitrary functions of 1 variable.Proof.
From the analysis above, the exterior derivative identities d ( d ( a )) = 0, d ( d ( a )) = 0, d ( d ( c )) =0, d ( d ( c )) = 0, d ( d ( b )) = 0 give 5 compatibility equations while the remaining independent derivativecoefficients at this step are { a , a , c , c , c ; b } . An inspection shows that the resulting structureequation is in involution with the last nonzero Cartan character s = 5. (cid:3) Vanishing cubic differential Ψ . In this sub-section, we consider the class of Legendrian surfaceswith vanishing cubic differential Ψ, a fourth order invariant (2.15).
Definition 2.29.
Let
M ֒ → P be a nondegenerate Legendrian surface. M is a Ψ-null surface if thefourth order cubic differential Ψ defined at 4-adapted frame vanishes. The following analysis shows that a Ψ-null surface necessarily has flat asymptotic web, and that the localmoduli space of Ψ-null surfaces is finite dimensional.From (2.23), Ψ vanishes when(2.30) a = − a , a = − a . Differentiating these equations, one gets a = b , a = − b − b + 3 b ,b = − b + b , a = 10 b − b . Exterior derivatives d ( d ( a )) = 0 , d ( d ( a )) = 0 with these relations give b = 8 b + 85 a − a a + 152 b − b ,b = − b + b − a + 85 c + 196125 a a − c + 42125 a . Differentiating b = − b + b , one gets b = 2225 a − c − a a + 32 c − a ,b = 2 c − a − a a + 5150 a − c . Exterior derivatives d ( d ( b )) = 0 , d ( d ( b )) = 0 with these relations give c = c + 685 b a + 585 b a − a b − b a ,c = 43 c + 26615 b a + 6 b a − b a − a b . ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 11
The identities from d ( d ( c )) = 0 , d ( d ( c )) = 0 determine the derivative of c by dc + 103 c ω = ( 6807875 a − c a − b + 9993875 a + 20 b b − a a − b + 63835 c a − c a − a a + 44735 c a ) ω + ( 10119875 a − c a + 62 b + 13016875 a − b b − a a + 54 b + 1657 c a − c a − a a + 77135 c a ) ω . Differentiating this equation again, one gets a compatibility equation of the form(2.31) ( a − a ) c = [ a i , b j , c k ] , where the right hand side is a polynomial in the variables a i , b j , c k . At this juncture, the analysis dividesinto two cases. Case a − a = 0. It turns out that the condition a − a = 0 is not compatible with the vanishingof Ψ, and there is no nondegenerate Legendrian surfaces with Ψ ≡
0, and a − a = 0. Some of theexpressions for the analysis of this case are long. Let us explain the relevant steps of differential analysis,and omit the details of the long and non-essential terms.From (2.31), solve for c . Differentiating this, one gets a set of two equations, from which one solves for c , c . Differentiating these equations, one gets another set of two equations which imply b = b = 0.Differentiating these equations again, one finally gets two quadratic equations for a , a , which force a = a = 0, a contradiction. Case a − a = 0. From Section 2.2, this is the case when the asymptotic 3-web is flat. Successivederivatives of the equation a − a = 0 imply the following. b = b ,c = c ,c = 2 a b . Furthermore, these equations are compatible, i.e., d = 0 is an identity.The remaining independent coefficients at this step are { a , b , c } . Let us remove the sub-script,and denote { a , b , c } = { a, b, c } . The structure equations for these coefficients are reduced to da = − a ω + b ( ω − ω ) , (2.32) db = − b ω −
12 (3 a + c )( ω − ω ) ,dc = − c ω + 2 ab ( ω − ω ) . The Maurer-Cartan form φ takes the form(2.33) φ = ω − aω − aω c ( ω + ω ) b ( ω + ω ) − b ( ω + ω ) ω ω · b ( ω + ω ) − aω a ( ω + ω ) ω · ω − b ( ω + ω ) a ( ω + ω ) − aω · · · − ω − ω − ω · ω ω + ω aω − ω ·· ω + ω ω aω · − ω . Proposition 2.34.
Let
M ֒ → P be a nondegenerate, Ψ -null Legendrian surface. The asymptotic 3-webof M is necessarily flat. The Maurer-Cartan form of the 5-adapted frame of M is reduced to (2.33) ,and the structure coefficients { a, b, c } satisfy the equation (2.32) . The local moduli space of Ψ -nullLegendrian surfaces has general dimension 1.Proof. Let F → M be the canonical bundle of 5-adapted frames from Proposition 2.22. (2.32) showsthat the invariant map ( a, b, c ) : F → C generically has rank two. From the general theory of geometricstructures with closed structure equation, [Br2], the local moduli space of this class of Legendrian surfaceshas general dimension dim( C ) − rank( a, b, c ) = 1. A Legendrian surface in this class necessarily possessesa minimum 1-dimensional local group of symmetry. The line field h ω , ω − ω i ⊥ is tangent to the fibersof the invariant map ( a, b, c ), and it generates a local symmetry. (cid:3) Isothermally asymptotic.
In this sub-section, we consider the class of Legendrian surfaces whichare the analogues of the classical isothermally asymptotic surfaces in P , [Fe]. Definition 2.35.
Let
M ֒ → P be a nondegenerate Legendrian surface. M is isothermally asymptotic ifthe fourth order cubic differential Ψ , (2.23) , is a multiple of the second order cubic differential Φ , (2.8) . The following analysis shows that the differential equation describing the isothermally asymptotic Leg-endrian surfaces is in good form and admits arbitrary function worth solutions locally.From (2.23), Ψ ≡ a + 2 a = 2 a + a . We wish to give an analysis of the compatibility equations derived from this condition.Differentiating a + 2 a = 2 a + a , one gets a = 5 a + 10 b − b − b ,a = 5 a − b − b − b . The identities from d ( d ( a )) = 0 , d ( d ( a )) = 0 determine the derivative of a by da + 2 a ω = ( − b + 8 b − b + 275 b − a a + 825 a a ) ω + ( − b + 35 b + 3 b − b + 85 a − a a + 1225 a a − a a ) ω . Exterior derivative d ( d ( a )) = 0 with these relations then gives b = 5 b + b + 5 b + 465 a a − a a − c − a + 20 c + 5 a + 215 a − a a . At this step, we interrupt the differential analysis and invoke a version of Cartan-K¨ahler theorem.
Proposition 2.36.
The structure equation for the nondegenerate isothermally asymptotic Legendriansurfaces is in involution with the general solution depending on five arbitrary functions of 1 variable.Proof.
From the analysis above, the exterior derivative identities d ( d ( b )) = 0, d ( d ( b )) = 0, d ( d ( b )) = 0, d ( d ( c )) = 0, d ( d ( c )) = 0, d ( d ( a )) = 0 give 6 compatibility equations while the re-maining independent derivative coefficients at this step are { b , b , b , c , c , c ; a } . A shortanalysis shows that the resulting structure equation becomes involutive after one prolongation with thelast nonzero Cartan character s = 5. Since the prolonged structure equation does not enter into ouranalysis in later sections, the details shall be omitted. (cid:3) ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 13
Isothermally asymptotic with flat asymptotic web.
Consider the class of isothermally asymptoticLegendrian surfaces which have flat asymptotic 3-web. From (2.23) and (2.25), this is equivalent to thecondition(2.37) a = a , a = a . The following analysis shows that the differential equation describing such Legendrian surfaces is still ingood form and admits arbitrary function worth solutions locally. Note that a Ψ-null surface is necessarilyisothermally asymptotic with flat asymptotic web.This is in contrast with the Ψ-null surface case, where the defining equation (2.30) is also a set of twolinear equations among a i ’s and yet the resulting structure equations close up to admit solutions withfinite dimensional moduli. This reflects the subtle well-posedness of the equation (2.37). The discoveryof this class of Legendrian surfaces is perhaps most unexpected of the analysis in this section.We wish to give an analysis for the compatibility equations derived from (2.37). Differentiating thegiven equations a = a , a = a , one gets a = 4 b + 3 b , b = − b + b ,a = 4 b − b , a = − b . Exterior derivatives d ( d ( a )) = 0 , d ( d ( a )) = 0 , d ( d ( a )) = 0 with these relations give b = b ,b = − b + 43 b + 13 a − a − c + 43 c ,b = − b + 3 c − c + a − a . Successively differentiating b = − b + b , one gets c = c ,c = c , c = c + 4 b a + 4 b a . The identities from d ( d ( b )) = 0 , d ( d ( b )) = 0 determine the derivative of b by db = − b ω + (2 b a + 3 b a )( ω − ω ) . Moreover, d ( d ( b )) = 0 is an identity.At this step, we invoke a version of Cartan-K¨ahler theorem. Proposition 2.38.
The structure equation for the nondegenerate isothermally asymptotic Legendriansurfaces with flat asymptotic web is in involution with the general solution depending on one arbitraryfunction of 1 variable.Proof.
From the analysis above, the exterior derivative identity d ( d ( c )) = 0 gives 1 compatibility equa-tion while the remaining independent derivative coefficients at this step are { c ; b } . By inspection,the resulting structure equation is in involution with the last nonzero Cartan character s = 1. (cid:3) Asymptotically ruled.
In this sub-section, we consider the class of Legendrian surfaces for whichthe asymptotic Legendrian P -field defined at 3-adapted frame is constant along the corresponding as-ymptotic foliation. Definition 2.39.
Let
M ֒ → P be a nondegenerate Legendrian surface. Let F be an asymptotic folia-tion(one of the three) defined at 2-adapted frame. F is ruled if the corresponding Legendrian P -fielddefined at 3-adapted frame is constant along F . With an abuse of terminology, we call the leaf-wise constant P -field rulings of the asymptotic foliation.We wish to give a differential analysis for Legendrian surfaces with three, two, or one ruled asymptoticfoliations in turn. Recall ω = − ( ω + ω ). From the structure equation (2.3), de ≡ e ω , mod e ; ω , (2.40) de ≡ f ω , mod e , e ; ω ,df ≡ e ( a ω ) , mod e , e , f ; ω ,de ≡ e ω , mod e ; ω ,de ≡ f ω , mod e , e ; ω ,df ≡ e ( a ω ) , mod e , e , f ; ω ,de ≡ ( e − e ) ω , mod e ; ω ,d ( e − e ) ≡ ( f + f ) ω , mod e , ( e − e ); ω ,d ( f + f ) ≡ e ( − ( a + 2 a ) + (2 a + a )) ω , mod e , ( e − e ) , ( f + f ); ω . Tri-ruled.
This is the class of surfaces for which all of the three asymptotic foliations are ruled.The following differential analysis shows that the local moduli space of tri-ruled Legendrian surfacesconsists of two points.Assume that each of the three Legendrian P -fields [ e ∧ e ∧ f ] , [ e ∧ e ∧ f ], and [ e ∧ ( e − e ) ∧ ( f + f )]is constant along the asymptotic foliations defined by ω = 0 , ω = 0, and ω = 0 respectively. By (2.40),this implies a = 0 , a = 0 , a = a = a. A tri-ruled Legendrian surface has flat asymptotic 3-web, and it is also isothermally asymptotic. Thecubic differential Ψ vanishes when a = 0, and a non-flat tri-ruled surface is distinct from Ψ-null surfacesdiscussed in Section 2.3.A differential analysis shows that a tri-ruled surface necessarily has b i = 0 , c = c = a (we omit thedetails). Maurer-Cartan form φ is reduced to(2.41) φ = ω aω aω a ( ω + ω ) · · ω ω · · · a ( ω + ω ) ω · ω · a ( ω + ω ) ·· · · − ω − ω − ω · ω ω + ω − aω − ω ·· ω + ω ω − aω − ω , with(2.42) da = − a ω . Proposition 2.43.
Let
M ֒ → P be a nondegenerate, tri-ruled Legendrian surface. M is necessarilyisothermally asymptotic with flat asymptotic web. The Maurer-Cartan form of the 5-adapted frame of M is reduced to (2.41) , and the single structure coefficient { a } satisfies the equation (2.42) . The localmoduli space of tri-ruled Legendrian surfaces consists of two points.Proof. From (2.42), the moduli space is divided into two cases; a ≡
0, or a = 0. (cid:3) A differential geometric characterization of tri-ruled surfaces is presented in Section 5.
ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 15
Doubly-ruled.
Assume that each of the two Legendrian P -fields [ e ∧ e ∧ f ] and [ e ∧ e ∧ f ]is constant along the asymptotic foliations defined by ω = 0 and ω = 0 respectively. By (2.40), thisimplies(2.44) a = 0 , a = 0 . Note that such a doubly ruled surface with flat asymptotic web is necessarily tri-ruled.Successively differentiating (2.44), one gets b = 0 , b = 0 ,a = a = 0 , b = b = b = 0 ,b = − c + 35 a + 25 a a . Exterior derivative d ( d ( b )) = 0 with these relations gives c = 445 b a − a a + 165 b a − a a . The identities from d ( d ( a )) = 0 , d ( d ( a )) = 0 determine the derivative of a by da = − a ω + ( − c + 4825 a + 6425 a a + 8825 a ) ω + (8 c + 12 c + 8825 a − a a − a ) ω . Differentiating this equation again, one gets c = c − a a + 12 a a − b a − b a . Exterior derivatives d ( d ( c )) = 0 , d ( d ( c )) = 0 with these relations determine the derivative of c (theexact expression for dc is long, and shall be omitted). Differentiating this equation, d ( d ( c )) = 0 finallygives c = 14( a − a ) ( − a c − a a + 8 a a b − a b + 80 c b + 9 a a − c b + 44 a b − a a a + 15 c a ) . Here we assumed that a = a , or equivalently that the Legendrian surface is not tri-ruled.Differentiating this equation again, and comparing with the formula for dc , one gets two polynomialcompatibility equations for six coefficients { a , a , b , a , c , c } . Successive derivatives of these equa-tions generate a sequence of compatibility equations for a nondegenerate Legendrian surface to admitexactly two asymptotic P -rulings.Partly due to the complexity of the polynomial compatibility equations, our analysis is incomplete.We suspect that if there do exist nondegenerate, doubly-ruled(and not tri-ruled) Legendrian surfaces, themoduli space of such surfaces is at most discrete.2.5.3. Singly-ruled.
Assume that the Legendrian P -field [ e ∧ ( e − e ) ∧ ( f + f )] is constant along theasymptotic curves defined by ω = 0. From (2.40), this implies a + 2 a = 2 a + a . Note the equivalence relations.
Singly ruled and flat asymptotic web = Singly ruled and isothermally asymptotic , = Isothermally asymptotic and flat asymptotic web . An analysis shows that the structure equation for a nondegenerate, singly-ruled Legendrian surfacebecomes involutive after one prolongation with the general solution depending on five arbitrary functionsof 1 variable. We omit the details of differential analysis for this case. Second order deformation
Definition 3.1.
Let x :
M ֒ → P be a nondegenerate Legendrian surface. Let x ′ : M ֒ → P be aLegendrian deformation of x . x ′ is a k -th order deformation if there exists a map g : M → Sp C suchthat for each p ∈ M , the k -adapted frame bundles of x ′ and g ( p ) ◦ x are isomorphic at p . When theapplication map g is constant, the deformation x ′ is trivial , and x ′ is congruent to x up to motionby Sp C . Two deformations x ′ and x ′ are equivalent if there exists an element g ∈ Sp C such that x ′ = g ◦ x ′ . A ’deformation’ would mean an ’equivalence class of deformations modulo Sp C action’ forbrevity. It follows from the construction of adapted frames in Section 2 that a k -th order deformation is a ( k + 1)-th order deformation when for each p ∈ M , the application map g ( p ) ∈ Sp C not only preserves the k -adapted frame at p , but also the first order derivatives of the k -adapted frame at p (this is a vagueexplanation, but the meaning is clear).The definition of k -th order deformation indicates a way to uniformize the various geometric conditionsthat naturally occur in the theory of deformation and rigidity of submanifolds in a homogeneous space.Take for an example the familiar case of surfaces in three dimensional Euclidean space with the usual1-adapted tangent frame of the group of Euclidean motions. One surface is a first order deformation ofthe other if they have the same induced metric, and it is a second order deformation if they also have thesame second fundamental form. By Bonnet’s theorem, a second order deformation is a congruence, [Sp].Fubini, and Cartan studied the problem of third order deformation of projective hypersurfaces in P n +1 ,[Ca] and the reference therein. For n ≥
3, a third order deformation of a hypersurface with nondegeneratesecond fundamental form is necessarily a congruence, [JM] for a modern proof. For n = 2, Cartan showedthat a generic surface does not admit a nontrivial third order deformation, but that there exist two specialclasses of surfaces which admit maximum three parameter family of deformations.The purpose of this section is to lay a foundation for generalizing Cartan’s work on projective de-formation of surfaces in P to deformation of Legendrian surfaces in P . As a preparation, we firstconsider the second order deformation. By applying a modified moving frame method, we determine thefundamental structure equation for the second order deformation of a nondegenerate Legendrian surface.The analysis shows that the resulting structure equation is in involution, and admits arbitrary functionworth solutions locally. This implies that there is no local obstruction to second order deformation of aLegendrian surface.The structure equation established in this section will be applied to the projective deformation ofLegendrian surfaces with geometric constraints in Section 4.3.1. Structure equation.
Let x :
M ֒ → P be a nondegenerate Legendrian surface. Let F → M be theassociated canonical bundle of 5-adapted frames with the induced sp C -valued Maurer-Cartan form φ .The pair ( F, φ ) satisfies the properties described in Proposition 2.22. Let x ′ : M ֒ → P be a secondorder deformation of x. Let F ′ → M be the associated canonical bundle with the induced Maurer-Cartanform π . From the definition of second order deformation, F ′ can be considered as a graph over F whichagrees with F up to 2-adapted frame. By pulling back π on F, we regard π as another sp C -valuedMaurer-Cartan form on F.Set(3.2) π = φ + δφ. The components of δφ are denoted by δφ = (cid:18) δω δηδθ − δω t (cid:19) , ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 17 where δθ t = δθ, δη t = δη . Maurer-Cartan equations for π and φ imply the fundamental structureequation for the deformation δφ ;(3.3) d ( δφ ) + δφ ∧ φ + φ ∧ δφ + δφ ∧ δφ = 0 . Differentiating the components of δφ from now on would mean applying this structure equation.We employ the method of moving frames to normalize the frame bundle F ′ based at F. In effect, onemay adopt the following analysis as the constructive definition of ( F ′ , π ). The equivariant reductionprocess for F ′ in this section can be considered as the derivative of the one applied for F in Section 2. Toavoid repetition, some of the details of non-essential terms in the analysis below shall be omitted.
1, and 2-adapted frame . Let ( e ′ , f ′ ) and ( e, f ) denote the 5-adapted Sp C -frames of F ′ and Frespectively, (2.3). The condition of second order deformation and the definition of 2-adapted frameimply that there exist frames such that e ′ = e , (3.4) ( e ′ , e ′ ) ≡ ( e , e ) mod e . We take this identification as the initial circuit for the algorithmic process of moving frame computation.From the general theory of moving frames, (3.4) shows that one may adapt F ′ to normalize δω = 0 , δω = 0 , (3.5) δθ = 0 , δθ = 0 , δθ = 0 . Differentiating these equations, one gets (cid:18) δω − δω δω δω δω − δω (cid:19) ∧ (cid:18) ω ω (cid:19) = 0 , (cid:18) δθ δθ δθ δθ (cid:19) ∧ (cid:18) ω ω (cid:19) = 0 . By Cartan’s lemma, there exist coefficients δs ijk , δt ijk ; i, j, k = 1 ,
2, such that δω ij − δ ij δω = δs ijk ω k , where δs ijk = δs ikj ,δθ ij = δt ijk ω k , where δt ijk fully symmetric in indices . The coefficients { δt ijk } depend on the second order jet of the immersion x ′ . By the assumption ofsecond order deformation, the cubic differentialΦ ′ = ( θ ij + δθ ij ) ( ω i + δω i )( ω j + δω j ) , = Φ + δt ijk ω i ω j ω k , by (3.5) , must be a nonzero multiple of Φ. One may thus use the group action that corresponds to δω to scaleso that(3.6) δt ijk = 0 . . On the 2-adapted frame satisfying (3.6), set δω = δs k ω k . There are 6 + 2 = 8independent coefficients in { δs ijk = δs ikj , δs k } . Differentiating δθ ij = 0 , i, j = 1 ,
2, one gets 3 linearrelations among them. By the group action that corresponds to { δω , δω , δη , δη , δη } , one maytranslate the remaining 5 coefficients so that δω = 0 , (3.7) δω ij = 0 , for i, j = 1 , . At this step, the deformation δφ is reduced to(3.8) δφ = · δω δω δη δη δη · · · δη δη δη · · · δη δη δη · · · · · ·· · · − δω · ·· · · − δω · · . Note that since all of the third order terms { δs ijk = δs ikj , δs k } are absorbed by frame adapta-tions, a second order deformation of a nondegenerate Legendrian surface is automatically a third orderdeformation , see remark below Definition 3.1. . On the 3-adapted frame satisfying (3.7), set δω i = δh ik ω k , for i = 1 , ,δη ij = δη ji = δh ijk ω k , for i, j = 1 , . There are 10 independent coefficients in { δh ij , δh ijk } . Differentiating (3.7), one gets 5 linear relationsamong them. By the group action that corresponds to { δη , δη } , one may translate δh = 0 , δh =0. The structure coefficients can be normalized accordingly so that δω = ( u + u ) ω , δω = ( u + u ) ω , (3.9) ( δη ij ) = (cid:18) u ω u ( ω + ω ) u ( ω + ω ) u ω (cid:19) , for 3 coefficients { u , u , u } .Note that the first equation of (3.9) implies that the web curvature of the asymptotic 3-web is invariantunder the second order deformation. Let us denote the covariant derivatives of { u , u i } by du = − u ω + u k ω k ,du i = − u i ω + u ik ω k , for i = 1 , . . On the 4-adapted frame satisfying (3.9), set δη i = δh i k ω k , for i = 1 , . By the group action that corresponds to { δη } , one may translate δh + δh = 0. Introduce variables { v , v , v ; w , w } , and put δη = v ω + v ω , (3.10) δη = v ω − v ω ,δη = w ω + w ω . At this step, no more frame adaptation is available. The reduction process of moving frame method stopshere. Let us denote the covariant derivatives of { v i , w i } by dv i = − v i ω + v ik ω k ,dw i = − w i ω + w ik ω k , for i = 1 , . ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 19
The normalization we chose for the 4, and 5-adapted frame implies a set of compatibility equationsamong the deformation coefficients { u , u i ; v , v i ; w i } . Differentiating { δω , δω } from (3.9), one gets u = − u − v + v , (3.11) u = − u + 2 v + v . Differentiating { δη , δη , δη } , one gets v = 12 ( − v + v ) , (3.12) u = − v ,u = − v . The identity from the exterior derivative d ( d ( u )) = 0 implies(3.13) v = 2 v + v − v + 85 ( a − a ) u . Differentiating { δη , δη } from (3.10) with these relations, one gets v = 3 v − w + 8 w − u + 165 a u − a u + 2 a u + 2 a u + 2 u u , (3.14) v = 3 v + 6 w − w − u + 125 a u − a u + 2 a u + 2 a u + 2 u u . Differentiating { δη } from (3.10), one finally gets w = w + (2 v − v + 2 b − b ) u + (2 v + 2 b ) u + ( − v − b ) u (3.15) + ( − a − a ) v + (2 a + 2 a ) v . The fundamental structure equation (3.3) for δφ is now an identity. Proposition 3.16.
Let x :
M ֒ → P be a nondegenerate Legendrian surface. Let x ′ : M ֒ → P be asecond order Legendrian deformation of x . Let π = φ + δφ be the induced Maurer-Cartan form of x ′ , (3.2) , where φ is the induced Maurer-Cartan form of x . There exists a 5-adapted frame for x ′ such thatthe coefficients of δφ satisfy the structure equations (3.8) through (3.15) . These equations furthermoreimply that; a) The structure equation for second order deformation becomes involutive after one prolongation withthe general solution depending on five arbitrary functions of 1 variable. b) The second order deformation x ′ is necessarily a third order deformation. If x ′ is a fourth orderdeformation of x , x ′ is congruent to x .Proof. a) We show that the structure equation for deformation becomes involutive after a partial pro-longation. The identities from exterior derivatives d ( d ( v )) = 0 , d ( d ( v )) = 0 determine the derivativeof v by(3.17) dv ≡ ( − w − w + 3 w ) ω + ( w − w − w ) ω , mod ω ; u , u ; u , u i , v i , w i . Note the relation du du dw dw ≡ u ·· u w w w w (cid:18) ω ω (cid:19) mod ω ; u , u i , v i , w i . (3.18)By inspection, the structure equations (3.17) and (3.18) are in involution with the last nonzero Cartancharacter s = 5.b) For the first part, see the remark at the end of . For the second part, the conditionfor the fourth order deformation implies u = u = u = 0. The compatibility equations (3.12) and (3.14) then force the remaining deformation coefficients to vanish so that δφ = 0. The rest follows from theuniqueness theorem of ODE, [Gr]. (cid:3) We shall examine the second order Legendrian deformation with the additional condition that itpreserves the fourth order differential Ψ, or that it preserves both Ψ and the fifth order differential χ .The primary object of our analysis will be to give characterization of such surfaces that support maximumparameter family of nontrivial deformations.4. Ψ -deformation In this section, we apply the fundamental structure equation for second order deformation to thegeometric situation where the deformation is required to preserve a part of fourth order invariants of aLegendrian surface.
Definition 4.1.
Let x :
M ֒ → P be a nondegenerate Legendrian surface. Let x ′ : M ֒ → P be a secondorder deformation of x . x ′ is a Ψ-deformation if the application map g : M → Sp C for the secondorder deformation x ′ is such that for each p ∈ M , the fourth order cubic differential Ψ ′ of x ′ and Ψ of g ( p ) ◦ x are isomorphic at p . As noted in Proposition 3.16, there is no local obstruction for the second order deformation of aLegendrian surface, whereas if one requires the second order deformation to preserve all of the fourthorder invariants, the deformation is necessarily a congruence. The idea is to impose a condition thatbalances between these two extremes.Let us give a summary of results in this section. The condition for a second order deformation tobe a Ψ-deformation is expressed as a pair of linear equations on the deformation coefficients, (4.3). Amore or less basic over-determined PDE analysis of these equations shows that the resulting structureequation for Ψ-deformation closes up admitting at most three parameter family of solutions, (4.7). Theclass of isothermally asymptotic surfaces with flat asymptotic web discussed in Section 2.4.1 are examplesof such surfaces admitting maximum parameter family of Ψ-deformations, which we call D -surfaces ,(4.15). Analysis of the structure equation shows that there exist another class of surfaces with finite localmoduli that admit maximum parameter family of Ψ-deformations, which we call D -surfaces , (4.18). D -surfaces and D -surfaces account for the set of maximally Ψ-deformable Legendrian surfaces, Theorem4.21. Further analysis shows that there exist subsets called S -surfaces and S -surfaces which admitΨ-deformations that also preserve the fifth order differential χ .We continue the analysis of Section 3.4.1. Structure equation.
Let x :
M ֒ → P be a nondegenerate Legendrian surface. Let x ′ : M ֒ → P be a Ψ-deformation of x. Let π = φ + δφ be the induced Maurer-Cartan form of x ′ , where φ is theinduced Maurer-Cartan form of x. From (3.9), the deformation of the invariant differentials Ψ and χ are given by δ Ψ = δ ( η ij ω i ω j ) , (4.2) = ( u + 2 u ) ( ω ) ω + ( u + 2 u ) ω ( ω ) ,δχ = δ ( η ω + η ω ) , = v ( ω ) + ( v + v ) ω ω − v ( ω ) . The condition for the deformation to preserve Ψ is expressed by the pair of linear equations(4.3) u + 2 u = 0 , u + 2 u = 0 . We wish to give an analysis of the compatibility equations for the deformation δφ derived from (4.3). ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 21
Differentiating (4.3), one gets v = − v , (4.4) u = 2 v ,u = − v . Since v = ( − v + v ), we observe that a Ψ -deformation leaves χ invariant when v = v = 0.Differentiating v + v = 0, one gets w = w + 65 ( − a + a ) u , (4.5) v = − w − u + ( a + a + 85 a − a ) u . Differentiating the first equation of (4.5) for w , one gets w = w + 65 ( a − a ) v + (9 b + 12 b − a ) u ,w = w + 4 v u + ( − a − a − a − a ) v + ( − b − b + 4 b ) u . The identity from the exterior derivative d ( d ( v )) = 0 with these relations finally gives(4.6) w = − u v + ( − a + 225 a + a + a ) v + (24 b − b + 15 b + a + a − a ) u . At this step, the remaining independent deformation coefficients are { u , v , w } . Moreover, theysatisfy a closed structure equation, i.e., their derivatives are expressed as functions of themselves and donot involve any new variables. Let us record the structure equations for { u , v , w } . du + 43 u ω = v ( − ω + ω ) , (4.7) dv + 2 v ω = ( 32 u + ( 35 a − a − a − a ) u + 12 w ) ω + ( − u + ( a + a + a ) u − w ) ω ,dw + 83 w ω = w ω + w ω , where w is in (4.6) and w = 2 u v + ( − a + 185 a − a − a ) v + (36 b − b + 26 b + a + a − a ) u . Exterior derivative d ( d ( w )) = 0 gives a universal integrability condition for Ψ-deformation;(4.8) ( − b + 8 b − b − a − a + 10 a ) v + 125 ( a − a ) w ≡ u . The full expression for the right hand side of (4.8) is given byRHS of (4.8) = − a − a ) u + ( − c + 8 c − b + 10 b − b − b + 3215 a a − a a − a + 12325 a − a a − a a + 3215 a a + 3 a a − a + 2 a − a − a + 13 a + 13 a ) u , where a ijk denote the covariant derivative of a ij as before. Proposition 4.9.
Let
M ֒ → P be a nondegenerate Legendrian surface. A Ψ -deformation of M isdetermined by three parameters { u , v , w } by (4.3) , (4.4) , and (4.5) . These three deformation pa-rameters satisfy a closed structure equation (4.7) . The structure equation, and the universal integrabilitycondition (4.8) imply that; a) A nondegenerate Legendrian surface admits at most three parameter family of Ψ -deformations. b) If the asymptotic web of M is not flat, M admits at most two parameter family of Ψ -deformations. c) Assume the asymptotic web of M is flat. If the structure coefficients of M do not satisfy the differ-ential relation ( a + a + 4 b − b ) = 0 , M admits at most one parameter family of Ψ -deformations. d) Assume the asymptotic web of M is flat and the structure coefficients satisfy the relation ( a + a + 4 b − b ) = 0 . If the structure coefficients of M do not satisfy the additional relation c = c , M does not admit nontrivial Ψ -deformations. e) A nondegenerate Legendrian surface M admits maximum three parameter family of Ψ -deformationsif, and only if M has flat asymptotic web, and the structure coefficients of M satisfy the followingrelations. a = a ,b − b = 14 ( a + a ) ,c = c . Proof. a) It follows from the uniqueness theorem of ODE, [Gr].b) If the web curvature (2.25) of the asymptotic web does not vanish identically, one can solve (4.8)for w on a dense open subset of M .c) The asymptotic web is flat when a − a = 0. By the structure equations from Section 2.2, (4.8) isreduced to v ( a + a + 4 b − b ) ≡ u . Under the assumption of c), one can solve for v as a function of u . Differentiating this, (4.7) impliesthat w is also determined as a function of u .d) and e) When a − a = 0 and a + a + 4 b − b = 0, (4.8) is reduced to u ( c − c ) = 0 . If c − c = 0, the structure equation (4.7) is compatible and admits solutions with maximum threedimensional moduli. If c − c does not vanish identically, u = 0. The structure equation (4.7) thenimplies v = w = 0. (cid:3) Example 4.10.
The analysis of Section 2 shows that the following classes of Legendrian surfaces admitmaximum three parameter family of nontrivial Ψ -deformations. a) Ψ-null surfaces, Section 2.3b) Isothermally asymptotic surfaces with flat asymptotic web, Section 2.4.1c) Tri-ruled surfaces, Section 2.5.1 Note that a) and c) are subsets of b) . It is evident that a generic nondegenerate Legendrian surface does not admit any nontrivial Ψ-deformations. In consideration of the main theme of the paper, to understand Legendrian surfaceswith special characteristics, we do not pursue to formulate the explicit criteria for Ψ-rigidity.4.2.
Surfaces with maximum ∞ Ψ -deformations. The structure of the moduli space of solutionsto the deformation equation (4.7) depends on the geometry of the base Legendrian surface. Among thevariety of cases, we consider in this subsection the class of Legendrian surfaces that admit maximum threeparameter family of Ψ-deformations. The rationale for this choice comes from the fact that Kummer’squartic surface constitutes an example of Cartan’s maximally third order deformable surfaces in P , [Fe]. ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 23
Let
M ֒ → P be a nondegenerate Legendrian surface with maximum three parameter family of Ψ-deformations. From e) of Proposition 4.9, such surfaces are characterized by the following three relationson the structure coefficients; a − a = 0 , (4.11) b − b = 14 ( a + a ) ,c − c = 0 . We wish to give an analysis of the compatibility conditions derived from these relations, and determinethe structure equation for the maximally Ψ-deformable surfaces.Since M has flat asymptotic web, let us assume the results of Section 2.2 and continue the analysisfrom that point on. Differentiating the third equation of (4.11), one gets c = c , (4.12) c = c + (4 a + 4 a ) b + ( − a + 2 a ) b . The remaining undetermined derivative coefficients at this step are { a , c ; b } . The identities fromexterior derivatives d ( d ( a )) = 0 , d ( d ( a )) = 0 determine the derivative of a by(4.13) da + 2 a ω = 2 ( b − a + c + a a )( ω + ω ) + 4 b ω . Moreover, d ( d ( a )) = 0 is an identity.Exterior derivative d ( d ( b )) = 0, (2.27), with these relations gives the universal integrability conditionto admit maximum ∞ family of Ψ-deformations.(4.14) ( a − a ) c + 2( a − a ) b + 3( − b + b )( a + 2 b ) + ( a − a )( a a − a ) = 0 . At this juncture, the analysis divides into two cases.
Case a − a = 0. This is the case of isothermally asymptotic surfaces with flat asymptotic web. Asnoted in Example 4.10, this class of surfaces satisfy the defining relations (4.11) and admit maximumthree parameter family of Ψ-deformations. Definition 4.15. A Legendrian D -surface is an immersed, nondegenerate Legendrian surface in P which is isothermally asymptotic with flat asymptotic 3-web. Let us record the full structure equation for D -surfaces. da + 43 a ω = − b ω + (4 b − b ) ω , (4.16) da + 43 a ω = ( − b + 3 b ) ω + ( − b + 3 b ) ω ,db + 2 b ω = b ( ω − ω ) ,db + 2 b ω = ( − b + a − c − a )( ω + ω ) − b ω ,dc + 83 c ω = c ( ω + ω ) + 4 b ( a + a ) ω ,db + 83 b ω = b (2 a + 3 a )( ω − ω ) . The induced Maurer-Cartan form φ takes the following form.(4.17) φ = ω ( a + a ) ω ( a + a ) ω c ( ω + ω ) b ω + b ω ( − b + b ) ω − b ω ω ω · b ω + b ω a ω a ( ω + ω ) ω · ω ( − b + b ) ω − b ω a ( ω + ω ) a ω · · · − ω − ω − ω · ω ω + ω − ( a + a ) ω − ω ·· ω + ω ω − ( a + a ) ω · − ω Case a − a = 0. The structure equation closes up in this case. First, solve (4.14) for c . Differenti-ating this, one can solve for c . At this step, the structure equation for this class of surfaces closes upwith 7 independent structure coefficients { a , a , a , b , b ; a , b } . Moreover, an analysis shows thatthe resulting structure equation is compatible, i,e,, d = 0 is an identity and does not impose any newcompatibility conditions. Definition 4.18. A Legendrian D -surface is an immersed, nondegenerate Legendrian surface in P which satisfies the following conditions.a) it is not isothermally asymptotic,b) it has flat asymptotic 3-web, and satisfies the differential relations (4.11) , (4.12) , (4.13) , and (4.14) . Let us record the full structure equation for D -surfaces. da + 43 a ω = − b ω + ( − a − b + 4 b ) ω , (4.19) da + 43 a ω = ( − b + 3 b ) ω + ( − b + 3 b ) ω ,da + 43 a ω = a ω + (4 b − b ) ω ,db + 2 b ω = b ( ω − ω ) ,db + 2 b ω = − ( b − a + c + a a )( ω + ω ) − b ω ,da + 2 a ω = 2 ( b − a + c + a a )( ω + ω ) + 4 b ω ,db + 83 b ω = −
14 ( − b a + 4 b a + a a − b a − b a + a a )( ω − ω ) . The induced Maurer-Cartan form φ takes the following form.(4.20) φ = ω ( a + a ) ω ( a + a ) ω c ( ω + ω ) b ω + b ω ( − b + b ) ω − b ω ω ω · b ω + b ω a ω a ( ω + ω ) ω · ω ( − b + b ) ω − b ω a ( ω + ω ) a ω · · · − ω − ω − ω · ω ω + ω − ( a + a ) ω − ω ·· ω + ω ω − ( a + a ) ω · − ω , where c is given by (4.14). Theorem 4.21.
The set of nondegenerate Legendrian surfaces in P which admit maximum three pa-rameter family of Ψ -deformations fall into two categories; Legendrian D -surfaces, or Legendrian D -surfaces. A general Legendrian D -surface depends on one arbitrary function of 1 variable, whereas ageneral Legendrian D -surface depends on four constants. ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 25
Proof.
The generality of solutions for the structure equation for D -surfaces is treated in Proposition2.38. For the generality of D -surfaces, consider the invariant map I = ( a , a , a , b , b ; a , b ) :F → C , where F is the canonical bundle of 5-adapted frames. Since I generically has rank 3, the localmoduli space of D -surfaces has general dimension dim ( C ) − rank( I ) = 4. (cid:3) The analogy of Theorem 4.21 with Cartan’s classification of maximally third order deformable surfacesin P is obvious, [Ca]. Cartan’s classification is also divided into two cases; one case with infinitedimensional local moduli, and the other case with finite dimensional local moduli. This analogy in a wayconversely justifies our choice of Ψ-deformations.4.3. (Ψ , χ ) -deformations. In this subsection, we examine which of the maximally Ψ-deformable sur-faces admit deformations that leave invariant both Ψ and the fifth order quadratic differential χ , (2.15). Definition 4.22.
Let x :
M ֒ → P be a nondegenerate Legendrian surface. Let x ′ : M ֒ → P be a Ψ -deformation of x . x ′ is a (Ψ , χ )-deformation if the application map g : M → Sp C for the Ψ -deformation x ′ is such that for each p ∈ M , the fifth order quadratic differentials χ ′ of x ′ and χ of g ( p ) ◦ x are isomorphic at p . Let us give a summary of results in this subsection. The condition for a Ψ-deformation to be a(Ψ , χ )-deformation is expressed by a single linear equation on the deformation coefficients, (4.23). Anover-determined PDE analysis of this equation shows that the resulting structure equation for (Ψ , χ )-deformation closes up admitting at most one parameter family of solutions, (4.24). The structure equa-tion for the subset of maximally Ψ-deformable surfaces which admit one parameter family of (Ψ , χ )-deformations is then determined, Theorem 4.34.We continue the analysis of Section 4.2, specifically from (4.14).Let x :
M ֒ → P be a nondegenerate, maximally Ψ-deformable Legendrian surface. Let x ′ : M ֒ → P be a Ψ-deformation of x. Let π = φ + δφ be the induced Maurer-Cartan form of x ′ , where φ isthe induced Maurer-Cartan form of x. From (4.2), the condition for the deformation to preserve χ isexpressed by the single linear equation(4.23) v = 0 . We wish to give an analysis of the compatibility equations for the Ψ-deformation δφ derived from (4.23).Differentiating (4.23), one gets(4.24) w = u ( − u + 2 a + 2 a + 2 a ) . Since v = 0 and w is a function of u , there exists at most one parameter family of (Ψ , χ ) -deformations .Differentiating the equation for w again, one gets the integrability equation u ( a − b + 2 b ) = 0 . If a − b + 2 b = 0, this forces u = 0 and the deformation is trivial. Hence we must have(4.25) a − b + 2 b = 0 . Remark 4.26.
A similar analysis shows that for a general nondegenerate Legendrian surface, either itadmits maximum one parameter family of (Ψ , χ ) -deformations, or it does not admit any such deforma-tions. The Legendrian surfaces which admit (Ψ , χ ) -deformations are characterized by the following tworelations on the structure coefficients; a − a = 0 ,a − a = 4 b . Successively differentiating (4.25), one gets a set of three compatibility equations. b = 0 , b = 0 , (4.27) ( a − a ) b = 0 , ( a − a )( c + a a − a ) = 0 . At this juncture, the analysis divides into two cases.
Case a − a = 0. This is a subset of D -surfaces. It is easily checked that the structure equation(4.16) remains in involution with the additional condition b = b = 0. Definition 4.28. A Legendrian S -surface is a Legendrian D -surface for which the structure coefficientssatisfy the additional relation b = b = 0 . Let us record the full structure equation for S -surfaces. da + 43 a ω = − b ( ω + ω ) , (4.29) da + 43 a ω = 3 b ( ω + ω ) ,db + 2 b ω = − ( c + a − a )( ω + ω ) ,dc + 83 c ω = c ( ω + ω ) . The induced Maurer-Cartan form φ takes the following form.(4.30) φ = ω ( a + a ) ω ( a + a ) ω c ( ω + ω ) b ω b ω ω ω · b ω a ω a ( ω + ω ) ω · ω b ω a ( ω + ω ) a ω · · · − ω − ω − ω · ω ω + ω − ( a + a ) ω − ω ·· ω + ω ω − ( a + a ) ω · − ω . Note that the subset of Ψ-null surfaces with the structure coefficient b = 0, and tri-ruled surfaces areexamples of Legendrian S -surfaces. Case a − a = 0. This is a subset of D -surfaces. It is easily checked that the structure equation(4.16) remains compatible with the additional condition b = b = a = b = 0, c = a − a a . Definition 4.31. A Legendrian S -surface is a Legendrian D -surface for which the structure coefficientssatisfy the additional relation b = b = a = b = 0 , and c = a − a a . Let us record the full structure equation for D -surfaces. da + 43 a ω = 0 , (4.32) da + 43 a ω = 0 ,da + 43 a ω = 0 . ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 27
The induced Maurer-Cartan form φ takes the following form.(4.33) φ = ω ( a + a ) ω ( a + a ) ω ( a − a a )( ω + ω ) · · ω ω · · a ω a ( ω + ω ) ω · ω · a ( ω + ω ) a ω · · · − ω − ω − ω · ω ω + ω − ( a + a ) ω − ω ·· ω + ω ω − ( a + a ) ω · − ω . Note that when a = a , the structure equation for S -surfaces degenerates to the structure equation for S -surfaces with the additional condition b = 0 , c = − a + a . Theorem 4.34.
The set of maximally Ψ -deformable Legendrian surfaces in P which admit one pa-rameter family of (Ψ , χ ) -deformations fall into two categories; Legendrian S -surfaces, or Legendrian S -surfaces. A general Legendrian S -surface depends on one arbitrary function of 1 variable, whereas ageneral Legendrian S -surface depends on two constants.Proof. The structure equation for S -surfaces is in involution with the last nonzero Cartan characters = 1(we omit the details). For S -surfaces, consider the invariant map I = ( a , a , a ) : F → C , whereF is the canonical bundle of 5-adapted frames. Since I generically has rank 1, the local moduli space of S -surfaces has general dimension dim ( C ) − rank( I ) = 2. (cid:3) Examples
In this final section, we give a differential geometric characterization of tri-ruled surfaces, Section 2.5.1,which are examples of Legendrian S -surfaces. In Section 5.1, the flat case is characterized as a part ofa Legendrian map from P blown up at three distinct collinear points. In Section 5.2, the non-flat caseis characterized as a part of a Legendrian embedding from P blown up at three non-collinear points. Inboth cases, the Legendrian map is given by a system of cubics through the three points.Let ( X , X , X , Y , Y , Y ) be the standard adapted coordinate of C such that the symplectic 2-form ̟ = dX ∧ dY + dX ∧ dY + dX ∧ dY .5.1. Flat surface.
This is the class of surface for which all the structure coefficients vanish; a i = b j = c k = 0.Since dω = 0, take a section of the frame for which ω = 0, and dω = dω = 0 consequently.Introduce a local coordinate ( x, y ) such that ω = dx, ω = dy , and express the Maurer-Cartan form φ = A dx + B dy for constant coefficient matrices
A, B . Since dφ = − φ ∧ φ = 0, A and B commute and g = exp( Ax + By ) is a solution of the defining equation g − dg = φ. The exponential can be computed, and by definition of φ in Section 2, the first column of g gives thefollowing local parametrization of the flat Legendrian surface.x ( x, y ) = X X X Y Y Y = xy − xy ( x + y ) xy + y xy + x . Theorem 5.1.
Let π : M → P be the rational surface obtained by blowing up P at three distinctcollinear points { p , p , p } . Let L be the line through p i ’s, E i = π − ( p i ) be the exceptional divisor,and let H be the linear divisor of P . Let b L be the -2-curve, the proper transform of L . There exists asix dimensional proper subspace W of the linear system [ π ∗ (3 H ) − E − E − E ] which gives a Legendrianmap b x : M → P . b x is an embedding on M − b L , and it degenerates to a point on b L . a) A flat Legendrian surface is locally equivalent to a part of b x( M − ∪ E i ) . b) The system W for b x is a six dimensional subspace of the proper transform of the set of cubicsthrough p i ’s. Each -1-curve E i is mapped to a line under b x . c) The asymptotic web is given by the proper transform of the three pencils of lines through p i ’s. Proof of theorem is presented below in four steps.
Step 1.
Consider the birational map x : P P associated to x ( x, y ) defined by(5.2) x([ x, y, z ]) = X X X Y Y Y = z xz yz − xy ( x + y )( xy + y ) z ( xy + x ) z , where [ x, y, z ] is the standard projective coordinate of P . It is undefined at three points p = [1 , , ,p = [0 , , ,p = [1 , − , . At p , introduce the parametrization of the blow up by [ x, y, z ] = [1 , λ z, z ] for the blow up parameter λ . The birational map becomes(5.3) x([ x, y, z ]) = X X X Y Y Y = z z λ z − λ z (1 + λ z )( λ + λ z ) z ( λ z + ) z = z zλ z − λ (1 + λ z )( λ + λ z ) z ( λ z + ) → − λ , as z → . Similar formulae for p , p show that the exceptional divisors E , E are mapped to(5.4) E → − λ , E → λ − − . Let L = { z = 0 } ⊂ P be the line through p i ’s. By definition, x( L ) = [0 , , , , ,
0] = x ∈ P . Onemay check that x : P − L → P is an embedding, and that the image x( P − L ) is disjoint from theexceptional loci (5.3), (5.4).A computation with (5.3) at p , and similar computations at p , p show that the associated lift b x : M → P is well defined and holomorphic, and that b x : M − b L → P is a smooth embedding. ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 29
Step 2.
Consider alternatively the following polynomial equations satisfied by x.3 X Y + X Y + X Y = 0 ,X Y + 12 X X ( X + X ) = 0 ,X Y − X X − X = 0 ,X Y − X X − X = 0 , (cid:18) X ( Y − Y ) − X ( Y − Y ) (cid:19) Y = Y Y ( Y − Y ) . By a direct computation, one can verify that this set of equations have rank 3 on b x( M ) = x( P − L ),except at x . One can also check that at x , b x( M ) is not smooth and has a second order branch typesingularity(we omit the details. Note that b L is a -2-curve and it cannot be blown down). Step 3.
Let D the hyperplane section D = b x − ( { Y = 0 } ). From (5.2), the divisor consists of theproper transform of three lines { L , L , L } = { y = 0 , or x = 0 , or x + y = 0 } ⊂ P . Hence the linearsystem [ D ] = [ X i ( π ∗ ( H ) − E i )] = [ π ∗ (3 H ) − E − E − E ] . By definition, W is a subspace of the linear system [ π ∗ (3 H ) − E − E − E ], the proper transform ofcubics through p i ’s. Finally, h D, E i i = 1, and each E i is mapped to a line. Step 4.
One may check by direct computation that the asymptotic web is given by the foliations dy = 0 , dx = 0 , dx + dy = 0, which represent three pencils of lines through p , p , p respectively.5.1.1. Generalization.
The construction of flat surface admits a straightforward generalization.Let f k ( x, y ) be a homogeneous polynomial of degree k for k = 3 , , ... m , such that the top degree f m ( x, y ) has no multiple factors (product of m mutually non-proportional linear functions in x, y ).Consider the associated birational map x : P P defined byx([ x, y, z ]) = X X X Y Y Y = z m xz m − yz m − − P mk =3 f k z m − k P mk =3 1 k − ( ∂∂x f k ) z m − k +1 P mk =3 1 k − ( ∂∂y f k ) z m − k +1 . A direct computation shows that x is Legendrian.x is undefined at m points { z = 0 , f m ( x, y ) = 0 } ⊂ P . Let π : M m → P be the rational surfaceobtained by blowing up P at these points. Let E i = π − ( p i ) , i = 1 , , ..m , be the exceptional divisor,and let b L be the proper transform of the line { z = 0 } . An analysis similar as above shows that x admitsa well defined smooth lift b x : M m → P . But when m ≥
4, the singular locus of b x consists of b L , and onepoint from each E i − b L .This class of Legendrian surfaces were first introduced in [Bu1].5.2. Tri-ruled surface.
This is the class of surface with the induced Maurer-Cartan form (2.41). Sincethe flat case is already treated, we examine the case a = 0.From (2.42), one may scale a = 1. Then ω = 0, and dω = dω = 0 consequently. Introduce alocal coordinate ( s, t ) such that ω = i √ ds, ω = i √ dt , where i = −
1, and express the Maurer-Cartanform φ = A ds + B dt for constant coefficient matrices
A, B . As in Section 5.1, the equation g − dg = φ can be integrated and one gets the following local parametrization of a tri-ruled Legendrian surface upto conformal symplectic transformation.x ( s, t ) = X X X Y Y Y = sin( s + t )sin( s )sin( t ) − cos( s + t )cos( s )cos( t ) . By a conformal symplectic transformation, we mean a linear transformation of C that preserves thesymplectic form up to nonzero scale, e.g., a linear transformation ( X, Y ) → ( l X, l Y ) for nonzero l , l .An analysis shows that this local parametrization gives rise to a Legendrian embedding of P × P = Q × Q ⊂ P × P , the product of two conics, blown up at two points. Since this surface is isomorphic to P blown up at three non-collinear points, consider the associated Legendrian birational map x : P P defined by(5.5) x([ x, y, z ]) = X X X Y Y Y = ( x − y ) z ( y − z ) x ( z − x ) y ( x + y ) z ( y + z ) x ( z + x ) y , where [ x, y, z ] is the standard projective coordinate of P , Lemma 5.12. It is undefined at three points p = [1 , , , (5.6) p = [0 , , ,p = [0 , , . Theorem 5.7.
Let π : N → P be the rational surface obtained by blowing up P at three non-collinearpoints { p , p , p } . Let L k be the line through ( p i , p j ) , ( ijk ) = (123) , E i = π − ( p i ) be the exceptionaldivisor, and let H be the linear divisor of P . Let b L k be the proper transform of L k . There exists a sixdimensional proper subspace W of the linear system [ π ∗ (3 H ) − E − E − E ] which gives a Legendrianembedding b x : N ֒ → P . a) A non-flat tri-ruled Legendrian surface is locally equivalent to a part of b x( N − ∪ E i ∪ b L i ) . b) The system W for b x is a six dimensional subspace of the proper transform of the set of cubicsthrough p i ’s. Each of the six -1-curves E i and b L k is mapped to a line under the embedding. c) The asymptotic web is given by the proper transform of the three pencils of lines through p i ’s. Proof of theorem is presented below in four steps.
Step 1. b x is an immersion:By a direct computation, it is verified that x is an immersion on P − ∪ p i .At p , introduce the parametrization of the blow up by [ x, y, z ] = [1 , λ z, z ] for the blow up parameter λ . The birational map becomes(5.8) x([ x, y, z ]) = X X X Y Y Y = ( x − y ) z ( y − z ) x ( z − x ) y ( x + y ) z ( y + z ) x ( z + x ) y = (1 − λ z )( λ − z ( z − λ (1 + λ z )( λ + 1) z ( z + 1) λ → − λ λ , as z → . ROJECTIVELY DEFORMABLE LEGENDRIAN SURFACES 31
Similar computations for p , p show that x admits a lift b x : N → P such that b x( N ) = x( P − ∪ p i ).The exceptional divisors are respectively mapped to E → − λ λ , E → − λ λ , for blow up parameters λ , λ .From (5.8), one may check that the three vectors x , ∂ x ∂z , ∂ x ∂λ are independent at z = 0. Similarcomputations for p , p show that b x is an immersion on the exceptional divisors E i . Hence b x is animmersion on N . Step 2. b x is injective:It is clear that b x is injective on ∪ E i , and that b x( ∪ E i ) is disjoint from x( P − ∪ p i ). It suffices to showthat x is injective on P − ∪ p i .Suppose x([ x, y, z ]) = x([ x ′ , y ′ , z ′ ]). Then x y = µ x ′ y ′ , x z = µ x ′ z ′ ,y z = µ y ′ z ′ , y x = µ y ′ x ′ ,z x = µ z ′ x ′ , z y = µ z ′ y ′ , for a nonzero scaling parameter µ .Case x = 0; y, z = 0. Then x ′ y ′ = x ′ z ′ = 0. If x ′ = 0, then y ′ z ′ = 0 = yz , a contradiction. Hence x ′ = 0. The remaining equations then show that yz = y ′ z ′ .Case x, y, z = 0. One has xy = x ′ y ′ , yz = y ′ z ′ , zx = z ′ x ′ . Hence [ x, y, z ] = [ x ′ , y ′ , z ′ ]. Step 3.
Let D the hyperplane section D = b x − ( { X + X + X = 0 } ). From (5.5), X + X + X =( x − y )( y − z )( z − x ). Hence the linear system[ D ] = [ X i ( π ∗ ( H ) − E i )] = [ π ∗ (3 H ) − E − E − E ] .W is a subspace of the linear system of the proper transform of cubics through p i ’s, and b x is not normal.Since [ b L k ] = [ π ∗ ( H ) − E i − E j ] , ( ijk ) = (123), one has h D, E i i = h D, b L k i = 1, and each E i and b L k is mapped to a line. Step 4.
The equations for the asymptotic web can be checked on the affine chart [ x, y,
1] by a directcomputation. We omit the details. Let C be a line on P that passes through exactly one of p i ’s. Theproper transform b C of C has the divisor class π ∗ ( H ) − E i . Hence h D, b C i = 2, and each leaf of theasymptotic foliations is mapped to a linear P ⊂ P which is necessarily Legendrian from the definingproperties of the tri-ruled surfaces. (cid:3) Remark 5.9.
The linear system of conics through three non-collinear points gives the classical quadratictransformation of P . As the three points degenerate to become collinear, b x( N ) degenerates to the flat surface in Section5.1. The isolated singularity of the flat Legendrian surface thus admits a smoothing.It is not known if every del Pezzo surface admits a Legendrian embedding. Legendrian embeddings ofa set of degree 4 del Pezzo surfaces were constructed in [Bu2].Note the algebraic equations satisfied by b x(this is not a complete intersection). X − Y = X − Y = X − Y , (5.10) X ( X X + Y Y ) + Y ( X Y + Y X ) = 0 . Legendrian surface b x( N ) can thus be considered as a complexification of the homogeneous special Leg-endrian torus with parallel second fundamental form, [HL].5.2.1. Generalization.
The construction of tri-ruled surface admits a straightforward generalization.Let m, n be a pair of positive integers. Let f m , g m be the homogeneous polynomials of degree m oftwo variables which represent sin( ms ) , cos( ms );sin( ms ) = f m (sin( s ) , cos( s )) = X ≤ k ≤ m − ( − k (cid:18) m k + 1 (cid:19) sin k +1 ( s ) cos m − (2 k +1) ( s ) , cos( ms ) = g m (sin( s ) , cos( s )) = X ≤ k ≤ m ( − k (cid:18) m k (cid:19) sin k ( s ) cos m − k ( s ) . Consider the following local parametrization of a Legendrian surface.(5.11) x ( s, t ) = X X X Y Y Y = sin( ms + nt ) √ m sin( s ) √ n sin( t ) − cos( ms + nt ) √ m cos( s ) √ n cos( t ) = f m ( u , u ) g n ( v , v ) + g m ( u , u ) f n ( v , v )( − i) m + n − √ m u u m − v n ( − i) m + n − √ n v u m v n − f m ( u , u ) f n ( v , v ) − g m ( u , u ) g n ( v , v )( − i) m + n − √ m u u m − v n ( − i) m + n − √ n v u m v n − , where u + u + u = 0 , v + v + v = 0. Lemma 5.12.
The local parametrization (5.11) is equivalent to the following Legendrian birational map x : P P up to conformal symplectic transformation. (5.13) x([ x, y, z ]) = X X X Y Y Y = x m z − y n z m − n +1 √ m ( z − x ) x m − y n z m − n √ n ( y − z ) x m y n − z m − n x m z + y n z m − n +1 √ m ( z + x ) x m − y n z m − n √ n ( y + z ) x m y n − z m − n . Proof.
Let [ x, y, z ] be the homogeneous coordinate of P . Take the following birational map ϕ : P Q × Q ⊂ P × P ; ϕ ([ x, y, z ]) = ([ u , u , u ] , [ v , v , v ]) , = ([2 xz, z − x , i( z + x )] , [2 yz, − z + y , i( z + y )]) . Lemma follows from de Moivre’s formula,( u ± i u ) m ( v ± i v ) n = ( g m ( u , u ) ± i f m ( u , u ))( g n ( v , v ) ± i f n ( v , v )) . (cid:3) Consider the case m = n . (5.13) is undefined at the three points of (5.6). Let b x : N ֒ → P be theinduced lift. It is never an immersion except when ( m, n ) = (1 , m ≥
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