Rigidity properties of holomorphic Legendrian singularities

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold.


Introduction
Singularities of Legendrian varieties in contact manifolds have been studied in singularity theory or symplectic/contact geometry, often in differentiable or real analytic categories. In this article, we study them in the complex-analytic category. Methods of algebraic geometry can be applied more efficiently to holomorphic Legendrian singularities. Using this approach, we present two rigidity results on Legendrian singularities. Let us start with precise definitions of the terms we use. is called a Legendrian singularity. Two Legendrian singularities x 1 ∈ Z 1 ⊂ (M 1 , D 1 ) and x 2 ∈ Z 2 ⊂ (M 2 , D 2 ), where Z i is a Legendrian subvariety in a contact manifold (M i , D i ), i = 1, 2, are contactomorphic, if there exist open neighborhoods U i ⊂ M i of x i , i = 1, 2, and a contactomorphism ϕ : U 1 → U 2 such that ϕ(x 1 ) = x 2 and ϕ(Z 1 ∩ U 1 ) = Z 2 ∩ U 2 .
Are there many interesting examples of Legendrian singularities? The following construction provides lots of them. Example 1.3. For a complex manifold X, the projectivized cotangent bundle M = PT * X has a natural contact structure (see e.g. Example 1.2 B of [AG] or Example 2.2 of [LB2]). For any complex analytic subvariety Y ⊂ X, its conormal variety Z Y ⊂ M, the closure of the projectivized conormal bundle of the smooth locus of Y , is a Legendrian subvariety. The conormal variety Z Y is usually (but not always) singular when Y is singular. When Y ⊂ X is a hypersurface, the conormal variety Z Y is the Nash blowup of Y .
In Section 3, we give another class of examples of Legendrian singularities, those arising from Lagrangian cones.
Our first rigidity result is in terms of tangent cones. Recall (see Chapter 3, Section 3 of [Mu]) that for an analytic subvariety Z of a complex manifold M and a point x ∈ Z, if I ⊂ O M,x is the ideal of the germ of Z at x, then the tangent cone T C Z,x is the subscheme of the Zariski tangent space T Z,x defined by the ideal generated by lowest-order terms of the Taylor expansions of elements of I at x. Roughly speaking, the tangent cone of a singular variety is the lowest order approximation of the singularity. It seldom determines the singularity. Remarkably, a Legendrian singularity is determined by the tangent cone, if the tangent cone is reduced.
Theorem 1.4. A Legendrian singularity is contactomorphic to the germ at the origin of its tangent cone if the tangent cone is reduced. More precisely, a Legendrian singularity is a Lagrangian cone singularity (in the sense of Definition 3.4) if and only if its tangent cone is reduced. This is proved in Section 4. Of course, the reducedness of the tangent cone is a strong requirement. There are many examples of Legendrian singularities with non-reduced tangent cones: for instance, cuspidal Legendrian curves discussed in Section 4 of [Zh]. One motivation for Theorem 1.4 comes from the study of Fano contact manifolds. In his investigation [Ke] of Fano contact manifolds, Kebekus studied a certain Legendrian singularity x ∈ Z (Z = locus(H x ) in the notation of [Ke]). He showed that the projectivized tangent cone PT C Z,x is nonsingular and asserted that the singularity z ∈ Z is biholomorphic to the germ of a Lagrangian cone at 0. We believe that the latter assertion, if it is true, would have significant consequences in the study of Fano contact manifolds. But its proof given in Section 6.1 of [Ke] had a gap. Theorem 1.4 has grown out of our attempt to remedy this gap. But it is not strong enough to fix it, as the smoothness of PT C Z,x does not imply that T C Z,x is reduced. A technical difficulty here arises from the fact that Z is (a priori) not normal.
In fact, Legendrian singularities are usually not normal and their normalizations cannot be realized as Legendrian singularities. This can be seen from the following result on the deformation-rigidity of normal Legendrian singularities, which is proved in Section 5. Theorem 1.5. Let ∆ be a neighborhood of the origin 0 in C. Let (M, D) be a contact manifold and consider {Z t ⊂ M, t ∈ ∆} a holomorphic family of Legendrian subvarieties parametrized by ∆. Assume that Z t is normal for every t ∈ ∆. Then for any x 0 ∈ Z 0 , there exist a neighborhood 0 ∈ ∆ ⊂ ∆ and a holomorphic arc Theorem 1.5 suggests that it might be possible to classify normal Legendrian singularities to some extent. In fact, normal Legendrian singularities are not easy to find. Some normal Legendrian singularities are described in Example 3.8.
Normal Legendrian singularities are interesting from another viewpoint. The following theorem says that a Legendrian singularity, unless it is nonsingular, has nonzero torsion differentials. This is a special case of a stronger result, Theorem 2.5 in [Zh]. For the reader's convenience, we give an elementary proof (different from the one in [Zh]) at the end of Section 2. Theorem 1.6. For a Legendrian singularity x ∈ Z ⊂ (M, D), let θ be a germ of 1-form at x ∈ M defining D. Then x is a nonsingular point of Z if and only if θ| Z is zero in the space of Kähler differentials Ω Z,x .
By Theorem 1.6, normal Legendrian singularities provide examples of normal singularities with explicit nonzero torsion differentials. We mention that some examples of normal singularities with nonzero torsion differentials were given in [GR] by cohomological methods. One of their examples, the cone over the twisted cubic curve (d = 3 in Proposition 4.1 of [GR]), is a Legendrian singularity in Example 3.8.
Then θ = 0 defines a contact structure on C 2m+1 , which we call the standard contact structure.
By Darboux theorem (Chapter 4, Section 1.1 of [AG]), any contact structure is locally equivalent to the standard contact structure. Thus when studying a Legendrian singularity x ∈ Z ⊂ (M, D), we may assume that M is a neighborhood of C 2m+1 and D is the standard contact structure. We remark that in many references (like [AG] or [Kb]) the form m i=1 x i dx m+i − dx 2m+1 is used as the standard form. When algebrogeometric tools are used, however, our choice θ is more convenient because it is the expression of a contact structure on P 2m+1 in affine coordinates.
The following is a standard result in contact geometry. It is essentially given in p. 79 of [AG] or pp. 30-31 of [Kb]. As our standard form θ is slightly different from theirs, we recall the proof for readers' convenience.
Theorem 2.2. In Notation 2.1, let U be a neighborhood of 0 ∈ C 2m+1 . For a holomorphic function f on U , let F ⊂ U be the hypersurface defined by f = 0 and let v f be the holomorphic vector field on U defined by .
(iii) v f is zero at a point y ∈ F if and only if F is singular at y or θ(T F,y ) = 0; (iv) v f is tangent to the hypersurface F; (vi) for any nonsingular point y ∈ F and any tangent vector w ∈ T F,y satisfying θ(w) = 0, we have dθ(v f (y), w) = 0 ; and (vii) v f is tangent to any Legendrian subvariety Z ⊂ U contained in the hypersurface F.
Proof. (i), (ii), (iii) can be checked by straightforward calculation. (iv) is immediate from (ii). (v) can be checked from Cartan formula, and dθ = 2 m k=1 dx m+k ∧ dx k . (vi) follows from Cartan formula again: It remains to prove (vii). Pick a nonsingular point z ∈ Z. The 2-form dθ induces a nondegenerate 2-form on the vector space D z by the definition of the contact structure. The tangent space T Z,z is an isotropic subspace of D z with respect to this 2-form dθ| D z and it is contained in T F,z from Z ⊂ F. By (vi), the vector v f (z) ∈ D z satisfies dθ(v f (z), T Z,z ) = 0. Thus the linear span v f (z), T Z,z is a subspace of the 2m-dimensional vector space D z and is isotropic with respect to the nondegenerate 2-form dθ| D z . Since To see the geometric meaning of Theorem 2.2 (vi), it is convenient to recall the notion of Cauchy characteristic of a distribution. Definition 2.3. Let D ⊂ T X be a vector subbundle of corank 1 on a complex manifold and denote by σ : ∧ 2 D → T X /D the Frobenius bracket tensor. For each x ∈ X, the Cauchy characteristic of D at x is In particular, the subbundle D is a contact structure if and only if Ch(D) x = 0 for each x ∈ X.
The following is standard. (1) is straightforward to check and (2) is a special case of Theorem 2.2 in Chapter 2 of [BCG].

Lemma 2.4. Let (M, D) be a contact manifold and let
In particular, the Cauchy characteristic Ch(D) determines a foliation of rank 1 on X o .
(2) For each x ∈ X o , choose a neighborhood O ⊂ X o of x equipped with a holomorphic submersion ψ : O → B whose fibers are leaves of the foliation in (1). Then there exists a contact structure D on B such that Theorem 2.2 (vi) says that the leaves of v f are the foliation given by Lemma 2.4 applied to the nonsingular locus X of the hypersurface F. This is used to prove the next proposition, which is a direct translation of Proposition 1 in [Gi] in symplectic geometry into the setting of contact geometry.
(3) a submersion ψ : F → M whose fibers are leaves of the vector field v f in the sense of Theorem 2.2, such that is a Legendrian subvariety of (M , D ) and Z = ψ −1 (ψ(Z)).

Proof.
The assumption D 0 T Z,0 implies the existence of f and F in (1). Using the vector field v f of Theorem 2.2, we obtain a submersion ψ : F → M whose fibers are leaves of v f . By (vi) of Theorem 2.2 and Lemma 2.4, there exists a contact structure D on M satisfying (a). It is clear from (vii) of Theorem 2.2 that Z ∩ U = ψ −1 (ψ(Z ∩ U )) and ψ(Z ∩ U ) is a Legendrian subvariety of (M , D ).
By Proposition 2.5, the proof of Theorem 1.6 is reduced to the next theorem.
Theorem 2.6. Let 0 ∈ Z ⊂ C 2m+1 be the germ of a Legendrian subvariety with respect to the standard contact structure such that D 0 ⊂ T Z,0 . Then the differential θ| Z ∈ Ω Z,0 is not zero.
Proof. We define a weight function wt on O C 2m+1 ,0 and Ω C 2m+1 ,0 in the following way. Set Define the weight wt(f ) of a function f ∈ O C 2m+1 ,0 as the weight of the monomial of lowest weight in the Taylor series of f at 0 and define the weight of elements of The space of Kähler differentials of Z at 0 is given by (see e.g. Definition 1.109 of [GLS]) The condition D 0 ⊂ T Z,0 implies that all elements of I and dI have weight at least 2. Thus all elements of I · Ω C 2m+1 ,0 have weight at least 3. Since wt(θ) = 2, the lowest order term of θ must be the lowest order term of some element in O C 2m+1 ,0 · dI. To have weight 2, the lowest order term must be d-exact. But θ is homogeneous and not d-exact. A contradiction.

Lagrangian cones as Legendrian varieties
We use the following terms regarding cones.
Definition 3.1. Let V be a complex vector space, which we regard as an affine space, and let Sym • V * be the ring of polynomial functions on V . An affine cone in V is a subscheme Y of the affine space V defined by a homogeneous ideal I ⊂ Sym • V * . The corresponding projective subscheme PY ⊂ PV will be called the projectivization of Y . If the subscheme Y is reduced, i.e., the ideal I is radical ( I = √ I ), we will call it a reduced affine cone. If Y is reduced, then so is its projectivization PY ⊂ PV . But the converse is not always true.
Definition 3.2. Let (V , ω) be a symplectic vector space, i.e., a vector space V equipped with a nondegenerate anti-symmetric 2-form ω ∈ ∧ 2 V * . Let 2m be the dimension of V . An m-dimensional reduced affine cone 0 ∈ Y ⊂ V is called a Lagrangian cone if the restriction of ω to the nonsingular locus of Y is zero. Lemma 3.3. In Notation 2.1, let C 2m be the hyperplane defined by (x 2m+1 = 0) and equipped with the symplectic form Let Z ⊂ C 2m be an m-dimensional subvariety. When regarded as a subvariety of C 2m+1 equipped with the standard contact structure, the variety Z is a Legendrian subvariety if and only if Z is a Lagrangian cone in (C 2m , dθ| C 2m ) in the sense of Definition 3.2.
Proof. We use the radial vector field on C 2m ).
In terms of Theorem 2.2, the radial vector field R is the restriction of −2v x 2m+1 to C 2m . Assume that Z ⊂ C 2m is a Lagrangian cone. As it is an affine cone, the radial vector field R is tangent to the smooth locus of Z. It is straightforward to check that the contraction R dθ| C 2m is a constant multiple of the 1-form θ| C 2m . It follows that the restriction of θ to the smooth locus of Z is zero. Thus Z is Legendrian in C 2m+1 with respect to θ.
Conversely, if a subvariety Z ⊂ C 2m is a Legendrian subvariety of C 2m+1 with respect to θ, then the radial vector field R = −2v x 2m+1 | C 2m is tangent to Z by Theorem 2.2 (vii). Thus Z is a reduced affine cone. Since dθ vanishes on the smooth locus of Z, it is a Lagrangian cone with respect to dθ| C 2m .
Definition 3.4. We say that a Legendrian singularity x ∈ Z ⊂ (M, D) is a Lagrangian cone singularity, if it is contactomorphic to the germ at 0 of a Lagrangian cone in (C 2m , dθ| C 2m ) regarded as a Legendrian subvariety of (C 2m+1 , θ) as in Lemma 3.3.
We skip the proof of the following elementary lemma. There is another way that Lagrangian cones in (V , ω) give rise to Legendrian subvarieties of a contact manifold. The symplectic form ω provides the projective space PV with the following contact structure (Chapter 4, Section 1.2, Example A in [AG], Example 2.1 in [LB2], Section E.1 in [Bu2]).
Then the subbundle D ω ⊂ T PV is a contact structure on PV .
Proposition 3.7. In Definition 3.6, for a reduced affine cone Y ⊂ V , its projectivization PY ⊂ PV is a Legendrian subvariety with respect to D ω if and only if Y is Lagrangian with respect to ω.
Example 3.8. Subadjoint varieties (see Theorem 11 in [LM]) are Legendrian subvarieties of (PV , D ω ) that are homogeneous under the action of the symplectic automorphisms of (V , ω). There exists one subadjoint variety corresponding to each complex simple Lie algebra, as listed in Table 1 of [Bu]. For example, the twisted cubic curve in P 3 is the subadjoint variety corresponding to the simple Lie algebra of type G 2 . As subadjoint varieties are projectively normal, their affine cones become normal Legendrian subvarieties.
Example 3.9. Landsberg-Manivel [LM] and Buczynski (Chapters G, H, I of [Bu2]) have discovered many examples of nonsingular Legendrian subvarieties in PV , different from the subadjoint varieties of Example 3.8. They are not projectively normal, so the Legendrian singularities of their affine cones are not normal.
For later use, we recall the following proposition, which is just a reformulation of Corollary 5.5 and Lemma 5.6 of [Bu]. Proof. The symplectic form ω gives an isomorphism such that for a symmetric bilinear form Q ∈ Sym 2 V * , the endomorphism A : , u) for all u ∈ V . Corollary 5.5 and Lemma 5.6 of [Bu] say that ω identifies quadrics vanishing on Y with elements of sp(V ) which are tangent to Y . Our homomorphism ω † is, up to a scalar multiple, just an expression of ω in terms of linear vector fields on V , thus the proposition follows.

Legendrian singularities with reduced tangent cones
In this section, we prove Theorem 1.4. The following local result in contact geometry is a key step of the proof. To prove Theorem 4.1, we use the following two classical results. The first one is Poincaré's result on the normal forms of holomorphic vector fields (see Ch.4, Sec. 2.1 in [AI]) and the second one is Arnold-Givental's relative Darboux theorem (see Ch. 4, Section 1.3, Theorem A in [AG] or Theorem 1.1 in [Zh]).

Theorem 4.2. Let v be a germ of holomorphic vector fields at the origin in
Viewing h| H as a function h(z 1 , . . . , z 2m ) and using (4.1), the restriction of 2v f to the hypersurface H becomes Since h ∈ m 3 0 , the linear part of 2v f | H has eigenvalues λ 1 = · · · = λ 2m = 1. By Theorem 4.2, we can find a new coordinate system (w 1 , . . . , w 2m ) on a neighborhood of 0 in H such that up to replacing H by a smaller open subset, ).
In particular, the vector field v f does not vanish on H \ 0. By Theorem 2.2 (iii), this implies that the restriction D| H\0 is a vector subbundle of T H\0 with rank 2m − 1. Let X be the germ at 0 of the hyperplane x 2m+1 = 0 in C 2m+1 . Consider the biholomorphic map ϕ from X to H defined by ϕ * w i = x i , 1 ≤ i ≤ 2m. Then ϕ sends the vector field to the vector field 2v f | H . Let α : Bl 0 (X) → X and β : Bl 0 (H) → H be the blowups of the hypersurfaces at 0. We have submersions α : Bl 0 (X) → PD 0 and β : Bl 0 (H) → PD 0 whose fibers are the leaves of the radial vector fields v −x 2m+1 | X and v f | H , respectively. By Theorem 2.2 (vi) and Lemma 2.4, the distribution D| X\0 descends by α to a contact structure D X on PD 0 and the distribution D| H\0 descends by β to a contact structure D H on PD 0 . Recall that any two contact structures on P 2m−1 are related by a projective linear transformation (Proposition 2.3 in [LB2]). Thus by a linear coordinate change of w 1 , . . . , w 2m , we may assume that D X = D H , i.e., the biholomorphism ϕ o := ϕ X\0 sends D| X\0 to D| H\0 . Then ϕ * o (θ| H\0 ) = g · θ| X\0 for some nowhere-vanishing holomorphic function g on X \ 0. By Hartogs extension, we can assume that g is a nowhere-vanishing holomorphic function on F such that So the condition of Theorem 4.3 is satisfied and the germs of X and H at 0 are contactomorphic.
To prove Theorem 1.4, we need the following two propositions.
Proposition 4.4. Let 0 ∈ Z ⊂ (C 2m+1 , D = (θ = 0)) be a Legendrian singularity whose projectivized tangent Proof. Recall that the projectivized tangent cone PT C Z,0 is the exceptional divisor of the blowup of Z at 0 (see e.g. [Mu] Ch.3, Sec.3). Thus we can realize each point of PT C Z,0 as the limit of the tangent lines to an arc {x t ∈ Z, t ∈ ∆}, where ∆ ⊂ C is an open neighborhood of 0 ∈ C, such that x 0 = 0 ∈ Z and x t is a nonsingular point of Z if t ∈ ∆ \ {0} (see Exercise 20.3 in [Ha]). Since Z is Legendrian, T Z,x t ⊂ D x t for all t 0. It follows that the limit of the tangent lines to the arc at t = 0 is contained in PD 0 . Thus each (closed) point of PT C Z,0 is contained in PD 0 . As PT C Z,0 is reduced, this implies the proposition.
Proposition 4.5. Let 0 ∈ Z ⊂ (C 2m+1 , D = (θ = 0)) be a Legendrian singularity whose tangent cone T C Z,0 ⊂ T C 2m+1 ,0 is reduced. Then the tangent cone T C Z,0 is contained in D 0 ⊂ T C 2m+1 ,0 and is a Lagrangian cone with respect to the symplectic form ω = dθ| D 0 . Consequently, the projectivized tangent cone PT C Z,0 is a Legendrian subvariety of PD 0 with respect to D ω .
Proof. Let us regard all tangent spaces of Z ⊂ C 2m+1 as affine subspaces in C 2m+1 .
Since T C Z,0 is reduced, it is contained in D 0 from Proposition 4.4. To show that it is a Lagrangian cone, denote by σ x := dθ| D x the symplectic form on D x ⊂ T C 2m+1 ,x for each x ∈ C 2m+1 . To show that T C Z,0 is a Lagrangian cone, it suffices by Proposition 3.7 to show that σ 0 (u, v) = 0 for a general point u ∈ T C Z,0 and any vector v ∈ T T C Z,0 ,u ⊂ D 0 , v Cu.
Note that P(Cu + Cv) is a tangent line to PT C Z,0 ⊂ PD 0 at the nonsingular point [u] ∈ PT C Z,0 . Let β : Bl 0 (Z) → Z be the blowup of Z at 0. Identify PT C Z,0 with the exceptional divisor of β (see e.g. [Mu] Ch.3, Sec.3). The assumption that PT C Z,0 is reduced implies that Bl 0 (Z) is nonsingular at the point [u] ∈ PT C Z,0 . Thus we can find an arc {x t ∈ Z, t ∈ ∆} for a neighborhood ∆ ⊂ C of 0 ∈ C such that (2) x t is a nonsingular point of Z if t 0; and (3) the derivative ∂ ∂t | t=0 x t ∈ Cu. The vector v gives a tangent vector v 0 ∈ T PT C Z,0 , [u] ⊂ T Bl 0 (Z), [u] . Since both PT C Z,0 and Bl 0 (Z) are nonsingular at [u], we can find a holomorphic family of tangent vectors be the corresponding tangent vector to Z. When s 0, the plane is tangent to the smooth locus of Z ⊂ C 2m+1 . Thus σ x s ( ∂ ∂t | t=s x t , v s ) = 0 for all s 0 because Z is Legendrian. Then by continuity, we obtain σ 0 (u, v) = 0.
Proof of Theorem 1.4. It is immediate that the tangent cone of a Lagrangian cone Y ⊂ C 2m at 0 (in Definition 3.2) is isomorphic to itself. In particular, the tangent cone of a Lagrangian cone at 0 is reduced. So one direction of Theorem 1.4 is trivial.
To prove the other direction, let us use the notation of Proposition 4.5. We are to show that the germ 0 ∈ Z ⊂ C 2m+1 is a Lagrangian cone singularity in the sense of Definition 3.4, assuming that its tangent cone T C Z,0 is reduced.
To start with, we can assume that the tangent cone T C Z,0 spans D 0 . For otherwise, the Zariski tangent space T Z,0 , which is the linear span of the tangent cone, does not contain D 0 . Then we can choose a nonsingular hypersurface F containing Z such that T F,0 D 0 and apply Proposition 2.5 to obtain a a submersion ψ : F → M to a manifold of dimension 2m − 1 with a contact structure D such that ψ(Z) is a Legendrian subvariety of (M , D ) and Z = ψ −1 (ψ(Z)). This implies that the tangent cone of ψ(Z) at ψ(0) is reduced. Thus by induction, we can assume that ψ(0) ∈ ψ(Z) is a Lagrangian cone singularity. Thus by Lemma 3.5, the Legendrian singularity 0 ∈ Z is a Lagrangian cone singularity. Thus from now, we assume that T C Z,0 spans D 0 . By Proposition 4.5, the germ of 0 ∈ Z is contained in a germ of a nonsingular hypersurface Γ with T Γ ,0 = D 0 . We can view Γ as the graph of an element of O C 2m ,0 , where C 2m = (x 2m+1 = 0). In other words, in the notation of Theorem 4.1, there exists a homogeneous quadratic polynomial q(x 1 , . . . , x 2m ) and a holomorphic function h(x 1 , . . . , x 2m ) ∈ m 3 0 such that is the defining equation of Γ . Set In terms of the coordinates (z 1 , . . . , z 2m ) on Γ defined by z i = x i | Γ , 1 ≤ i ≤ 2m, the same computation as in the proof of Theorem 4.1 gives Let Bl 0 (Γ ) be the blowup of Γ at 0. Since v f | Γ vanishes at 0, it induces a vector field v on the blowup Bl 0 (Γ ). The first line in the expression of 2v f | Γ induces a vector field on Bl 0 (Γ ) that vanishes on the exceptional divisor. In fact, the vector field on the blowup induced by z k ∂ ∂z k + z m+k ∂ ∂z m+k obviously vanishes on the exceptional divisor. The restriction of v to the exceptional divisor PT Γ ,0 ⊂ Bl 0 (Γ ) comes thus from the second line in the expression of 2v f | Γ , which is precisely ω † (q) in the notation of Proposition 3.10. Since v f is tangent to Z by Theorem 2.2, we know that v is tangent to Bl 0 (Z) ⊂ Bl 0 (Γ ) and also tangent to PT C Z,0 ⊂ PT Γ ,0 = PD 0 . Thus by Proposition 3.10 and Proposition 4.5, the quadratic polynomial q must vanish on the Legendrian subvariety PT C Z,0 of PD 0 .
Let m Γ ,0 be the maximal ideal of the local ring O Γ ,0 and let J ⊂ O Γ ,0 be the ideal of Z inside Γ . Recall that the projective tangent cone PT C Z,0 ⊂ PT Γ ,0 is defined by the homogeneous ideal J * ⊂ Sym • T * Γ ,0 generated by lowest order terms of elements of J. Since the projective scheme PT C Z,0 ⊂ PD 0 is reduced, the homogeneous ideal J * defining the scheme PT C Z,0 is a radical ideal, which implies that q ∈ J * . Since we have assumed that T Z,0 spans D 0 = T Γ ,0 , the homogeneous ideal J * contains no elements of degree 1. Thus q is an element of minimal degree in J * . Consequently, it is the leading term of some element of J, i.e., there exists some Thus the hypersurface in C 2m+1 defined by contains Z. By Theorem 4.1, this hypersurface is contactomorphic to the hyperplane x 2m+1 = 0. Thus the germ at 0 of Z is contactomorphic to that of a Lagrangian cone by Lemma 3.3.

Deformation-rigidity of normal Legendrian singularities
In this section, we prove Theorem 1.5. Throughout, we fix a linear coordinate t on C and denote a point of C simply by its coordinate t ∈ C. Sometimes, we use another linear coordinate τ on C to distinguish it from t. We begin with recalling a few standard facts on time-dependent vector fields. As some algebraic geometers may not be familiar with them, we will provide most of the proofs. (i) When W is a complex manifold and Ψ : W ×∆ → M ×∆ is a holomorphic map, define for each t ∈ ∆ and w ∈ W , They define a holomorphic map Ψ t : W → M and a holomorphic sectionΨ t ∈ H 0 (W , Ψ * t T M ).
(ii) Let B be a holomorphic vector field on M × ∆ such that dπ ∆ ( B) = 0. For each t ∈ ∆, denote by B t ∈ H 0 (M, T M ) the vector field dπ M • B| M×{t} .
The following lemma on time-independent vector fields is a straightforward holomorphic translation of the standard result on differentiable manifolds (see e.g. Theorem 8.1 of [St]).
Lemma 5.2. Let X be a complex manifold and let A ∈ H 0 (X, T X ) be a holomorphic vector field. Then for each x ∈ X, there exist neighborhoods O x ⊂ X of x and ∆ x ⊂ C of 0 with a holomorphic map where − → A denotes the vector field on X × ∆ x that is sent to A by the projection π X and τ is the restriction of the coordinate t to ∆ x ⊂ C. Moreover, the germ of Φ A at (x, 0) is uniquely determined by the above properties.
Moreover, the germ of Ψ B at (z, 0) is uniquely determined by the properties (a), (b) and (c).
Proof. Equip X := M × ∆ (resp. X × ∆) with the subbundle D ⊂ T X (resp. D ⊂ T X×∆ ) given by where π M : M × ∆ → M and π X : X × ∆ → X are the natural projections. Then the vector field B preserves D and so do ∂ ∂t and A := B + ∂ ∂t . Applying Lemma 5.2 with x = (z, 0), we obtain two holomorphic maps for a suitable choice of the neighborhoods U z and ∆ z . It is easy to see that the conditions (a) and (b) are satisfied. The condition (c) follows from (5.1), (5.3) and Finally, (d) is from dΨ B (D) ⊂ D, which follows from (5.2) and (5.4). The uniqueness of the germ of Ψ B follows from the uniqueness theorem on solutions of ordinary differential equations.
The next lemma is a direct consequence of the uniqueness of the integral curve of a vector field through a given point on a manifold, applied to the vector field B + ∂ ∂t .
Lemma 5.5. In Lemma 5.4, assume that there exists a complex submanifold W ⊂ M and a holomorphic map F : W × ∆ → M × ∆, which is biholomorphic over its image, such that In other words, for any y ∈ W in a neighborhood of z and any t ∈ ∆ in a neighborhood of 0, we have Definition 5.6. Let M be a complex manifold and S ⊂ M × ∆ be a submanifold such that the restriction π ∆ | S is submersive with connected fibers. Then we can view S as a family of submanifolds {S t ⊂ M, t ∈ ∆} parametrized by ∆. For each t ∈ ∆, there exists a sectionṠ t ∈ H 0 (S t , N S t ) of the normal bundle N S t of S t ⊂ M, called the infinitesimal deformation of the family of submanifolds at t, which can be described in the following way for t close to 0 (this is a reformulation of the standard definition, e.g., pp. 148-150 of [Kd]). For a point z ∈ S 0 , we can pick neighborhoods W z ⊂ S 0 of z and ∆ z ⊂ ∆ of 0 with a holomorphic map G : W z × ∆ z → S ∩ (M × ∆ z ), which is biholomorphic over its image, such that (1) G t (w) ∈ S t , i.e., G(w, t) ∈ S ∩ (M × {t}) for any w ∈ W z and t ∈ ∆ z ; and (2) G| W z ×{0} = Id W z ×{0} , i.e., G 0 = Id W z .
Then for any w ∈ W z and t ∈ ∆ z , the valueṠ t of the sectionṠ t ∈ H 0 (S t , N S t ) at the point y = G t (w) ∈ S t is given byṠ This is independent of the choice of G.
Lemma 5.7. In Definition 5.6, assume that there exists a vector field B on M × ∆ such that dπ ∆ ( B) = 0 anḋ S t (y) = B t (y) modulo T S t ,y for each t ∈ ∆ and y ∈ S t .
For z ∈ S 0 ⊂ M, let Ψ B : U z × ∆ z → M × ∆ z be the map defined in Lemma 5.4. Then Ψ B t (y) ∈ S t for any t ∈ ∆ z close to 0 and any y ∈ S 0 close to z.
Proof. For a fixed z ∈ S 0 , let G : W z × ∆ z → S be as in Definition 5.6. By the assumption onṠ t , we can find a holomorphic vector field E on W z × ∆ z satisfying dπ ∆ ( E) = 0 and in other words, on the image of G. Let us apply Lemma 5.4 to the vector field E on W z × ∆ z . We have neighborhoods W ⊂ W z of z ∈ W z and ∆ z ⊂ ∆ z of 0 ∈ ∆ z with a holomorphic map Ψ E : W × ∆ z → W z × ∆ z such that (5.6) We claim that F satisfies the conditions of Lemma 5.5. The conditions (i) and (ii) of Lemma 5.5 are immediate from the properties of (1) and (2)  = B + ∂ ∂t by (5.5).
By the claim, we can apply Lemma 5.5 to see Ψ B t (y) = F t (y) ∈ S t for any t ∈ ∆ z close to 0 and any y ∈ S 0 close to z ∈ S 0 .
To prove Theorem 1.5, we recall some basic facts from contact geometry. The following lemma is from Example 1.2.C in Chapter 4 of [AG].
Lemma 5.8. Let S be a complex manifold and let L be a line bundle on S. Then the underlying complex manifold of the 1-jet bundle J 1 S L satisfying the exact sequence has a natural contact structure. Moreover, a section Σ ⊂ J 1 S L is a Legendrian submanifold if and only if it is the 1-jet of the section j(Σ) ∈ H 0 (S, L).
The next lemma is classical. See Theorem 7.1 in [Kb] for (i) and Lemma 7.1 in [LB] for (ii).