[.] [.]
Darboux charts around holomorphic Legendrian curves and applications
Antonio Alarcón and Franc Forstnerič
Abstract In this paper we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold . By using such a chart, we show that every holomorphic Legendrian immersion from an open Riemann surface can be approximated on relatively compact subsets of by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion from a compact bordered Riemann surface is a uniform limit of topological embeddings such that is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds in complex contact manifolds.
Keywords complex contact manifold, Darboux chart, Legendrian curve, isotropic submanifold
MSC (2010): 53D10; 32E30, 32H02, 37J55
Date: February 2, 2017. This version: August 24, 2018
1. Introduction and main results
A complex contact manifold is a pair , where is a complex manifold of (necessarily) odd dimension and is a completely noninvolutive holomorphic hyperplane subbundle (a contact subbundle) of the holomorphic tangent bundle . Locally near any point of , a contact subbundle is the kernel of a holomorphic form satisfying ; such is called a holomorphic contact form. (Globally we have for a holomorphic form with coefficients in the line bundle ; see Sec. 5.) By a fundamental theorem of Darboux [14] from 1882, there are local coordinates at any point of in which the contact structure is given by the standard contact form
(1.1) 
The proof of Darboux’s theorem given by Moser [30] (see also [20, p. 67]) easily adapts to the holomorphic case; see [5, Theorem A.2]. This shows that a holomorphic contact structure has no local invariants, and hence all interesting problems are of global nature.
Let be a complex contact manifold of dimension . A smooth immersed submanifold is said to be isotropic if
holds for all . 
If , this is equivalent to . The contact condition implies ; the immersion is said to be Legendrian if is of maximal dimension . (See Sec. 5 for more details.) It is easily seen that the image of a smooth Legendrian immersion is necessarily a complex submanifold of (see Lemma 5.1). Since we shall mainly consider the case when is an open Riemann surface and is an isotropic holomorphic curve, we will use the term Legendrian curve also when , with the exception of Sec. 5 where we consider also higher dimensional complex isotropic submanifolds.
In this paper we prove that there exists a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve, and also around some higher dimensional isotropic Stein submanifolds, in an arbitrary complex contact manifold. The following is our first main result; it is proved in Section 2.
Theorem 1.1.
Let be a complex contact manifold of dimension . Assume that is an open Riemann surface, is a nowhere vanishing holomorphic form on , and is a holomorphic Legendrian immersion. Then there are an open neighborhood of and a holomorphic immersion (embedding if is an embedding) such that and the contact structure on is given by the contact form
(1.2) 
where are complex coordinates on .
Recall that any holomorphic line bundle on an open Riemann surface is holomorphically trivial according to the OkaGrauert principle (see [17, Theorem 5.3.1, p. 190]); in particular, admits a nowhere vanishing holomorphic form. Furthermore, by the GunningNarasimhan theorem (see [22] or [17, Corollary 8.12.2, p. 386]) there exist plenty of holomorphic immersions , i.e., holomorphic functions without critical points. Taking and replacing by , the normal form (1.2) changes to
(1.3) 
This is formally the same as (1.1), the difference being that is now a holomorphic immersion (and not just a local coordinate function), and the normal form (1.3) is valid globally in a tube around the immersed Legendrian curve .
The existence of global holomorphic Darboux charts, given by Theorem 1.1, has many applications, some of which are presented in the sequel. It shows that the contact structure has no local invariants along a noncompact holomorphic Legendrian curve, and any such curve extends to a local complex Legendrian submanifold of maximal dimension ; in the Darboux chart (1.1) this extension is provided by . In the case , we see that all small Legendrian perturbations of the Legendrian curve in with are of the form and , where and is a holomorphic function on .
Before proceeding, we wish to briefly address the question that might be asked by the reader at this point: How many complex contact manifolds are there? Examples and constructions of such manifolds can be found in the papers [7, 9, 12, 23, 26, 27, 28, 24, 33, 34], among others. Many of these constructions mimic those in smooth contact geometry (for the latter, see e.g. Cieliebak and Eliashberg [13] and Geiges [20]). On the other hand, some of the constructions in the complex world have no analogue in the real one.
If is a holomorphic contact structure on a complex manifold , then its normal line bundle satisfies where is the canonical bundle of (see [27, 28] and (5.1) below for a discussion of this topic). Assuming that is compact and is a holomorphic line bundle satisfying this condition, the space of all contact structures on with normal bundle isomorphic to is a (possibly empty) connected complex manifold (see LeBrun [27, p. 422]). It follows from Gray’s theorem [21] that all these contact structures are contactomorphic to each other. If is simply connected, it admits at most one isomorphism class of the root , and hence at most one complex contact structure up to contactomorphisms (see [27, Proposition 2.3]). Only two general constructions of compact complex contact manifolds are known:

The projectivized cotangent bundle of any complex manifold of dimension at least 2. Recall that carries a tautological form , given in any local holomorphic coordinates on and associated fibre coordinates on by . Considering as projective coordinates on , we get the contact structure on .

Let be a simple complex Lie group with the Lie algebra . The adjoint action of on the projectivization of has a unique closed orbit which is contained in the closure of every other orbit; this orbit is a contact Fano manifold. See the papers by Boothby [10], Wolf [33] and Beauville [7, 8]. The simplest example of this type is the contact structure on the projective spaces .
It is conjectured that any projective contact manifold is of one of these two types (see Beauville [9, Conjecture 6]). For projective threefolds, this holds true according to Ye [34]. Demailly proved [15, Corollary 2] that if a compact Kähler manifold admits a contact structure, then its canonical bundle is not pseudoeffective (and hence not nef), and in particular the Kodaira dimension of equals . (The latter fact was also shown by Druel [16, Proposition 2].) If in addition then is projective and hence , not being pseudoeffective, is negative, i.e., is a Fano manifold [15, Corollary 3]. Together with the results by Kebekus et al. [23, Theorem 1.1] it follows that a projective contact manifold is Fano with or of type (1); see [15, Corollary 4] and Peternell [31, Theorem 2.9]. Since a homogeneous Fano contact manifold is known to be of type (2), the above conjecture reduces to showing that every Fano contact manifold is homogeneous.
The situation is more flexible on noncompact complex manifolds, and especially on Stein manifolds. A holomorphic form which is contact at some point of is contact in the complement of a closed complex hypersurface (possibly empty) given by the vanishing of the form ; this holds for a generic holomorphic form on a Stein manifold. If is a holomorphic symplectic manifold and is a holomorphic vector field on satisfying (such is called a Liouville vector field; here, denotes the Lie derivative), then the restriction of the form to any smooth complex hypersurface transverse to is a contact form on . For example, letting (the standard symplectic form on ) and yields the standard contact structure on any hyperplane ), and hence it induces a complex contact structure on .
We now present the applications of Theorem 1.1 obtained in this paper.
In Section 3 we prove the following general position result whose proof combines Theorem 1.1 and the arguments from [5, proof of Lemma 4.4].
Theorem 1.2.
Let be a complex contact manifold. Every holomorphic Legendrian immersion from an open Riemann surface can be approximated, uniformly on any relatively compact subset , by holomorphic Legendrian embeddings .
In general one cannot approximate a holomorphic Legendrian immersion uniformly on compacts in by holomorphic Legendrian embeddings of the whole Riemann surface. For example, if is a proper holomorphic immersion with a single double point, then by choosing to be an open neighborhood of which is very thin near infinity we can ensure that is the only nonconstant complex line in up to a reparametrization. Approximation of Legendrian embeddings by global ones is possible in the model contact space ; moreover, there exist proper holomorphic Legendrian embeddings from any open Riemann surface (see Alarcón et al. [5, Theorem 1.1]). The latter result also holds for the special linear group endowed with its standard contact structure (see Alarcón [2]). On the other hand, there exist examples of (Kobayashi) hyperbolic complex contact structures on for any (see Forstnerič [18]); in particular, these structures do not admit any nonconstant holomorphic Legendrian maps from or .
The proof of Theorem 1.2 also provides local deformation theory of noncompact holomorphic Legendrian curves. In particular, the space of Legendrian deformations of a holomorphic Legendrian curve normalized by a bordered Riemann surface is an infinite dimensional complex Banach manifold (see Remark 3.2).
Another application of Theorem 1.1 is that we can uniformly approximate a holomorphic Legendrian curve with smooth boundary in an arbitrary complex contact manifold by complete holomorphic Legendrian embeddings bounded by Jordan curves. In order to formulate this result, we need to recall the following notions. Assume that is a Riemannian metric on . An immersion is said to be complete if the induced metric on is a complete metric. A compact bordered Riemann surface, , is the same thing as a compact smoothly bounded domain in an open Riemann surface . A map from such a domain is said to be holomorphic if it extends to a holomorphic map on an open neighborhood of in .
Theorem 1.3.
Let be a complex contact manifold, and let be a compact bordered Riemann surface. Every holomorphic Legendrian immersion can be approximated uniformly on by topological embeddings such that is a complete holomorphic Legendrian embedding.
Since is a compact subset of , the notion of completeness of complex curves uniformly close to is independent of the choice of a metric on .
Theorem 1.3 may be compared with the results on the CalabiYau problem in the theory of conformal minimal surfaces in and null holomorphic curves in ; see Alarcón et al. [3, 6] and the references therein for recent developments on this subject.
Theorem 1.3 is proved in Section 4. The special case with the model contact space (see (1.1)) was obtained in [5, Theorem 1.2]. The Darboux charts, furnished by Theorem 1.1, make it possible to extend this result to any complex contact manifold.
In Section 5 we consider isotropic complex submanifolds of higher dimension in a complex contact manifold, and we prove a contact neighborhood theorem in the case when is a Stein submanifold (see Theorem 5.3). In particular, we obtain the following result.
Theorem 1.4.
Let be complex contact manifolds of the same dimension. If are biholomorphic Legendrian (i.e., isotropic and of maximal dimension) Stein submanifolds such that is trivial over for , then and have holomorphically contactomorphic neighborhoods.
Moreover, in the special case when the complex isotropic submanifold is Stein and contractible, we find a Darboux chart around similar to those furnished by Theorem 1.1; see Theorem 5.6.
It seems that the results in this paper are the first of their kind in the holomorphic case. On the other hand, contact neighborhood theorems of isotropic submanifolds are well known in the smooth case. For example, two smooth diffeomorphic isotropic submanifolds with isomorphic conformal symplectic normal bundles have contactomorphic neighborhoods (see Geiges [20, Theorem 2.5.8]). In particular, diffeomorphic closed Legendrian submanifolds (i.e., isotropic submanifolds of maximal dimension) have contactomorphic neighborhoods (see [20, Corollary 2.5.9]). For example, if is a Legendrian knot in a smooth contact 3manifold, then with a coordinate along and coordinates in slices transverse to , the contact form
provides a model for a contact neighbourhood of in (see [20, Example 2.5.10]). By the proof of Theorem 1.1 we can also get a contact neighborhood with the form . However, a crucial difference appears between the real and the complex case: there is no smooth immersion , so only has the meaning as a nonvanishing form on . In particular, the form is not contact for any smooth function , and there are no smooth contact neighborhoods (1.3) of a smooth Legendrian knot.
It is natural to ask what could be said about contact neighborhoods of compact holomorphic Legendrian curves and, more generally, of higher dimensional compact isotropic complex submanifolds. According to Bryant [11, Theorem G] (see also Segre [32]), every compact Riemann surface embeds as a complex Legendrian curve in . The first question to answer is which closed Legendrian curves in admit Darboux type neighborhoods. One major obstacle is that the tubular neighborhood theorem fails in general for compact complex submanifolds. In another direction, the deformation theory of certain compact complex Legendrian submanifolds has been studied by Merkulov [29] by using Kodaira’s deformation theory approach [25]. He showed that a compact complex Legendrian submanifold of with is contained in a complete analytic family of compact complex Legendrian submanifolds of .
2. Normal form of a contact structure along an immersed Legendrian curve
In this section we prove Theorem 1.1. We shall repeatedly use the following known lemma; we include a sketch of proof for the sake of completeness.
Lemma 2.1.
Let be an open Riemann surface, and let be a holomorphic matrixvalued function on , with , which has maximal rank at every point of . Then there exists a holomorphic map such that holds for all , where is the identity matrix.
Proof.
Denote the rows of by for ; these are holomorphic maps such that the vectors are linearly independent at every point . We must find holomorphic maps such that the matrix function with the rows is invertible at each point; then satisfies the lemma.
Recall that every holomorphic vector bundle on an open Riemann surface is trivial by the OkaGrauert principle (see [17, Theorem 5.3.1, p. 190]), and every holomorphic vector subbundle of a holomorphic vector bundle over a Stein manifold splits , i.e., we have where is another holomorphic vector subbundle of (this follows from Cartan’s Theorem B, see [17, Corollary 2.4.5, p. 54]). Let , and let be the holomorphic rank subbundle spanned by the rows of the matrix at each point . Then where is a trivial bundle of rank ; thus it is generated by global holomorphic sections . This proves the lemma. ∎
Remark 2.2.
The conclusion of Lemma 2.1 holds for any Stein manifold on which every complex vector bundle is topologically trivial; for instance, on a contractible Stein manifold. Indeed, by the OkaGrauert principle it follows that every holomorphic vector bundle on is holomorphically trivial, and hence the proof of Lemma 2.1 applies verbatim.
Proof of Theorem 1.1.
The normal bundle of the immersion is a holomorphic vector bundle of rank over , hence a trivial bundle by the OkaGrauert principle (see [17, Theorem 5.3.1]). By the DocquierGrauert tubular neighborhood theorem (see [17, Theorem 3.3.3]), there are a Stein open neighborhood of and a holomorphic immersion with . Furthermore, can be chosen to have convex fibers, so it is homotopy equivalent to . Hence, every holomorphic vector bundle on is holomorphically trivial by the OkaGrauert principle. In particular, the complex line bundle is trivial, and the quotient projection with is a holomorphic form on defining the contact structure . (Compare with (5.3).) The contact condition is that
We shall find a holomorphic change of coordinates in a neighborhood of which fixes pointwise and reduces to the form (1.2). To simplify the notation, the neighborhood in question will always be called , but the reader should keep in mind that it is allowed to shrink around during the proof.
Let denote points in , and let be complex coordinates on . Along we have that
for some holomorphic functions without common zeros (since is the contact hyperplane at ). The form does not appear in the above expression since is a Legendrian curve. Let . We introduce new coordinates , where the holomorphic map satisfies for all ; such exists by Lemma 2.1. Dropping the primes, this transforms along to the constant form . Geometrically speaking, this amounts to rotating the contact plane for to the constant position given by . Denoting the variable by , we have at all points of .
We now consider those terms in the Taylor expansion of along which give a nontrivial contribution to the coefficient function of the form . Since the coefficient of equals on , it is a nowhere vanishing holomorphic function in a neighborhood of this set, and we simply divide by it. We thus have
(2.1) 
where the coefficients and are holomorphic functions on . The form (the remainder) contains all terms , terms whose coefficients are of order in the variables , or terms that contain the variable; such terms disappear in at all points of .
We claim that the functions in (2.1) have no common zeros in . Indeed, at a common zero of these functions, the form at the point does not contain the term and hence vanishes, a contradiction. Write . Applying Lemma 2.1 with the row matrix gives a holomorphic change of coordinates of the form
such that the coefficient of becomes , and hence
(2.2) 
for some new coefficients . Note that contains the factor (since a nontrivial differential in the direction does not appear in any other way). Hence, the term with and all terms containing with in (2.2) can be placed into the remainder since they do not contribute to . Renaming the variable by we thus have
(2.3) 
If (i.e., ), we are finished with the first part of the proof and proceed to the second part given below.
Assume now that . We begin by eliminating the variable from the coefficients of the differentials by the shear
This ensures that the functions in the coefficient of in (2.3) have no common zeros on (since at such point would not contain ). Applying Lemma 2.1 we change the coefficient of to by a linear change of the variables with a holomorphic dependence on . Set and . By the same argument as in the previous step, we can move the term with , as well as all terms containing in the subsequent differentials , to the remainder . This gives
(2.4) 
It is clear that this process can be continued, and in finitely many steps we obtain
where is the normal form (1.2).
We now complete the proof by applying Moser’s method [30] in order to get rid of the remainder . Consider the following family of holomorphic forms on :
Note that , , and for all we have
The second identity holds because, by the construction, contains only terms which do not contribute to . Hence, is a contact form in a neighborhood of , still denoted , determining a contact structure for every , and vanishes on . (The dot indicates the derivative.) We shall find a timedependent holomorphic vector field on a neighborhood of that vanishes on and whose flow satisfies
(2.5) 
and the initial condition
(2.6) 
At time we shall then get in an open neighborhood of , thereby completing the proof of the theorem.
Let denote the Reeb vector field of contact form (see [20, p. 5]), i.e., the unique holomorphic vector field satisfying the conditions
A vector field whose flow satisfies conditions (2.5), (2.6) is sought in the form
(2.7) 
where is a smooth family of holomorphic functions and is a smooth family of holomorphic vector fields tangent to on a neighborhood of . Then,
Differentiating the equation (2.5) on gives
where denotes the Lie derivative. By using Cartan’s formula we see that must satisfy the equation
(2.8) 
Contracting this form with the Reeb vector field and noting that gives
(2.9) 
This is a parameter family of linear holomorphic partial differential equations for the functions . Note that the contact plane field is tangent to the hypersurface along . Since , is noncharacteristic for the Reeb vector field along for every . It follows that the equation (2.9) has a unique local solution satisfying the initial condition for all . Since the right hand side of (2.9) vanishes on , the solutions satisfy
(2.10) 
This choice of ensures that the component of the form vanishes. Since the form is nondegenerate on , the equation (2.8) has a unique holomorphic solution tangent to . In view of (2.10) we have
Thus, we see from (2.8) that the vector field vanishes along , and hence so does (2.7) in view of (2.10). It follows that the flow of exists for all in some neighborhood of and it satisfies conditions (2.5) and (2.6). This reduces to the normal form (1.2) on a neighborhood of . ∎
3. Local analysis near a Legendrian curve
Let be an open Riemann surface and be a nowhere vanishing holomorphic form on . We consider holomorphic Legendrian curves in the manifold endowed with the contact form (1.2):
(3.1) 
Here, denotes the variable in and the other coordinates are Euclidean coordinates on . By Theorem 1.1, this corresponds to the local analysis near an immersed Legendrian curve in a complex contact manifold . When applying these results in conjuction with Theorem 1.1, we consider only Legendrian curves in an open neighborhood of which corresponds to a Darboux patch in . The following lemma is seen by an obvious calculation; see [5, Lemma 3.1] for the case .
Lemma 3.1.
Let be the contact form (3.1) on . Given a holomorphic map , the map with
(3.2) 
is a holomorphic Legendrian disc satisfying
The same is true if is replaced by a bordered Riemann surface provided that the holomorphic form has vanishing periods over all closed curves in .
Proof of Theorem 1.2.
Let be a holomorphic Legendrian immersion from an open Riemann surface to a complex contact manifold of dimension . It suffices to show that for every compact smoothly bounded domain the restriction can be approximated in the norm by a Legendrian embedding of class . Fix such a domain . After shrinking around , Theorem 1.1 provides a holomorphic immersion , where is the unit ball in and , such that and the contact structure is given by the form (3.1). Note that the nonisotropic dilation
(3.3) 
for preserves the contact structure (3.1), so we may assume that .
For simplicity of notation, we present the proof in the case , so , and
Let be the complex subvariety of defined by
(3.4) 
where denotes the diagonal of and is the union of the irreducible components of disjoint from . Since is an immersion, we have or else ; in the latter case, (and hence ) is an embedding and there is nothing to prove.
Let denote the inclusion map . Since is an immersion, there is an open neighborhood of the diagonal such that . To prove the theorem, we must find arbitrarily close to in the norm a holomorphic Legendrian map such that the associated map
satisfies the condition
(3.5) 
Assuming that is close enough to , this condition ensures that is a holomorphic Legendrian embedding which approximates the initial Legendrian immersion . Indeed, (3.5) implies that for all , while for the same holds provided that is close to in the norm.
To find Legendrian maps satisfying (3.5), we apply the transversality method together with the technique of controlling the periods. The argument is similar to the one in [5, Lemma 4.4] in the case when , and . It suffices to construct a holomorphic map , where is the unit ball in for some big integer and , satisfying the following conditions:

is the inclusion map ,

is a holomorphic Legendrian immersion for every , and

the map , defined by
is a submersive family of maps on , in the sense that
Assume for a moment that such exists. By compactness of , it follows from (c) that the partial differential is surjective on for some . For such , the map is transverse to any complex subvariety of , in particular, to (see (3.4)). It follows that for a generic choice of the map is transverse to on , and hence it does not intersect by dimension reasons. (See Abraham [1] or [17, Section 7.8] for the details of this argument.) Choosing sufficiently close to gives a holomorphic Legendrian embedding close to in the norm, so is a Legendrian embedding as explained above.
It remains to explain the construction of . It suffices to find for any given pair of points a holomorphic spray as above, with , such that
(3.6) 
Since submersivity of the differential is an open condition and is compact, we then obtain a spray satisfying conditions (a)–(c) by composing finitely many sprays satisfying (3.6) (cf. [4, proof of Theorem 2.4]).
Let be a nowhere vanishing holomorphic vector field on , and let denote the flow of with the initial condition . Every sufficiently close to the zero function determines a holomorphic map defined by
Note that is the identity map, and the Taylor expansion in any local coordinate on is
Fix a pair of distinct points in . Choose a smooth embedded arc connecting to . Let be closed curves forming a basis of the homology group such that and the compact set is Runge in . Let be the period map with the components
(3.7) 
defined for all close enough to and all . Here, denotes the pullback of the form by the map .
Let be a small number whose value will be determined later. Choose holomorphic functions satisfying the following conditions:

(here, is the Kronecker symbol);

and for all and ;

, , ;

;

for all and .
Functions with these properties are easily found by first constructing suitable smooth functions on the curves and applying Mergelyan’s approximation theorem. Let and . Consider the function
(3.8) 
Note that for all and for all . Condition (i) implies that
for , and hence
(3.9) 
The implicit function theorem then shows that the period vanishing equation
(3.10) 
(which trivially holds at and ) can be solved on in a neighborhood of