1 Introduction

[.] [.]

H-principle for complex contact structures

on Stein manifolds

Franc Forstnerič

Abstract  In this paper we prove that every formal complex contact structure on a Stein manifold is homotopic to a holomorphic contact structure on a Stein domain which is diffeotopic to . We also prove a parametric h-principle in this setting, analogous to Gromov’s h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-principle for germs of holomorphic contact structures along totally real submanifolds of class in arbitrary complex manifolds.

Keywords  Stein manifold, complex contact structure, h-principle

MSC (2010)  37J55; 53D10, 32E10, 32E30

1. Introduction

A complex contact manifold is a pair , where is a complex manifold of (necessarily) odd dimension and is a completely nonintegrable holomorphic hyperplane subbundle (a contact bundle) of the holomorphic tangent bundle , meaning that the O’Neill tensor , , is nondegenerate. Note that where is a holomorphic -form on with values in the complex line bundle (the normal bundle of ) which realises the quotient projection

 (1.1) 0⟶ξ\lhook\joinrel\relbar\joinrel→TX\lx@stackrelα⟶L⟶0.

Thus, is a holomorphic section of the twisted cotangent bundle . The contact condition is equivalent to at every point of . A theorem of Darboux [9] says that, locally at any point, is holomorphically contactomorphic to the standard contact bundle on given by the -form where are complex coordinates on . (See also [36] or [23, p. 67] for the real case, and [2, Theorem A.2] for the holomorphic case.)

We denote by the space of all holomorphic contact forms on , endowed with the compact-open topology. In this paper we consider the existence and homotopy classification of complex contact forms on Stein manifolds of dimension .

We begin by recalling a few general observations due to LeBrun and Salamon [33, 34]. If , then is a holomorphic -form on with values in the line bundle , i.e., an element of where is the canonical bundle of . Being nowhere vanishing, defines a holomorphic trivialisation of the line bundle , so we conclude that

 (1.2) K−1X=K∗X≅Ln+1.

Similarly, is a nowhere vanishing section of the line bundle (i.e., is an -valued complex symplectic form on the bundle ), so we have that

 (1.3) Λ2nξ≅Ln=(TX/ξ)n.

In particular, on a contact -fold we have . It is easily seen that conditions (1.2) and (1.3) are equivalent to each other. These observations impose strong restrictions on the existence of complex contact structures, especially on compact manifolds; see the survey by Beauville [3] and the introduction to [1].

Assume now that is a Stein manifold of dimension . For a generic holomorphic -form on , the equation defines a (possibly empty) complex hypersurface , and is a contact form on the Stein manifold . This observation shows that there exist a plethora of Stein contact manifolds, but does not answer the question whether a given Stein manifold (or a given diffeomorphism class of Stein manifolds) admits a contact structure. More precisely, when is a complex hyperplane subbundle satisfying (1.3) homotopic to a holomorphic contact subbundle?

The following notion is motivated by Gromov’s h-principle for real contact structures on smooth open manifolds (see [30]).

Definition 1.1.

Let be a complex manifold of dimension . A formal complex contact structure on is a pair , where is a smooth -form on with values in a complex line bundle satisfying (1.2), is a smooth -form on with values in , and

 (1.4) α∧βn=α∧nβ∧⋯∧β≠0holds at each point of X.

Note that is a nowhere vanishing section of the vector bundle of rank ; such always exists if is a Stein manifold of dimension . A -form satisfying (1.4) is an -valued complex symplectic form on the complex -plane bundle , and is a topological trivialisation of .

We denote by the space of all formal contact structures on , endowed with the compact-open topology. We have the natural inclusion

 (1.5) Conthol(X)\lhook\joinrel\relbar\joinrel→Contfor(X),α↦(α,dα).

The following is our first main result; it is proved in Sect. 6. (See also Theorem 6.1.)

Theorem 1.2.

Let be a Stein manifold of odd dimension. Given , there is a Stein domain , diffeotopic to , and a homotopy such that and . Furthermore, if are connected by a path in , they are also connected by a path of holomorphic contact forms on some Stein domain diffeotopic to .

A domain is said to be diffeotopic to if there is a smooth family of diffeomorphisms such that and . If denotes the complex structure operator on , then is a homotopy of complex structures on with and .

By Cieliebak and Eliashberg [8, Theorem 8.43 and Remark 8.44], the domain and the diffeotopy in Theorem 1.2 can be chosen such that the domain is Stein (equivalently, the manifold with is Stein) for every .

We also prove a parametric version of Theorem 1.2 (see Theorem 6.1) which says that a continuous compact family of formal complex contact structures on can be deformed to a continuous family of holomorphic contact structures on a Stein domain diffeotopic to , and the deformation may be kept fixed for those values of the parameter for which the given formal structure is already a holomorphic contact structure.

For real contact structures, Gromov’s h-principle [30] says that the inclusion (1.5) of the space of smooth contact forms into the space of formal contact forms is a weak homotopy equivalence on any smooth open manifold. (See also Eliashberg and Mishachev [14, Sect. 10.3].) The situation is more complicated for closed manifolds as was discovered later by Bennequin [4] and Eliashberg [11, 13]. In particular, the h-principle for real contact structures fails on the -sphere, but it holds for the class of overtwisted contact structures on any compact orientable -manifold; see [11, Theorem 1.6.1]. This was extended to manifolds of dimensions by Borman, Eliashberg, and Murphy in 2015 [6].

Our results in the present paper seem to be the first analogues in the holomorphic category of the above mentioned Gromov’s h-principle. At this time we are unable to construct holomorphic contact forms on the whole Stein manifold under consideration. The main, and seemingly highly nontrivial problem arising in the proof, is the following. (The analogous approximation problem for integrable holomorphic subbundles — holomorphic foliations — is also open in general; see [17, Problem 9.16.8].)

Problem 1.3.

Given a holomorphic contact form on an open neighbourhood of a compact convex set , is it possible to approximate uniformly on by holomorphic contact forms on ? Is such approximation also possible for any continuous family of holomorphic contact forms with parameter in a compact Hausdorff space?

This issue does not appear in the smooth case since one can pull back a contact structure on a neighbourhood of a compact convex set to a contact structure on by a diffeomorphism which equals the identity near .

By following the proof of Theorem 1.2 and using the gluing lemma for biholomorphic maps [17, Theorem 9.7.1]), we obtain the following result which is proved in Sect. 6.

Theorem 1.4.

If Problem 1.3 has an affirmative answer, then every formal complex contact structure on a Stein manifold is homotopic to a holomorphic contact structure on . Furthermore, if the parametric version of Problem 1.3 has an affirmative answer, then the inclusion (1.5) is a weak homotopy equivalence.

We now consider more carefully the case when is a Stein manifold with . Let be a holomorphic line bundle on satisfying (1.2), i.e., such that is a trivial line bundle. (Recall that every complex vector bundle on a Stein manifold carries a compatible structure of a holomorphic vector bundle by the Oka-Grauert principle; see [17, Theorem 5.3.1].) Note that admits a nowhere vanishing holomorphic section (see [17, Corollary 8.3.2]), i.e., an -valued holomorphic -form on . Let . Then, . Since by the assumption, we see that is a trivial bundle (see (1.3)). A trivialisation of is a -form on with values in such that is a trivialisation of , i.e., . Hence, the necessary condition (1.2) for the existence of an -valued formal contact structure on is also sufficient when is Stein and .

We denote by the subset of given by pairs of -valued forms . Clearly, is a union of connected components of . We claim that the connected components of coincide with the homotopy classes of topological trivialisations of . One direction is obvious: given a homotopy with , the family is a homotopy of trivialisations of . Conversely, assume that and there is a homotopy of trivialisations of with and . Since and is Stein, it is homotopy equivalent to a -dimensional CW complex. A simple topological argument in the line of [17, proof of Corollary 8.3.2] then shows that and can be connected by a homotopy of nowhere vanishing sections of . Let for . Then, where is a trivialisation of and . At we have , and it follows that . This proves the claim.

Recall that the isomorphism classes of complex (or holomorphic) line bundles on a Stein manifold are in bijective correspondence with the elements of by Oka’s theorem (see [17, Theorem 5.2.2]). The above observations yield the following homotopy classification of formal complex contact structures on Stein threefolds.

Proposition 1.5.

If is a Stein manifold of dimension , then the connected components of the space of formal complex contact structures on are in one-to-one correspondence with the following pairs of data:

1. an isomorphism class of a complex line bundle on satisfying , i.e., an element with , and

2. a choice of a homotopy class of trivialisations of the line bundle , that is, an element of .

In particular, if and then the space is connected; this holds in particular for .

Theorem 1.2 and Proposition 1.5 imply the following corollary.

Corollary 1.6.

Let be a Stein manifold of dimension . Given a holomorphic line bundle on such that , there is a Stein domain diffeotopic to and a holomorphic contact subbundle such that . Furthermore, given a pair of holomorphic -valued contact forms on such that is null homotopic, there is a Stein domain as above and a homotopy connecting to .

Since in the above corollary we must pass to a Stein subdomain of when constructing contact structures and their homotopies, the following problem remains open.

Problem 1.7.

Let be a Stein manifold of dimension with . Is the space connected? In particular, is connected?

Remark 1.8.

Corollary 1.6 gives a homotopy classification of contact forms on Stein -folds, but not necessarily of contact bundles. A holomorphic contact bundle on is determined by a holomorphic -form up to a nonvanishing factor . Since , this changes the trivialisation of by . (More generally, if then the trivialisation of given by changes by the factor .) Hence, a homotopy class of holomorphic contact bundles on a Stein -fold is uniquely determined by a pair , where satisfies and . By Corollary 1.6, every such pair is represented by a holomorphic contact bundle on a Stein domain diffeotopic to . ∎

We do not have a comparatively good classification results for on Stein manifolds of dimension five or more. Granted the necessary conditions (1.2), (1.3) for the normal bundle , the existence and classification of complex symplectic forms on the -plane bundle amounts to the analogous problem for sections of an associated fibre bundle with the fibre . We do not pursue this issue here.

One may wonder to what extent is it possible to control the choice of the domain in Theorem 1.2 and Corollary 1.6. In our proof, arises as thin Stein neighbourhood of an embedded CW complex in which represents its Morse complex, so it carries all topology of . However, since a Mergelyan type approximation theorem is used in the construction, we do not know how big can be. We describe the construction more precisely at the end of this introduction and supply references. The method actually gives much more. Assume that is an odd dimensional complex manifold (not necessarily Stein) and is a tamely embedded CW complex of dimension at most . (A suitable notion of tameness was introduced by Gompf [25, 26].) Let be a formal contact structure on . After a small topological adjustment of in , there is a holomorphic contact form on a Stein thickening of such that is homotopic to in .

This is illustrated most clearly by looking at holomorphic contact structures in neighbourhoods of totally real submanifolds. A real submanifold of class in a complex manifold is said to be totally real if the tangent space at any point (a real vector subspace of ) does not contain any complex line. By Grauert [27], such admits a basis of tubular Stein neighbourhoods in , the so called Grauert tubes. Note that every smooth -manifold is a totally real submanifold of a Stein -manifold: take the compatible real analytic structure on , let be its complexification, and choose to be a Grauert tube around in .

The following is the -parametric h-principle for germs of complex contact structure along a totally real submanifold; see Theorem 4.1 for the fully parametric case.

Theorem 1.9.

Let be a totally real submanifold of class in a complex manifold . Every formal complex contact structure is homotopic in to a holomorphic contact structure in a tubular Stein neighbourhood of in . Furthermore, any two holomorphic contact forms in a neighbourhood of which are formally homotopic along are also homotopic by a family of holomorphic contact forms in a Stein neighbourhood of .

In dimension we have the following simpler statement in view of Proposition 1.5.

Corollary 1.10.

Let be a -dimensional complex manifold and be a totally real submanifold of class . Then, germs of complex contact forms on along are classified up to homotopy by pairs consisting of a complex line bundle over a neighbourhood of satisfying and an element of .

When is a totally real submanifold of maximal dimension in a complex -manifold , we have (since the complex structure operator on induces an isomorphism of the tangent bundle onto the normal bundle of in ). Replacing by a Grauert tube around , it follows that , so the canonical class of only depends on . We shall see in Example 1.12 that this is not the case in general for totally real submanifolds of lower dimension.

Example 1.11.

Let be a Grauert tube around the -sphere in its complexification. Then, and . By Corollary 1.10 there is a unique homotopy class of germs of complex contact structures around in . We get it for instance by taking a totally real embedding of into (see [22, Theorem 1.4] or [29, p. 193]) and using the standard complex contact form on .

It was shown by Eliashberg [11] that there exist countably many homotopy classes of smooth contact structures on . By choosing them real analytic, we can complexity them to obtain holomorphic contact structures in neighbourhoods of in . By what has been said above, these are homotopic to each other as holomorphic contact bundles. ∎

Example 1.12.

Let be a Grauert tube around the -sphere in its complexification. We have which is trivial, so is holomorphically trivial by the Oka-Grauert principle. Let be a holomorphic line bundle; the isomorphism classes of such bundles correspond to the elements of . Considering as the zero section of , we can view as the normal bundle of in . Since is trivial, the adjunction formula for the canonical bundle gives

 KX|Y≅KY⊗(NY,X)−1=X−1.

For each choice of the bundle with even Chern number , has a unique holomorphic square root with . By Corollary 1.10 there is a holomorphic -valued contact form on a neighbourhood of in . A Stein tube around in the trivial bundle can be represented as a domain in , for example, as a tube around the standard -sphere . The examples with nonzero Chern classes clearly cannot be represented as domains in . ∎

Example 1.13.

Let be a -dimensional Grauert tube around an embedded circle . In this case , and by Corollary 1.10 the homotopy classes of holomorphic contact forms along are classified by . We can see them explicitly in the following model case. Let be the complex coordinates on . Set and . For each let

 αk=⎧⎨⎩dz+1k+1xk+1dyif k≠−1,1√2(1xdz+xdy)if k=−1.

Then for every , so the homotopy class of the corresponding framing of the trivial bundle equals . By Remark 1.8, the contact bundle on is homotopic to if is even, and to is is odd. The bundles and are not homotopic to each other through contact bundles.

Note that the form for is the pullback of (the standard contact form on ) by the covering map , . In order to understand , consider the contact form on given by

 α=cosxdz+sinxdy.

Let denote the universal covering map . A calculation shows that where is the contact form on given by

 α′=12(x+1x)dz+12i(x−1x)dy,α′∧dα′=1ixdx∧dy∧dz.

Then, is homotopic to through the family of contact forms on defined by

 σt=1√2(1+t2)((tx+1x)dz+(x−tx)e−iπt/2dy),t∈[0,1].

We have , , and for all . ∎

Example 1.14.

The previous example can be generalised to and which are complexifications of the -torus and the -torus, respectively. Let us consider the latter. Denote by the -dimensional torus, the product of copies of . The domain is a Stein tube around the standard totally real embedding onto the distinguished boundary of the polidisc. We have and (see Rotman [37, p. 404]). Clearly, is trivial, and since is a free abelian group, it’s only square root is the trivial bundle. Hence by (1.2) all contact forms on have values in the trivial bundle, and we have -many homotopy classes of trivialisations. Consider the following family of contact forms on , where :

 αk,l,m={zmdz+1k+1xk+1yldyif k≠−1,12xzmdz+xyldyif k=−1,

We have that , so this family provides all possible homotopy classes of framings of the trivial bundle . ∎

The above examples suggest that in many natural cases one can find globally defined holomorphic contact forms representing all homotopy classes in Proposition 1.5.

Problem 1.15.

Is it possible to represent every homotopy class of formal complex contact structures on an affine algebraic manifold by an algebraic contact form?

Our proofs of Theorems 1.9 and 4.1 proceed by triangulating the manifold and inductively deforming a formal contact structure to an almost contact structure along . We show that the open partial differential relation of first order, controlling the almost contact condition on a totally real disc, is ample in the coordinate directions; see Lemma 2.1. Hence, Gromov’s h-principle [29, 31] can be applied to extend an almost contact structure from the boundary of a cell to the interior, provided that it extends as a formal contact structure; see Lemma 2.2. Finally, approximating an almost contact form on sufficiently closely in the fine topology by a holomorphic -form ensures that is a contact form in a neighbourhood of in . The same arguments apply to families of such structures, thereby yielding the parametric h-principle stated in Theorem 4.1.

A similar method is used to prove Theorems 1.2 and 6.1 (see Sect. 6). The inductive step amounts to extending a holomorphic contact form from a neighbourhood of a compact strongly pseudoconvex domain in across a handle whose core is a totally real disc attached with its boundary sphere to . More precisely, , the attachment is -orthogonal along where denotes the almost complex structure on , and is a Legendrian submanifold of the strongly pseudoconvex hypersurface with its smooth contact structure given by complex tangent planes.) The union then admits a basis of tubular Stein neighbourhoods (see [12] and [18]). Assuming that extends to as a formal contact structure, Lemma 4.3 furnishes an almost contact extension. Finally, by Mergelyan’s theorem we can approximate in the topology on by a holomorphic contact form on a Stein neighbourhood of .

With these analytic tools in hand, Theorems 1.2 and 6.1 are proved by following the scheme developed by Eliashberg [12] in his landmark construction of Stein manifold structures on any smooth almost complex manifold with the correct handlebody structure. (The special case is rather different and was explained by Gompf [24, 25, 26], but this is not relevant here.) A more precise explanation of Eliashberg’s construction was given by Slapar and the author [20, 21] in their proof of the soft Oka principle for maps from any Stein manifold to an arbitrary complex manifold . Expositions are also available in the monographs by Cieliebak and Eliashberg [8, Chap. 8] and the author [17, Secs. 10.9–10.11].

Finally, the proof of Theorem 1.4 (see Sect. 6) follows the induction scheme used in Oka theory; see [17, Sect. 5]. Besides the tools already mentioned above, an additional ingredient is a gluing lemma for holomorphic contact forms (see Lemma 6.2).

2. Germs of complex contact structures on domains in R2n+1⊂C2n+1

We denote the complex variables on by with for , where . We shall consider as the standard real subspace of .

Let be a compact set in which is the closure of a domain with piecewise boundary. In this section we consider the problem of approximating a complex contact structure , defined on a neighbourhood of a compact subset , by a complex contact structure defined on a neighbourhood of in , provided that admits a formal contact extension to in the sense of Definition 1.1. (For applications in this paper, it suffices to consider the case when is the standard handle of some index and , where and are closed unit balls in the respective spaces, and is the attaching set of the handle.) We will show that the parametric h-principle holds in this problem (see Lemma 2.2).

We begin with preliminaries. Let , and let be a closed set in a complex manifold . A function of class on an open neighbourhood of is said to be -flat to order on if the jet of of order vanishes at each point of . In any system of local holomorphic coordinates on centred at a point , this means that the value and all partial derivatives of order up to of the functions vanish at each point . In particular, such satisfies the Cauchy-Riemann equations at every point :

 ∂f∂zj(x)=∂f∂xj(x)=−i∂f∂yj(x),j=1,…,n.

If is smooth of class and the above holds for all , then is said to be -flat (to infinite order) on .

Assume now that is a compact domain with piecewise boundary in . It is classical (see e.g. [32, Lemma 4.3] or [8, Proposition 5.55]) that every function of class extends to a function which is -flat to order on . When is of class , we can obtain such an extension explicitly by first extending to a smooth function on and setting

 F(x+iy)=∑|I|≤l1I!∂|I|f∂xI(x)i|I|yI=f(x)+i2n+1∑i=1∂f∂xi(x)yi+O(|y|2).

Here, , , , and . If is only of class then a -flat extension is obtained by applying Whitney’s jet-extension theorem [39] to the jet on the right hand side above.

A smooth differential -form

 (2.1) α=2n+1∑i=1ai(z)dzi

in a neighbourhood of in is said to be -flat to order on is every coefficient function is such. Every smooth -form defined on extends to a -flat -form on by taking -flat extensions of its coefficient. Assume that is such. In view of the CR equations we have for each that

 (2.2) dα(x)=∂α(x)=∑1≤i

Write and set

 (2.3) βi,j(x):=pj,i(x)−pi,j(x)=∂aj∂xi(x)−∂ai∂xj(x).

With this notation, we have for all that

 (2.4) dα(x)=:β(x)=∑1≤i

and

 (2.5) (dα)n(x)=βn(x)=2n+1∑i=1bi(x)dz1∧⋯ˆdzi⋯∧dz2n+1,

where indicates that this term is omitted. Every coefficient in (2.5) is a homogeneous polynomial of order in the coefficients of (2.4), obtained as follows. Let be a partition of the set into a union of pairs , with . Then,

 (2.6) bi(x)=n!∑P∏(ik,jk)∈Pβik,jk(x)=n!∑P∏(ik,jk)∈P(pjk,ik(x)−pik,jk(x))

for all . Finally, from (2.4) and (2.5) we obtain for all that

 (2.7) α(x)∧(dα)n(x)=α(x)∧βn(x)=(2n+1∑i=1(−1)i−1ai(x)bi(x))dz1∧⋯∧dz2n+1.

A smooth -form on , defined on a neighbourhood of and -flat on to the first order, is said to be an almost contact form on if

 (2.8) α∧(dα)n≠0  at every point of D.

Note that . Approximating sufficiently closely in the topology on by a holomorphic -form gives a holomorphic contact structure on a neighbourhood of in . If the coefficients of are real analytic, then the complexification of defines a holomorphic contact structure near .

We see from (2.3), (2.6), and (2.7) that the condition (2.8) depends only on the first order jet of the restrictions of the coefficients of to , so it defines an open set in the space of -jets of -forms on . More precisely, we may view as a smooth section of the trivial bundle . Let be the bundle of -jets of sections of . The fibre of over a point (with consists of all matrices . A section is a map , where and . Such a section is said to be holonomic if is the -jet of for each , that is, for all . Let be the open subset of defined by

 (2.9) R={(x,a,p)∈E(1):2n+1∑i=1(−1)i−1aibi≠0},

where each is determined by according to the formula (2.6) (ignoring the base point ). Thus, is an open differential relation of first order in which controls the contact condition for -flat 1-forms along .

Lemma 2.1.

The partial differential relation defined by (2.9) is ample in the coordinate directions (in the sense of M. Gromov [31, 29]).

Proof.

Choose an index . Write and for . Consider a restricted -jet of the form where the vector is omitted. Set

 (2.10) Re={pi∈C2n+1:(x,a,p1,…,pi−1,pi,pi+1,…,p2n+1)∈R}.

The differential relation is said to be ample in the coordinate directions if every set of this type is either empty, or else the convex hull of each of its connected components equals . In the case at hand, we see from (2.6) and (2.7) that the function

 h(a,p)=2n+1∑j=1(−1)i−1ajbj(p),

where is determined by (2.6), is affine linear in . Indeed, every appears at most once in each of the products in (2.6). Since

 Re={pi∈C2m+1:h(a,p1,…,pi,…,p2n+1)≠0},

it follows that is either empty or else the complement of a complex affine hyperplane in ; in the latter case its convex hull equals . This proves Lemma 2.1. ∎

In order to apply this lemma, we need the following observation. Let be a -form (2.1) with smooth coefficients , and let

 (2.11) β(x)=∑1≤i

be a smooth -form on . (At this point we consider forms with values in the trivial line bundle.) Note that the linear projection