Abstract
For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of “overtwistedness”. We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.
Similar content being viewed by others
Notes
Our use of the term “contactization” is slightly nonstandard, as the word is typically used in the literature to mean a product of a Liouville domain with ℝ instead of with \({\mathbb {S}}^{1}\). In this paper, we shall go back and forth between both meanings of the term—it should always be clear from context which one is meant.
Our proof of Theorem C owes a considerable debt to Yves Benoist, who explained to us how to use number theory to find lattices in the groups considered by Geiges in [23].
Actually this construction provides infinitely many examples with pairwise distinct fundamental groups. We thank Gaëtan Chenevier for arithmetic discussions clarifying this.
We are deeply indebted to Bruno Sévennec and JeanClaude Sikorav for discussions that led to the proof of Theorem D.
In Theorem F and several other results in this paper, we write the word “semipositive” in parentheses: this means that the condition is presently necessary for technical reasons, but should be removable in the future using the polyfold technology of HoferWysockiZehnder, cf. [36].
References
Albers, P., Bramham, B., Wendl, C.: On nonseparating contact hypersurfaces in symplectic 4manifolds. Algebr. Geom. Topol. 10, 697–737 (2010). http://dx.doi.org/10.2140/agt.2010.10.697
Bennequin, D.: Entrelacements et équations de Pfaff. In: Third Schnepfenried Geometry Conference, vol. 1, Schnepfenried, 1982. Astérisque, vol. 107, pp. 87–161. Soc. Math. France, Paris (1983)
Bourgeois, F.: A MorseBott approach to contact homology. ProQuest LLC, Ann Arbor, MI (2002). Thesis (Ph.D.), Stanford University. http://proquest.umi.com/pqdlink?did=726452491&Fmt=7&clientId=79356&RQT=309&VName=PQD
Bourgeois, F.: Odd dimensional tori are contact manifolds. Int. Math. Res. Not. 2002(30), 1571–1574 (2002). http://dx.doi.org/10.1155/S1073792802205048
Bourgeois, F.: Contact homology and homotopy groups of the space of contact structures. Math. Res. Lett. 13(1), 71–85 (2006)
Bourgeois, F.: A survey of contact homology. In: New Perspectives and Challenges in Symplectic Field Theory. CRM Proc. Lecture Notes, vol. 49, pp. 45–71. Amer. Math. Soc., Providence (2009)
Bourgeois, F., van Koert, O.: Contact homology of lefthanded stabilizations and plumbing of open books. Commun. Contemp. Math. 12(2), 223–263 (2010). http://dx.doi.org/10.1142/S0219199710003762
Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003) (electronic). http://dx.doi.org/10.2140/gt.2003.7.799
Cieliebak, K., Volkov, E.: First steps in stable Hamiltonian topology. Preprint (2010). arXiv:1003.5084v3
Colin, V., Giroux, E., Honda, K.: Finitude homotopique et isotopique des structures de contact tendues. Publ. Math. Inst. Hautes Études Sci. 109(1), 245–293 (2009). http://dx.doi.org/10.1007/s102400090022y
Ding, F., Geiges, H.: Contact structures on principal circle bundles. Preprint (2011). arXiv:1107.4948. Bull. Lond. Math. Soc. (to appear)
Dragnev, D.: Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Commun. Pure Appl. Math. 57(6), 726–763 (2004). http://dx.doi.org/10.1002/cpa.20018
Eliashberg, Y.: Classification of overtwisted contact structures on 3manifolds. Invent. Math. 98(3), 623–637 (1989). http://dx.doi.org/10.1007/BF01393840
Eliashberg, Y.: Filling by holomorphic discs and its applications. In: Geometry of LowDimensional Manifolds, 2, Durham, 1989. London Math. Soc. Lecture Note Ser., vol. 151, pp. 45–67. Cambridge Univ. Press, Cambridge (1990). http://dx.doi.org/10.1017/CBO9780511629341.006
Eliashberg, Y.: On symplectic manifolds with some contact properties. J. Differ. Geom. 33(1), 233–238 (1991). http://projecteuclid.org/getRecord?id=euclid.jdg/1214446036
Eliashberg, Y.: Unique holomorphically fillable contact structure on the 3torus. Int. Math. Res. Not. 1996(2), 77–82 (1996). http://dx.doi.org/10.1155/S1073792896000074
Eliashberg, Y., Gromov, M.: Convex symplectic manifolds. In: Several Complex Variables and Complex Geometry, Part 2. Santa Cruz, CA, 1989. Proc. Sympos. Pure Math., vol. 52, pp. 135–162. Amer. Math. Soc., Providence (1991)
Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. In: GAFA 2000, Tel Aviv, 1999. Geom. Funct. Anal., pp. 560–673 (2000). Special Volume, Part II
Etnyre, J.: Planar open book decompositions and contact structures. Int. Math. Res. Not. 2004(79), 4255–4267 (2004). http://dx.doi.org/10.1155/S1073792804142207
Etnyre, J., Honda, K.: Tight contact structures with no symplectic fillings. Invent. Math. 148(3), 609–626 (2002). http://dx.doi.org/10.1007/s002220100204
Etnyre, J., Pancholi, D.: On generalizing Lutz twists. J. Lond. Math. Soc. 84(3), 670–688 (2011). http://dx.doi.org/10.1112/jlms/jdr028
Gay, D.: Fourdimensional symplectic cobordisms containing threehandles. Geom. Topol. 10, 1749–1759 (2006) (electronic). http://dx.doi.org/10.2140/gt.2006.10.1749
Geiges, H.: Symplectic manifolds with disconnected boundary of contact type. Int. Math. Res. Not. 1994(1), 23–30 (1994). http://dx.doi.org/10.1155/S1073792894000048
Geiges, H.: Examples of symplectic 4–manifolds with disconnected boundary of contact type. Bull. Lond. Math. Soc. 27(3), 278–280 (1995). http://dx.doi.org/10.1112/blms/27.3.278
Geiges, H.: An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008). http://dx.doi.org/10.1017/CBO9780511611438
Ghiggini, P., Honda, K.: Giroux torsion and twisted coefficients. Preprint (2008)
Ghiggini, P., Van HornMorris, J.: Tight contact structures on the Brieskorn spheres −Σ(2,3,6n−1) and contact invariants. Preprint (2009)
Giroux, E.: Convexité en topologie de contact. Comment. Math. Helv. 66(4), 637–677 (1991). http://dx.doi.org/10.1007/BF02566670
Giroux, E.: Une structure de contact, même tendue, est plus ou moins tordue. Ann. Sci. École Norm. Sup. (4) 27(6), 697–705 (1994). http://www.numdam.org/item?id=ASENS_1994_4_27_6_697_0
Giroux, E.: Une infinité de structures de contact tendues sur une infinité de variétés. Invent. Math. 135(3), 789–802 (1999). http://dx.doi.org/10.1007/s002220050301
Giroux, E.: Structures de contact en dimension trois et bifurcations des feuilletages de surfaces. Invent. Math. 141(3), 615–689 (2000). http://dx.doi.org/10.1007/s002220000082
Giroux, E.: Géométrie de contact: de la dimension trois vers les dimensions supérieures. In: Proceedings of the International Congress of Mathematicians, vol. II, Beijing, pp. 405–414. Higher Ed. Press, Beijing (2002)
Giroux, E.: Sur la géométrie et la dynamique des transformations de contact (d’après Y. Eliashberg, L. Polterovich et al.), Séminaire Bourbaki, volume 2008/2009. Exposés 997–1011. Astérisque, no. 332, Exp. No. 1004, pp. viii, 183–220 (2010)
Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985). http://dx.doi.org/10.1007/BF01388806
Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114(3), 515–563 (1993). http://dx.doi.org/10.1007/BF01232679
Hofer, H.: A general Fredholm theory and applications. In: Current Developments in Mathematics. 2004, pp. 1–71. Int. Press, Somerville (2006)
Hofer, H., Wysocki, K., Zehnder, E.: Applications of polyfold theory I: Gromov–Witten theory. Preprint (2011). arXiv:1107.2097
Lancaster, P., Rodman, L.: Canonical forms for symmetric/skewsymmetric real matrix pairs under strict equivalence and congruence. Linear Algebra Appl. 406, 1–76 (2005). http://dx.doi.org/10.1016/j.laa.2005.03.035
Latschev, J., Wendl, C.: Algebraic torsion in contact manifolds. Geom. Funct. Anal. 21(5), 1144–1195 (2011). With an appendix by M. Hutchings., http://dx.doi.org/10.1007/s0003901101383
Lerman, E.: Contact cuts. Isr. J. Math. 124, 77–92 (2001). http://dx.doi.org/10.1007/BF02772608
Marcus, D.A.: Number Fields. Universitext. Springer, New York (1977)
Massot, P.: Geodesible contact structures on 3manifolds. Geom. Topol. 12(3), 1729–1776 (2008)
McDuff, D.: Symplectic manifolds with contact type boundaries. Invent. Math. 103(3), 651–671 (1991). http://dx.doi.org/10.1007/BF01239530
McDuff, D., Salamon, D.: JHolomorphic Curves and Symplectic Topology. Colloquium Publications, vol. 52. American Mathematical Society, Providence (2004)
Mitsumatsu, Y.: Anosov flows and nonStein symplectic manifolds. Ann. Inst. Fourier (Grenoble) 45(5), 1407–1421 (1995). http://www.numdam.org/item?id=AIF_1995__45_5_1407_0
Mori, A.: Reeb foliations on S ^{5} and contact 5manifolds violating the ThurstonBennequin inequality. Preprint (2009). arXiv:0906.3237
Niederkrüger, K.: The plastikstufe—a generalization of the overtwisted disk to higher dimensions. Algebr. Geom. Topol. 6, 2473–2508 (2006). http://dx.doi.org/10.2140/agt.2006.6.2473
Niederkrüger, K., Presas, F.: Some remarks on the size of tubular neighborhoods in contact topology and fillability. Geom. Topol. 14(2), 719–754 (2010). http://dx.doi.org/10.2140/gt.2010.14.719
Niederkrüger, K., van Koert, O.: Every contact manifold can be given a nonfillable contact structure. Int. Math. Res. Not. 23, 22 (2007). Art. ID rnm115. http://dx.doi.org/10.1093/imrn/rnm115
Niederkrüger, K., Wendl, C.: Weak symplectic fillings and holomorphic curves. Ann. Sci. École Norm. Sup. 44(5), 801–853 (2011). arXiv:1003.3923
Polterovich, L.: The surgery of Lagrange submanifolds. Geom. Funct. Anal. 1(2), 198–210 (1991). http://dx.doi.org/10.1007/BF01896378
Presas, F.: A class of nonfillable contact structures. Geom. Topol. 11, 2203–2225 (2007). http://dx.doi.org/10.2140/gt.2007.11.2203
Salamon, D.: Lectures on Floer homology. In: Symplectic Geometry and Topology, Park City, UT, 1997. IAS/Park City Math. Ser., vol. 7, pp. 143–229. Amer. Math. Soc., Providence (1999)
Tischler, D.: On fibering certain foliated manifolds over S ^{1}. Topology 9, 153–154 (1970)
van der Geer, G.: Hilbert Modular Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16. Springer, Berlin (1988)
Wendl, C.: Nonexact symplectic cobordisms between contact 3manifolds. Preprint (2010). arXiv:1008.2456
Wendl, C.: Strongly fillable contact manifolds and J–holomorphic foliations. Duke Math. J. 151(3), 337–384 (2010). http://dx.doi.org/10.1215/001270942010001
Acknowledgements
We are grateful to Bruno Sévennec and JeanClaude Sikorav for emails leading to the proof of Theorem D, Yves Benoist for conversations which were crucial for the proof of Theorem C, Sylvain Courte for his proof of Lemma 5.5, Yves de Cornulier for his proof of Lemma 9.15, Helmut Hofer and Joel Fish for explaining to us some details of the polyfold machinery, and Paolo Ghiggini for many helpful discussions at the beginning of this project. The idea that some modification of Mori’s ideas in [46] might lead to a notion of Giroux torsion in higher dimensions was first suggested to us by John Etnyre. We would also like to thank the mathematics department in Nantes for creating a pleasant working environment which hosted several meetings of the authors, and a very careful anonymous referee whose comments on the original version of this article have led to several improvements in the exposition. The first and second author were partially supported by the ANR grant ANR10JCJC 0102. The third author was supported by an Alexander von Humboldt Foundation fellowship.
Author information
Authors and Affiliations
Corresponding author
Appendix: Cotamed complex structures: existence and convexity
Appendix: Cotamed complex structures: existence and convexity
1.1 A.1 Contractibility of the space of cotamed almost complex structures
To go from the linear situation to global existence results on a manifold we will need the following result.
Proposition 2.1
(Sévennec)
The space of complex structures on a finite dimensional vector space tamed by two given symplectic forms is either empty or contractible.
Using the fact that the space of complex structures tamed by a symplectic form is nonempty (which follows for instance by the linear Darboux theorem), and applying the proposition above twice to the same symplectic form, we recover as a special case the classical result of Gromov that states that the space of tamed complex structures is contractible. The proof of the proposition uses the following two lemmas, of which the first is more or less standard.
Lemma A.1
(Cayley, Sévennec)
Let V be a real finite dimensional vector space and \({\mathcal{J}}(V)\) the space of complex structures on V. We can define for any fixed \(J_{0} \in {\mathcal{J}}(V)\) a map
which is a diffeomorphism from
to
The inverse of this map is given by \(\mu_{J_{0}}^{1} \colon A \mapsto (A I)J_{0}(A  I)^{1}\).
Proof
One can view \(\mathcal{A}^{*}_{J_{0}}(V)\) as the set of J _{0}complex antilinear maps that do not have any eigenvalue equal to 1. Using the equations (J−J _{0})J _{0}=−J(J−J _{0}) and (J+J _{0})J _{0}=J(J+J _{0}), one sees that the image of \(\mu_{J_{0}}\) consists of J _{0}complex antilinear maps, and \(\mu_{J_{0}}(J)  I = 2 (J + J_{0})^{1} J_{0}\) is invertible. □
Lemma A.2
(Sévennec)
Let (V,ω) be a finite dimensional symplectic vector space and denote by \({\mathcal{J}}_{t}(\omega) \subset {\mathcal{J}}(V)\) the space of complex structures tamed by ω. Choosing any \(J_{0} \in {\mathcal{J}}_{t}(\omega)\), it follows that \({\mathcal{J}}_{t}(\omega)\) lies in \({\mathcal{J}}_{J_{0}}^{*}(V)\), and the image of \({\mathcal{J}}_{t}(\omega)\) under the associated map \(\mu_{J_{0}}\) is a convex domain in \(\mathcal{A}_{J_{0}}^{*}(V)\).
We first explain how to prove Proposition 2.1 using the above lemma. Suppose there is a complex structure J _{0} tamed by ω _{0} and ω _{1}. The space of cotamed complex structures \({\mathcal{J}}_{t}(\omega_{0}) \cap {\mathcal{J}}_{t}(\omega_{1})\) is then diffeomorphic under the map \(\mu_{J_{0}}\) to the intersection of the convex subsets given by the lemma. This intersection is again convex and hence contractible.
Proof of Lemma A.2
For any complex structure J tamed by ω, the endomorphism J+J _{0} is invertible because for any nonzero w, we have ω(w,(J+J _{0})w)>0, so in particular (J+J _{0})w is not zero. This proves the first part of the lemma.
Now fix a nonzero vector v∈V, and let C _{ v } be the set of \(A\in \operatorname{End}(V)\) that anticommute with J _{0}, and that satisfy
We now prove that \(C_{v}\subset \operatorname{End}(V)\) is convex. Every segment A _{ s }=(1−s)A _{0}+sA _{1} with s∈[0,1] for arbitrary A _{0},A _{1}∈C _{ v } defines a polynomial of degree 2
and the above inequality corresponds to checking that P(s) is positive for all values s∈[0,1]. The leading coefficient −ω((A _{1}−A _{0})v,J _{0}(A _{1}−A _{0})v) of P(s) is never positive, because J _{0} tames ω, so that P(s) is either a line or a parabola facing downward. In both cases P(s)≥min{P(0),P(1)}>0 for all s∈(0,1) so the inequality holds for the whole segment A _{ s }.
Note that C _{ v }≠∅ since 0∈C _{ v }. Define the intersection
which is a nonempty convex subset of \(\operatorname{End}(V)\). In fact, one has \(C^{*} \subset \mathcal{A}_{J_{0}}^{*}(V)\), because if there were a matrix A∈C ^{∗} with det(A−I)=0, then A would have an eigenvector w∈V with eigenvalue 1, but then −ω((A−I)w,J _{0}(A+I)w)=0 so that A∉C _{ w }.
Since C ^{∗} lies in the domain of \(\mu_{J_{0}}^{1}\) and \({\mathcal{J}}_{t}(\omega)\) lies in the domain of \(\mu_{J_{0}}\), we have \(C^{*} = \mu_{J_{0}}({\mathcal{J}}_{t}(\omega))\), so that the image of the complex structures tamed by ω is convex as we wanted to show. □
1.2 A.2 Existence of a cotamed complex structure
In this appendix, we prove Proposition 2.2, which we now recall:
Proposition 2.2
Let V be a finite dimensional real vector space equipped with two symplectic forms ω _{0} and ω _{1}. The following properties are equivalent:

(1)
the segment between ω _{0} and ω _{1} consists of symplectic forms

(2)
the ray starting at ω _{0} and directed by ω _{1} consists of symplectic forms

(3)
there is a complex structure J on V tamed both by ω _{0} and by ω _{1}.
The equivalence between (1) and (3) was explained to us by JeanClaude Sikorav. It relies on the simultaneous reduction of symplectic forms. Specifically, we need [38, Theorem 9.1] which we shall state (in a slightly weakened form) and reprove (in its full force) below as Proposition A.3, since the very general context of [38] makes it hard to read for people interested only in the symplectic case.
Recall that according to the linear Darboux theorem, any symplectic form on a 2ndimensional vector space is represented in some basis by the standard matrix
We now want to understand what can be said for a pair of symplectic structures. Below we give an approximate normal form which is sufficient for our purposes and more pleasant to state than the precise result (cf. [38, Theorem 9.1]), though the precise result can also be extracted from the proof that we will give at the end of this section.
Proposition A.3
Let ω _{0} and ω _{1} be symplectic forms on a finite dimensional vector space V. There exists a matrix A _{1} that splits into blocks of the form
for λ,ν≠0 with the following property: for any ε>0, there is a basis of V such that ω _{0} is represented by a block diagonal matrix with standard blocks Ω _{2k }, and ω _{1} is represented by a matrix which is εclose to A _{1}.
If the linear segment between ω _{0} and ω _{1} consists of symplectic forms, then the coefficients λ in the 2×2blocks of A _{1} described above cannot be negative.
The relation with cotamed complex structures will come from the following.
Proposition A.4

(a)
Let V=ℝ^{2} with two antisymmetric bilinear forms ω _{0} and ω _{1} defined by ω _{ j }(v,w)=v ^{t} A _{ j } w, where
If λ>0, then is tamed by both forms.

(b)
Let V=ℝ^{4}, and let ω _{0} and ω _{1} be antisymmetric bilinear forms defined by the matrixes
with μ≠0. Then there exists a complex structure J on ℝ^{4} that is tamed by both forms.
Proof
We only need to prove (b). For simplicity write V as ℂ^{2}, and the matrices A _{0} and A _{1} as
with z=λ+iμ=re ^{iψ}. The matrices
define complex structures on V, and it follows that is positive definite if cosϕ<0, and is positive definite if cos(ψ−ϕ)<0. As long as ψ≠π (which we have excluded by requiring that μ≠0), it follows that we can choose ϕ such that ϕ∈(π/2,3π/2) and ϕ−ψ∈(π/2,3π/2)+2πℤ. □
Proof of Proposition 2.2
We first explain the easy equivalence between (1) and (2). The (open) ray starting at ω _{0} and directed by ω _{1} and the open interval between ω _{0} and ω _{1} span the same cone in the space of antisymmetric bilinear forms. Since being symplectic is invariant under nonzero scalar multiplication, we have the equivalence.
The implication (3) \(\implies\) (1) is also direct because, for any t∈[0,1], we have
which is positive whenever v∈V is nonzero. So in particular, no such v can be in the kernel of an element of the segment between ω _{0} and ω _{1}.
To prove (1) \(\implies\) (3), we use the fact that by Proposition A.3, there is a matrix \(A_{1}'\) that splits into certain standard blocks, such that we can find for any ε>0 a basis of V for which ω _{0} is in canonical form, and for which ω _{1} is represented by a matrix that is εclose to \(A_{1}'\).
If condition (1) holds, then the blocks of \(A_{1}'\) correspond to the ones described in Proposition A.4, and we obtain the existence of a complex structure J on V that is tamed both by the standard symplectic form and by \(A_{1}'\). By choosing ε>0 sufficiently small, it follows that J is also tamed by ω _{0} and ω _{1}, because tameness is an open condition. □
Proof of Proposition A.3
The proof will proceed in several steps.
Decomposition into generalized eigenspaces. In the first step we shall decompose V into suitable subspaces that are both ω _{0} and ω _{1}orthogonal.
Let φ _{ r }:V→V ^{∗} for r=0,1 be the isomorphisms defined by φ _{ r }(v):=ω _{ r }(v,⋅). We consider the endomorphism \(B = \varphi_{0}^{1} \circ \varphi_{1}\) of V so that ω _{1}(v,w)=ω _{0}(Bv,w). The endomorphism B is invertible and it is ω _{0}symmetric since:
To define the generalized eigenspaces of B, complexify the vector space V to obtain V ^{ℂ}, and extend the ω _{ r } to sesquilinear forms \(\omega_{r}^{{\mathbb {C}}}\). A computation analogous to the preceding one shows that B is \(\omega_{0}^{{\mathbb {C}}}\)symmetric and we still have \(\omega_{0}^{{\mathbb {C}}}(v,Bw) = \omega_{1}^{{\mathbb {C}}}(v,w)\).
The characteristic polynomial of B splits over ℂ as \(P(X) = \prod_{\lambda}(X  \lambda)^{m_{\lambda}}\), so we can decompose V ^{ℂ} into generalized eigenspaces
Lemma A.5
If λ and μ are eigenvalues of B such that \(\lambda \neq \bar{\mu}\), then \(E^{{\mathbb {C}}}_{\lambda}\) and \(E^{{\mathbb {C}}}_{\mu}\) are both \(\omega^{{\mathbb {C}}}_{0}\) and \(\omega^{{\mathbb {C}}}_{1}\)orthogonal.
Proof
We prove by induction on k and l that ker(B−λ)^{k} and ker(B−μ)^{l} are orthogonal.
To start the induction, note that if v _{ λ }∈ker(B−λ), and v _{ μ }∈ker(B−μ), then
thus since \(\lambda \ne \bar{\mu}\), it follows that \(\omega_{0}^{{\mathbb {C}}}(v_{\lambda}, v_{\mu}) = 0\). Similarly, \(\omega_{1}^{{\mathbb {C}}}(v_{\lambda}, v_{\mu}) = \omega_{0}^{{\mathbb {C}}}(v_{\lambda}, B v_{\mu}) = \mu \omega_{0}^{{\mathbb {C}}}(v_{\lambda}, v_{\mu}) = 0\).
Assume now it has already been shown for the integers k and l that ker(B−λ)^{k} and ker(B−μ)^{l} are both \(\omega_{0}^{{\mathbb {C}}}\) and \(\omega_{1}^{{\mathbb {C}}}\)orthogonal. Choose a vector \(v_{\lambda}' \in \ker (B  \lambda)^{k+1}\) and use the fact that \(Bv_{\lambda}' = \lambda v_{\lambda}' + w\) for some w∈ker(B−λ)^{k}. Then we obtain for any v _{ μ }∈ker(B−μ)^{l},
and also \(\omega_{1}^{{\mathbb {C}}}(v_{\lambda}', v_{\mu}) = \omega_{0}^{{\mathbb {C}}}(B v_{\lambda}', v_{\mu}) = \bar{\lambda}\omega_{0}^{{\mathbb {C}}}(v_{\lambda}', v_{\mu}) + \omega_{0}^{{\mathbb {C}}}(w, v_{\mu}) = 0\), which proves the induction step from (k,l) to (k+1,l). Since λ and μ have completely symmetric roles, this also explains how to go to (k,l+1). □
We now relate this decomposition of V ^{ℂ} to the initial real vector space V. For a real eigenvalue λ, the intersection \(V\cap E_{\lambda}^{{\mathbb {C}}}\) defines a real subspace E _{ λ } with \(\dim_{{\mathbb {R}}}E_{\lambda}= \dim_{{\mathbb {C}}}E_{\lambda}^{{\mathbb {C}}}\). Complex conjugation defines an isomorphism \(E_{\lambda}^{{\mathbb {C}}}\to E_{\bar{\lambda}}^{{\mathbb {C}}}, v_{\lambda}\mapsto \bar{v}_{\lambda}\), and we can write \(V\cap (E_{\lambda}^{{\mathbb {C}}}\oplus E_{\bar{\lambda}}^{{\mathbb {C}}})\) for λ∈ℂ∖ℝ as the direct sum of real subspaces \(E_{\{\lambda,\bar{\lambda}\}} = \{v+\bar{v} \mid v\in E_{\lambda}^{{\mathbb {C}}}\} \oplus \{i (v\bar{v}) \mid v\in E_{\lambda}^{{\mathbb {C}}}\}\).
This way we find a decomposition of V into pairwise ω _{0} and ω _{1}orthogonal subspaces
with μ _{1},…,μ _{ k }∈ℝ∖{0}, and λ _{1},…,λ _{ l }∈ℂ∖ℝ.
Blocks with real eigenvalue. For the following considerations, we restrict to one of the subspaces \(E_{\lambda_{j}}\) with λ _{ j }∈ℝ, and denote λ _{ j } for simplicity just by λ. We will construct a basis of E _{ λ } such that ω _{0} and ω _{1} have the particularly nice form described in the proposition. Note that ω _{0} and ω _{1} are both nondegenerate on E _{ λ }.
Let k+1 be the nilpotency index of B−λ, i.e. (B−λ)^{k+1}=0 and (B−λ)^{k}≠0. Let v _{0} be an element of E _{ λ } not in ker(B−λ)^{k}. We set v _{ j }:=ε ^{−j}(B−λ)^{j} v _{0} to define a collection of vectors v _{0},…,v _{ k }. Choose now a vector w _{ k }∈E _{ λ } with ω _{0}(v _{ k },w _{ k })=1 and ω _{0}(v _{ j },w _{ k })=0 for every j≠k, and define inductively w _{ j−1}:=ε ^{−1}(B−λ)w _{ j }, or equivalently
for j≥1.
Lemma A.6
The vectors v _{0},…,v _{ k },w _{0},…,w _{ k } are linearly independent and satisfy the relations ω _{ r }(v _{ j },v _{ j′})=ω _{ r }(w _{ j },w _{ j′})=0 for all r=0,1, and j,j′, and
Proof
We start by proving ω _{ r }(v _{ j },v _{ j′})=0. For this we will use an induction on j−j′. If j−j′=0 then the statement follows directly from the antisymmetry of ω _{ r }. Suppose that the claim is true for j−j′≤m and consider any j and j′ with j−j′=m+1 (in particular j≥1). We have
by the induction hypothesis. Using the fact that Bv _{ j′}=εv _{ j′+1}+λv _{ j′}, we compute
The first term is zero by the induction hypothesis and the second one is zero because of the preceding computation. The proof of ω _{ r }(w _{ j },w _{ j′})=0 follows the same lines, and will be omitted.
Note that
and in particular this implies that v _{0},…,v _{ k },w _{0},…,w _{ k } are linearly independent vectors with respect to which ω _{0} has standard form.
The remaining relation for ω _{1} can be obtained by
□
If we restrict ω _{0} and ω _{1} to the subspace \(E = \operatorname{span}(v_{0},\dotsc,v_{k}, w_{0},\dotsc,w_{k})\) and represent them in this basis, we now find that ω _{0} is in standard form Ω _{2k } and ω _{1} is represented by a matrix εclose to λΩ _{2k }.
To continue the proof, restrict ω _{0}, ω _{1}, and B to the ω _{0}symplectic complement E′ of the space E. Note that E′ is stable under B because for u∈E′,
and similarly for ω _{0}(w _{ j },Bu)=0. We can thus proceed as before to reduce all eigenspaces E _{ λ } with λ∈ℝ to ω _{0}symplectic blocks in normal form.
Blocks with complex eigenvalue. We proceed now to the generalized complex eigenspace \(E^{{\mathbb {C}}}_{\lambda}\) with λ∈ℂ∖ℝ. Let k be the largest integer for which \(E^{{\mathbb {C}}}_{\lambda}\ne \ker (B\lambda)^{k}\), and construct as before a chain of vectors \(v_{0},\dotsc,v_{k} \in E^{{\mathbb {C}}}_{\lambda}\) by starting with an element \(v_{0} \in E^{{\mathbb {C}}}_{\lambda}\setminus \ker (B\lambda)^{k}\), and defining inductively
Using complex conjugation, we also find a chain \(\bar{v}_{0},\dotsc, \bar{v}_{k}\) that lies in \(E^{{\mathbb {C}}}_{\bar{\lambda}}\). Since B is the complexification of a real linear map, \(\bar{v}_{j+1} := \varepsilon^{1} (B \bar{\lambda}) \bar{v}_{j}\) holds.
Next, we define two chains w _{0},…,w _{ k } in \(E^{{\mathbb {C}}}_{\bar{\lambda}}\) and \(\bar{w}_{0},\dotsc, \bar{w}_{k}\) in \(E^{{\mathbb {C}}}_{\lambda}\) by starting with a vector \(w_{k} \in E^{{\mathbb {C}}}_{\bar{\lambda}}\) with \(\omega^{{\mathbb {C}}}_{0}(v_{k},w_{k}) = 1\) and \(\omega^{{\mathbb {C}}}_{0}(v_{j},w_{k}) = 0\) for every j≠k, and defining \(w_{j1} := \varepsilon^{1} (B\bar{\lambda}) w_{j}\), or equivalently
for j≥1. Similarly, we obtain \(\bar{w}_{j1} = \varepsilon^{1} (B \lambda) \bar{w}_{j}\).
Lemma A.7

(a)
The space spanned by \(v_{0},\dotsc, v_{k  1}, \bar{v}_{0}, \dotsc, \bar{v}_{k  1}\) and the one spanned by \(w_{0},\dotsc, w_{k  1}, \bar{w}_{0}, \dotsc, \bar{w}_{k  1}\) are each isotropic with respect to both ω _{0} and ω _{1}.

(b)
The \(\omega_{0}^{{\mathbb {C}}}\)pairings for these vectors are given by

(c)
The \(\omega_{1}^{{\mathbb {C}}}\)pairings for these vectors are given by
Proof
To prove (a) note that since \(\lambda \ne \bar{\lambda}\), the spaces \(E^{{\mathbb {C}}}_{\lambda}\) and \(E^{{\mathbb {C}}}_{\bar{\lambda}}\) are both \(\omega_{0}^{{\mathbb {C}}}\) and \(\omega_{1}^{{\mathbb {C}}}\)isotropic, so we only need to show that \(\omega_{r}^{{\mathbb {C}}}(\bar{v}_{j}, v_{j'}) = \omega_{r}^{{\mathbb {C}}}(\bar{w}_{j}, w_{j'}) = 0\) for all j,j′, and for r=0,1. If j=j′, we write v _{ j } as v _{ x }+iv _{ y }, and we use sesquilinearity as follows:
By the same computation, \(\omega_{1}^{{\mathbb {C}}}(\bar{v}_{j}, v_{j}) = 0\).
If the statement is true for j′−j=m≥0, then
and
which finishes the induction. The argument for \(\omega_{r}^{{\mathbb {C}}}(\bar{w}_{j}, w_{j'})\) is identical.
To prove (b), note first that the second two equations are the complex conjugate of the first two. Since \(v_{j}, \bar{w}_{j'} \in E_{\lambda}^{{\mathbb {C}}}\), it also follows immediately that \(\omega_{0}^{{\mathbb {C}}}(\bar{v}_{j}, \bar{w}_{j'}) = 0\), so that we are only left with showing \(\omega_{0}^{{\mathbb {C}}}(v_{j}, w_{j'}) = \delta_{j,j'}\), but the required computation is identical to the one used to show the analogous relation in the proof of Lemma A.6.
The equalities for (c) follow similarly. □
We will now intersect the complex subspace spanned by the chains defined above with the initial real vector space V to finish the proof of the proposition. For this, define for all j≤k the real vectors
and
which all lie in \(E_{\lambda, \bar{\lambda}}\). Using the results deduced above, we obtain for all r=0,1, and j,j′ the equations \(\omega_{r}(v_{j}^{+}, v_{j'}^{\pm}) = \omega_{r}(v_{j}^{}, v_{j'}^{\pm}) = 0\) and \(\omega_{r}(w_{j}^{+}, w_{j'}^{\pm}) = \omega_{r}(w_{j}^{}, w_{j'}^{\pm}) = 0\), and finally
and similar computations for the other matrix elements, which prove the desired result with \(\mu = \operatorname{Re}\lambda\) and \(\nu = \operatorname{Im}\lambda\).
Sign of real eigenvalues. Assume that all 2forms in the family
for t∈[0,1] are nondegenerate. The λcoefficients in the 2×2blocks of \(A_{1}'\) correspond to the real eigenvalues of the map B, so that if λ<0 with eigenvector v, then we have ω _{1}(v,⋅)=ω _{0}(Bv,⋅)=λω _{0}(v,⋅), and it follows that ω _{ t }(v,⋅)=(1−t+tλ)ω _{0}(v,⋅) has to vanish for a certain value t _{0}∈(0,1), so that \(\omega_{t_{0}}\) is degenerate. □
Rights and permissions
About this article
Cite this article
Massot, P., Niederkrüger, K. & Wendl, C. Weak and strong fillability of higher dimensional contact manifolds. Invent. math. 192, 287–373 (2013). https://doi.org/10.1007/s0022201204125
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s0022201204125