Subcritical polarisations of symplectic manifolds have degree one

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni–Wood.


Donaldson hypersurfaces and symplectic polarisations
Let (M, ω) be a closed, connected, integral symplectic manifold, that is, the de Rham cohomology class [ω] dR lies in the image of the homomorphism H 2 (M ) → H 2 dR (M ) = H 2 (M ; R) induced by the inclusion Z → R. The cohomology classes in H 2 (M ) mapping to [ω] dR are called integral lifts, and by abuse of notation we shall write [ω] for any such lift.Following McDuff and Salamon [10, Section 14.5], we call a codimension 2 symplectic submanifold Σ ⊂ M a Donaldson hypersurface if it is Poincaré dual to d[ω] ∈ H 2 (M ) for some integral lift [ω] and some (necessarily positive) integer d.Donaldson [4] has established the existence of such hypersurfaces for any sufficiently large d.
The pair (M, Σ) is called a polarisation of (M, ω), and the number d ∈ N, the degree of the polarisation.Biran and Cieliebak [2] studied these polarisations in the Kähler case, where ω admits a compatible integrable almost complex structure J.In that setting, the complement (M \ Σ, J) admits in a natural way the structure of a Stein manifold.
As shown recently by Giroux [7], building on work of Cieliebak-Eliashberg, even in the non-Kähler case the complement of a symplectic hypersurface Σ ⊂ M found by Donaldson's construction admits the structure of a Stein manifold.Here, of course, the complex structure on M \ Σ does not, in general, extend over Σ. Complements of Donaldson hypersurfaces are also studied in [3].

Subcritical polarisations
The focus of Biran and Cieliebak [2] lay on subcritical polarisations of Kähler manifolds, which means that (M \ Σ, J) admits a plurisubharmonic Morse function ϕ all of whose critical points have, for dim M = 2n, index less than n.(They also assumed that ϕ coincides with the plurisubharmonic function defining the natural Stein structure outside a compact set containing all critical points of ϕ.) 2020 Mathematics Subject Classification.53D35, 57R17, 57R19, 57R95.This research is part of a project in the SFB/TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics, funded by the DFG (Project-ID 281071066 -TRR 191).[10, p. 504] propose the study of polarisations (M, Σ) where the complement M \ Σ is homotopy equivalent to a subcritical Stein manifold (of finite type).We relax this condition a little further and call (M, Σ) homologically subcritical if M \ Σ has the homology of a subcritical Stein manifold, that is, of a CW-complex containing finitely many cells up to dimension at most n−1.This means that there is some ℓ

More generally, McDuff and Salamon
Motivated by the many examples they could construct, Biran and Cieliebak [2, p. 751] conjectured that subcritical polarisations necessarily have degree 1.They suggested an approach to this conjecture using either symplectic or contact homology.A rough sketch of a proof along these lines, in the language of symplectic field theory, was given by Eliashberg-Givental-Hofer [5, p. 661].A missing assumption c 1 (M \ Σ) = 0 of that argument and a few more details -still short of a complete proof -were added by J.He [8, Proposition 4.2], who appeals to Gromov-Witten theory and polyfolds.
Here is our main result, which entails the conjecture of Biran-Cieliebak.Our proof is devoid of any sophisticated machinery.The assumption on (M, Σ) to be homologically subcritical guarantees the surjectivity of a certain homomorphism in homology described by Kulkarni and Wood [9]; this implies the claimed indivisibility.

The Kulkarni-Wood homomorphism
We consider a pair (M, Σ) consisting of a closed, connected, oriented manifold M of dimension 2n, and a compact, oriented hypersurface Σ ⊂ M of codimension 2. No symplectic assumptions are required in this section.
Write i : Σ → M for the inclusion map.The Poincaré duality isomorphisms on M and Σ from cohomology to homology, given by capping with the fundamental class, are denoted by PD M and PD Σ , respectively.
In their study of the topology of complex hypersurfaces, Kulkarni and Wood [9] used the following composition, which we call the Kulkarni-Wood homomorphism: Lemma 2. The Kulkarni-Wood homomorphism equals the cup product with the cohomology class σ Proof.Write νΣ for an open tubular neighbourhood of Σ in M .By homotopy, excision, duality, and the universal coefficient theorem we have where F and T denotes the free and the torsion part, respectively.This vanishes for 2n Similarly (or directly by Poincaré-Lefschetz duality) we have

Proof of Theorem 1
Under the assumptions of Theorem 1, the homomorphism Φ KW : . Thus, we can pick an even number k = 2m in this range.The free part of H 2m+2 (M ) is non-trivial, since this cohomology group contains the element [ω] m+1 of infinite order.
On the other hand, Φ KW is given by the cup product with d[ω], as shown in Lemma 2. If d[ω]/torsion were divisible, so would be all elements in the image of Φ KW in H 2m+2 (M )/torsion, and Φ KW would not be surjective.

Remark 4 .
The real Euler class of the circle bundle ∂(νΣ) equals d[ω] dR , and the natural Boothby-Wang contact structure on this bundle has an exact convex filling by the complement M \ νΣ, see [6, Lemma 3], [7, Proposition 5] and [3, Lemma 2.2].With [1, Theorem 1.2] the condition 'homologically subcritical' of Theorem 1 may be replaced by assuming the existence of some subcritical Stein filling of this Boothby-Wang contact structure.