1 Donaldson hypersurfaces and symplectic polarisations

Let \((M,\omega )\) be a closed, connected, integral symplectic manifold, that is, the de Rham cohomology class \([\omega ]_{\mathrm {dR}}\) lies in the image of the homomorphism \(H^2(M)\rightarrow H^2_{\mathrm {dR}}(M)=H^2(M;{\mathbb {R}})\) induced by the inclusion \({\mathbb {Z}}\rightarrow {\mathbb {R}}\). The cohomology classes in \(H^2(M)\) mapping to \([\omega ]_{\mathrm {dR}}\) are called integral lifts, and by abuse of notation, we shall write \([\omega ]\) for any such lift. Following McDuff and Salamon [11, Section 14.5], we call a codimension 2 symplectic submanifold \(\Sigma \subset M\) a Donaldson hypersurface if it is Poincaré dual to \(d[\omega ]\in H^2(M)\) for some integral lift \([\omega ]\) and some (necessarily positive) integer d. Donaldson [5] has established the existence of such hypersurfaces for any sufficiently large d.

The pair \((M,\Sigma )\) is called a polarisation of \((M,\omega )\), and the number \(d\in {\mathbb {N}}\), the degree of the polarisation. Biran and Cieliebak [2] studied these polarisations in the Kähler case, where \(\omega \) admits a compatible integrable almost complex structure J. In that setting, the complement \((M{\setminus }\Sigma ,J)\) admits in a natural way the structure of a Stein manifold.

As shown recently by Giroux [8], building on work of Cieliebak–Eliashberg, even in the non-Kähler case the complement of a symplectic hypersurface \(\Sigma \subset M\) found by Donaldson’s construction admits the structure of a Stein manifold. Here, of course, the complex structure on \(M{\setminus }\Sigma \) does not, in general, extend over \(\Sigma \). Complements of Donaldson hypersurfaces are also studied in [4].

2 Subcritical polarisations

The focus of Biran and Cieliebak [2] lay on subcritical polarisations of Kähler manifolds, which means that \((M{\setminus }\Sigma ,J)\) admits a plurisubharmonic Morse function \(\varphi \) all of whose critical points have, for \(\dim M=2n\), index less than n. (They also assumed that \(\varphi \) coincides with the plurisubharmonic function defining the natural Stein structure outside a compact set containing all critical points of \(\varphi \).)

More generally, McDuff and Salamon [11, p. 504] propose the study of polarisations \((M,\Sigma )\) where the complement \(M{\setminus }\Sigma \) is homotopy equivalent to a subcritical Stein manifold (of finite type). We relax this condition a little further and call \((M,\Sigma )\) homologically subcritical if \(M{\setminus }\Sigma \) 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 \(\ell \le n-1\) such that \(H_k(M{\setminus }\Sigma )\) vanishes for \(k\ge \ell +1\) and \(H_{\ell }(M{\setminus }\Sigma )\) is torsion-free.

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 [6, p. 661]. A missing assumption \(c_1(M{\setminus }\Sigma )=0\) of that argument and a few more details—still short of a complete proof—were added by J. He [9, Proposition 4.2], who appeals to Gromov–Witten theory and polyfolds.

Here is our main result, which entails the conjecture of Biran–Cieliebak.

Theorem 1

Let \((M,\omega )\) be a closed, integral symplectic manifold, and \(\Sigma \subset M\) a compact symplectic submanifold of codimension 2, Poincaré dual to the integral cohomology class \(d[\omega ]\) for some (positive) integer d. If \((M,\Sigma )\) is homologically subcritical, then \(d[\omega ]/\mathrm {torsion}\) is indivisible in \(H^2(M)/\mathrm {torsion}\). In particular, \(d=1\).

Notation

The free part of \(H^2(M)\) is not a well-defined subgroup of \(H^2(M)\), but the torsion part is, and so is the quotient group \(H^2(M)/\mathrm {torsion}\), which is free abelian. We write \(d[\omega ]/\mathrm {torsion}\) for the class represented by \(d[\omega ]\) in this quotient. This class is determined by \([\omega ]_{\mathrm {dR}}\) and does not depend on the choice of integral lift \([\omega ]\).

Our proof is devoid of any sophisticated machinery. The assumption on \((M,\Sigma )\) to be homologically subcritical guarantees the surjectivity of a certain homomorphism in homology described by Kulkarni and Wood [10]; this implies the claimed indivisibility.

3 The Kulkarni–Wood homomorphism

We consider a pair \((M,\Sigma )\) consisting of a closed, connected, oriented manifold M of dimension 2n, and a compact, oriented hypersurface \(\Sigma \subset M\) of codimension 2. No symplectic assumptions are required in this section.

Write \(i: \Sigma \rightarrow M\) for the inclusion map. The Poincaré duality isomorphisms on M and \(\Sigma \) from cohomology to homology, given by capping with the fundamental class, are denoted by \(\mathrm {PD}_M\) and \(\mathrm {PD}_{\Sigma }\), respectively.

In their study of the topology of complex hypersurfaces, Kulkarni and Wood [10] used the following composition, which we call the Kulkarni–Wood homomorphism:

$$\begin{aligned} \begin{array}{rl} \Phi _{\mathrm {KW}}: &{} H^k(M) {\mathop {\longrightarrow }\limits ^{i^*}} H^k(\Sigma ) {\mathop {\longrightarrow }\limits ^{\mathrm {PD}_{\Sigma }}} H_{2n-2-k}(\Sigma ) {\mathop {\longrightarrow }\limits ^{i_*}}\\ &{} {\mathop {\longrightarrow }\limits ^{i_*}} H_{2n-2-k}(M) {\mathop {\longrightarrow }\limits ^{\mathrm {PD}_M^{-1}}} H^{k+2}(M). \end{array} \end{aligned}$$

Lemma 2

The Kulkarni–Wood homomorphism equals the cup product with the cohomology class \(\sigma :=\mathrm {PD}_M^{-1}(i_*[\Sigma ])\in H^2(M)\).

Proof

For \(\alpha \in H^k(M)\), we compute

$$\begin{aligned} \begin{array}{rcl} \Phi _{\mathrm {KW}}(\alpha ) &{} = &{} \mathrm {PD}_M^{-1}\,i_*\,\mathrm {PD}_{\Sigma }\,i^*\alpha \; = \; \mathrm {PD}_M^{-1}\,i_*\bigl (i^*\alpha \cap [\Sigma ]\bigr )\\ &{} = &{} \mathrm {PD}_M^{-1}\bigl (\alpha \cap i_*[\Sigma ]\bigr ) \; = \; \mathrm {PD}_M^{-1}\bigl (\alpha \cap \mathrm {PD}_M(\sigma )\bigr )\\ &{} = &{} \mathrm {PD}_M^{-1}\bigl (\alpha \cap (\sigma \cap [M])\bigr ) \; = \; \mathrm {PD}_M^{-1}\bigl ((\alpha \cup \sigma )\cap [M]\bigr )\\ &{} = &{} \alpha \cup \sigma ; \end{array} \end{aligned}$$

all the formulae for cup and cap products used in this chain of identities can be found in [3, Theorem VI.5.2]. \(\square \)

Lemma 3

If the complement \(M{\setminus }\Sigma \) has the homology type of a CW-complex of dimension \(\ell \) for some \(\ell \le n-1\), then \(\Phi _{\mathrm {KW}}: H^k(M)\rightarrow H^{k+2}(M)\) is surjective in the range \(\ell -1\le k\le 2n-\ell -2\).

Proof

Write \(\nu \Sigma \) for an open tubular neighbourhood of \(\Sigma \) in M. By homotopy, excision, duality, and the universal coefficient theorem, we have

$$\begin{aligned} \begin{array}{rcl} H_k(M,\Sigma ) &{} \cong &{} H_k(M,\nu \Sigma ) \; \cong \; H_k(M{\setminus }\nu \Sigma ,\partial (\nu \Sigma ))\\ &{} \cong &{} H^{2n-k}(M{\setminus }\nu \Sigma )\\ &{} \cong &{} FH_{2n-k}(M{\setminus }\Sigma ) \oplus TH_{2n-k-1}(M{\setminus }\Sigma ), \end{array} \end{aligned}$$

where F and T denote the free and the torsion part, respectively. This vanishes for \(2n-k-1\ge \ell \), that is, for \(k\le 2n-\ell -1\). It follows that the homomorphism \(i_*: H_{2n-2-k}(\Sigma )\rightarrow H_{2n-2-k}(M)\) is surjective for \(2n-2-k\le 2n-\ell -1\), or \(k\ge \ell -1\).

Similarly (or directly by the Poincaré–Lefschetz duality), we have

$$\begin{aligned} H^k(M,\Sigma )\cong H_{2n-k}(M{\setminus }\Sigma ), \end{aligned}$$

which vanishes for \(2n-k\ge \ell +1\), that is, for \(k\le 2n-\ell -1\). Hence, the homomorphism \(i^*: H^k(M)\rightarrow H^k(\Sigma )\) is surjective for \(k+1\le 2n-\ell -1\), that is, for \(k\le 2n-\ell -2\). \(\square \)

4 Proof of Theorem 1

Under the assumptions of Theorem 1, the homomorphism \(\Phi _{\mathrm {KW}}: H^k(M)\rightarrow H^{k+2}(M)\) is surjective at least in the range \(n-2\le k\le n-1\); simply set \(\ell =n-1\) in Lemma 3. 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 \([\omega ]^{m+1}\) of infinite order.

On the other hand, \(\Phi _{\mathrm {KW}}\) is given by the cup product with \(d[\omega ]\), as shown in Lemma 2, so we have the commutative diagram

If \(d[\omega ]/\mathrm {torsion}\) were divisible, the map on free abelian groups at the bottom of this diagram would not be surjective, and neither would be \(\Phi _{\mathrm {KW}}\).

Remark 4

The real Euler class of the circle bundle \(\partial (\nu \Sigma )\) equals \(d[\omega ]_{\mathrm {dR}}\), and the natural Boothby–Wang contact structure on this bundle has an exact convex filling by the complement \(M{\setminus }\nu \Sigma \), see [7, Lemma 3], [8, Proposition 5], and [4, 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.