Subcritical polarisations of symplectic manifolds have degree one
aa r X i v : . [ m a t h . S G ] J a n SUBCRITICAL POLARISATIONS OF SYMPLECTICMANIFOLDS HAVE DEGREE ONE
HANSJ ¨ORG GEIGES, KEVIN SPORBECK, AND KAI ZEHMISCH
Abstract.
We show that if the complement of a Donaldson hypersurfacein a closed, integral symplectic manifold has the homology of a subcriticalStein manifold, then the hypersurface is of degree one. In particular, thisdemonstrates a conjecture by Biran and Cieliebak on subcritical polarisationsof symplectic manifolds. Our proof is based on a simple homological argumentusing ideas of Kulkarni–Wood. Donaldson hypersurfaces and symplectic polarisations
Let (
M, ω ) be a closed, connected, integral symplectic manifold, that is, the deRham cohomology class [ ω ] dR lies in the image of the homomorphism H ( M ) → H ( M ) = H ( M ; R ) induced by the inclusion Z → R . The cohomology classesin H ( M ) mapping to [ ω ] dR are called integral lifts , and by abuse of notation weshall 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 hypersur-face if it is Poincar´e dual to d [ ω ] ∈ H ( M ) for some integral lift [ ω ] and some(necessarily positive) integer d . Donaldson [4] has established the existence of suchhypersurfaces 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 polarisationsin the K¨ahler case, where ω admits a compatible integrable almost complex struc-ture J . In that setting, the complement ( M \ Σ , J ) admits in a natural way thestructure of a Stein manifold.As shown recently by Giroux [7], building on work of Cieliebak–Eliashberg, evenin the non-K¨ahler case the complement of a symplectic hypersurface Σ ⊂ M foundby Donaldson’s construction admits the structure of a Stein manifold. Here, ofcourse, the complex structure on M \ Σ does not, in general, extend over Σ. Com-plements of Donaldson hypersurfaces are also studied in [3].2.
Subcritical polarisations
The focus of Biran and Cieliebak [2] lay on subcritical polarisations of K¨ahlermanifolds, which means that ( M \ Σ , J ) admits a plurisubharmonic Morse function ϕ all of whose critical points have, for dim M = 2 n , index less than n . (They alsoassumed that ϕ coincides with the plurisubharmonic function defining the naturalStein structure outside a compact set containing all critical points of ϕ .) Mathematics Subject Classification.
Symplectic Structures in Geometry,Algebra and Dynamics , funded by the DFG (Project-ID 281071066 – TRR 191).
More generally, McDuff and Salamon [10, p. 504] propose the study of polarisa-tions ( M, Σ) where the complement M \ Σ is homotopy equivalent to a subcriticalStein 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 subcriticalStein manifold, that is, of a CW-complex containing finitely many cells up to di-mension at most n −
1. This means that there is some ℓ ≤ n − H k ( M \ Σ)vanishes for k ≥ ℓ + 1 and H ℓ ( M \ Σ) 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. Theysuggested an approach to this conjecture using either symplectic or contact homo-logy. A rough sketch of a proof along these lines, in the language of symplectic fieldtheory, was given by Eliashberg–Givental–Hofer [5, p. 661]. A missing assumption c ( M \ Σ) = 0 of that argument and a few more details — still short of a completeproof — were added by J. He [8, Proposition 4.2], who appeals to Gromov–Wittentheory and polyfolds.Here is our main result, which entails the conjecture of Biran–Cieliebak.
Theorem 1.
Let ( M, ω ) be a closed, integral symplectic manifold, and Σ ⊂ M a compact symplectic submanifold of codimension , Poincar´e dual to the integralcohomology class d [ ω ] for some (positive) integer d . If ( M, Σ) is homologicallysubcritical, then d [ ω ] / torsion is indivisible in H ( M ) / torsion . In particular, d = 1 . Our proof is devoid of any sophisticated machinery. The assumption on ( M, Σ)to be homologically subcritical guarantees the surjectivity of a certain homomor-phism in homology described by Kulkarni and Wood [9]; this implies the claimedindivisibility. 3.
The Kulkarni–Wood homomorphism
We consider a pair ( M, Σ) consisting of a closed, connected, oriented manifold M of dimension 2 n , 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 Poincar´e duality isomorphisms on M and Σ from cohomology to homology, given by capping with the fundamentalclass, 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 :Φ KW : H k ( M ) i ∗ −→ H k (Σ) PD Σ −→ H n − − k (Σ) i ∗ −→ H n − − k ( M ) PD − M −→ H k +2 ( M ) . Lemma 2.
The Kulkarni–Wood homomorphism equals the cup product with thecohomology class σ := PD − M ( i ∗ [Σ]) ∈ H ( M ) .Proof. For α ∈ H k ( M ) we computeΦ KW ( α ) = PD − M i ∗ PD Σ i ∗ α = PD − M i ∗ (cid:0) i ∗ α ∩ [Σ] (cid:1) = PD − M (cid:0) α ∩ i ∗ [Σ] (cid:1) = PD − M (cid:0) α ∩ PD M ( σ ) (cid:1) = PD − M (cid:0) α ∩ ( σ ∩ [ M ]) (cid:1) = PD − M (cid:0) ( α ∪ σ ) ∩ [ M ] (cid:1) = α ∪ σ. (cid:3) UBCRITICAL POLARISATIONS 3
Lemma 3.
If the complement M \ Σ has the homology type of a CW-complex ofdimension ℓ for some ℓ ≤ n − , then Φ KW : H k ( M ) → H k +2 ( M ) is surjective inthe range ℓ − ≤ k ≤ n − ℓ − .Proof. Write ν Σ for an open tubular neighbourhood of Σ in M . By homotopy,excision, duality, and the universal coefficient theorem we have H k ( M, Σ) ∼ = H k ( M, ν Σ) ∼ = H k ( M \ ν Σ , ∂ ( ν Σ)) ∼ = H n − k ( M \ ν Σ) ∼ = F H n − k ( M \ Σ) ⊕ T H n − k − ( M \ Σ) , where F and T denotes the free and the torsion part, respectively. This vanishesfor 2 n − k − ≥ ℓ , that is, for k ≤ n − ℓ −
1. It follows that the homomorphism i ∗ : H n − − k (Σ) → H n − − k ( M ) is surjective for 2 n − − k ≤ n − ℓ −
1, or k ≥ ℓ − H k ( M, Σ) ∼ = H n − k ( M \ Σ) , which vanishes for 2 n − k ≥ ℓ + 1, that is, for k ≤ n − ℓ −
1. Hence, the homo-morphism i ∗ : H k ( M ) → H k (Σ) is surjective for k + 1 ≤ n − ℓ −
1, that is, for k ≤ n − ℓ − (cid:3) Proof of Theorem 1
Under the assumptions of Theorem 1, the homomorphism Φ KW : H k ( M ) → H k +2 ( M ) is surjective at least in the range n − ≤ k ≤ n −
1; simply set ℓ = n − k = 2 m in this range. The freepart of H m +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 inLemma 2. If d [ ω ] / torsion were divisible, so would be all elements in the image ofΦ KW in H m +2 ( M ) / torsion, and Φ KW would not be surjective. Remark 4.
The real Euler class of the circle bundle ∂ ( ν Σ) equals d [ ω ] dR , andthe natural Boothby–Wang contact structure on this bundle has an exact convexfilling by the complement M \ ν Σ, see [6, Lemma 3], [7, Proposition 5] and [3,Lemma 2.2]. With [1, Theorem 1.2] the condition ‘homologically subcritical’ ofTheorem 1 may be replaced by assuming the existence of some subcritical Steinfilling of this Boothby–Wang contact structure.
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Mathematisches Institut, Universit¨at zu K¨oln, Weyertal 86–90, 50931 K¨oln, Ger-many
Email address : [email protected] Fakult¨at f¨ur Mathematik, Ruhr-Universit¨at Bochum, Universit¨atsstraße 150, 44780Bochum, Germany
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