Polarizations and symplectic isotopies

Abstract

The aim of this paper is to explain a link between symplectic isotopies of open objects such as balls and flexibility properties of symplectic hypersurfaces. We get connectedness results for spaces of symplectic ellipsoids or maximal packings of P 2 . IntroductionIn [3], Biran proved a decomposition theorem for rational Kähler manifolds which has proved useful in many situations such as symplectic packings [2, 20], Lagrangian embeddings [4] . . . This paper tries to add symplectic isotopies to this list of applications. In polarized symplectic manifolds -triples (M, ω, Σ) where ω ∈ H 2 (M, Z) and Σ is a symplectic hypersurface Poincaré-dual to a (necessarily positive) multiple kω of the symplectic form -, this decomposition result may be expressed as:Theorem 1 (Biran). In a polarized closed symplectic manifold (M, ω, Σ), there exists a zero-volume closed skeleton in M whose complement is a standard symplectic disc bundle supported by Σ.In fact, polarizations of sufficiently high degree exist on all closed (i.e., compact, without boundary) rational symplectic manifolds [7]. These manifolds split into a standard symplectic part -an explicit disc bundle over Σ -and a negligible skeleton, even isotropic in the Kähler case. Although no kind of uniqueness can be expected (see [3] for a discussion on the skeleton for instance), our next theorem explains how to construct many such decompositions, all of whose symplectic parts are isotopic. In order to state it, let us mention that both M \Σ and the complement of the zero-section L 0 in the symplectic disc bundle SDB(Σ, k) are exact symplectic manifolds, so they admit Liouville forms. In the statement below, λ 0 is a distinguished such form on SDB(Σ, k)\L 0 (see Section 1). Also, by a local embedding of (X, M ) into (Y, N ) -where X ⊂ M and Y ⊂ N -we always mean an 109

Packing stability for symplectic $4$-manifolds

“…To be precise we have the following. Moreover, arguing again as in [Ops13a], we see that these domains are the complement in P 2 of any such ellipsoids which are in a good position with the coordinate axis.…”

“…Since τ a and τ (a − 1) are less than a, Lemma 1.6 ensures that there exists a symplectic embedding ϕ of τ E(a − 1, a) ⊔ B(a i ) such that L 2 intersects τ E(a − 1, a) along its big axis (of size τ a) and avoids all the balls, while L 1 intersects τ E(a−1, a) along its small axis and avoids the balls. Arguing as in [Ops13a], Corollary 2.2, we can isotope ϕ(τ E(a− 1, a)) to the standard embedding ϕ 0 , keeping L 1 , L 2 fixed. As a result, we therefore get a packing of ψ : ⊔B(a i ) (a − 1, a))).…”

Ellipsoid embeddings and symplectic packing stability

“…Combining with work of Biran [2] and Opshtein [17], we also have a full filling result valid for general rational symplectic manifolds. Recall that the manifold (M, ω) is rational if [ω] ∈ H 2 (M, Q) ⊂ H 2 (M, R).…”

“…As a symplectic manifold, SDB(N, τ, m) is well defined up to isotopy. For more details see [2], section 2, or [17], section 1. Given the above, we can now state the Biran decomposition.…”

“…Theorem 1,[17],[2]). There exists a full filling SDB(N, τ, m) ֒→ (M, ω).By our induction hypothesis, there exits b 1 , .…”