Calabi–Yau caps, uniruled caps and symplectic fillings

Abstract

We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, they imply that any exact filling of the standard contact structure on the unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinite family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admitting minimal strong fillings of arbitrarily large b 2 .Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness properties and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero.

Genus one Lefschetz fibrations on disk cotangent bundles of surfaces

“…Moreover, we hope that our work maybe used towards settling Wendl's conjecture [22,Conjecture 9.23]: Any exact filling of (ST * Σ, ξ can ) is Liouville deformation equivalent to (DT * Σ, ω can ). This conjecture is true for g = 1 as shown by Wendl [21] and some evidence was obtained recently to verify this conjecture affirmatively by Li, Mak and Yasui [12] and also by Sivek and Van-Horn Morris [19], who showed that any exact filling must, at least topologically, bear some resemblance to DT * Σ for g ≥ 2.…”

Fillings of Genus–1 Open Books and 4–Braids

“…As illustrated by our examples, admitting a Calabi-Yau cap [24] does not impose finiteness on possible homology groups of Stein or minimal symplectic fillings of a contact 3-manifold. However, admitting a uniruled cap in the sense of [24] might be sufficient for this, which would extend the case of contact 3-manifolds with support genus zero we covered in Proposition 4.…”

“…(Note that this means that we can embed all our Stein fillings (X n , J n ) into K3 surface.) That is, the contact 3-manifold (Y, ξ) admits a Calabi-Yau cap, which by [24] implies that the Betti numbers of Stein fillings of (Y, ξ) is finite. In particular, it cannot admit arbitrarily large Stein fillings.…”

“…[29,6,5]), and the explicit monodromy of a genus-1 open book with three binding components supporting the standard Stein fillable contact structure on T 3 (Lemma 1 and Proposition 2; also [34]). The small topology of these fillings will then allow us to show that many of these contact 3-manifolds admit symplectic Calabi-Yau caps introduced in [24], so as to conclude that they can not admit arbitrarily large Stein fillings (Theorem 7). We moreover observe that there are contact 3-manifolds supported by planar spinal open books [25,8] which also admit infinitely many Stein fillings, but do not admit arbitrarily large ones (Remark 13).…”

Geometry of symplectic log Calabi–Yau pairs

“…Definition 5.5 ([13], [15]). Let (W, ω) be a concave symplectic 4-manifold with contact boundary (Y, ξ).…”