On Stein fillings of contact torus bundles

Abstract

We consider a large family MJX-tex-caligraphicscriptF of torus bundles over the circle, and we use recent work of Li–Mak to construct, on each Y∈F, a Stein fillable contact structure ξY. We prove that (i) each Stein filling of (Y,ξY) has vanishing first Chern class and first Betti number, (ii) if Y∈F is elliptic, then all Stein fillings of (Y,ξY) are pairwise diffeomorphic and (iii) if Y∈F is parabolic or hyperbolic, then all Stein fillings of (Y,ξY) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y∈F we exhibit non‐homotopy equivalent Stein fillings of (Y,ξY).

Geometry of symplectic log Calabi–Yau pairs

“…Lemma 2.4 (cf. Theorem 2.5 and Theorem 3.1 in [6]). Let D be a cycle of spheres in X and V = X − N D .…”

“…In this section we review some homological facts about topological divisors, especially cycles of spheres, and we refer to [20], [6] and [11] for details. We first introduce a pair of basic operations for topological divisors.…”

“…Golla and Lisca considered a large family F of torus bundles and showed that these torus bundles are equipped with contact structures arising from Looijenga D with b + (Q D ) = 1 (Theorem 2.5 in [6]). They also showed, for a subfamily of these torus bundles, such a contact structure is the unique universally tight contact structure with vanishing Giroux torsion (Theorem 1.2 in [6]). This led them to formulate the following conjecture.…”

“…This led them to formulate the following conjecture. Moreover, they investigated Stein (and symplectic) fillings and classified in many cases up to diffeomorphism (Theorems 3.1, 3.2, 3.5 in [6]). On the other hand, Ohta and Ono classified symplectic fillings of simple elliptic singularities up to symplectic deformation (Theorems 1, 1', 2 in [22]).…”

“…[6]). For a concave cycle D of symplectic spheres, the contact structure ξ D on Y D is universally tight.…”