Strong symplectic fillability of contact torus bundles

Abstract

Abstract. In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting π in S 1 -direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting π in S 1 -direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually overtwisted tight contact structure on the negative parabolic torus bundle with monodromy −T n (n < 0), we prove that it is Stein fillable. By the way, we give a partial answer to a conjecture of Golla and Lisca.

Local geometry of symplectic divisors with applications to contact torus bundles

“…Denote the original augmented graph by (Γ (1) , a (1) ) and the blown-up graph (Γ (2) , a (2) ). Note that Q Γ (2) z (2) = a (2) is still solvable after the augmented toric blow-up. If z (1) = (z 1 , z 2 , .…”

“…. ) satisfy Q Γ (1) z (1) = z (1) , then after blow-up of area a 0 , z (2) = (z 1 , z 1 + z 2 − 2πa 0 , z 2 , . .…”

“…So we could apply GS construction to both augmented graphs. In the following, we will denote the construction based on (Γ (1) , a (1) ) by GS-1 and denote the construction based on (Γ (2) , a (2) ) by GS-2.…”

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

“…. We do not have a closed-form description of f , but one can work out that f takes the values 0, 4,8,9,13, and 16 for t = 0, 1, 2, 3, 4, and 5 respectively. It is not hard to see that if we allow real values for d i , then the minimum of d 2 i occurs when m = 1 and d 1 = d, and satisfies d 2 − d = 2t.…”

The Kodaira Dimension of Contact 3-Manifolds and Geography of Symplectic Fillings

“…Remark 3.13. It was pointed out to us by Youlin Li that Etnyre has an argument to show that gluing a minimal strong symplectic filling and a Stein cobordism results in a minimal strong symplectic filling (See Proposition 2.3 in [6]). In general, it is not true that gluing a minimal symplectic filling with a minimal symplectic cobordism results in a minimal symplectic filling.…”