Symplectic fillings of asymptotically dynamically convex manifolds II–k-dilations

On fillings of contact links of quotient singularities

“…However, one should not expect the (iterated) planar property to hold in general, as there are already counterexamples in dimension 3 [8]. On the other hand, from the symplectic cohomology point of view in this paper, when the terminality condition is removed, we expect fillings of $$(S^{2n-1}/G,\xi _{\rm std})$$ to have $$k$$‐semi‐dilations [74], see [76] for the special case in complex dimension 2. The existence of $$k$$‐semi‐dilations, which is also an existence of holomorphic curves with a point constraint, leads to obstructions to co‐fillablity by a similar argument.…”

“…On the other hand, from the symplectic cohomology point of view in this paper, when the terminality condition is removed, we expect fillings of (𝑆 2𝑛−1 ∕𝐺, 𝜉 std ) to have 𝑘-semi-dilations [74], see [76] for the special case in complex dimension 2. The existence of 𝑘-semi-dilations, which is also an existence of holomorphic curves with a point constraint, leads to obstructions to co-fillablity by a similar argument.…”

On the intersection form of fillings

“…Based on "homological" foliations, various generalizations of the Eliashberg-Floer-McDuff theorem were obtained [1,5,10,17]. On the other hand, we studied the filling question from the perspective of Floer theories and obtained various uniqueness results [20,21,22,23,24,25]. In this note, we show the uniqueness of the integral intersection form for exact fillings of some flexibly fillable contact manifolds, which shall yield uniqueness of diffeomorphism types in some cases.…”

“…This notion was generalized by Lazarev [13] to the notion of asymptotically dynamically convex (ADC) manifolds to contain examples like flexibly fillable contact manifolds with vanishing first Chern class. Several structural maps on (S 1 -equivariant) symplectic cohomology of exact fillings of ADC manifolds are independent of topologically simple fillings [21,24]. Those topological conditions are used to get a Z grading for the symplectic cohomology generated by contractible orbits, as the ADC condition only requires that µ CZ (γ) + n − 3 > 0 for contractible Reeb orbits γ (which has a canonical Z-valued Conley-Zehnder index, as c 1 (ξ) = 0).…”