Symplectic fillings of asymptotically dynamically convex manifolds I

Abstract

We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that vanishing of symplectic cohomology and existence of symplectic dilation are properties independent of the filling for ADC manifolds. Using them, various topological applications on symplectic fillings are obtained. We use the structure maps to define the first symplectic obstructions to Weinstein fillability. In particular, we show that for all dimension 4k + 3, k ≥ 1, there exist infinitely many contact manifolds that are exactly fillable, almost Weinstein fillable but not Weinstein fillable. The invariance of the structure maps generalizes to strong fillings with vanishing first Chern class. We show that any strong filling with vanishing first Chern class of a class of manifolds, including (S 2n−1 , ξ std ), ∂(T * L × C n ) with L simply connected, must be exact and have unique diffeomorphism type. As an application of the proof, we show that the existence of symplectic dilation implies uniruledness. In particular any affine exotic C n with non-negative log Kodaira dimension is a symplectic exotic C n . Contents1. Introduction 1 2. Symplectic Cohomology 6 3. Invariance of restriction map and persistence of vanishing 13 4. Persistence of symplectic dilation 21 5. Uniruledness 33 6. Constructions of ADC manifolds 37 7. Obstructions to exact cobordisms 47 8. Nonexact fillings 49 Appendix A. Orientations 56 References 59

$(\mathbb{RP}^{2n-1},ξ_{std})$ is not exactly fillable for $n\ne 2^k$

“…As explained in [16], being terminal is equivalent to have positive minimal SFT degree for some contact forms, which is closely related to the concept of asymptotically dynamically convexity [14] 1 . In particular, the result here bears certain similarity with [20]. However, we do not assume the exact filling to have any topological properties (e.g.…”

“…In higher dimensions, the first, third and fourth inclusions were shown to be proper by Bowden, Crowley, and Stipsicz [4], Bowden, Gironella, and Moreno [5], Massot, Niederkrüger, and Wendl [15] respectively. See also [20] for exactly fillable, almost Weinstein fillable, but not Weinstein fillable examples. The situation in dimension three differs from higher dimensions in the sense that we have gauge theoretic tools as well as better holomorphic curve theories, but also face more topological constraints.…”

“…However, we do not assume the exact filling to have any topological properties (e.g. vanishing first Chern class and π 1 -injectivity) as in [20]. Our approach can be adapted to study more general quotient singularity C n /G.…”

Super-rigidity of certain skeleta using relative symplectic cohomology

“…The above difference is simply H(γ) − H(γ), where γ is seen as a point in X. Then (19) follows from ( 11), ( 13) and (14). Now assume γ is a non-constant orbit, then γ belongs to the neck where H i = f i , Hi = F i .…”

“…We do not know if this can be proved using purely algebro-geometric techniques (for example by computing their log Kodaira dimensions). A simpler proof of a more general version of this fact follows from uniruledness considerations by Zhou [19], see Theorem 5.10.…”