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 the positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that the vanishing of symplectic cohomology and the existence of symplectic dilations are properties independent of the filling for ADC manifolds. Using them, various topological applications on symplectic fillings are obtained, including the uniqueness of diffeomorphism types of fillings for many contact manifolds. 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 false(S2n−1,ξstdfalse),∂false(T∗L×Cnfalse) 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 Cn with nonnegative log Kodaira dimension is a symplectic exotic Cn.

Symplectic fillings of asymptotically dynamically convex manifolds I

“…Moreover, they give an infinity hierarchy on the complexity of symplectic manifolds, and also an infinity hierarchy on the complexity of ADC contact manifolds. Details of such construction will appear in the sequel paper [64].…”

“…One simple way of comparing complexity is by asking if one exact domain can be embedded into another. Due to the Viterbo transfer map, the vanishing of symplectic cohomology and existence of symplectic dilation are two levels of complexity, which are in fact the first two of the infinity many structures in [64]. For example, one can not embed an exact domain W with SH * (W ) = 0 to a flexible Weinstein domain.…”

“…In fact, there exists a whole hierarchy of structures after them called higher dilations, all of them have associated structure maps similar to δ ∂ , ∆ ∂ , which are also independent of the topologically simple exact filling for ADC manifolds. Details of the construction will appear in the sequel paper [64].…”

“…However the existence of symplectic dilation is very far from being equivalent to (1, Λ)-uniruledness. In fact, the existence of higher dilation in [64] will also imply uniruledness. On the other hand, if W is not 1-unruled, then SH * (W ) = 0.…”

Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres

“…To do this, one needs to specify the sets of Hamiltonian functions and almost complex structures to work with. We say that a time-dependent Hamiltonian H t : S 1 ˆM Ñ R is admissible if H t " H `Ft is the sum of an autonomous Hamiltonian H : M Ñ R which is quadratic at infinity, namely Hpr, yq " r 2 (77) on the cylindrical end M zM " p1, 8q ˆBM , where r P p1, 8q is the radial coordinate, and a time-dependent perturbation F t : S 1 ˆM Ñ R. We require that on M zM , we have that for any r 0 " 0, there exists an r ą r 0 such that F t vanishes in a neighborhood of the hypersurface tru ˆBM Ă M . For instance, one can take F t to be a function supported near non-constant orbits of X H , where it is modelled on a Morse function on S 1 .…”