# Symplectic fillings of asymptotically dynamically convex manifolds I

## TL;DR

This paper studies symplectic fillings of ADC manifolds, establishing invariance of certain structure maps, leading to topological classifications, obstructions to fillability, and implications for symplectic geometry and exotic structures.

## Contribution

It introduces invariant structure maps on positive symplectic cohomology for ADC manifolds, proving their independence from fillings and deriving new topological and fillability results.

## Key findings

- Invariance of structure maps implies filling-independent properties.
- Existence of infinitely many contact manifolds with specific fillability properties.
- Symplectic dilation implies uniruledness and exotic structures.

## 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 $4k+3, k\ge 1$, 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 $(S^{2n-1},\xi_{std}), \partial(T^*L \times \mathbb{C}^n)$ 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 $\mathbb{C}^n$ with non-negative log Kodaira dimension is a symplectic exotic $\mathbb{C}^{n}$.