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

TL;DR

This paper introduces the concept of $k$-(semi)-dilation for Liouville domains, shows its invariance for ADC manifolds, constructs examples, and uses the invariants to study embeddings and cobordisms, especially in Brieskorn varieties.

Contribution

It generalizes symplectic dilation to $k$-(semi)-dilation, proves invariance for ADC manifolds, and develops new invariants for Liouville domains and contact manifolds.

Findings

Existence of $k$-dilations without $(k-1)$-dilations for all $k extgreater 0$

Invariants called the order of (semi)-dilation are introduced

Determination of the order of (semi)-dilation for many Brieskorn varieties

Abstract

We introduce the concept of $k$-(semi)-dilation for Liouville domains, which is a generalization of symplectic dilation defined by Seidel-Solomon. We prove that the existence of $k$-(semi)-dilation is a property independent of fillings for asymptotically dynamically convex (ADC) manifolds. We construct examples with $k$-dilations, but not $k-1$-dilations for all $k\ge 0$. We extract invariants taking value in $\mathbb{N} \cup \{\infty\}$ for Liouville domains and ADC contact manifolds, which are called the order of (semi)-dilation. The order of (semi)-dilation serves as embedding and cobordism obstructions. We determine the order of (semi)-dilation for many Brieskorn varieties and use them to study cobordisms between Brieskorn manifolds.