Affine Varieties, Singularities and the Growth Rate of Wrapped Floer Cohomology

TL;DR

This paper investigates how the growth rate of wrapped Floer cohomology can distinguish contact and symplectic manifolds related to complex singularities and affine varieties, providing new invariants for classification.

Contribution

It introduces the growth rate of wrapped Floer cohomology as a tool to identify contactomorphic links of singularities and symplectomorphic affine varieties, with implications for manifold topology.

Findings

Contactomorphic links of singularities are rationally elliptic.

Symplectomorphic affine varieties imply certain Betti number bounds.

The invariant distinguishes specific classes of contact and symplectic manifolds.

Abstract

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if Q is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.