The growth rate of symplectic homology and affine varieties

TL;DR

This paper demonstrates that certain cotangent bundles with exponentially growing free loopspace homology cannot be symplectomorphic to smooth affine varieties, using the growth rate of symplectic homology as a key invariant.

Contribution

It introduces the use of the growth rate of symplectic homology to distinguish cotangent bundles from affine varieties, establishing new obstructions based on homology growth.

Findings

Cotangent bundles with exponential homology growth are not symplectomorphic to affine varieties.

Unit cotangent bundles of such manifolds are not Stein fillable by affine Stein domains.

Results apply to end connect sums of simply connected manifolds with multiple cohomology generators.

Abstract

We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.