Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications

TL;DR

This paper establishes symplectic criteria based on cohomology to determine when affine varieties contain uniruled subvarieties, with applications to birational geometry and the Minimal Model Program.

Contribution

It introduces new criteria using symplectic cohomology to identify uniruled subvarieties in affine varieties, connecting symplectic topology with algebraic geometry.

Findings

Criteria for uniruled subvarieties derived from symplectic cohomology

Vanishing properties of symplectic cohomology used to verify criteria

Applications to birational geometry and the Minimal Model Program

Abstract

We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of the Minimal Model Program.