|Dept.||Stony Brook University|
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.
Lecture 1: After briefly reviewing the Stein/Weinstein domain, Liouville domain, and Weinstein handle-decomposition, we will learn the definition of symplectic cohomology and some vanishing, invariance properties of symplectic cohomology. We will learn how symplectic cohomology changes under two algebro-geometric operations, Kaliman modification.