Floer cohomology, singularities, and birational geometry

Abstract

We explain a few recent results concerning the application of various Floer theories to topics in algebraic geometry, including singularity theory and birational geometry. We will also state conjectures and open problems related to these results. We start out with a purely dynamical interpretation of the minimal discrepancy of an isolated singularity and explain how Floer theory fits into this story. Using similar ideas, we show how one can prove part of the cohomological McKay correspondence by computing a Floer cohomology group in two different ways. Finally, we illustrate how Hamiltonian Floer cohomology can be used to prove that birational Calabi–Yau manifolds have the same small quantum cohomology algebras, and we speculate how this might extend to orbifolds.