Contractions of 3-folds: Deformations and invariants

Abstract

This paper discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper [Noncommutative deformation and flops, Duke Math. J. 165(8) (2016) 1397–1414], the author and Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterizes their geometric and homological properties in a unified manner, using the theory of noncommutative deformations. Toda has now clarified the enumerative significance of the contraction algebra for flopping curves, calculating its dimension in terms of Gopakumar-Vafa invariants [Noncommutative width and Gopakumar–Vafa invariants, Manuscripta Math. 148(3–4) (2015) 521–533]. Before reviewing these results, and others, the author gives a brief introduction to the rich geometry of flopping curves on 3-folds, starting from the resolutions of Kleinian surface singularities. This is based on a talk given at VBAC 2014 in Berlin.