Contractions of 3-folds: deformations and invariants

TL;DR

This paper explores new methods for studying flopping curves on 3-folds, introducing the contraction algebra as an invariant that captures their geometric and homological properties, and relates it to enumerative invariants.

Contribution

The paper introduces the contraction algebra as a 3-fold invariant for flopping curves and clarifies its enumerative significance via Gopakumar-Vafa invariants.

Findings

Contraction algebra characterizes properties of flopping curves.

Dimension of contraction algebra relates to Gopakumar-Vafa invariants.

Provides a unified approach to geometry and homology of flopping curves.

Abstract

This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterises 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. Before reviewing these results, and others, I give 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.