Contraction algebra and invariants of singularities

TL;DR

This paper demonstrates how the contraction algebra, introduced for flop curves in 3-folds, encodes key invariants of singularities and their resolutions, linking algebraic structures with geometric and enumerative invariants.

Contribution

It proves that the contraction algebra with its A_-structure recovers singularity invariants, including derived categories and Gopakuma-Vafa invariants, for smooth irreducible flopping curves.

Findings

Contraction algebra determines the derived category of singularities.

It recovers genus zero Gopakuma-Vafa invariants.

Establishes a link between algebraic and geometric invariants.

Abstract

In [7], Donovan and Wemyss introduced the contraction algebra of flop- ping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A_\infty-structure recovers various invariants associated to the underlying singularity and its small resolution, including the derived category of singularities and the genus zero Gopakuma-Vafa invariants.