Trivial extension DG-algebras, unitally positive $A_\infty$-algebras, and applications

TL;DR

This paper introduces a new class of $A_in$-algebras called unitally positive algebras, constructed from trivial extension DG-algebras associated with periodic modules, with applications to classifying 3-fold flops in birational geometry.

Contribution

It constructs the unitally positive $A_in$-algebra from trivial extension DG-algebras and applies this to prove the Donovan-Wemyss conjecture in birational geometry.

Findings

Provides a new algebraic framework for classifying 3-fold flops.

Simplifies the proof of the Donovan-Wemyss conjecture.

Establishes a construction applicable to DG-categories with mild assumptions.

Abstract

To any periodic module over any algebra, this paper introduces an associated trivial extension DG-algebra T. After first passing to a strictly unital $A_\infty$-minimal model, it then constructs a particular $A_\infty$-algebra N, called the unitally positive $A_\infty$-algebra, which roughly speaking describes the identity in degree zero and all the positive cohomology. The object N is fundamental, and can be constructed for any DG-category satisfying very mild assumptions. The main application is to birational geometry. When applied to contraction algebras, the construction gives a simple and direct proof of the Donovan-Wemyss conjecture, namely that smooth irreducible 3-fold flops are classified by their contraction algebras, and thus by noncommutative data.