Trivial extension DG-algebras, unitally positive $A_\infty$-algebras, and applications
Joseph Karmazyn, Emma Lepri, Michael Wemyss
TL;DR
The paper develops two derived constructions for periodic modules: a tractable trivial extension ${DG}$-algebra ${\mathcal{T}}$ and a unitally positive ${A_\infty}$-algebra ${\mathcal{N}}$, providing a direct, broadly applicable framework. Using these tools, it proves that for 3-fold flops, the contraction algebra ${A}_{con}$ (via a simple module $S$) determines the endomorphism DG-structure of a corresponding projective resolution up to ${A_\infty}$-equivalence, thereby recovering the base singularity and proving the Donovan–Wemyss conjecture in the single- and multi-curve cases. The main reconstruction result shows an ${A_\infty}$-isomorphism between the unitally positive algebra built from ${A}_{con}$ and the strictly unital minimal model of ${\mathop{End}}_{A}(\mathcal{Q})$, enabling classification via finite-dimensional data and Koszul duals. The authors further extend these ideas to a categorical upgrade, treating unitally positive ${A_\infty}$-categories and idempotents, and obtain a categorical version of the DW classification, including a multi-curve generalization.
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.