The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence
Gustavo Jasso, Bernhard Keller, Fernando Muro
TL;DR
This work outlines a proof of the Donovan–Wemyss Conjecture within the Homological MMP for threefolds, building on results of August, Hua–Wemyss, and the Derived Auslander–Iyama Correspondence. Central to the argument is the $2\mathbb{Z}$-derived contraction algebra $\mathbf{\Lambda}_{\mathrm{con}}(p)$ and the restricted universal Massey product $j^{*}\{m_{4}\}$, which together control the derived endomorphism algebra up to quasi-isomorphism. By proving that the $2\mathbb{Z}$-derived contraction algebra is determined by $H^{0}$ and the unit $j^{*}\{m_{4}\}$, and applying the Derived Auslander–Iyama correspondence, the authors show that derived equivalence of contraction algebras forces an isomorphism of the Tyurina data and hence of the isolated cDV singularities themselves. Consequently, the contraction algebras encode enough information to recover the underlying singularity, and the results yield a unique DG enhancement of the singularity category in this setting. The findings connect noncommutative deformation theory, Hochschild cohomology, and cluster-tilting theory to birational geometry, highlighting the deep interplay between derived categories, invariants of singularities, and flops in dimension three.
Abstract
We provide an outline of the proof of the Donovan--Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds. The proof relies on results of August, of Hua and the second-named author, Wemyss, and on the Derived Auslander--Iyama Correspondence -- a recent result by the first- and third-named authors.