Semisimple Deformations of Contraction Algebras for Smooth Threefold Flops

Building on Bryan--Katz--Leung, Hua--Toda and Kawamata, we study deformations of contraction algebras and the local geometry of the scheme of associative algebra laws. Let A be the basic contraction algebra of a completed projective flopping contraction of a smooth complex threefold. We construct a finite free polynomial family that is constant and semisimple away from the origin, with the original curve projectors fixed. For each restricted positive root β=(βᵢ), with coordinates indexed by the retained vertices, the split generic fibre has n_β matrix blocks of size ∑ᵢβᵢ, and the labelled projector eᵢ has rank βᵢ on each block. Here n_β is the corresponding genus-zero Gopakumar--Vafa invariant. If the geometric generic fibre of a finite free associative formal deformation of A is semisimple, it has the same matrix factors, and any lifts of the original projectors have the same ranks on those factors. The scheme of associative algebra laws is smooth at A. The closure of the corresponding semisimple orbit is its unique irreducible component through A. The proof uses the curve divisors of Katz--Morrison and Nabijou--Wemyss, presentations of the universal contraction quotient and its cocenter, and the orbit method of Jelisiejew--Smoktunowicz. These presentations prove the geometric Hua--Toda flatness assertions. In the geometric families considered here, the relative contraction algebra, its centre and its cocenter are finite free over arbitrary complete Noetherian local complex bases. On a smooth open neighbourhood of A in the scheme of algebra laws, the centres and cocenters form dual vector bundles compatible with arbitrary base change. For the type D algebras Dn,2 with n≥2, the centre morphism gives counterexamples to the smoothness assertion in Conjecture 1.4(2) of Jelisiejew--Smoktunowicz.