2026/01/09 by Atabey Kaygun · 1 voice
Mathematics · #math.AG
We study quadratic moduli schemes X of algebra laws on a fixed vector space W under the transport-of-structure action of GL(W) on Hom(W⊗ 2,W). We construct an intrinsic three-term deformation complex on X whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (à la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map κinc2,μ\colon H2inc(μ)→ H3inc(μ) that controls second-order lifts modulo isotriviality. If μ is smooth point in a reduced component and (κinc2,μ)-1(0)=\0\, then the G-orbit of μ is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish.