Obstruction Theory for Bigraded Differential Algebras
Jiahao Hu
TL;DR
This work develops an obstruction theory for Hirsch extensions of commutative bigraded bidifferential algebras (cbba’s) with twisted coefficients, enabling a systematic construction of minimal models and a clear criterion for their uniqueness. By translating additive homotopy theory into the cbba setting, the authors establish linear and quadratic-linear Hirsch extensions, define twisted homotopy classes, and formulate a robust obstruction framework that governs the existence and uniqueness of extensions and homotopies. They apply this theory to analyze automorphism groups of minimal cbba’s and to characterize formality through grading automorphisms, including how formality behaves under base-field extensions. The results unify and extend the cdga obstruction theory to the bigraded, twisted-coefficient context, with concrete consequences for minimal models, algebraic fibrations, and the structure of automorphism groups. The framework promises further insights into the homotopy theory of complex manifolds via bigraded de Rham algebras and their formality properties.
Abstract
We develop an obstruction theory for Hirsch extensions of cbba's with twisted coefficients. This leads to a variety of applications, including a structural theorem for minimal cbba's, a construction of relative minimal models with twisted coefficients, as well as a proof of uniqueness. These results are further employed to study automorphism groups of minimal cbba's and to characterize formality in terms of grading automorphisms.