Formal moduli and the splitting theory of supermanifolds

Abstract

We develop a formal moduli theory for the splitting problem of complex supermanifolds. Starting from Green's obstruction tower, we construct a finite-step filtered dg Lie algebra which controls splittings by filtered Maurer-Cartan theory. We prove that the classical successive obstruction classes are recovered as the leading terms of adapted Maurer-Cartan representatives, and we transfer the theory to a minimal filtered $L_\infty$-model whose higher brackets give the intrinsic Kuranishi relations among the obstruction coordinates. We also prove that, in a precise filtered sense, the affine Atiyah class contains the entire Green obstruction tower: the Donagi-Witten component gives the primary obstruction, while the higher obstructions arise as successive projected defects and residual classes of the same Atiyah cocycle. We then pass to families, by constructing the fixed-retract formal moduli problem with prescribed split model and by identifying the formal neighbourhood of the split section with the fibrewise deformation theory of the split model; under standard finiteness, base-change, and descent hypotheses this yields relative tangent-obstruction and Kuranishi presentations. Finally, we work out explicit examples of non-split supergeometries, including cases with residual higher obstructions and a first nonlinear Kuranishi relation. These examples illustrate the interaction between Green obstructions, Atiyah classes, and higher $L_\infty$-operations.