Coefficient Riemann--Hilbert for an Affine Singular Foliation

Abstract

For the affine foliation $⟨x{\partial }_x,y{\partial }_x⟩$, we prove that smooth coefficient integration is a strictly unital $A_{\infty }$ quasi-equivalence on every ordinary open subset of ${\mathbb{R}}^2$, for globally bounded, degreewise finite-projective real coefficient bodies, allowing internal cohomology of nonconstant rank. A horizontal-hull restriction of a finite Lie $2$-groupoid model admits a smooth simplicial section of the comparison from full Maurer--Cartan integration, with a homotopy relative to all unit-generated simplices. Strict differentiation and two relative Hom models yield a real-linear chain map whose internal-degree diagonal is the identity. Its inverse is a finite sum acting directly on the possibly rank-jumping body complex. Global acyclicity of the differentiation kernels gives invertible counits that are the identity on the body. Infinitely flat and nongeneric rank-jumping examples illustrate the nonregular theorem. Separately, regular two-term coefficients admit a residue classification and an exact ordinary-quotient descent criterion. Nonzero residue obstructs pullback through any strict map to an ordinary groupoid nerve with represented objects in the stated raw coefficient theory. For regular vertical families over $S\times {\mathbb{R}}^2$, we also prove a finite-projective bundle-valued theorem natural under smooth parameter pullback; its two-term invariants are bundle arrows, rather than pointwise rank profiles.

arXiv comment94 pages