Curved ∞-Local Systems And Projectively Flat Riemann-Hilbert Correspondence

2024/11/29 by Patrick Antweiler, Antweiler, Patrick
Mathematics · #Holomorphic and Operator Theory #Advanced Differential Equations and Dynamical Systems #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.2411.19595

Abstract

We generalize the higher Riemann-Hilbert correspondence in the presence of scalar curvature for a (possibly non-compact) smooth manifold M. We show that the dg-category of curved ∞-local systems, the dg-category of graded vector bundles with projectively flat \mathbb Z-graded connections and the dg-category of curved representations of the singular simplicial set of the based loop space of M are all A_∞-quasi equivalent. They provide dg-enhancements of the subcategory of the bounded derived category of twisted sheaves whose cohomology sheaves are locally constant and have finite-dimensional fibers. In the ungraded case, we reduce to an equivalence between projectively flat vector bundles and a subcategory of projective representations of π₁(M; x₀). As an application of our general framework, we also prove that the category of cohesive modules over the curved Dolbeault algebra of a complex manifold X is equivalent to a subcategory of the bounded derived category of twisted sheaves of \mathcal O_X-modules which generalizes a theorem due to Block to possibly non-compact complex manifolds.

Related

• A Riemann--Hilbert correspondence for infinity local systems
• Model Categories and the Higher Riemann-Hilbert Correspondence
• The microlocal Riemann-Hilbert correspondence for complex contact manifolds
• Derived Differentiable Manifolds
• The Riemann-Hilbert Correspondence for Algebraic Stacks