Patrick Antweiler | Searcharxiv

arXiv2025

Continuous Hochschild Cohomology and Formality

We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling deformations and prove formality theorems for the Fr\'echet algebras of smooth functions on a manifold, the de Rham algebra and for the Dolbeault algebra of a complex manifold. In the latter case, the Hochschild cohomology is equivalent to Kontsevich's extended deformation complex, the Hochschild cohomology of the derived category in case $X$ is a smooth projective variety and to Gualtieri's deformation complex of $X$ viewed as generalized complex manifold. We also compute the continuous Hochschild cohomology for various categories of matrix factorisations.

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arXiv2024

Curved $\infty$-Local Systems And Projectively Flat Riemann-Hilbert Correspondence

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 $\infty$-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 $A_\infty$-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 $\pi_1(M; x_0)$. 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.

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