2022/11/09 by A. F. Zakharov, Zakharov, Alexander
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2211.05033
openalex publication_date 2022/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we provide an explicit cdga that controls the rational homotopy type of the complement X-∪i Zi, where X is a smooth compact algebraic variety and \Zi\ is a collection of subvarieties such that all set-theoretical intersections are smooth. The model is given in terms of the cohomology of all intersections of Zi's, and the natural maps induced by the inclusions. Our construction is inspired by the work of J.Morgan, who covered the fundamental case where \Zi\ is a divisor with normal crossings, and it is built on developments of the theory of mixed Hodge diagrams by Cirici-Horel. We avoid any explicit reduction to the normal crossings divisor case, e.g. via the wonderful compactification of De Concini-Procesi. As an application of our approach we recover and generalize a few separate results on the complements of arrangements in a uniform manner. These include the Kritz-Totaro model for graph configuration spaces, Yuzvinsky's model for affine subspace arrangements and Dupont's model for complements of hypersurfaces with hyperplane-like intersection.