Rational Homotopy Type of Complements of Submanifold Arrangements

Abstract

In this work we provide an explicit cdga that controls the rational homotopy type of the complement $X-\cup_i Z_i$, where $X$ is a smooth compact algebraic variety and $\{Z_i\}$ 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 $Z_i$'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 $\{Z_i\}$ 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.