Rational Homotopy Type of Complements of Submanifold Arrangements

TL;DR

This paper constructs an explicit algebraic model to determine the rational homotopy type of complements of subvarieties in smooth algebraic varieties, generalizing several known models without relying on normal crossings.

Contribution

It introduces a new explicit cdga model for the rational homotopy type of subvariety complements, extending previous models to more general arrangements.

Findings

Unified framework for various arrangement complements

Generalizes models for graph configuration spaces and hyperplane arrangements

Avoids reduction to normal crossings via new construction

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.