Rational Homotopy Type of Complements of Submanifold Arrangements

Abstract: We will provide an explicit cdga controlling the rational homotopy type of the complement to a smooth arrangement $X-\cup_i Z_i$ in a smooth compact algebraic variety $X$ over $\mathbb{C}$. This generalizes the corresponding result of Morgan in case of a divisor with normal crossings to arbitrary smooth arrangements. The model is given in terms of the arrangement $Z_i$ and agrees with a model introduced by Chen-L\"u-Wu for computing the cohomology. As an application we reprove a formality theorem due to Feichtner-Yuzvinksy. Then we show that the Kritz-Totaro model computes the rational homotopy type in case of chromatic configuration spaces of smooth compact algebraic varieties.