Wonderful Blow-Ups of Weighted Building Sets and Configuration Spaces of Filtered Manifolds

TL;DR

The paper develops a comprehensive framework to blow up weighted submanifolds and weighted building sets in smooth manifolds, generalizing Fulton–MacPherson configuration spaces to filtered manifolds. By introducing weightings and their dual filtrations, it derives a robust theory of weighted normal/conormal bundles, alignment conditions, and smooth blow-up constructions (both spherical and projective). The main achievement is proving that the blow-up along a uniformly aligned, separated weighted building set yields a smooth manifold with corners, together with explicit charts and a stratification compatible with nests. The framework is then applied to configuration spaces for filtered manifolds, producing weighted Fulton–MacPherson spaces and enabling future work on jet configuration spaces; it also provides a novel invariant characterization of weightings via vanishing ideals and a dual cotangent viewpoint. Overall, this work furnishes a powerful, geometric toolkit for jet- and filtration-aware configuration spaces with rigorous smooth structures and locally explicit coordinates, paving the way for applications to jet spaces and sub-Riemannian contexts.

Abstract

Fulton and MacPherson famously constructed a configuration space that encodes infinitesimal collision data by blowing up the diagonals. We observe that when generalizing their approach to configuration spaces of filtered manifolds (e.g. jet spaces or sub-Riemannian manifolds), these blow-ups have to be modified with weights in order for the collisions to be compatible with higher-order data. In the present article, we provide a general framework for blowing up arrangements of submanifolds that are equipped with a weighting in the sense of Loizides and Meinrenken. We prove in particular smoothness of the blow-up under reasonable assumptions, extending a result of Li to the weighted setting. Our discussion covers both spherical and projective blow-ups, as well as the (restricted) functoriality of the construction. Alongside a self-contained introduction to weightings, we also give a new characterization thereof in terms of their vanishing ideals and prove that cleanly intersecting weightings locally yield a weighting. As our main application, we construct configuration spaces of filtered manifolds, including convenient local models. We also discuss a variation of the construction tailored to certain fiber bundles equipped with a filtration. This is necessary for the special case of jet configuration spaces, which we investigate in a future article.

Figures (10)

• Figure 1: Visualization of the blow-up of the origin of a disk. Points in the blow-up that are separated by an integer number of rotations are identified. The central spine thus can be identified with unit length vectors at the origin, and every point away from the origin has a unique preimage under the blow-down map.
• Figure 2: The elements of the weighted normal bundle can be visualized as arrows based at the support and curved according to the weights, drawn here for the standard weighting $\mathcal{E}^{(0,1,2)}$ along $N=\operatorname{supp}\mathcal{E}^{(0,1,2)}$. A weighted vector in $\nu\mathcal{E}^{(0,1,2)}$ with coordinates $(y_1,y_2,y_3)$ corresponds to the line $\{\lambda\cdot (y_1,y_2,y_3)\;|\;\lambda\in[0,1]\}$ traced out by the action associated with the weights. The coordinates correspond precisely to the end-point of the line, giving an identification $\nu{\mathcal{E}^{(0,1,2)}}\simeq\mathbb{R}^3$.
• Figure 3: Example of the map induced between the weighted normal bundles for weights $(1,1,2)$ and $(0,1,1)$ acting on an arbitrary blue weighted vector. The supports are the origin and the first axis, respectively, and highlighted in orange.
• Figure 4: We define three weightings supported over the coordinate subspaces $\textcolor{purple}{A}, \textcolor{ForestGreen}{B}$ and $\textcolor{olive}{C=\{0\}}$ as follows: $\textcolor{purple}{\mathcal{W}_A:=\mathcal{E}^{(0,0,2)}}$, $\textcolor{ForestGreen}{\mathcal{W}_B:=\mathcal{E}^{(0,2,1)}}$ and $\textcolor{olive}{\mathcal{W}_C:=R\cdot\mathcal{E}^{(1,1,2)}}$, where $R$ is a rotation of 45 degrees around the $x_1$-axis. These weightings intersect cleanly and have supports that intersect like coordinate subspaces, but are not aligned.
• Figure 5: Visualization of the blow-up of the origin of a disk equipped with weights $(1,2)$.

Theorems & Definitions (225)

• definition 1
• Lemma 2.1.3
• proof
• Lemma 2.1.4
• proof
• definition 2
• Lemma 2.2.1
• definition 3
• Proposition 2.2.4
• proof