Cellular $\mathbb{A}^1$-homology of wonderful models of subspace arrangements

Abstract

We compute the cellular $\mathbb{A}^1$-homology of De Concini--Procesi wonderful models of subspace arrangements. For a building set $\mathcal{G}$ over a field $k$, we identify the cellular $\mathbb{A}^1$-chain complex of $\mathbb{P}(\mathcal{G})$ with an $\eta$-twisted nested-set complex carrying Milnor--Witt coefficients and derived orientation data. The key geometric input is a motivic blow-up calculation: for a blow-up along a smooth center of codimension $c$, the relevant connecting class is $(c-1)_\epsilon\eta$, hence it is zero for $c$ odd and $\eta$ for $c$ even. This replaces the parity condition in the computation of Rains by a Milnor--Witt attaching class. As a consequence, the part of cellular $\mathbb{A}^1$-homology surviving after multiplication by $\eta$, and also the homology after inverting $\eta$, are expressed by the interval cohomology of the $2$-divisible subposet of the lattice generated by $\mathcal{G}$. For the braid arrangement, the condition becomes the odd-block condition on partitions, yielding explicit decompositions for the cellular $\mathbb{A}^1$-homology of $\overline{\mathcal M}_{0,N}$ and examples in low rank.