Vakil–Matchett Wood's homological stabilisation conjecture for symmetric complements

Let $i$ be a nonnegative integer, let $bb$ be a partition, and let $X$ be an irreducible smooth complex variety. For each $j$, let $\bar{w}^c_{1^j\lambda}(X)$ denote the complement in $\operatorname{Sym}^{j+n}X$ of the closure of the multiplicity stratum indexed by the partition $1^j\lambda$. Vakil–Matchett Wood's homological stabilisation conjecture. The limit

$$\lim_{j \rightarrow \infty} \dim H_i(\bar{w}^c_{1^j\lambda}(X), \mathbb{Q})$$

exists. This is the topological analogue of motivic stabilisation of symmetric powers; the cited work motivates the conjecture, while the paper proves homological stability for connected open oriented manifolds under different hypotheses.