\textbf{Abstract.} Let \(\mathrm{IEmb}\!\big(B^{4}(c),\mathbb{C}P^{2}\big)\) denote the space of unparameterized symplectic embeddings of \(k\) balls of capacities \((c_{1},\dots,c_{k})\), where \(1\le k\le 8\). It is known from the work of S.~Anjos, J.~Li, T.-J.~Li, and M.~Pinsonnault that the space of capacities decomposes into convex polygons called stability chambers, and that the homotopy type of \(\mathrm{IEmb}\!\big(B^{4}(c),\mathbb{C}P^{2}\big)\) depends solely on the stability chambers. Based on recent results of M.~Entov and M.~Verbitsky on Kähler-type embeddings, we show that for \(1\le k\le 8\), \(\mathrm{IEmb}\!\big(B^{4}(c),\mathbb{C}P^{2}\big)\) is homotopy equivalent to a union of strata \(F_{I}\) of the configuration space of the complex projective plane \(F(\mathbb{C}P^{2},k)\). The proof relies on constructing an explicit map from the space of K\"ahler-type embeddings to a generalized version of the configuration space that incorporates both configurations of points and compatible complex structures on \(\mathbb{C}P^{2}\).

We will study the syzygy order of equivariant cohomology of toric varieties by the means of regular sequences on the piecewise polynomial algebra associated to the underlying fan.