The essential skeleton of a product of degenerations

We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration X_R changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to X_K. Using the Kato fan, we define a skeleton Sk(X_R) when the model X_R is log-regular. We show that if X_R and Y_R are log-smooth, and at least one is semistable, then Sk(X_R ×_R Y_R) ≈ Sk(X_R) × Sk(Y_R). The essential skeleton Sk(X_K), defined by Mustaţǎ and Nicaise, is a birational invariant of X_K and is independent of the choice of R-model. We extend their definition to pairs, and show that if both X_K and Y_K admit semistable models, Sk(X_K ×_K Y_K) ≈ Sk(X_K)×Sk(Y_K). As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. We consider both the case of the Hilbert scheme of a semistable degeneration of K3 surfaces, and the generalized Kummer construction applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the 2n-dimensional degeneration is homeomorphic to a point, n-simplex, or ℂℙ_n, depending on the type of the degeneration.