The essential skeleton of a product of degenerations

Abstract

We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration$\mathscr{X}_{R}$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$\text{Sk}(\mathscr{X}_{R})$when the model$\mathscr{X}_{R}$is log-regular. We show that if$\mathscr{X}_{R}$and$\mathscr{Y}_{R}$are log-smooth, and at least one is semistable, then$\text{Sk}(\mathscr{X}_{R}\times _{R}\mathscr{Y}_{R})\simeq \text{Sk}(\mathscr{X}_{R})\times \text{Sk}(\mathscr{Y}_{R})$. The essential skeleton$\text{Sk}(X_{K})$, defined by Mustaţă 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,$\text{Sk}(X_{K}\times _{K}Y_{K})\simeq \text{Sk}(X_{K})\times \text{Sk}(Y_{K})$. 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$\mathbb{C}\mathbb{P}^{n}$, depending on the type of the degeneration.