The Essential Skeleton of a product of degenerations

Abstract

We study the problem of how the dual complex of the special fiber of an snc degeneration $\cX_R$ 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 $\Sk(\cX_R)$ when the model $\cX_R$ is log-regular. We show that if $\cX_R$ and $\cY_R$ are log-regular, and at least one is semistable, then $\Sk(\cX_R\times_R \cY_R) \simeq \Sk(\cX_R)\times \Sk(\cY_R)$. The essential skeleton $\Sk(X_K)$, defined by Musta\c{t}\u{a} 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, $\Sk(X_K\times_K Y_K) \simeq \Sk(X_K)\times \Sk(Y_K)$. As an application, we compute the homeomorphism type of the dual complex of some degenerations of hyper-K{\"a}hler 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 either a point, $n$-simplex, or $\mathbb{C}\mathbb{P}^n$, depending on the type of the degeneration.