Given a strict simple degeneration @math the first three authors previously constructed a degeneration @math of the relative degree @math Hilbert scheme of @math -dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of @math is at most @math . In this case we show that @math is a dlt model. This is even a good dlt model if @math has this property. We compute the dual complex of the central fibre @math and relate this to the Kontsevich-Soibelman skeleton. For a type II degeneration of @math surfaces we show that the stack @math carries a nowhere degenerate relative logarithmic @math -form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.
Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K "ahler setting, we can then deduce a finiteness statement for monodromy acting on @math , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-K "ahler manifolds, we prove that a finite monodromy projective degeneration of hyper-K "ahler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-K "ahler manifolds (e.g. Debarre--Voisin or Laza--Sacc a--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-K "ahler manifolds.