Semi-stable extensions over 1-dimensional bases

Abstract

Given a family of Calabi-Yau varieties over the punctured disc or over the field of Laurent series, we show that, after a finite base change, the family can be exended across the origin while keeping the canonical class trivial. More generally, we prove similar extension results for families whose log-canonical class is semi-ample. We use these to show that the Berkovich and essential skeleta agree for smooth varieties over C((t)) with semi-ample canonical class.

Tropical and non-Archimedean limits of degenerating families of volume forms

“…If X is toroidal, then X is lc, and X is dlt if and only if it is snc. Following [dFKX12,KNX15], we could say that an lc model X is qdlt (for quotient of dlt) if its lc centers are contained in a toroidal open subset U ⊂ X .…”

“…Its dimension d, which features as the exponent of the log term in the asymptotics of the mass, measures how "bad" the degeneration is. Further, the family X → D * admits a relative minimal model X , with certain mild (dlt) singularities [KNX15], and the essential skeleton can be identified with the dual complex of X [NX13]. In particular, d = 0 if and only if X can filled in with a central fiber X 0 which is a Calabi-Yau variety with klt singularities.…”

K-stability for Kähler manifolds

“…We denote the induced finite map X ′ → X also by u. Setting Ω ′ := u * Ω we have that (X ′ , Ω ′ ) is a test-configuration for (X, [ω]) which, for d sufficiently large and divisible, has reduced central fibre [41,Section 16]. Thus we need to compare the Donaldson-Futaki invariant and minimum norms of these two test-configurations.…”

Motivic zeta functions of degenerating Calabi–Yau varieties

“…2) The essential skeleton of X can also be computed from a minimal model of X in the sense of the Minimal Model Program. It is proven in Corollary 4 of [KNX15] that X has a minimal qdlt-model X min over R. One can still associate a Berkovich skeleton Sk(X min ) to such a qdlt-model, by [KNX15,§23]. It is shown in Theorem 24 of [KNX15] that Sk(X min ) coincides with the essential skeleton Sk(X).…”

“…It is proven in Corollary 4 of [KNX15] that X has a minimal qdlt-model X min over R. One can still associate a Berkovich skeleton Sk(X min ) to such a qdlt-model, by [KNX15,§23]. It is shown in Theorem 24 of [KNX15] that Sk(X min ) coincides with the essential skeleton Sk(X). If X is defined over an algebraic k-curve, rather than the field of Laurent series, one can even find a minimal dlt-model; in practice, it is often possible to reduce to this case by means of the approximation technique in [NX16a, §4.2].…”