[Paper Review] Berkovich skeleta and birational geometry

Johannes Nicaise|arXiv (Cornell University)|Sep 18, 2014

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper establishes a canonical Berkovich skeleton for smooth projective varieties over Laurent series fields by unifying two constructions: one via pluricanonical forms (Kontsevich-Soibelman) and another via minimal dlt-models from the Minimal Model Program. The key result is that under semi-ampleness of the canonical sheaf, both constructions yield the same essential skeleton, which is a strong deformation retract of the analytification, linking non-archimedean geometry to birational geometry.

ABSTRACT

We give a survey of joint work with Mircea Mustaţă and Chenyang Xu on the connections between the geometry of Berkovich spaces over the field of Laurent series and the birational geometry of one-parameter degenerations of smooth projective varieties. The central objects in our theory are the weight function and the essential skeleton of the degeneration. We tried to keep the text self-contained, so that it can serve as an introduction to Berkovich geometry for birational geometers.

Motivation & Objective

• To establish a canonical, intrinsic skeleton in the Berkovich analytification of a smooth projective variety over a Laurent series field.
• To unify two constructions of the skeleton: one via pluricanonical forms (Kontsevich-Soibelman) and another via minimal dlt-models from the Minimal Model Program.
• To prove that under semi-ampleness of the canonical sheaf, both constructions yield the same subspace, called the essential skeleton.
• To show that this essential skeleton is a strong deformation retract of the analytification, thus controlling its homotopy type.
• To provide a bridge between non-archimedean geometry and birational geometry, especially for degenerations of Calabi-Yau and general type varieties.

Proposed method

• Define the Berkovich skeleton of a proper sncd-model as the dual intersection complex of the special fiber, which is a strong deformation retract of the analytification.
• Introduce the weight function associated to a pluricanonical form, which measures the order of vanishing along components of the special fiber.
• Define the ω-essential skeleton as the union of faces in the skeleton where the weight function reaches its minimum for a given form ω.
• Construct the essential skeleton Sk(X) as the union of all ω-essential skeletons over non-zero pluricanonical forms.
• Use the Minimal Model Program to consider dlt-models, which generalize sncd-models by allowing mild singularities, and define their skeletons by restricting to the snc locus.
• Prove that minimal dlt-models exist if the canonical sheaf is semi-ample, and that their skeletons are independent of the choice of model.

Experimental results

Research questions

• RQ1 Can a canonical skeleton be defined in the Berkovich analytification of a smooth projective variety over a Laurent series field, even when minimal sncd-models do not exist?
• RQ2 Do the two constructions of the skeleton—via pluricanonical forms and via minimal dlt-models—yield the same subspace?
• RQ3 Is the essential skeleton a strong deformation retract of the analytification, and thus a homotopy invariant of the variety?
• RQ4 What is the topological nature of the essential skeleton when the canonical sheaf is trivial and the residue field is algebraically closed?
• RQ5 Can the deformation retraction to the skeleton be understood via a gradient flow of the weight function?

Key findings

• The essential skeleton Sk(X) constructed via pluricanonical forms coincides with the skeleton of any minimal dlt-model when the canonical sheaf is semi-ample.
• The essential skeleton Sk(X) is a strong deformation retract of the Berkovich analytification X^an, thus controlling its homotopy type.
• The skeleton of any minimal dlt-model is independent of the choice of model, due to birational crepant equivalence.
• When the canonical sheaf is trivial and the residue field is algebraically closed, the essential skeleton is a closed pseudo-manifold of maximal dimension.
• The skeleton can be obtained from any sncd-model skeleton via a sequence of elementary collapses, as shown in [dFKX12], preserving the deformation retraction.
• The weight function associated to a volume form ω achieves its minimum precisely on the essential skeleton Sk(X), suggesting a potential gradient flow structure.