Applications of Berkovich spaces

Abstract

This thesis applies the techniques of non-archimedean geometry to the study of degenerations and compactifications of algebraic varieties. The central object we investigate is the so-called essential skeleton, a combinatorial object that lies embedded in non-archimedean spaces and encodes an important part of the geometry of the space. This originates in the work of Kontsevich and Soibelman on mirror symmetry, an important development in algebraic geometry that has its roots in mathematical physics. The interplay of the theory of Berkovich spaces, the ideas of mirror symmetry and the tools of birational geometry gives form and meaning to the study of the essential skeleton. Chapters 3 and 4 are built on the research paper The essential skeleton of a product of degenerations, in collaboration with Morgan Brown [BM19]. We establish the behaviour of the essential skeleton under some natural operations, and we merge the language of logarithmic geometry into the construction of Berkovich skeletons. As main application, we compute the essential skeleton of some degenerations of hyperkähler varieties. We consider Hilbert schemes of a semistable degeneration of K3 surfaces, and generalised Kummer constructions 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 CPn, depending on the type of the degeneration and in accordance with the predictions of mirror symmetry. Chapters 5 to 7 are based on the joint work Essential skeletons of pairs and the geometric P=W conjecture with Mirko Mauri and Matthew Stevenson [MMS18]. We introduce and study an explicit formulation of the weight function, a key tool to define the essential skeleton, in the case of varieties defined over a non-archimedean trivially-valued field. As a result, we employ these techniques to compute the dual boundary complexes of certain character varieties: this provides the first evidence for the geometric P=W conjecture in the compact case, and the first application of Berkovich geometry in non-abelian Hodge theory.