The geometry of dual complexes

Abstract

In this thesis we examine some aspects of the topology of dual complexes under three different viewpoints: birational, non-archimedean and symplectic. Our main source of inspiration is the question raised by Kollár and Xu, whether the dual complex of a log Calabi–Yau pair is a quotient of a sphere. Complete proofs exist only in low dimension, smaller than 4, or conditionally in dimension 5. We contribute to providing new evidence in any dimension. First, using the connectivity theorems of the Minimal Model Program, we answer the question positively for Mori fibre spaces of low Picard numbers or of low-dimensional bases. No assumption on the dimension of the total space is imposed. We also compute the dual complex of special log Calabi–Yau pairs of great interest in non-abelian Hodge theory. This provides the first non-trivial evidence for the geometric P=W conjecture of Katzarkov–Noll–Pandit–Simpson in the compact case. To this end, we define the essential skeleton of a pair over a trivially-valued field as the minimality locus of weight functions on the Berkovich analytification. In analogy to the case of degenerations, we show that the weight functions are defined in terms of the classical notion of log discrepancy, and more intrinsically, in terms of a canonical metric, called Temkin’s metric. Finally, we provide a general technique for constructing Lagrangian torus fibrations on affine varieties, whose bases are the skeletons of one of their compactifications. In particular, we provide a Lagrangian version of Mikhalkin’s tropicalisation of the pair-of-pants, and a Lagrangian fibration on the 3-fold negative vertex with codimension 2 discriminant locus, answering an old question of Gross. We also construct an analogue of the non-archimedean SYZ fibration studied by Nicaise, Xu and Yu.