2025/09/09 by Collins, Tristan C., Guenancia, Henri
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.07508
We study various geometric properties of log Calabi-Yau manifolds, i.e. log smooth pairs (X,D) such that KX+D=0. More specifically, we focus on the two cases where X is a Fano manifold and D is either smooth or has two proportional components. Despite the existence of a complete Ricci flat Kähler metric on X∖ D in both cases, we will show that the geometric properties of the pair (X,D) are vastly different, e.g. validity of Bochner principle, local triviality of the quasi-Albanese map, polystability of TX(-log D) and compactifiability of the universal cover of X∖ D. When D has two components we show that the universal cover of X∖ D is a Calabi-Yau manifold of infinite topological type, and we describe the geometry at infinity from a Riemannian point of view.