2023/10/12 by Memariansorkhabi, Soheil
#14E25 #14G05 #32M15 #32Q40 (secondary) #32Q45 (primary) #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.08060
Let X=Γ\backslash \mathbbBn be an n-dimensional complex ball quotient by a torsion-free non-uniform lattice Γ whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of X are controlled by the systole of X, which is the length of the shortest closed geodesic of X. There are a number of arithmetic and geometric consequences: the systole of X controls the growth rate of rational points on X, uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle K_X, where X is the toroidal compactification of X. These results follow from the bound we find for the Seshadri constant of K_X in terms of the systole.