Volumes of subvarieties of complex ball quotients and sparsity of rational points

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>∖</m:mo> <m:msup> <m:mi mathvariant="double-struck">B</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> X=\Gamma\setminus\mathbb{B}^{n} 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 a 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mover accent="true"> <m:mi>X</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> </m:msub> </m:math> K_{\overline{X}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>X</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> </m:math> \overline{X} is the toroidal compactification of 𝑋. These follow from the bound we find for the Seshadri constant of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mover accent="true"> <m:mi>X</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> </m:msub> </m:math> K_{\overline{X}} in terms of the systole.