Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces

The first goal of this paper is to study the question of finiteness of integral points on a cofinite non-compact complex two-dimensional ball quotient defined over a number field. Along the process we will also consider some negatively curved compact surfaces. Using some fundamental results of Faltings, the question is to reduce to a conjecture of Borel about existence of virtual holomorphic one-forms on cofinite non-cocompact complex ball quotients. The second goal of this paper is to study the conjecture on such non-compact surfaces.