Generalized integral points on abelian varieties and the Geometric Lang-Vojta conjecture

Let $A$ be an abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and an effective horizontal divisor $\mathcal{D} \subset \mathcal{A}$. We study $(S, \mathcal{D})$-integral sections $\sigma$ of $\mathcal{A}$ where $S \subset B$ is arbitrary. These sections $\sigma$ are algebraic and satisfy the geometric condition $f(\sigma(B) \cap \mathcal{D})\subset S$. Developing the idea of Parshin, we formulate a hyperbolic-homotopic height of such sections as a substitute for intersection theory to establish new results concerning the finiteness and the polynomial growth of large unions of $(S, \mathcal{D})$-integral points where $S$ is only required to be finite in a thin analytic open subset of $B$. Such results are out of reach of purely algebraic methods and imply new evidence and interesting phenomena to the Geometric Lang-Vojta conjecture.