Abstract:
Motivated by the arithmetic Bombieri–Lang conjecture, we propose a program toward a geometric Bombieri–Lang type finiteness theory for moduli spaces of algebraic varieties. A crucial ingredient is a global form of Shing-Tung Yau’s Schwarz lemma over moduli spaces. We study the distribution of non-rigid loci in moduli spaces, inspired by the philosophy of the André–Oort conjecture. In particular, we discuss how hyperbolicity, curvature properties, and Hodge-theoretic structures may lead to geometric finiteness phenomena analogous to those predicted by the Bombieri–Lang conjecture in arithmetic geometry. We also compare this geometric Bombieri–Lang finiteness framework with recent work of Junyi Xie and Xinyi Yuan on geometric Bombieri–Lang type problems, and suggest the possibility of a unified approach. Based on joint work with K. Chen, T.Z. Hu, R.R. Sun and C.L. Yu.