Second Main Theorem on the Moduli Spaces of Polarized Varieties

Abstract

Let (X, D) be a smooth log pair over C such that the complement U := X \ D carries a maximally varied family of polarized manifolds. We prove a version of second main theorem on (X, D) by using the Viehweg-Zuo construction of the family and McQuillan's tautological inequality. As an application, we generalize a classical result of Nadel about the distribution of entire curves in the (compactified) base space of polarized families.

Holomorphic curves in moduli spaces of polarized Abelian varieties

“…We first introduce an inequality of curvature currents proved by Sun [10], who obtained this inequality by Viehweg-Zuo's construction [12,13].…”

“…With the previous notations, we state a curvature current inequality. The details can be found in [10], Proposition 2.4. Lemma 5.1 (Curvature current inequality, [10]).…”

“…The details can be found in [10], Proposition 2.4. Lemma 5.1 (Curvature current inequality, [10]). Suppose m is the maximal length of iteration, i.e., the largest integer such that ζ m = 0.…”

“…where d is the fiber dimension of a family, see Theorem B in [10]. In further, Sun used this result to obtain a Second Main Theorem of holomorphic curves into moduli spaces of polarized Abelian varieties, see Theorem A in [10].…”

“…Similarly, (E γ , θ γ ) is defined. Over Ŝ, we can have the following commutative diagram (see (2.2) in [10])…”