SFB-Tag (Transregio 45): Borel's theorem for the moduli space of canonically polarized varieties

Let X be a complex algebraic locally symmetric variety. Borel and Kobayashi-Ochiai proved that, for all complex algebraic varieties S, every holomorphic map S^{an}->X^{an} is algebraic. In particular, if S is a complex algebraic variety, then any holomorphic family of polarized abelian varieties over S^{an} is, in fact, (the analytification of) an algebraic family of polarized abelian varieties over S. This seemingly transcendental statement is used for instance in the work of Deligne on Weil's conjecture for K3 surfaces, and the work of Andre, Charles, Madapusi-Pera, and Maulik on Tate's conjecture for K3 surfaces. We prove the analogous statement for the moduli space of canonically polarized varieties, i.e., we show that any holomorphic family of  canonically polarized varieties over S^{an} is algebraic. As a first simple application of our results, we obtain a new proof of the Brody hyperbolicity of the moduli space of canonically polarized varieties. Moreover, we also deduce that the algebraic structure on the analytic moduli space of canonically polarized varieties is unique. This is joint work with Robert Kucharczyk, Ruiran Sun, and Kang Zuo.