Higher-dimensional Shafarevich conjecture for canonically polarized manifolds

Let $U$) be a smooth variety, let $B$ be a smooth compactification of $U$ such that $\Delta:=B\setminus U$ is a global normal crossing divisor, and fix a Hilbert polynomial $h$. For a family $f:X\to U$ of canonically polarized manifolds with Hilbert polynomial $h$, write $\operatorname{Var} f$ for the dimension of the image of its moduli map. The higher-dimensional Shafarevich expectations include the following.

Higher-dimensional Shafarevich conjecture. (B) Families of canonically polarized manifolds over $U$ with Hilbert polynomial $h$ fall into finitely many deformation equivalence classes. (R) No good comprehensive conjecture is known; Viehweg's rigidity conjecture asserts that if $f:X\to U$ is a family of projective manifolds with $\Omega_{X/U}$ relatively ample, then $f$ is rigid. (H) For a family $f:X\to U$ of canonically polarized manifolds, Viehweg's hyperbolicity conjecture asserts that if $\operatorname{Var} f=\dim B$, then $\omega_B(\Delta)$ is big, equivalently $\kappa(B,\Delta)=\dim B$; the Kebekus--Kovács conjecture gives the conditions $\kappa(B,\Delta)<-\infty$ and $\operatorname{Var} f<\dim Y$, or $\kappa(B,\Delta)\geq0$ and $\operatorname{Var} f\leq k(B,\Delta)$; and Campana's conjecture asserts that if $(B,\Delta)$ is special, then $f$ is isotrivial, where special means that for every $p$ and every line bundle $\mathcal L\subseteq\Omega_B^p(\log\Delta)$, $k(\mathcal L)<p$.

These statements generalize the finiteness, rigidity, and hyperbolicity parts of the classical Shafarevich conjecture from families of curves to families of higher-dimensional canonically polarized manifolds. The source explicitly notes that no comprehensive rigidity conjecture is known and presents several distinct hyperbolicity conjectures.