Iitaka–Severi conjecture on dominant rational maps to manifolds of general type

Let $X$ be a compact complex manifold. Consider compact complex manifolds $Y$ of general type such that there exists a dominant rational map

$$X\dashrightarrow Y.$$

Iitaka–Severi conjecture. The number of birational equivalence classes of such manifolds $Y$ is finite. This conjecture generalizes the de Franchis–Severi finiteness theorem for maps from a fixed compact complex curve. The paper explains that the principal difficulty in higher dimensions is the absence of uniform birational embedding theorems for manifolds with big canonical line bundle.