Towards the Global Torelli Theorem

In this paper, we prove finiteness results for K-trivial varieties in all dimensions. In dimension 2, these results are corollaries of the Global Torelli Theorem for polarized K3 surfaces. In particular, we establish a sufficient condition when two fibers of a projective family of K-trivial varieties are isomorphic via the Hodge theory. As applications, we prove that if a projective family of K-trivial varieties has a dense subset of birationally equivalent fibers, then all fibers are isomorphic.