| This talk surveys results on the singularities of the Minimal Model Program in a non-technical manner and discusses applications to the study of varieties with trivial canonical class. A classical Decomposition Theorem splits any compact Kähler manifolds with numerically trivial canonical bundle into a product of a torus, a Calabi-Yau- and a holomorphic-symplectic manifold. Building on extension theorems for differential forms on singular spaces, we prove an infinitesimal version for singular varieties. In view of recent progress in minimal model theory, this can be seen as a first step towards a structure theory for manifolds of Kodaira dimension zero. In the second part of the talk we compare the étale fundamental group of a klt variety with that of its smooth locus. As first major application, we show that any flat holomorphic bundle, defined on the smooth part of a projective klt variety is algebraic and extends across the singularities. This allows to generalise a famous theorem of Yau, which states that any Ricci-flat Kähler manifold with vanishing second Chern class is an étale quotient of a torus. This is joint work with Daniel Greb and Thomas Peternell. |