Bogomolov’s decomposition theorem

For decades (complex) algebraic geometry and (complex) differential geometry had being going going side-by-side in studying more or less the same objects. In spite of the difference between motivations, definitions and methods that they’re using, the results obtained by one of these branches often turned out to be valuable in another. One of the examples illustrating this principle is given by the Bogomolov’s decomposition theorem, which states that any compact Kähler manifold with trivial canonical class can, after passing to a finite covering, be splitted as a direct product of a complex torus, a number of simple Calabi–Yau manifolds and a number of irreducible holomorphically symplectic manifolds.
The original proof is based on a collection of deep theorems from differential and Riemannian geometry.
I am going to show the context, in which Calabi–Yau manifolds arise in Riemannian geometry (“Berger’s list of irreducible holonomies”), provide a connection between differential geometric and algebro-geometric definitions of Calabi–Yau manifolds and explain how to deduce the proof of Bogomolov’s decomposition theorem using different differential geometric tools: Calabi–Yau theorem, Cheeger–Gromoll splitting theorem, Bochner’s principle etc.