We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following the same steps as in the general case, treated next.
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