Rational, log canonical, Du Bois singularities: On the conjectures of Kollar and Steenbrink

Let X be a proper complex variety with Du Bois singularities. Then H-i(X, C ) --> H-i (X, O-X) is surjective for all i. This property makes this class of singularities behave well with regard to Kodaira type vanishing theorems . Steenbrink conjectured that rational singularities are Du Bois and Kollar conjectured that log canonical singularities are Du Bois. Kollar also conj ectured that under some reasonable extra conditions Du Bois singularities a re log canonical. In this article Steenbrink's conjecture is proved in its full generality, Kollar's first conjecture is proved under some extra condi tions and Kollar's second conjecture is proved under a set of reasonable co nditions, and shown that these conditions cannot be relaxed.