Ricci-flat holonomy: A classification

The reductive holonomy algebras for a torsion-free affine connection are analysed, with the goal of establishing which ones can correspond to a Ricci-flat connection with the same properties. Various families of holonomies are eliminated through different algebraic means, and examples are constructed (in this paper and in ‘Projective Holonomy II: Cones and Complete Classification’, by the same author) in the remaining cases, thus solving this problem completely, for reductive holonomy.